Laws in Quantitative Linguistics - User contributions [en]

Morphological productivity

2007-06-29T08:15:49Z

Altmann:

'''1. Problem and history'''

In the lexicon of a language there are two processes working:

(a) Birth of new lexemes by creating or borrowing

(b) Death by elimination of lexemes

Birth means either creation of new words or constructing them by morphological processes like derivation, reduplication, compounding etc. The construction is a case of (multidi-mensional) ''diversification'' (<math>rightarrow</math>). At the same time it is a case of specification of meaning and word prolongation. Birth and death take place in rates called birth-rate <math>\lambda_x</math>and death-rate <math>\mu_x</math> respectively.
Since up to now only a case of derivation has been examined, the hypothesis must be restricted but we assume that it holds for other kinds of word building, too. It holds, of course, only under the condition that the given way of word buidling is actual in a language.
As far as known, the only model was proposed by Wimmer and Altmann (1995) in which the random variable x is the number of derivates built from a stem (x = 0,1,2,…) and <math>f_x</math> is the number of stems having x derivates.

'''2. Hypothesis'''

''Morphological productivity is the result of a birth-and-death process in which stems build new derivates and some derivates are eliminated. The probability of the number of stems having x derivates follows the Pólya distribution''.

'''3. Derivation'''

The creation of new morphological constructs is proportional to the class size x and can be generally symbolized as a + cx. Since creation cannot be infinite, there must be a limit n to it. The community controls the creation in the same linear way symbolized as b + (n-x-1)c. This results in the birth rate <math>\lambda = \frac{a+cx}{b+(n-x-1)c}\quad , x = 0, 1, ...n-1</math>. The death rate is similar, depending only on the class size and the morphological nature of language. It can be sym-bolized as <math>\mu = \frac{n}{n-x+1}\quad, x = 1, 2, ...,n</math>
Inserting the above birth-rate and death-rate in the balancing birth-and-death equations and reparametrizing

p = a/(a+b), q = b/(a+b), s = c/(a+b)

we obtain the Pólya distribution (Wimmer, Altmann 1995)

(1)<math> P_x = \frac{{-p/s \choose x}{-q/s \choose n-x}}{{-1/s \choose n}}\quad, x= 0, 1, ..., n, s>0, =\le p\le1, q=1-p, n \epsilon N_0</math>
<math> \frac{p}{s}, \frac{q}{s}\ge = 0</math> or <math> \frac{p}{s}, \frac{q}{s}\le 0</math>

'''Example''': Indonesian stem productivity

Wimmer and Altmann (1995) studied the productivity of stems in the Indonesian dictionary (Echols, Shadily 1963). There were 6970 stems having no derivates, 1961 stems having each 1 derivate etc. The data and the results of fitting are presented in Table 1 and Fig. 1.

<div align="center">[[Image:Tabelle11_MP.jpg]]</div>

(((Den Graph mit Excell machen, da 0 fehlt)
Fig. 1. Fitting the Pólya distribution to the data in Table 1

'''4. Authors:''' U. Strauss, G. Altmann

'''5. References'''

'''Altmann, G., Bagheri, D., Goebl, H., Köhler, R., Prün, C'''. (2002). ''Einführung in die quantitative Lexikologie''. Götingen: Peust & Gutschmidt.

'''Echols, J.M., Shadily, H.''' (1963). ''An Indonesian-English dictionary''. Ithaca: Cornell University Press.

'''Wimmer, G., Altmann, G.''' (1995). A model of morphological productivity. ''J. of Quantitative Linguistics 2, 212-216.''

[[Category:Unfertig]]

Morph length

2007-06-26T08:24:04Z

Altmann:

'''1. Problem and history'''

Morph length is a specal case of length or word length (<math>\rightarrow</math>) research. The segmentation of a text in morphs is a problem depending on language and grammar (cf. Best 2001 for German).
Morph lengths are focussed in Quantitative Linguistics in different ways:
Gerlach (1982) tested the interaction of word length with morph length in a German lexicon; the hypothesis was: the more morphs a word has, the shorter the morphs are. The test results were very good. Without any data Saporta (1963: 70) proposes the hypothesis: “The mean length of morphs will be inversely related to the number of phonemes in the inventory.” This idea is said to go back to R. Jakobson (Saporta 1963: 72, footnote 15).
Here we are confronted with another hypothesis: morph lengths in texts abide by a law in the same way as word lengths and other entities do.
Since up to now only a small number of data has been collected (Best 2000a, 2001a) the only distribution found rather inductively but belonging to the family of length-distributions (cf. Wimmer, Altmann 1996) was the 1-displaced Hyperpoisson distribution.
In a paper concerning morph segmentation Creutz (2003: 282) proposes the gamma distribution to be a model for morph lengths in the lexicon; but there is no proof of it. As soon as 1963 Saporta (²1966: 69) presents a little overview over morph lengths (in phonemes) in Spanish; testing the data (1679 morphs) the binomial distribution can be shown to be an acceptable model (C = 0.0166). He “cannot help wondering whether or not such a distribution is universal and, if not, what other factors correlate with different distributions.”
However, in Lakota the morphs have a specific form and their frequency distribution must be modelled by means of a difference equation of second order.

'''2. Hypothesis'''

''2.1. Morph length in texts abides by a regular probability distribution derived form the unified theory, namely the 1-displaced Hyperpoisson distribution.''
''2.2. If morphs have specific forms, the distribution is multimodal and must be modeled by an appropriate approach''.

'''3. Derivation'''

3.1. Substituting <math>a_0 = -1, a_1 = a, b_1 = b, a_2 = 0</math> in formula (10) of unified theory (→) and solving with displacement one obtains the Hyperpoisson distribution

(1)<math> P_x = \frac{a^{x-1}}{b^{x-1} _1 F_1 (1; b; a)}, \quad x = 1, 2, 3, ...; a, b > 0</math>

'''Example''': Morph length distribution in German

Best (2001a) used a text from Eichsfelder Tageblatt (6.3.1997, p.8): „Sieben Deutsche in Jemen entführt)“ and counting the length of morphemes he obtained the results in Table 1.

<div align="center">[[Image:Tabelle111_ML.jpg]]</div>

<div align="center">[[Image:Grafik1_ML.jpg]]</div>
<div align="center">Fig. 1. Fitting the 1-displaced Hyperpoisson distribution to morph length data</div>

3.2. In Lakota, morphemes usually consist of even number of syllables. In that case the usual approach must be extended and the distribution must be modeled by a difference equation of second order i.e.

(2)<math> P_x = g(x)P_{x-1}+h(x)P_{x-2}\quad</math>.

Pustet and Altmann (2005) set g(x) = a(k + x – 1)/x and h(x) = b(2k + x – 2)/x and obtained as solution the Gegenbauer distribution given as

(3)<math> P_x = (n)=\begin{cases} (1-a-b)^k,\quad x=0 \\ p_0 \sum_{j=0}^{[x/2]}\frac{b^j k^{(x-j)}a^{x-2j}}{j!(x-2j)!}, \quad x = 1, 2, 3,...\end{cases} </math>

where [.] is the integer part and k(x - j) is the ascending factorial function.

'''Example'''. Morpheme length in Lakota

Pustet and Altmann (2005) modeled the morpheme length distribution in Lakota and obtained the results in Table 2 and Fig. 2.

<div align="center">[[Image:Tabelle222_ML.jpg]]</div>

<div align="center">[[Image:Grafik2_ML.jpg]]</div>
<div align="center">Fig. 2. Fitting the Gegenbauer distribution to the Lakota data</div>

'''4. Authors: U. Strauss, G. Altmann, K.-H. Best'''

'''5. References'''

'''Best, K.-H.''' (2000a). Morphlängen in Fabeln von Pestalozzi. ''Göttinger Beiträge zur Sprachwissenschaft 3, 19-30.''

'''Best, K.-H'''. (2001a). Zur Länge von Morphen in deutschen Texten. In: Best, K.-H. (ed.), ''Häufigkeitsverteilungen in Texten: 1-14''. Göttingen: Peust & Gutschmidt.

'''Best, K.-H.''' (2001b). Probability distributions of language entities. ''J. of Quantitative Linguistics 8, 1-11.''

'''Best, K.-H'''. (2005). Morphlänge. In: Köhler, R., Altmann, G., Piotrowski, R. (eds.), ''Quantitative Linguistik - Quantitative Linguistics. Ein internationales Handbuch: 255-260''. Berlin/ N.Y.: de Gruyter

'''Creutz, M'''. (2003). Unsupervised segmentation of words using prior distributions of morph length and frequency. ''Proceedings of ACL-03, The 41st Annual Meeting of th Association of Computational Linguistics: 280-287''. Sapporo, Japan, 7-12 July.

'''Gerlach, R.''' (1982). Zur Überprüfung des Menzerathschen Gesetzes im Bereich der Morphologie. ''Glottometrika 4, 95-102''.

'''Gorot´, E.I.''' (1990). Izomorfnye i otličitel´nye čerty morfemy i sloga v raspredelenii dliny. In: ''Kvantitativnaja lingvistika i avtomatičeskij analiz tekstov: 32-36''. Tartu.

'''Krott, A'''. (1996). Some remarks on the relation between word length and morpheme length. ''J. of Quantitative Linguistics 3, 29-37''.

'''Nikonov, V.A'''. (1978). Dlina slova. ''Voprosy jazykoznanija 6, 104-111''.

'''Pustet, R., Altmann, G'''. (2005). Morpheme length distribution in Lakota. ''J. of Quantitative Linguistics 12(1), 53-63''.

'''Saporta, S'''. (1963, ²1966). Phoneme distribution and language universals. In: Greenberg, J.H. (ed.), ''Universals of language. Second edition. Report of a conference held at Dobbs Ferry, New York, April 13-115, 1961: 61-72''. Cambridge, Mass. & London: The M.I.T. Press.

'''Wimmer, G., Altmann, G'''. (1996). The theory of word length: some results and generalizations. ''Glottometrika 15, 112-133''.

[[Category:Unfertig]]

Hierarchic relations

2007-06-25T07:51:34Z

Altmann:

'''1. Problem and history'''

In different domains of language one observed the fact that the length of a construct influences the length of its consitutents. Usually the constituents get smaller with increasing length of the construct but not in all cases. The problem is to find a theoretical model encompassing all dependencies of this kind. The constructs whose length is the independent variable are: hreb, sentence, rhythmic unit, word, syllable; the constituents whose length or duration is the dependent variable are: sentence, clause, word, syllable, morph, sound. There is also the possibility to consider two independent variables, e.g. word (measured in number of syllables) and syllable (measured in number of sounds) , while the dependent variable is the syllable duration.
Up to now the following particular cases have been examined (see Table 1)

<div align="center">[[Image:Tabelle11_HR.jpg]]</div>

The origin of the problem can be found in 19th century, in phonetics, probably for the first time with Sievers (1876, 1901) who measured the syllable duration in rhythmic units (Sprechakte). A number of phoneticians tested different hypotheses, a part of which corroborated, another part falsified it. The isochrony hypothesis in English is a special case of this problem. The problem was generalized by Menzerath who stated that ''the greater the whole the smaller its parts'' (1954: 101). Different researchers proposed some empirical formulas (Fónagy, Magdics 1960; Nooteboom 1972, 1973; Landblom, Rapp 1972), Altmann (1980) set up the pertinent differential equation and called the result Menzerath´s law. Hřebíček (1992, 1995, 1997) showed that the whole hierarchy of textual levels is based on this dependence and called it ''Menzerath-Altmann´s law''.

There is a great number of individual examinations in different domains of languge. In phonetics the most exhaustive is Weber (1998), in textology the works by Hřebíček (see above) and a mixture of problems including biology and sociology can be found in Altmann, Schwibbe (1989). Bohn (1998) analyzed the relationship between Chinese characters and the complexity of composing graphemes, length of words and simplicity of characters, clause length and word length, sentence length and clause length (cf. also Menzel 2005).
The law has a strong corroboration not only within linguistics but displays analogies to other sciences. Thus the simple form of Menzerath´s law is identical with the allometric law in biology and with the power laws current in different sciences. Its correspondences can be found in (i) molecular biology, (ii) sociology of baboons, (iii) in the domain of self-organized criticality, (iv) in chaos research, (v) in the theory of fractals, (vi) in information theory.
It has been observed that construct length is not always the only cause of shortening of the constituents. Also accent, vowel quality, syllable structure, frequency etc. can intervene (cf. Weber 1998). In that case more complex formulas must be used.
The law has several consequences, all of which must still be tested (cf. Altmann, Schwibbe 1989: 8-14):
1. In longer words more phonetic changes occur than in shorter ones.

2. In languages with greater average word length more phonetic/phonemic changes occur than within the same time interval in languages with smaller average word length

3. The adding of an affix to a word evokes the tendency to reduce the inventory of consonants of the word.

4. The shortening of average syllable length in Hypothesis 3 can also be achieved by inserting epenthetic vowels between the stem and affix (or compounding stem).

5. Partial reduplication is more frequent in natural languages than full reduplication.

6. Short roots/morphemes/stems build more compounds or derived words than long ones.

7. The more elements there are in a compound the shorter they are (see the hypotheses on compounds <math>\leftarrow</math>)

8. Fenk-Fenk (to be inserted)

There are different interpretations of the law:

1. In general, long constructs contain more redundancy than short ones. In order to prevent excessive growth of redundancy one can reduce the size of the constituents. The size, the place and the time of this reduction is not known and cannot be predicted. The analogy to self-organized criticality of sand-piles is evident (cf. Bak 1996).

2. Köhler (1989) shows that mechanism of shortening is a consequence of restrictions of the memory: the longer the construct, the more place must be reserved for the structural information between the constituents, thus the size of the constituents must be reduced.

'''2. Hypothesis'''

''The size of the components is a function of the construct size''.

'''3. Derivation'''

The average size of constituents changes with the increase of the size of the construct. It is assumed that the relative rate of change of the size of components is proportional to the rate of change of the size of constructs, the proportionality function being

<math> g(x) = a_0 + \frac{a_1}{x} + \frac{a_2}{x^2}</math>.

Thus in case that all other variables other than construct size are subsumed under the ceteris paribus condition one obtains

(1)<math> \frac{dy}{y-d}= \left(a_0 +\frac{a_1}{x}+ \frac{a_2}{x^2}\right)</math>.

where d is the minimal value y can attain.
In case that there is another independent variable, z, one starts from

(2)<math>\frac{dy}{dx}\frac{1}{y-d}=a_0 + \frac{a_1}{x}+\frac{a_2}{x^2}\quad</math> and <math>\frac{dy}{dz}\frac{1}{y-d}=b_0 + \frac{b_1}{z}+\frac{b_2}{z^2}</math>

which can be extended to any number of variables.
The solution of (1) yields

(3)<math> y= Cx^{a_1}e^{a_0 x-a_2/x} + d</math>

the combining (2) the solution is

(4)<math> y= Cx^{a_1}z^{b_1}e^{a_0 x + b_0 z-a_2 /x-b_2 /z} + d</math>

In different applications the following special cases of (3) have been used (d > 0)

(1a) <math> y=ax^{-b}(+d), \quad x= 1, 2, 3, ...; \quad a, b > 0</math>

(1b)<math> y=ax^b e^{cx}(+d), \quad x= 1, 2, 3, ...; \quad a > 0</math>

(1c)<math> y=ae^{-cx}(+d), \quad x= 1, 2, 3, ...; \quad a, c > 0</math>

(1d) <math> y=ae^{c/x}(+d), \quad x= 1, 2, 3, ...; \quad a, c > 0</math>

Solution (4) is still very seldom. Both approaches are special cases of the unified theory (<math>\leftarrow</math>).

'''Examples''':

'''4. Authors: U. Strauss, G. Altmann'''

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'''Altmann, G., Beöthy, E., Best, K.-H.''' (1982). Die Bedeutungskomplexität der Wörter und das Menzerathsche Gesetz. ''Zeitschrift für Phonetik, Sprachwissenschaft und Kommunikationsforschung 35, 537-543''.

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[[Category:Unfertig]]

Hierarchic relations

2007-06-25T07:47:01Z

Altmann:

'''1. Problem and history'''

In different domains of language one observed the fact that the length of a construct influences the length of its consitutents. Usually the constituents get smaller with increasing length of the construct but not in all cases. The problem is to find a theoretical model encompassing all dependencies of this kind. The constructs whose length is the independent variable are: hreb, sentence, rhythmic unit, word, syllable; the constituents whose length or duration is the dependent variable are: sentence, clause, word, syllable, morph, sound. There is also the possibility to consider two independent variables, e.g. word (measured in number of syllables) and syllable (measured in number of sounds) , while the dependent variable is the syllable duration.
Up to now the following particular cases have been examined (see Table 1)

<div align="center">[[Image:Tabelle11_HR.jpg]]</div>

The origin of the problem can be found in 19th century, in phonetics, probably for the first time with Sievers (1876, 1901) who measured the syllable duration in rhythmic units (Sprechakte). A number of phoneticians tested different hypotheses, a part of which corroborated, another part falsified it. The isochrony hypothesis in English is a special case of this problem. The problem was generalized by Menzerath who stated that ''the greater the whole the smaller its parts'' (1954: 101). Different researchers proposed some empirical formulas (Fónagy, Magdics 1960; Nooteboom 1972, 1973; Landblom, Rapp 1972), Altmann (1980) set up the pertinent differential equation and called the result Menzerath´s law. Hřebíček (1992, 1995, 1997) showed that the whole hierarchy of textual levels is based on this dependence and called it ''Menzerath-Altmann´s law''.

There is a great number of individual examinations in different domains of languge. In phonetics the most exhaustive is Weber (1998), in textology the works by Hřebíček (see above) and a mixture of problems including biology and sociology can be found in Altmann, Schwibbe (1989). Bohn (1998) analyzed the relationship between Chinese characters and the complexity of composing graphemes, length of words and simplicity of characters, clause length and word length, sentence length and clause length (cf. also Menzel 2005).
The law has a strong corroboration not only within linguistics but displays analogies to other sciences. Thus the simple form of Menzerath´s law is identical with the allometric law in biology and with the power laws current in different sciences. Its correspondences can be found in (i) molecular biology, (ii) sociology of baboons, (iii) in the domain of self-organized criticality, (iv) in chaos research, (v) in the theory of fractals, (vi) in information theory.
It has been observed that construct length is not always the only cause of shortening of the constituents. Also accent, vowel quality, syllable structure, frequency etc. can intervene (cf. Weber 1998). In that case more complex formulas must be used.
The law has several consequences, all of which must still be tested (cf. Altmann, Schwibbe 1989: 8-14):
1. In longer words more phonetic changes occur than in shorter ones.

2. In languages with greater average word length more phonetic/phonemic changes occur than within the same time interval in languages with smaller average word length

3. The adding of an affix to a word evokes the tendency to reduce the inventory of consonants of the word.

4. The shortening of average syllable length in Hypothesis 3 can also be achieved by inserting epenthetic vowels between the stem and affix (or compounding stem).

5. Partial reduplication is more frequent in natural languages than full reduplication.

6. Short roots/morphemes/stems build more compounds or derived words than long ones.

7. The more elements there are in a compound the shorter they are (see the hypotheses on compounds <math>\leftarrow</math>)

8. Fenk-Fenk (to be inserted)

There are different interpretations of the law:

1. In general, long constructs contain more redundancy than short ones. In order to prevent excessive growth of redundancy one can reduce the size of the constituents. The size, the place and the time of this reduction is not known and cannot be predicted. The analogy to self-organized criticality of sand-piles is evident (cf. Bak 1996).

2. Köhler (1989) shows that mechanism of shortening is a consequence of restrictions of the memory: the longer the construct, the more place must be reserved for the structural information between the constituents, thus the size of the constituents must be reduced.

'''2. Hypothesis'''

''The size of the components is a function of the construct size''.

'''3. Derivation'''

The average size of constituents changes with the increase of the size of the construct. It is assumed that the relative rate of change of the size of components is proportional to the rate of change of the size of constructs, the proportionality function being

<math> g(x) = a_0 + \frac{a_1}{x} + \frac{a_2}{x^2}</math>.

Thus in case that all other variables other than construct size are subsumed under the ceteris paribus condition one obtains

(1)<math> \frac{dy}{y-d}= \left(a_0 \frac{a_1}{x}+ \frac{a_2}{x^2}\right)</math>.

where d is the minimal value y can attain.
In case that there is another independent variable, z, one starts from

(2)<math>\frac{dy}{dx}\frac{1}{y-d}=a_0 + \frac{a_1}{x}+\frac{a_2}{x^2}\quad</math> and <math>\frac{dy}{dz}\frac{1}{y-d}=b_0 + \frac{b_1}{z}+\frac{b_2}{z^2}</math>

which can be extended to any number of variables.
The solution of (1) yields

(3)<math> y= Cx^{a_1}e^{a_0 x-a_2/x} + d</math>

the combining (2) the solution is

(4)<math> y= Cx^{a_1}z^{b_1}e^{a_0 x + b_0 z-a_2 /x-b_2 /z} + d</math>

In different applications the following special cases of (3) have been used (d > 0)

(1a) <math> y=ax^{-b}(+d), \quad x= 1, 2, 3, ...; \quad a, b > 0</math>

(1b)<math> y=ax^b e^{cx}(+d), \quad x= 1, 2, 3, ...; \quad a > 0</math>

(1c)<math> y=ae^{-cx}(+d), \quad x= 1, 2, 3, ...; \quad a, c > 0</math>

(1d) <math> y=ae^{c/x}(+d), \quad x= 1, 2, 3, ...; \quad a, c > 0</math>

Solution (4) is still very seldom. Both approaches are special cases of the unified theory (<math>\leftarrow</math>).

'''Examples''':

'''4. Authors: U. Strauss, G. Altmann'''

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[[Category:Unfertig]]

Complexity of syntactic constructions

2006-12-04T16:34:42Z

Altmann:

'''1. Problem and history'''

The complexity of a syntactic construct is measured in terms of the number of its immediate constituents. The partitioning in immediate constituents can be performed on the basis of any grammar.
The first model seems to be that of Köhler and Altmann (2000).

'''2. Hypothesis'''

''The complexity of syntactic constructions follows the hyper-Pascal distribution''.

'''3. Derivation'''

The complexity depends on following quantities (Köhler, Altmann 2000:192):

minX – the requirement of minimization of the complexity of a syntactic construction in order to decrease memory effort in processing the construction;

maxH – the requirement of maximazing compactness. This enables us diminishing the complexity of the subordinated level of embedding by embedding constituents into the given level… minX on the level m corresponds to the requirement maxH on the level m+1;

E – a variable representing the average degree of fullness, the default value of complexity;

I(K) – the size of inventory of constructions.

Assumptions: The number of constructions with complexity x+1 is proportional to that with complexity x. maxH increases the probability of a higher complexity, minX decreases it. E and I(K) are inversely proportional to each other: the greater the inventory, the less complexity is needed. The relation E/I(K) yield a constant, say q.
Putting these assumptions together, we obtain

(1)<math> P_{x+1}= \frac{maxH + x}{minX + x} \frac{E}{I(K)}P_x, \quad x= 1, 2, ...</math>

Setting maxH = k-1, minX = m-1 and E/I(K) = q yields

(2)<math> P_{x+1} = \frac{k+x-1}{m+x-1}qP_x, \quad x=1, 2, ...</math>

resulting in

(3)<math> P_X = \frac{{k+x-2 \choose x-1}}{{m+x-2 \choose x-1}}q^{x-1}P_1
, \quad x=1,2,3...</math>

where<math> P_1^{-1}= _2F_1 (k,1;m;q)</math>.

'''Example''': Complexity of syntactic constructions in the Negra corpus (Brants 1999)
Köhler and Altmann (2000) fitted (3) to the complexity of syntactic constructions in the Negra corpus. The result is presented in Table 1 and Fig. 1.

<div align="center">[[Image:Tabelle11_CoSC.jpg]]</div>

Since the number of observations is too great, the use of the chi-square is problematic. The authors use the contingency coefficient <math>C = X^2/N</math> which is acceptable. It would be advisable to use single texts instead of corpora.

<div align="center">[[Image:Grafik1_CoSC.jpg]]</div>
<div align="center">Fig. 1. Distribution of syntactic complexity in the Negra corpus</div>

'''4. Authors: G. Altmann'''

'''5. References'''

'''Brants, T'''. (1999). ''Tagging and parsing with cascaded Markov models. Automation of corpus annotation''. Saarbrücken: Universität der Saarlandes.

'''Köhler, R., Altmann, G.''' (2000). Probability distributions of syntactic units and properties. ''J. of Quantitative Linguistics 7, 189-200''.

Frequency and polytextuality

2006-08-11T09:38:59Z

Altmann:

'''1. Problem and history'''

Under polytextuality one understands the number of environments of a linguistic entity. The entity can be syllable, mora, morphem, word and other units. The environment for syllable, mora and morphem is the word, the environment of the word are other words. Usually the number of different environments is called number of types. The frequency of the given entity in all its environments in, say, a corpus, is considered as the number of tokens. The question is, whether there is some relationship between the number of types (environments) and the number of tokens (frequency) of units of the given level.

The relationship between frequency and polytextuality has been launched by R. Köhler (1986) as a complement to Zipfian properties, in order to enlarge his control cycle. Since the computation of data is very laborious and the erroneous identification with another “type-token” problem (<math>\rightarrow</math>) lead to confusion, one can find this relationship also under the name “(morphological) productivity” (cf. Baayen 2001) which in turn represents a slightly different problem (cf. Wimmer, Altmann 1995). The relationship appeared in different works on language synergetics (cf. e.g. Gieseking 2002), Köhler (2005) reformulated the pertinent part of his control cycle and Tamaoka, Altmann (2005) showed by means of Japanese morae that the unified theory (<math>\rightarrow</math>) leads to the identical result.

Usually one considers frequency as the spiritus movens, the independent variable of many relationships, but Köhler (1986) assumed here an inverse relationship.

'''2. Hypothesis'''

''The frequency of linguistic units depends on their polytextuality.''

'''3. Derivation'''

Since in most cases linguistic properties are related by their relative rates of change, Tamaoka (2007), taking into account some ceteris paribus factors, and leaning against the unfied theory (<math>\rightarrow</math>) set up the equation

(1)<math> \frac{dy}{y}= \left( c+\frac{b}{x}\right)dx</math>

where x is polytexty, y is frequency and c are some additional factors. They considered Japanese morae, their polytexty and frequency in a Japanese corpus. The resulting equation is

(2)<math>y = ax^b e^{cx}\quad</math>.

Using Köhlers model (Fig. 1) one can write the relationships as follows:

(3) ln(F) = R ln(Appl) + B ln(PT) – C exp(ln(PT))

i.e.
ln(F) = R ln(Appl) + B ln(PT) – C (PT)\quad

from which

<math>F = Appl^R PT^B e^{-c({PT})}\quad</math>

follows. Since <math>Appl^R</math> can, in the framework of a synchronic study, be considered as a constant, say A, PT = x, and F = y, we obtain

<math>y = A x^b e^{-cx}\quad</math>

whih is identical with the above result of the differential equation.

<div align="center">[[Image:Figur1_Freq.jpg]]</div>
<div align="center">Fig. 1. The relationship between polytextuality and frequency in general</div>

Thus Köhler´s model explains also the additional factors.

'''Example 1'''. Types and tokens of Japanese morae

Tamaoka and Makioka (2004) computed the frequencies of 103 Japanese morae in a corpus containing 341,771 different words with total frequency 287,792,797. For each mora its frequency and the contexts (different words) were ascertained. Tamaoka and Altmann (2005) showed that the best fit to these data (in logarithmic transformation) can be obtained by the curve

<math>y = 26.57366832x^{1.31502554}exp(-0.0000125937521x)</math>.

yielding a determination coeffciient D = 0.92. The result of fitting is displayd in Table 1 and graphically presented in Fig. 2.

<div align="center">[[Image:Tabelle11_Freq.jpg]]</div>

<div align="center">[[Image:Grafi1_Freq.jpg]]</div>
<div align="center">Fig. 2. Relation between types and tokens of Japanese morae</div>

'''4. Authors: R. Köhler, G. Altmann'''

'''5. References'''

'''Baayen, R.H.''' (2001). ''Word frequency distributions''. Dordrecht: Kluwer.

'''Gieseking, K.''' (2002). Untersuchungen zur Synergetik der englischen Lexik. In: Köhler, R. (ed.), ''Korpuslinguistische Untersuchungen in die quantitative und systemtheoretische Linguistik: 387-433''. http://ubt.opus.hbz-nrw.de/volltexte/2004/279/

'''Köhler, R.''' (2006). Frequenz, Kontextualität und Länge von Wörtern. Eine Erweiterung des synergetisch-linguistischen Modells. In: Rapp, R., Sedlmeier, P., Zunker-Rapp, G. (eds.), ''Perspectives on Cognition''. Lengerich, Berlin, Bremen, Miami et al: Pabst Science Publishers, 327-338.

'''Tamaoka, K.''' (2007). On the relation between types and tokens of Japanese morae. In: ''Script problems (in print)''.

'''Tamaoka, K., Makioka, Sh.''' (2004). Frequency of occurrence for units of phonemes, morae, and syllables appearing in a lexical corpus of a Japanese newspaper. ''Behavior Research Methods, Instruments & Computers 36(3), 531-547''.

'''Wimmer, G., Altmann, G.''' (1995). A model of morphological productivity. ''J. of Quantitative Linguistics 2, 212-216.''

[[Category:Unfertig]]

Frequency and polytextuality

2006-08-11T07:46:10Z

Altmann:

'''1. Problem and history'''

Under polytextuality one understands the number of environments of a linguistic entity. The entity can be syllable, mora, morphem, word and other units. The environment for syllable, mora and morphem is the word, the environment of the word are other words. Usually the number of different environments is called number of types. The frequency of the given entity in all its environments in, say, a corpus, is considered as the number of tokens. The question is, whether there is some relationship between the number of types (environments) and the number of tokens (frequency) of units of the given level.

The relationship between frequency and polytextuality has been launched by R. Köhler (1986) as a complement to Zipfian properties, in order to enlarge his control cycle. Since the computation of data is very laborious and the erroneous identification with another “type-token” problem (<math>\rightarrow</math>) lead to confusion, one can find this relationship also under the name “(morphological) productivity” (cf. Baayen 2001) which in turn represents a slightly different problem (cf. Wimmer, Altmann 1995). The relationship appeared in different works on language synergetics (cf. e.g. Gieseking 2002), Köhler (2005) reformulated the pertinent part of his control cycle and Tamaoka, Altmann (2005) showed by means of Japanese morae that the unified theory (<math>\rightarrow</math>) leads to the identical result.

Usually one considers frequency as the spiritus movens, the independent variable of many relationships, but Köhler (1986) assumed here an inverse relationship.

'''2. Hypothesis'''

''The frequency of linguistic units depends on their polytextuality.''

'''3. Derivation'''

Since in most cases linguistic properties are related by their relative rates of change, Tamaoka and Altmann (2006), taking into account some ceteris paribus factors, and leaning against the unfied theory (<math>\rightarrow</math>) set up the equation

(1)<math> \frac{dy}{y}= \left( c+\frac{b}{x}\right)dx</math>

where x is polytexty, y is frequency and c are some additional factors. They considered Japanese morae, their polytexty and frequency in a Japanese corpus. The resulting equation is

(2)<math>y = ax^b e^{cx}\quad</math>.

Using Köhlers model (Fig. 1) one can write the relationships as follows:

(3) ln(F) = R ln(Appl) + B ln(PT) – C exp(ln(PT))

i.e.
ln(F) = R ln(Appl) + B ln(PT) – C (PT)\quad

from which

<math>F = Appl^R PT^B e^{-c({PT})}\quad</math>

follows. Since <math>Appl^R</math> can, in the framework of a synchronic study, be considered as a constant, say A, PT = x, and F = y, we obtain

<math>y = A x^b e^{-cx}\quad</math>

whih is identical with the above result of the differential equation.

<div align="center">[[Image:Figur1_Freq.jpg]]</div>
<div align="center">Fig. 1. The relationship between polytextuality and frequency in general</div>

Thus Köhler´s model explains also the additional factors.

'''Example 1'''. Types and tokens of Japanese morae

Tamaoka and Makioka (2004) computed the frequencies of 103 Japanese morae in a corpus containing 341,771 different words with total frequency 287,792,797. For each mora its frequency and the contexts (different words) were ascertained. Tamaoka and Altmann (2005) showed that the best fit to these data (in logarithmic transformation) can be obtained by the curve

<math>y = 26.57366832x^{1.31502554}exp(-0.0000125937521x)</math>.

yielding a determination coeffciient D = 0.92. The result of fitting is displayd in Table 1 and graphically presented in Fig. 2.

<div align="center">[[Image:Tabelle11_Freq.jpg]]</div>

<div align="center">[[Image:Grafi1_Freq.jpg]]</div>
<div align="center">Fig. 2. Relation between types and tokens of Japanese morae</div>

'''4. Authors: R. Köhler, G. Altmann'''

'''5. References'''

'''Baayen, R.H.''' (2001). ''Word frequency distributions''. Dordrecht: Kluwer.

'''Gieseking, K.''' (2002). Untersuchungen zur Synergetik der englischen Lexik. In: Köhler, R. (ed.), ''Korpuslinguistische Untersuchungen in die quantitative und systemtheoretische Linguistik: 387-433''. http://ubt.opus.hbz-nrw.de/volltexte/2004/279/

'''Köhler, R.''' (2006). Frequenz, Kontextualität und Länge von Wörtern. Eine Erweiterung des synergetisch-linguistischen Modells. In: Rapp, R., Sedlmeier, P., Zunker-Rapp, G. (eds.), ''Perspectives on Cognition''. Lengerich, Berlin, Bremen, Miami et al: Pabst Science Publishers, 327-338.

'''Tamaoka, K., Altmann, G.''' (2006). On the relation between types and tokens of Japanese morae. ''Glottometrics 13, (in print)''.

'''Tamaoka, K., Makioka, Sh.''' (2004). Frequency of occurrence for units of phonemes, morae, and syllables appearing in a lexical corpus of a Japanese newspaper. ''Behavior Research Methods, Instruments & Computers 36(3), 531-547''.

'''Wimmer, G., Altmann, G.''' (1995). A model of morphological productivity. ''J. of Quantitative Linguistics 2, 212-216.''

[[Category:Unfertig]]

Polysemy

2006-08-05T12:05:12Z

Altmann:

'''1. Problem and history'''

The simplest problem connected with polysemy is its frequency distribution in the dictionary. One can easily see that in a monolingual dictionary lexical items have different number of explications. Sometimes the explications are numbered and indicate meaning difference or nuance. This is the only source of approaching the problem. The number of meanings of a word is sometimes called semantic size or semantic volume (cf. Tuldava 1998: 119). Some reserchers stated that the mean semantic volume of different word classes differs significantly (cf. Višnjakova 1976; Tuldava 1979; Ufimceva 1968: 89)
Historically, G.K. Zipf (1935, 1945a, 1949) was probably the first who has shown that meaning is associated with frequency, with length and other properties analyzed in synergetic linguistics (see Vol. 2). Nevertheless, the distribution of the semantic volume can be studied separately (see Introduction, Chapter 2), since it represents a loop in the system. Another question is the frequency of polysemic words in text.
The first trials to find a theoretical distribution can be found in Krylov, Jakubovskaja (1977), Krylov (1982) who derived the geometric distribution, Tuldava (1979) who used on empirical grounds a kind of exponential function and Levickij, Drebet, Kiiko (1999) who used rather the mixed geometric distribution, assuming that the words in the dictionary are of a mixed character. A good survey of problems can be found in Hoffmann (2001).

'''2. Hypothesis'''

''The number of words (<math>f_x</math>) with x meanings is a function of x.''

'''3. Derivation'''

'''3.1. Waring distribution'''

Wimmer and Altmann (1999a) considered the building of polysemy as a simple Poissonian birth-and-death process and set up the equations

(1)<math>\lambda_0 P_0 = \mu_1 P_1\quad</math>

<math>(\lambda _x + \mu_x)P_x = \lambda_{x-1}P_{x-1} + \mu_{x+1}P_{x+1}, \quad x= 1, 2,...</math>

Inserting <math>\lambda_x = a+x, \mu_x = a+b+x</math> and displacing the distribution one step to the right – because zero-semic words are not taken into account - they obtained the Waring distribution

(2)<math> P_x = \frac{ba^{(x-1)}}{(a+b)(a+b+1)^{(x-1)}},\quad x=1, 2, ...</math>

'''3.2. Bissinger-geometric distribution'''

Since the Waring distribution can be presented by a backward recurrence formula, Wimmer and Altmann (1999a) consider also the possibility of forward dependence using a parent distribution {Px* }x≥0 and building partial sums in the form

(3)<math> P_0= f_0 (P*_1 + P*_2 + P*_3 + P*_4 + ...)</math>

<math> P_1 = f_1 (P*_2 + P*_3 + P*_4 +...)</math>

<math> P_2 = f_2 (P*_3 + P*_4 +...)</math>

From the great number of possibilities (cf. Wimmer, Altmann 2000) they chose the simple form

(4)<math> f_1 = (P*_{i+1} + P*_{i+2} + ...) = c \sum_{j=i+1}^\infty\frac{P*_1}{j}, \quad i=0, 1, ...</math>

yielding <math> c = \frac{1}{1-P*_0}</math> as a normalizing constant.. Using the geometric distribution as parent distribution and displacing the result one step to the right they obtain

(5)<math>P_x = \frac{p}{q}\sum_{j=x}^\infty\frac{q^j}{j}, \quad x= 1, 2, 3, ...</math>

known as Bissinger-geometric distribution (cf. Wimmer, Altmann 1999).

'''Example:''' Polysemy in Maori (New Zealand)

Using the data ascertained by V. Krupa, Wimmer and Altmann (1999a) fitted the Waring and the Bissinger-geometric distributions to Maori dictionary data as shown in Table 1.

<div align="center">[[Image:Tabelle1_P.jpg]]</div>

[[[Please complete:

'''3.3. Tuldava´s version'''
<math>y_x = a \exp (-bx^c), \quad y_x </math> = number of words with x meanings

'''3.4. Krylov-Jakubovskaja´s version''' (

<math> y_x = e^{-bx}</math>]]]

'''4. Authors:''' U. Strauss, G. Altmann

'''5. References'''

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Wörterbuch zum deutschen Sprachgebrauch.''
(Hrsg. E. Agricola unter Mitwirkung von H. Görner
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'''Altmann, G.''' (1985). Semantische Diversifikation. ''Folia Linguistica 19, 177-200''.
Altmann, G., Bagheri, D., Goebl, H., Köhler, R., Prün, C. (2002). ''Einführung in die quantitative Lexikologie''. Götingen: Peust & Gutschmidt.

'''Altmann, G.,Beöthy, E., Best, K.-H.(1982), Die Bedeutungskomplexität der Wörter
und das Menzerathsche Gesetz. ''Zeitschrift für
Phonetik, Sprachwissenschaft und Kommunikationsforschung
35 (5), 537-543''.

'''Altmann, G., Best, K.-H., Kind, B.''' (1987). Eine Verallgemeinerung des Gesetzes der semantischen Diversifikation. ''Glottometrika 8, 130-139''.

'''Andreevskaja, A.V.''' (1990). Kvantitativnoe issle-dovanie polisemii kornevych slov russkogo jazyka
XI-XX vekov. ''Ucenye Zapiski Tartuskogo
Universiteta 912, 3-11''.

'''Andrukovic, P.F., Korol’ov, E.I. (1977). O statisticeskich i leksikogrammaticeskich svojstvach slov.
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1-9.''

'''Arapov, M.V.''' (1987). Upotrebitel’nost’ i
mnogoznacnost’ slova. In: ''Ucenye Zapiski Tartuskogo
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'''Drebet, V.V.''' (1996). Stilistische Kennzeichen der
polysemen Substantive in der deutschen Gegenwartssprache.
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'''Drebet, V.V., Levickij, V.V., Cherubim, D. (1996).
Morphologische Faktoren bei der Polysemie der
deutschen Adjektive. ''Naukovy Visnyk Cerniveckoho
Universytetu 1, 29-32.''

'''Fickermann, I., Markner-Jäger, B., Rothe, U.(1984). Wortlänge und Bedeutungskomplexität. Glottometrika 6, 115-126.'' Bochum: Brockmeyer.

'''Gindin, S.I.''' (1982). Castota slova i jego znacimost’
v sisteme jazyka. In: ''Lingvostatistika i vycislitelnaja
lingvistika 8. Tartu, 22-53.''

'''Guiraud, P.''' (1954). ''Les caractèrs statistique du
vocabulaire''. Paris: Press universitaires.

'''Hammerl, R'''. (1991). ''Untersuchungen zur Struktur der Lexik: Aufbau eines lexikalischen Basismodells.'' Trier, WVT.

'''Hoffmann, Ch'''. (2001). Polylexie lexikalischer Einheiten in Texten. In: Uhlířova, L., Wimmer, G., Altmann, G., Köhler, R. (Eds.), ''Text as a linguistic paradigm: levels, constituents, constructs. Festschrift in honour of Ludek Hřebíček: 76-97''. Trier: WVT.

'''Kapatruk, M.D.''' (1980). Metody vyvcennja osnovnoho znacennja slova. ''Movoznavstvo 5, 75-77'''.

'''Kijko, S.V., Kijko, J.J.''' (1996a). Kvantytatyvne dos-lidžennja polisemii dijesliv sucasnoji nimec’koji
movy. ''Naukovy Visnyk Cerniveckoho Universytetu
1, 32-38.''

'''Kijko, J.J., Kijko, S.V.''' (1996b). Polysemie der Verben und ihre stilistische Markierung. ''Naukovy
Visnyk Cerniveckoho Universytetu 2, 60-64''.

'''Köhler, R.''' (1986). ''Zur linguistischen Synergetik.
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'''Krott, A.''' (2002). Ein funktionalanalytisches Modell der Wortbildung. In: Köhler, R. (ed.), ''Korpuslinguistische Untersuchungen in die quantitative und systemtheoretische Linguistik: 75-126''. http://ubt.opus.hbz-nrw.de/volltexte/2004/279/

'''Krylov, Ju.K'''. (1982a). Ob odnoj paradigme lingvostatističeskich raspredelenij. ''Acta et Commentationens Universitatis Tartuensis 628, 80-102''.

'''Krylov, Ju.K'''. (1982b). Eine Untersuchung statistischer Gesetzmäßigkeiten auf der paradigmatischen Ebene der Lexik natürlicher Sprachen. In: Guiter, H., Arapov, M.V. (eds.), ''Studies on Zipf´s law: 234-262''. Bochum: Brockmeyer.

'''Krylov, Ju.K., Jakubovskaja, M.D.''' (1977), Statisticeskij analiz polisemii kak jazykovoj
universalii i problema semanticeskogo toždestva slova. ''Naucno-techniceskaja informacija, ser. 2,(3), 1-6.'''

'''Kucera, H., Francis, W. N.''' (eds.)(1967). ''Computational analysis of present-day American English.'' Providence, N.J.: Brown University Press.

'''Lehrer, A.''' (1974). Homonymy and polysemy:
measuring similarity of meaning. ''Language
Sciences 32, 33-39.''

'''Levickij, V. V.''' (1985). Opyt eksperimental’nogo
razgranicenija leksiceskoj polisemii i
omonimii. In: ''Psicholingvisticeskie issledovanija.
Leksika. Fonetika: 4-14''. Kalinin: Kalininskij Universitet.

'''Levickij, V'''. (2005). Polysemie. In: Köhler, R., Altmann, G., Piotrowski, R.G. (eds.), ''Handbook of Quantitative Linguistics: 458-464.'' Berlin: de gruyter.

'''Levickij, V.V., Drebet, V.V., Kijko, S.V.''' (1999). Some quantitative characteristics of polysemy of verbs, nouns and adjectives in the German language. ''J. of Quantitative Linguistics 6, 172-187''.

'''Levickij, V.V., Drebet, V.V., Kijko S.V.''' (1999).
Some Quantitative Characteristics of Polysemy of
Verbs, Nouns and Adjectives in the German Language.
''Journal of Quantitative Linguistics 6(2),
192-197.''

'''Levickij, V.V., Kijko, J.J., Spolnicka, S.V.'''
(1996). Quantitative analysis of verb polysemy in
modern German. ''Journal of Quantitative Linguistics
3 (2), 132-135.''

'''Malov, A. V. (1988). Rangovye polisemicskie ras-predelenija leksiki tolkovych slovarej russkogo i
anglijskogo jazykov. ''Ucenye Zapiski Tartuskogo
Universiteta 827, 111-115.''

'''Moskovic, V.A.''' (1969). ''Statistika i semantika.''
Moskva: Nauka.

''Muravycka, M.P.''' (1975). Psycholinhvistycnyj analiz
leksycnoji omonimiji. ''Movoznavstvo 3, 59-67.''

'''Obuchova, N.V.''' (1986). O specifike raspredelenija
mnogoznacnosti leksiceskich jedinic v kitajskom
jazyke. ''Ucenye Zapiski Tartuskogo Universiteta
745, 119-128''.

'''Olšanskij, J.G., Skiba, V.P.''' (1987). ''Leksiceskaja
polisemija v sisteme jazyka i tekste.'' Kišynjov: Štiinca.

'''Papp, F.O. (1967)., O nekotorych kolicestvennych
charakteristikach slovarnogo sostava jazyka. ''Slavia, vii, 51-58''.

'''Polikarpov, A.A.''' (1987). Polisemija. Sistemno-kvantitativnye aspekty. In: Tuldava, J.A. (ed.), ''Kvantitativnaja lingvistika i avtomatičeskij analiz tekstov: 135-154. Tartu: Učenye zapiski Tartuskogo gosudarstvennogo universiteta 774.''

'''Polikarpov, A. A.''' (1990). Leksi¼ñêôœ`eskaja polisemija v evolucionnom aspekte. ''Ucenye Zapiski
Tartuskogo Universiteta 974, 77-86''. Tartu.

'''Polikarpov, A.A., Krjukova, O.S.''' (1989). O
sistemnom sootnošenii kratkogo i srednego tolkovych
slovarej russkogo jazyka. ''Ucenye Zapiski
Tartuskogo Universiteta 872, 111-125.''

'''Polikarpov, A.A., Kurlov, V.J.''' (1994). Stilistika,
semantika, grammatika: opyt analiza sistemnych
vzaimosvjazej (po dannym tolkovogo slovarja).
''Voprosy jazykoznanija 1, 62-82.''

'''Rothe, U.''' (1983). Wortlänge und Bedeutungsmenge. Eine Untersuchung zum Menzerathschen Gesetz an drei romanischen Sprachen. ''Glottometrika 5, 101-112''.

'''Sambor, J.''' (1984). Menzerath’s law and the
polysemy of words. ''Glottometrika 6, 94-114''. Bochum: Brockmeyer.

'''Schierholz, S.J.''' (1991). ''Lexikologische Analysen
zur Abstraktheit, Häufigkeit und Polysemie
deutscher Substantive''. Tübingen: Niemeyer.

'''Tuldava, J. '''(1979). O nekotorych kvantitativno-
sistemnych charakteristikach polisemii. ''Ucenye Zapiski Tartuskogo Universiteta 502, 107-141.''

'''Tuldava, J.''' (1987). ''Problemy i metody kvanti-tativno-sistemnogo issledovanija leksiki''. Tallinn:
Valgus.

'''Tuldava, J.''' (1998). ''Probleme und Methoden der quantitativ-systemischen Lexikologie''. Trier: WVT.

'''Višnjakova, S.M.''' (1976). Opyt statisti¼ñêôœ`eskogo issledovanija mnogozna¼ñêôœ`nosti slov anglijskogo jazyka.
''Vycislitel’naja lingvistika. 168-178.''

'''Wimmer, G., Altmann, G'''. (1999a). Rozdelenie polysémie v maorijčine. In: Ondrejovič, S., Genzor, J. (eds.), ''Pange lingua. Zborník na počest´ Viktora Krupu: 17-24''. Bratislava: Veda.

'''Wimmer, G., Altmann, G'''. (2000). On the generalization of the STER distribution applied to generalized hypergeometric parents. ''Acta Universitatis Palackiensis Olomouciensis, Facultats rerum naturalium Mathematica 39, 215-247''.

'''Wolff, D'''. (1972). Bedeutungshäufigkeit und ihr statistisches Verhalten. ''Beiträge zur Linguistik und Informationsverarbeitung 22, 33-44''.

'''Zipf, G.K.''' (1935).''The psycho-biology of language''. Boston: Houghton Mifflin.

'''Zipf, G.K'''. (1945a). The meaning-frequency relationship of words. ''J. of General Psychology 33, 251-255''.

'''Zipf, G.K.''' (1949). ''Human behavior and the principle of least effort''. Cambridge: Addison-Wesley.

Repeat rate and inventory of phonemes

2006-07-26T14:36:51Z

Altmann:

'''1. Problem and history'''

The repeat rate is a measure expressing the diversity of relative frequencies of the elements of a closed system. For computing it, the data must be normalized, i.e. the functions used in ”Phoneme frequencies” (<math>rightarrow</math>) must be considered probability functions. Its expression is

(1)<math> R = \sum_{x=1}^K p_x^2</math>

weher <math>p_x = f_x/N</math>, N being the number of phonemes in the sample, fx is the absolute frequency of phoneme x, K is the number of phonemes in the inventory (inventory size). The problem is to find whether R is associated with the inventory size K.

Lehfeldt and Altmann (1983) have shown that R is lawlikely joined with K if the frequencies are distributed according to the right truncated geometric distribution. Zörnig and Altmann (1983) have shown that the relation holds if the frequencies follow the Zipf-Mandelbrot distribution (see Ranking).

'''2. Hypothesis'''

''The repeat rate of phonemes R is a monotone decreasing function of the phoneme invetory K.''

'''3. Derivation'''

'''3.1. From the geometric distribution'''.
Let the rank-ordered phoneme frequencies follow the 1-displaced right truncated geometric distribution, defined as

(2)<math> P_x = aq^{x-1},\quad x = 1, 2, 3, ..., K</math> .

with

(3)<math> a = \frac{p}{1-q^K}</math>

from which

(4)<math> q^K = \frac{a-1+q}{a}</math> .

For the Repeat rate we have

(5)<math> R = \sum_{x=1}^K (aq^{x-1})^2 = \frac{a^2(1-q^{2K})}{1-q^2}</math>.

Using (4) from which <math>q^{2K} = (a-1+q)^2/a^2</math> one obtains

(6)<math>R = \frac{2a+q-1}{1+q}</math>.

The first approximation to a theoretical value of R is given by setting

(7)<math> q = \frac{K-2}{K+2}</math> .

Inserting (3) and (7) in (6) and neglecting <math>q^K</math> which is very small as compared to the other values one obtains

(8)<math> R =\frac{2}{k}</math>,

i.e. the Repaet rate is a very simple function of the inventory size.

'''Example'''. Repeat rate for 63 languages

Altmann and Lehfeldt (1980: 154-156) collected the R-values for 63 languages from the published literature and compared them with the theoretical value (8) . The result is shown in Table 1 and Figure 1. Some of the points concern letters, some are for phoneme in texts and in dictionary, in some cases alternative counting has been performed (e.g. considering long vowels as one or two phonemes, etc.).

<div align="center">[[Image:Tabelle1_RRAIOP.jpg]]</div>

'''3.2. From the Zipf-Mandelbrot distribution'''

Zörnig and Altmann (1983) used the Zipf-Mandelbrot distribution to derive R, i.e.

(9)<math> P_x = \frac{A}{(B+x)^c}, \quad x = 1, 2, ..., K</math>.

Since <math> R = A^2 \sum_{x=1}^K (B+x)^{-2c}</math>

where A is the normalizing constant. Using an approximation of the sum by an appropriate integral they obtained

(10)<math> R = \begin{cases} \frac{(1-c)^2\lbrack (B+K)^{1-2c} - (B+1)\rbrack ^{1-2c}}{(1-2c) \lbrack (B+K)^{1-c} - (B+1)^{1-c}\rbrack ^2},\quad c \ne 1, c\ne 0.5 \\ \frac{(1-c)^2}{\lbrack (B+K)^{1-c}-(B+1)^{1-c}\rbrack ^2}\ln \frac{B+K}{B+1},\quad c = 0.5 \\\frac{(B+1)^{-1}-(B+K)^{-1}}{\ln{B+K \choose B+1}^2},\quad c=1 \end{cases}</math>

By an iterative procedure they have shown that the best fitting to the existing data is the third formula in (10) i.e. with c = 1 and B = 0.61, thus

(11)<math> R = \frac{1.61^{-1}-(0.61 + K)^{-1}}{\left( \ln \frac{0.61 + K}{1.61} \right)^2}</math>

which can be seen in Fig. 2. The results of fitting are in Table 2. The parameter B could not be interpreted as yet. It must be computed anew at any enlarging of the stock of examined languages.

<div align="center">[[Image:Tabelle2_RRAIOP.jpg]]</div>

Both results are satisfactory. However, other formulas in Phoneme frequency (<math>\rightarrow</math>) can be tested, too. Of course, the best fit follows from <math>R = aK^{-b}</math> but considering the fact that the data are very mixed (letters, phonemes, texts, dictionaries, different interpretations etc.) the result is a preliminary stimulus for further research.

<div align="center">[[Image:Grafik1_RRAIOP.jpg]]</div>
<div align="center">Figure 2. Fitting formulas (8) –––––– and (11) -------- to Repeat rates in 63 languages</div>

'''4. Authors:''' U. Strauss, G. Altmann, V. Kromer

'''5. References'''

'''Altmann, G., Lehfeldt, W'''. (1980). ''Einführung in die quantitative Phonologie''. Buchum: Brockmeyer.

'''Zörnig, P., Altmann, G.''' (1983). The repeat rate of phoneme frequencies and the Zipf-Mandel-brot law. ''Glottometrika 5, 205-211''.

'''Zörnig, P., Altmann, G'''. (1984). The entropy of phoneme frequencies and the Zipf-Mandelbrot law. ''Glottometrika 6, 41-47''.

Polysemy and synonymy of words

2006-07-26T14:35:45Z

Altmann:

'''1. Problem and history'''

The problem is very simple: If a word obtains new meanings, i.e. if its polysemy increases, then it penetrates into the semantic domain of other words. Consequently, these words become its synonyms. Thus the more polysemic a word the more synonyms it has.
In dictionaries the different meanings are usually denoted with numbers or letters. Counting them is the simplest way of measurement. Synonym can be found in a dictionary of synonyms.
The hypothesis was set up by Köhler (1990:8) in the framework of synergetic linguistics. Its testing was performed by Ziegler and Altmann (2001).

'''2. Hypothesis'''

''The greater the polysemy of a word the more synonyms it has''.

'''3. Derivation'''

The relative increase of synonymy (y) is proportional to the relative increase of polysemy (x), i.e.

(1)<math> \frac{dy}{y} = \frac{b}{x}dx</math>

resulting in the power law

(2)<math> y = Ax^b\quad</math> .

In Köhler´s synergetic sense

<div align="center">[[Image:Grafik11_PaS.jpg]]</div>

where L-Polysemy and L-Synonymy are given in logarithmic units, b is a constant and u is a linguistic factor which gets a special interpretation. Thus we have

L-Synonymy = u + b(L-Polysemy).

Taking antilogarithms we obtain the same result as in (2), wehere <math>A = e^u</math>.

'''Example'''. Italian dictionary

Ziegler and Altmann (2001) measured the polysemy of 1079 words from the Italian dictionary by Deveto and Oli (1995) using the presentation in the dictionary. For each of these words they counted the number of synonyms in the synonymy dictionary by Stopelli (1998) and obtained the data in Table 1.

<div align="center">[[Image:Tabelle1_PaS.jpg]]</div>

In Table 2 one finds the mean number of synonyms for x-polysemnatic words and the theoretical curve (2). Since data for x  9 are not represntative, they were not taken into account.

<div align="center">[[Image:Tabelle2_PaS.jpg]]</div>

The theoretical curve is y = 0.9690x1.4192 and D = 0.97. Further languages must be analyzed. The empirical points and the theoretical curve are in Fig. 1

<div align="center">[[Image:Grafik1_PaS.jpg]]</div>
<div align="center">Fig. 1. Dependence of mean synonymy on polysemy</div>

'''4. Authors:''' U. Strauss, G. Altmann

'''5. References'''

'''Deveto, G., Oli, G.C'''. (1995). ''Il dizionario della lingua italiana''. Firenze: Le Monnier.

'''Köhler, R.''' (1990). Linguistische Analyseebenen, Hierarchisierung und Erklärung im Modell der sprachlichen Selbstregulation. ''Glottometrika 11, 1-18''.

'''Stopelli, P'''. (1998). ''Sinonimi e contrari con generici, spezifici, analoghi, inversi''. Milano: Garzanti.

'''Ziegler, A., Altmann, G'''. (2001). Die Beziehung zwischen Synonymie und Polysemie. In: Ondrejovič, S., Považaj, M. (eds.), ''Lexicographica ´99: 226-229''. Bratislava: Veda

Polysemy and age

2006-07-26T14:34:39Z

Altmann:

'''1. Problem and history'''

The longer a word exists the more meanings it can acquire. It does not hold generally for any word but it holds on the average for ensembles of words that came into existence in the same time interval. The greatest problem is that even if one has a historical dictionary, the time of birth is merely approximate, since dictionaries vonsider the first appearing in texts.
The first count has been performed by Wollf (1972) who counted the present number of meanings of English words that appeared in different centuries and periods of the evolution of English. The fate of particular Japanese words has been considered by Sanada (1999) but there are merely graphs, no numbers. A theoretical approach has been made by Strauss and Altmann (2003) which is in agreement with the “unified derivation” proposed by Wimmer and Altmann (2005).

'''2. Hypothesis'''

''The older an ensemble of words coming into existence in the same time interval, the more meanings have the words on the average.''

'''3. Derivation'''

The relative rate of polysemy increase is proportional to the relative rate of time interval change. Since words are not differentiated according to frequency, length, polytexty etc. all these factors are considered a constant ceteris paribus condition. In the simplest case the proportionality factor and the ceteris paribus constant are equal, yielding the equation

(1)<math> \frac{dy}{y} = \begin{pmatrix} b + \frac{b}{x}\end{pmatrix}dx</math>

where x is the time interval and y is the polysemy. The solution is

(2)<math> y= ax^b e^{bx}\quad</math>

where <math>a = e^C</math>, C being the integration constant. The curve goes to infinity but Wolff found even words with 95 meanings. There is surely a finite limit being in agreement with the principles of synergetic linguistics but it cannot preliminarily be deduced.

'''Example''': Age and polysemy in English

Wolff (1972) considered the age of English words and their polysemy, unfortunately truncating the polysemy above 10. Strauss and Altmann (2003) computed the mean polysemy for individual time intervals and fitted (2) to the data. The results are presented in Table 1, where the time index shows the time depth. The constant a ≈ 1. The graphical representation can bee seen in Fig. 1.

<div align="center">[[Image:Tabelle11_PaA.jpg]]</div>

<div align="center">[[Image:Grafik1_PaA.jpg]]</div>
<div align="center">Fig. 1. Empirical and computed values of the age-polysemy relation</div>

'''4. Authors:''' U. Strauss, G. Altmann

'''5. References'''

'''Sanada, H'''. (1999). Analysis of Japanese vocabulary by the theory of synergetic linguistics. '''J. of Quantitative Linguistics 6, 239-251'''.

'''Strauss, U., Altmann, G.''' (2003). Age and polysemy of words. ''Glottometrics 6, 61-64''.

'''Wimmer, G., Altmann, G.''' (2005). Unified derivation of some linguistic laws. In: Peter Grzybek (ed.), ''Contributions to the Science of Language. Word Length Studies and Related Issues'' (in print). Dordrecht, NL: Kluwer.

'''Wolff, D.''' (1972). Bedeutungshäufigkeit und ihr statistisches Verhalten. ''Beiträge zur Linguistik und Informationsverarbeitung 22, 33-44''.

[[Category:Unfertig]]

Morphological productivity

2006-07-26T14:32:50Z

Altmann:

'''1. Problem and history'''

In the lexicon of a language there are two processes working:

(a) Birth of new lexemes by creating or borrowing

(b) Death by elimination of lexemes

Birth means either creation of new words or constructing them by morphological processes like derivation, reduplication, compounding etc. The construction is a case of (multidi-mensional) ''diversification'' (<math>rightarrow</math>). At the same time it is a case of specification of meaning and word prologation. Birth and death take place in rates called birth-rate <math>\lambda_x</math>and death-rate <math>\mu_x</math> respectively.
Since up to now only a case of derivation has been examined, the hypothesis must be restricted but we assume that it holds for other kinds of word building, too. It holds, of course, only under the condition that the given way of word buidling is actual in a language.
As far as known, the only model was proposed by Wimmer and Altmann (1995) in which the random variable x is the number of derivates built from a stem (x = 0,1,2,…) and <math>f_x</math> is the number of stems having x derivates.
Another kind of productivity examined by H. Baayen will be refered in the next volume.

'''2. Hypothesis'''

''Morphological productivity is the result of a birth-and-death process in which stems build new derivates and some derivates are eliminated. The probability of the number of stems having x derivates follows the Pólya distribution''.

'''3. Derivation'''

The creation of new morphological constructs is proportional to the class size x and can be generally symbolized as a + cx. Since creation cannot be infinite, there must be a limit n to it. The community controls the creation in the same linear way symbolized as b + (n-x-1)c. This results in the birth rate <math>\lambda = \frac{a+cx}{b+(n-x-1)c}\quad , x = 0, 1, ...n-1</math>. The death rate is similar, depending only on the class size and the morphological nature of language. It can be sym-bolized as <math>\mu = \frac{n}{n-x+1}\quad, x = 1, 2, ...,n</math>
Inserting the above birth-rate and death-rate in the balancing birth-and-death equations (see Appendix) and reparametrizing

p = a/(a+b), q = b/(a+b), s = c/(a+b)

we obtain the Pólya distribution (Wimmer, Altmann 1995)

(1)<math> P_x = \frac{{-p/s \choose x}{-q/s \choose n-x}}{{-1/s \choose n}}\quad, x= 0, 1, ..., n, s>0, =\le p\le1, q=1-p, n \epsilon N_0</math>
<math> \frac{p}{s}, \frac{q}{s}\ge = 0</math> or <math> \frac{p}{s}, \frac{q}{s}\le 0</math>

'''Example''': Indonesian stem productivity

Wimmer and Altmann (1995) studied the productivity of stems in the Indonesian dictionary (Echols, Shadily 1963). There were 6970 stems having no derivates, 1961 stems having each 1 derivate etc. The data and the results of fitting are presented in Table 1 and Fig. 1.

<div align="center">[[Image:Tabelle11_MP.jpg]]</div>

(((Den Graph mit Excell machen, da 0 fehlt)
Fig. 1. Fitting the Pólya distribution to the data in Table 1

'''4. Authors:''' U. Strauss, G. Altmann

'''5. References'''

'''Altmann, G., Bagheri, D., Goebl, H., Köhler, R., Prün, C'''. (2002). ''Einführung in die quantitative Lexikologie''. Götingen: Peust & Gutschmidt.

'''Echols, J.M., Shadily, H.''' (1963). ''An Indonesian-English dictionary''. Ithaca: Cornell University Press.

'''Wimmer, G., Altmann, G.''' (1995). A model of morphological productivity. ''J. of Quantitative Linguistics 2, 212-216.''

[[Category:Unfertig]]

Lexicon growth

2006-07-26T14:31:27Z

Altmann:

'''1. Problem and history'''

Every lexicon grows. New lexemes are added by creation, by combining of existing elements and by borrowing. The problem is to find how the whole or its special parts change.

Historically, this research goes back to Piotrovskij, Bektaev, Piotrovskaja (1977), Beöthy, Altmann (1982) and Tuldava (1984, 1988). A lot of testing has been performed in Best, Kohlhase (1983).

Best (2003a:117) has shown that in some cases lexicon reduction abides by the same law.

'''2. Hypothesis'''

''The growth of the vocabulary is a time-dependent process''.

“Growth” means the increase of vocabulary in terms of numbers of items in general or of special items like borrowings or terminology.
“Vocabulary” can be the vocabulary of a child, that of a field, that of the whole language, etc.

'''3. Derivation'''

'''3.1. The Piotrovskij et al. approach (1977), also Tuldava (1984)'''

The relative rate of change of the vocabulary size (L) in a time period t is constant, or the rate of change of the vocabulary is proportional to its actual state, i.e.

<math> \frac{dL}{L}= a dt</math>

yielding

(1)<math> L_t = L_0 e^{at}\quad</math> , t = time period,

<math> \quad L_t</math> = vocabulary size at the end of t

<math> \quad L_0</math> = vocabulary size at the beginning of the measurement

Though this approach is realistic, it cannot be effectively tested beacause (a) there are no complete dictionaries encompassing all fields of human activity, (b) personal vocabularies cannot be infinite.

'''3.2. The Beöthy – Altmann – Tuldava approach'''

Beöthy, Altmann (1982) used this approach to borrowings, Tuldava (1984) to the increase of the vocabulary of literary language. The rate of change of the vocabulary size is proportional to its present state (<math>L_n</math>) and to the difference to the limiting boundary (Ln) (the limiting boundary can be the capacity of the memory or that of printed dictionaries), i.e.

<math> \frac{dL_t}{dt}= aL_t(L_n - L_t)</math>

yielding

(2)<math> L-t = \frac{L_n}{1+be^{-ct}} </math>

where a nd b are parameters. This is the logistic curve used also with language change (<math>\rightarrow</math>)

'''Example''': The growth of the Estonian literary lexicon

Tuldava (1998: 140) shows the increase of the Estonian literary lexicon in the 17th to 20th century based on normative dictionaries. He transforms t = (t´- 1600)/100 (see Table 1 and Fig. 1).

<div align="center">[[Image:Tabelle111_LG.jpg]]</div>

<div align="center">[[Image:Grafik1_LG.jpg]]</div>
<div align="center">Fig. 1. Growth of Estonian literary vocabulary (Tuldava 1998)</div>

Evidently, formula (2) yields somewhat better results (in this case) and is more realistic.

'''Example''': Borrowing of Latin words in Hungarian

Beöthy and Altmann (1982) show the extent of borrowing of Latin words in Hungarian in Table 2 and Fig. 2.

<div align="center">[[Image:Tabelle22_LG.jpg]]</div>

<div align="center">[[Image:Grafik2_LG.jpg]]</div>
<div align="center">Fig. 2. Increase of Latin borrowings in Hungarian (Beöthy, Altmann 1982)</div>

The fit is almost perfect.

'''4. Authors''': U. Strauss, G. Altmann, K.-H. Best

'''5. References'''

'''Beöthy, E., Altmann, G'''. (1982). Das Piotrowski-Gesetz und der Lehnwortschatz. ''Zs. für Sprachwissenschaft 1, 171-178''.

'''Best, K.-H.''' (2001). Ein Beitrag zur Fremdwortdiskussion. In: S. J. Schierholz, E. Fobbe, S. Goes, R. Knirsch (eds.), ''Die deutsche Sprache in der Gegenwart. Festschrift f. Dieter Cherubim zum 60. Geburtstag: 263-270''. Frankfurt u.a.: Lang.

'''Best, K.-H.''' (2001a). Wo kommen die deutschen Fremdwörter her? ''Göttinger Beiträge zur Sprachwissenschaft 5, 6-20''.

'''Best, K.-H'''. (2001b). Der Zuwachs der Wörter auf ''-ical'' im Deutschen. ''Glottometrics 2, 11-16''.

'''Best, K.H'''. (2003a). ''Quantitative Linguistik. Eine Annäherung. 2.,'' überarbeitete und erweiterte Auflage. Göttingen: Peust & Gutschmidt.

'''Best, K.-H'''. (2003b). Anglizismen - quantitativ. ''Göttinger Beiträge zur Sprachwissenschaft 8, 7-23.''

'''Best, K.-H'''. (2003c). Slawische Entlehnungen im Deutschen. In: S. Kempgen, U. Schweier, T. Berger (eds.), ''Rusistika × Slavistika × Lingvistika. Festschrift für Werner Lehfeldt: 464-473'' München: Vlg. Otto Sagner.

'''Best, K.-H.''' (2003d). Spracherwerb, Sprachwandel und Wortschatzwachstum in Texten. Zur Reichweite des Piotrowski-Gesetzes. ''Glottometrics 6, 9-34''.

'''Best, K.-H'''. (2003e). Zur Entwicklung von Wortschatz und Redefähigkeit bei Kindern. ''Göttinger Beiträge zur Sprachwissenschaft 9, 7-20''.

'''Best, K.-H.''' (2003f). Wie verläuft Sprachwandel? ''Naukovij Visnik Černivec´kogo Univeristetu 155, 86-94.''

'''Best, K.-H.''' (2004). Das Fremdwort aus der Sicht der Quantitativen Linguistik. In: S. Wichter, O. Stenschke, M. Tants (eds.), ''Theorie, Steuerung und Medien des Wissenstransfers: 89-99''. Frankfurt: Lang.

'''Best, K.-H'''. (2004a). Zur Ausbreitung von Wörtern arabischer Herkunft im Deutschen. ''Glottometrics 8, 74-77''.

'''Best, K.-H., Altmann, G'''. (1986). Untersuchungen zur Gesetzmäßigkeit von Entlehnungsprozessen im Deutschen. ''Folia Linguistica Historica 31-41''.

'''Best, K.-H., Kohlhase, J'''. (eds.) (1983). ''Exakte Sprachwandelforschung''. Göttingen, Herodot.

'''Hentschel, G'''. (1995). Zur ‚Seuche‘ des deutschen Lehnwortes im Polnischen und zu den ‚Selbstheilungskräften‘ dagegen. In: A. Bochnakowa et S. Wid»ak (eds.), ''Munus amicitiae. Studia Linguistica in honorem Witoldi Mańczak k septuagenarii: 69-78''. Universitas Iagellonica, Ser. Varia CCCLVI, Cracowiae.

'''Kempgen, S'''. (1990). Zur Modellierung von Lehnbeziehungen. In: W. Breu (Hrsg.), ''Slavistische Linguistik 1989: 99-116''. München: Sagner.

'''Körner, H'''. (2001). Der Zuwachs der Wörter auf -ion im Deutschen. ''Glottometrics 2, 82-86''.

'''Körner, H'''. (2003). ''Wortschatzentwicklung im Deutschen. Untersuchung zur Überprüfung des Piotrowski-Gesetzes''. Göttingen: Magisterarbeit.

'''Körner, H'''. (2004). Zur Entwicklung des deutschen (Lehn)Wortschatzes. ''Glottometrics 7, 25-49''.

'''Lazard, G'''. (1965). Les empruntes arabes dans la prose persane du X-e au XII-e siècle : aperçu statistique. ''Revue de l´Ecole National des language orientales 2, 53-67.''

'''Müller-Hasemann, W'''. (1983). Das Eindringen englischer Wörter ins Deutsche ab 1945. In: Best, Kohlhase (eds.) 1983: 143-160.

'''Piotrovskij, R.G., Bektaev, K.B., Piotrovskaja, A.A'''. (1977). ''Matematičeskaja lingvistika''. Moskva: Nauka.

'''Piotrowski, R.G., Bektaev, K.B., Piotrovskaja, A.A'''. (1985). ''Mathematische Linguistik''. Bochum: Brockmeyer.

'''Tuldava, J'''. (1987). ''Problemy i metody kvantitativno-sistemnogo issledovanija leksiki''. Tallinn: Valgus.

'''Tuldava, J'''. (1998). ''Probleme und Methoden der quantitativ-systemischen Lexikologie.'' Trier: WVT.

'''Wagner, K.R., Altmann, G., Köhler, R'''. (1986). Zum Gesamtwortschatz der Kinder. In: Wagner, K.R. (ed.), ''Wortschatz-Erwerb:'' 128-142. Bern: Lang.

Hreb length

2006-07-26T14:29:48Z

Altmann:

'''1. Problem and history'''

Up to now two kinds of “hrebs” have been established:

''Sentence hreb'' is a set of sentences of the text containing the same word, the same denotation or the same reference. Hřebíček calls them “(sentence) aggregates” or “larger context”.

Denotation hreb is a set of elements of a text having the same denotation. There can be word hrebs, considering merely the denotation of words, morpheme hrebs considering the reference of all morphemes or phrase hrebs.

The criteria for establishing hrebs are not yet unique.

The discoverer of these units is Hřebíček (1990, 1992, 1993, 1995, 1995a, 1997) who examined their length especially in Turkish texts and used exclusively the Zipf-Alekseev distribution. Schwarz (1995) extended the research to German texts and used also the Waring distribution. Ziegler and Altmann (2002) developed some theoretical issues concerning denotation hrebs and baptized these entities to “hrebs”.

'''2. Hypothesis'''

''Hreb length abides by the usual length distribution following from the unified theory (<math>\rightarrow</math>) or the Zipf-Alekseev distribution used in word association research (<math>\rightarrow</math>).''

'''3. Derivation'''

In the usual “length”-approach, <math>P_{x+1}</math> = <math>g(x)P_x</math>, one inserts g(x) = (n+x-1)/(b+n+x) and obtains the 1-displaced version of the Waring distribution

(1)<math>P_x = \frac{bn^{x-1}}{(b+n)(b+n+1)^{x-1}}, \quad x = 1, 2, ...</math>

For the derivation of the Zipf-Alekseev distribution see “Word associations” (<math>\rightarrow</math>).

'''Example'''. Sentence hreb length in a German text (Schwarz 1995)

Schwarz analyzed a German text (F. Schirmacher, “Dem Druck des härteren, strengeren Lebens standhalten.“ Frankfurter Allgemeine Zeitung 2.6.1990, Nr. 127) and obtained the results in Table 1 and Figures 1 and 2.

<div align="center">[[Image:Tabelle11_H-R.jpg]]</div>

In this case, the Zipf-Alekseev distribution yields sowhat better results, inspite of the fact that it has 4 paramaters. However, bothe fittings are satisfactory.

<div align="center">[[Image:Grafik1_HR.jpg]]</div>
<div align="center">Fig. 1. Fitting the Waring distribution to hreb length of German data (Schwarz 1995)</div>

<div align="center">[[Image:Grafik2_HR.jpg]]</div>
<div align="center">Fig. 2. Fitting the modified Zipf-Aslekseev distribution to German data (Schwarz 1995)</div>

'''Example''': Denotation hreb length in an Early New High German text (Ziegler, Altmann 2002).

Ziegler and Altmann examined a letter form an Archive in Bratislava written in Early New High German segmented to denotation hrebs. The results are given in Table 2. This time the Waring distribution is slightly better.

<div align="center">[[Image:Tabelle22_H-R.jpg]]</div>

'''4. Authors:'''U. Strauss, G. Altmann

'''5. References:'''

'''Hřebíček, L.''' (1992). ''Text in communication: Supra-sentence structure''. Bochum, Brockmeyer.

'''Hřebíček, L'''. (1993a). Text as a construct of aggregations. In: Köhler, R., Rieger, B. (eds.), ''Contributions to quantitative linguistics. Dordrecht: Kluwer: 33-39.''

'''Hřebíček, L.''' (1995). ''Text levels. Language constructs, constituents and Menzerath-Altmann law.'' Trier: WVT.

'''Hřebíček, L.''' (1996). Word associations and text. ''Glottometrika 15, 12-17''.

'''Hřebíček, L.''' (1997). ''Lectures on text theory''. Prague: Oriental Institute.

'''Schwarz, C.''' (1995). The distribution of aggregates in texts. ''ZeT – Zeitschrift für empirische Textforschung 2, 62-66''

'''Ziegler, A., Altmann, G'''. (2002). ''Denotative Textanalyse''. Wien: Edition Praesens.

'''Uhlirova 2003;''' (Rez.), Lehfeldt 2003 (Rez.), Andersen 2003 Rez.; Grzybek 2003 Rez.; (Ergänzen später)

Change in language

2006-07-26T12:00:56Z

Altmann:

'''1. Problem and history'''

Everything in language changes. The complete complex of causes can not be ascertained; we are interested in the general processes of change and its course, whatever the entity con¬cerned. There are four aspects treated in this volume, for which models could be set up:

Qualitative change

(i) change of individual entities, which is the object of this chapter

(ii) sound change (<math>\rightarrow</math>)

Volume change

(iii) lexicon growth (<math>\rightarrow</math>)

(iv) lexicon decay or glottochronology (<math>\rightarrow</math>)

The research in quantitative form most probably began with different hypotheses of G. K. Zipf concerning age and frequency, age and length, etc. (1946, 1947, 1949, cf. Prün 1985). Measurement concerning individual phenomena can be found in Piotrowski (1960), Graudina (1964), Lazard (1965), whose results have been empirically fitted by Piotrovskaja, Piotrowski (1974) using an arctangent function. The theoretical derivation has been performed by Beö¬thy, Altmann (1982), Altmann, von Buttlar, Rott, Strauß (1983) combining Piotrowski´s find¬ings with an assumption of Weinreich, Labov, Herzog (1968). Altmann (1983) derived the three possible variants of the law shown below. A number of corroborations was brought by Best (1983), Best, Kohlhase (1983, 1983a), Imsiepen (1983), Kohlhase (1983), Müller-Hase¬mann (1983), Best, Altmann (1986), Kroch (1989a,b, 2001), Best, Beöthy, Altmann (1990), Tuldava (1998), Best (2001), Bresnan, Dingare, Manning (2001), Vulanović (2003). The law is called Piotrowski law or Piotrowski-Altmann law and is used for modelling phenomena like the increase of the number of borrowings, changes in morphology, etc.

'''2. Hypothesis'''

Everything in language changes as a result of interaction between old forms and new forms.

The independent variable is time, given usually in form of a transformed time index.
The dependent variable is the proportion of new forms.

'''3. Derivation'''

The interaction can be presented in the form

(1) <math>dp_t=k_tp_t(C-p_t)dt\quad\quad</math>,

<math>p_t</math> = proportion of new forms

<math>pk_t</math> = a function of time (can also be a constant)

C = limit of change

t > 0, time

<math>dp_t</math> = change of the proportion

telling that the change of the proportion of new forms is proportional to the interaction of new and old forms. The solution yields three variants:

'''(a) Complete change''' if C = 1 and <math>k_t = b</math> is constant

(2) <math>p=\frac{1}{1+ae^{-bt}}</math>

where a is the integration constant. The result in (2) represents the so-called logistic curve used in different disciplines for modeling growth phenomena.

'''Example'''. Since years represent large numbers hindering the fitting, one usually transforms the time intervals in a time variable beginning with t = 1. Often the cumulative frequencies are changed to cumulative proportions, or the proportions are ascertained from the occurrence of rival forms.
Complete change: The replacement of –{t} by –{st} in German 2nd person singular indicative present time with the verb “wollen”, shown by Best (2003a). The result of fitting is presented in Table 1 and Fig. 1. Here ft is the relative frequency of –{st}, pt is the computed relative frequency according to (2).

<div align="center">[[Image:Figur11_CiL.jpg]]</div>

The fitting is excellent. The curve was fitted to the proportion of –{st} found in the sources.

<div align="center">[[Image:Grafik1_CiL.jpg]]</div>

<div align="center">Fig. 1. The result presented in Table 1</div>

(b) '''Partial change''' if <math>k-t = b</math> is constant, C is the asymptote

(3) <math>p=\frac{C}{1+ae^{-bt}}</math>

and a is the integration constant.

Example. Partial change: Borrowings from Latin in Hungarian
Beöthy and Altmann examined the borrowing from different langugages in Hungarian. The fate of Latin words is shown in Table 2 and Fig. 2. The fitting was performed for the cumulative values.

<div align="center">[[Image:Figur22_CiL.jpg]]</div>

<div align="center">[[Image:Grafik2_CiL.jpg]]</div>
<div align="center">Fig. 2. Fitting formula (3) to data in Table 2</div>

(c) '''Reversible change''' if <math>k_t</math> = a´ - b´t, C = constant

(4)<math>p_t=\frac{C}{1+ae^{-bt+ct^2}}</math>

where a, b, c are simple functions of a´, b´and C.

Example 3. Reversible change: Epithesis with strong verbs in German. Imsiepen (1983) observed that the epithesis of /e/ with strong verbs (1st and 3rd person sg. Past tense) is a reversible process in German. Altmann (1983) has shown some estimation procedures for this curve, but since several observed values are very unreliable, Best, Beöthy and Altmann (1990) considered smoothed values (moving average of 7 values) and obtained the data in Table 3. They took the value of C into consideration and used formula (4).

<div align="center">[[Image:Tabelle33_CiL.jpg]]</div>

The fitting is very good.

<div align="center">[[Image:Grafik4_CiL.jpg]]</div>

'''4. Authors:''' U. Strauss, G. Altmann

'''5. References'''

'''Altmann, G.''' (1983a). Das Piotrowski-Gesetz und seine Verallgemeinerungen. In: Best, K.-H., Kohlhase, J. (Hrsg.), ''Exakte Sprachwandelforschung: 54-90''. Göttingen: Herodot.

'''Altmann, G.''' (1985). On the Dynamic Approach to Language. In: Ballmer, T. T. (ed.), ''Linguistic Dynamics: 181-189''. Berlin/ New York: de Gruyter.

'''Altmann, G.''' (1992). Piotrowski’s Law of Language Change. In: Saukkonen, P. (ed.), ''What is Language Synergetics? 34-35.'' Oulu: Acta Universitatis Ouluensis, Series B: Humaniora, 16.

'''Altmann, G., Bagheri, D., Goebl, H., Köhler, R., Prün, C.''' (2002). ''Einführung in die quantitative Lexikologie.'' Götingen: Peust & Gutschmidt.

'''Altmann, G., v. Buttlar, H., Rott, W., Strauß, U.''' (1983). A law of change in language. In: Brainerd, B. (ed.), ''Historical linguistics: 104-115''. Bochum: Brockmeyer.

'''Bailey, Ch.J.N.''' (1973). ''Variation and linguistic theory.'' Arlington: Center for Applied Lin-guistics.

'''Beöthy, E., Altmann, G.''' (1982). Das Piotrowski-Gesetz und der Lehnwort¬schatz. ''Zs. für Sprachwissenschaft 1, 171-178.''

'''Best, K.-H.''' (1983). Zum morphologischen Wandel einiger deutscher Verben. In: Best, Kohlhase (eds.) 1983: 107-118.

'''Best, K.-H.''' (2000). Der Zuwachs der Wörter auf -ical im Deutschen. ''Glottometrics 2, 11-16.''

'''Best, K.-H.''' (2001). Ein Beitrag zur Fremdwortdiskussion. In: Schierholz, S.J., Fobbe, E., Goes, S., Knirsch, R. (eds.), ''Die deutsche Sprache der Gegenwart. Festschrift für Dieter Cherubim zum 60. Geburtstag: 263-270.'' Frankfurt: Lang.

'''Best, K.-H.''' (2002). Satzlängen im Deutschen: Verteilungen, Mittelwerte, Sprachwandel. ''Göttinger Beiträge zur Sprachwissenschaft 7, 7-31.''

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'''Best, K.-H.''' (2003c). Spracherwerb, Sprachwandel und Wortschatzwachstum in Texten. Zur Reichweite des Piotrowski-Gesetzes. ''Glottometrics 6, 9-34.''

'''Best, K.-H.''' (2003d). Zum Wandel von Idiolekten. ''Naukovyj Visnyk Černivec’koho Universytetu, Vypusk 165-166, 36-43.''

'''Best, K.-H.'''(2003e). Zur Entwicklung von Wortschatz und redefähigkeit bei Kindern. ''Göttinger Beiträge zur Sprachwissenschaft 9, 7-20.''

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'''Best, K.H., Kohlhase, J.''' (eds.) (1983). ''Exakte Sprachwandelforschung.'' Göttingen, Herodot.

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'''Best, K.-H., Zhu, Jinyang''' (2006). Sprachwandel im Chinesischen. ''Archiv orientální 74, 203-214.''

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'''Sankoff, D.''' (1971). Stochastic models for glottochronology. In: Hodson, F.R. et al. (eds.), ''Anglo-Romanian Conference on Mathematics in the Archaeological and Historical Sciences. Mamaia, Romania, 1970.'' Edinburgh: Edinburgh Univ. Press, 1971. S.381-386.

'''Sankoff, D.''' (1972). Lexical replacement processes. ''Computer Studies in the Humanities and Verbal Behaviour 4, 208-212.''

'''Tuldava, J.''' (1998). ''Probleme und Methoden der quantitativ-systemischen Lexikologie''. Trier: WVT.

'''Vulanović, R.''' (2003). Fitting periphrastic do in affirmative declaratives. QUALICO 2003, Atlanta.

'''Weinreich, U., Labov, W., Herzog, M.E.''' (1968). Empirical foundations for a theory of language change. In: Lehmann, W.P., Malkiel, Y. (eds.), ''Directions for historical linguistics: 95-188.'' Austin: University of Texas Press.

'''Winter, W.''' (1971). Formal frequency and linguistic change. Some preliminary comments. ''Folia Linguistica 5, 55-61''.

'''Zipf, G.K.''' (1946). Cultural-chronological strata in speech. ''Journal of abnormal and social psychology 41, 351-355.''

'''Zipf, G.K.''' (1947). Prehistoric `cultural strata´ in the evolution of Germanic: The case of Gothic. ''Modern language notes 62, 522-530''.

'''Zipf, G.K.''' (1949). ''Human behavior and the principle of least effort''. Cambridge, Mass: Addison-Wesley.

Change in language

2006-07-26T11:58:36Z

Altmann:

'''1. Problem and history'''

Everything in language changes. The complete complex of causes can not be ascertained; we are interested in the general processes of change and its course, whatever the entity con¬cerned. There are four aspects treated in this volume, for which models could be set up:

Qualitative change

(i) change of individual entities, which is the object of this chapter

(ii) sound change (<math>\rightarrow</math>)

Volume change

(iii) lexicon growth (<math>\rightarrow</math>)

(iv) lexicon decay or glottochronology (<math>\rightarrow</math>)

The research in quantitative form most probably began with different hypotheses of G. K. Zipf concerning age and frequency, age and length, etc. (1946, 1947, 1949, cf. Prün 1985). Measurement concerning individual phenomena can be found in Piotrowski (1960), Graudina (1964), Lazard (1965), whose results have been empirically fitted by Piotrovskaja, Piotrowski (1974) using an arctangent function. The theoretical derivation has been performed by Beö¬thy, Altmann (1982), Altmann, von Buttlar, Rott, Strauß (1983) combining Piotrowski´s find¬ings with an assumption of Weinreich, Labov, Herzog (1968). Altmann (1983) derived the three possible variants of the law shown below. A number of corroborations was brought by Best (1983), Best, Kohlhase (1983, 1983a), Imsiepen (1983), Kohlhase (1983), Müller-Hase¬mann (1983), Best, Altmann (1986), Kroch (1989a,b, 2001), Best, Beöthy, Altmann (1990), Tuldava (1998), Best (2001), Bresnan, Dingare, Manning (2001), Vulanović (2003). The law is called Piotrowski law or Piotrowski-Altmann law and is used for modelling phenomena like the increase of the number of borrowings, changes in morphology, etc.

'''2. Hypothesis'''

Everything in language changes as a result of interaction between old forms and new forms.

The independent variable is time, given usually in form of a transformed time index.
The dependent variable is the proportion of new forms.

'''3. Derivation'''

The interaction can be presented in the form

(1) <math>dp_t=k_tp_t(C-p_t)dt\quad\quad</math>,

<math>p_t</math> = proportion of new forms

<math>pk_t</math> = a function of time (can also be a constant)

C = limit of change

t > 0, time

<math>dp_t</math> = change of the proportion

telling that the change of the proportion of new forms is proportional to the interaction of new and old forms. The solution yields three variants:

'''(a) Complete change''' if C = 1 and <math>k_t = b</math> is constant

(2) <math>p=\frac{1}{1+ae^{-bt}}</math>

where a is the integration constant. The result in (2) represents the so-called logistic curve used in different disciplines for modeling growth phenomena.

'''Example'''. Since years represent large numbers hindering the fitting, one usually transforms the time intervals in a time variable beginning with t = 1. Often the cumulative frequencies are changed to cumulative proportions, or the proportions are ascertained from the occurrence of rival forms.
Complete change: The replacement of –{t} by –{st} in German 2nd person singular indicative present time with the verb “wollen”, shown by Best (2003a). The result of fitting is presented in Table 1 and Fig. 1. Here ft is the relative frequency of –{st}, pt is the computed relative frequency according to (2).

<div align="center">[[Image:Figur11_CiL.jpg]]</div>

The fitting is excellent. The curve was fitted to the proportion of –{st} found in the sources.

<div align="center">[[Image:Grafik1_CiL.jpg]]</div>

<div align="center">Fig. 1. The result presented in Table 1</div>

(b) '''Partial change''' if <math>k-t = b</math> is constant, C is the asymptote

(3) <math>p=\frac{C}{1+ae^{-bt}}</math>

and a is the integration constant.

Example. Partial change: Borrowings from Latin in Hungarian
Beöthy and Altmann examined the borrowing from different langugages in Hungarian. The fate of Latin words is shown in Table 2 and Fig. 2. The fitting was performed for the cumulative values.

<div align="center">[[Image:Figur22_CiL.jpg]]</div>

<div align="center">[[Image:Grafik2_CiL.jpg]]</div>
<div align="center">Fig. 2. Fitting formula (3) to data in Table 2</div>

(c) '''Reversible change''' if <math>k_t</math> = a´ - b´t, C = constant

(4)<math>p_t=\frac{C}{1+ae^{-bt+ct^2}}</math>

where a, b, c are simple functions of a´, b´and C.

Example 3. Reversible change: Epithesis with strong verbs in German. Imsiepen (1983) observed that the epithesis of /e/ with strong verbs (1st and 3rd person sg. Past tense) is a reversible process in German. Altmann (1983) has shown some estimation procedures for this curve, but since several observed values are very unreliable, Best, Beöthy and Altmann (1990) considered smoothed values (moving average of 7 values) and obtained the data in Table 3. They took the value of C into consideration and used formula (4).

<div align="center">[[Image:Tabelle33_CiL.jpg]]</div>

The fitting is very good.

<div align="center">[[Image:Grafik4_CiL.jpg]]</div>

'''4. Authors:''' U. Strauss, G. Altmann

'''5. References'''

'''Altmann, G.''' (1983a). Das Piotrowski-Gesetz und seine Verallgemeinerungen. In: Best, K.-H., Kohlhase, J. (Hrsg.), ''Exakte Sprachwandelforschung: 54-90''. Göttingen: Herodot.

'''Altmann, G.''' (1985). On the Dynamic Approach to Language. In: Ballmer, T. T. (ed.), ''Linguistic Dynamics: 181-189''. Berlin/ New York: de Gruyter.

'''Altmann, G.''' (1992). Piotrowski’s Law of Language Change. In: Saukkonen, P. (ed.), ''What is Language Synergetics? 34-35.'' Oulu: Acta Universitatis Ouluensis, Series B: Humaniora, 16.

'''Altmann, G., Bagheri, D., Goebl, H., Köhler, R., Prün, C.''' (2002). ''Einführung in die quantitative Lexikologie.'' Götingen: Peust & Gutschmidt.

'''Altmann, G., v. Buttlar, H., Rott, W., Strauß, U.''' (1983). A law of change in language. In: Brainerd, B. (ed.), ''Historical linguistics: 104-115''. Bochum: Brockmeyer.

'''Bailey, Ch.J.N.''' (1973). ''Variation and linguistic theory.'' Arlington: Center for Applied Lin-guistics.

'''Beöthy, E., Altmann, G.''' (1982). Das Piotrowski-Gesetz und der Lehnwort¬schatz. ''Zs. für Sprachwissenschaft 1, 171-178.''

'''Best, K.-H.''' (1983). Zum morphologischen Wandel einiger deutscher Verben. In: Best, Kohlhase (eds.) 1983: 107-118.

'''Best, K.-H.''' (2000). Der Zuwachs der Wörter auf -ical im Deutschen. ''Glottometrics 2, 11-16.''

'''Best, K.-H.''' (2001). Ein Beitrag zur Fremdwortdiskussion. In: Schierholz, S.J., Fobbe, E., Goes, S., Knirsch, R. (eds.), ''Die deutsche Sprache der Gegenwart. Festschrift für Dieter Cherubim zum 60. Geburtstag: 263-270.'' Frankfurt: Lang.

'''Best, K.-H.''' (2002). Satzlängen im Deutschen: Verteilungen, Mittelwerte, Sprachwandel. ''Göttinger Beiträge zur Sprachwissenschaft 7, 7-31.''

'''Best, K.H.''' (2003a). “Spracherwerb, Sprachwandel und Wortschatzwachstum in Texten. Zur Reichweite des Piotrowski-Gesetzes.” ''Glottometrics 6, 9-34.''

'''Best, K.-H.''' (2003b). Wie verläuft Sprachwandel? ''Naukovij Visnik Černivec´kogo Univeristetu 155, 86-94.''

'''Best, K.-H.''' (2003c). “Spracherwerb, Sprachwandel und Wortschatzwachstum in Texten. Zur Reichweite des Piotrowski-Gesetzes.” ''Glottometrics 6, 9-34.''

'''Best, K.-H.''' (2003d). Zum Wandel von Idiolekten. ''Naukovyj Visnyk Černivec’koho Universytetu, Vypusk 165-166, 36-43.''

'''Best, K.-H.'''(2003e). Zur Entwicklung von Wortschatz und redefähigkeit bei Kindern. ''Göttinger Beiträge zur Sprachwissenschaft 9, 7-20.''
'''Best, K.-H.''' (2004). Kürzungstendenzen im Deutschen aus der Sicht der Quantitativen Linguistik. In: Bär, J. A., Roelcke, T., & Steinhauer, A. (Hrsg.), ''Sprachliche Kürze. Konzeptuelle, strukturelle und pragmatische Aspekte.'' Berlin/ New York: de Gruyter. (Im Druck)

'''Best, K.-H., Altmann, G.''' (1986). Untersuchungen zur Gesetzmäßigkeit von Entlehnungs¬prozessen im Deutschen. ''Folia Linguistica Historica 31-41.''

'''Best, K.-H., Beöthy, E., Altmann, G.''' (1990). Ein methodischer Beitrag zum Piotrowski-Gesetz. ''Glottometrika 12, 115-124.''

'''Best, K.H., Kohlhase, J.''' (eds.) (1983). ''Exakte Sprachwandelforschung.'' Göttingen, Herodot.

'''Best, K.-H., Kohlhase, J.''' (1983a). Der Wandel von ''ward'' zu ''wurde''. In: Best, Kohlhase (eds.) 1983: 91-102.

'''Best, K.-H., Zhu, Jinyang''' (2006). Sprachwandel im Chinesischen. ''Archiv orientální 74, 203-214.''
'''Brainerd, B.''' (1983). A stochastic model for language change. In: Brainerd, B. (ed.): ''Historical linguistics: 25-49''. Bochum: Brockmeyer.

'''Brainerd, B.''' (ed.) (1983). ''Historical linguistics''. Bochum : Brockmeyer.

'''Bresnan, J., Dingare, S., Manning, C.D.''' (2001). Soft constraints mirror hard constraints: voice and person in English and Lummi. In: Butt, M., King, T.H. (eds.), ''Proceedings of the LFG01 Conference: 13-22.'' Stanford: CSLI (http://cslipublications.stanford.edu/LFG/6(lfg01.pdf)

'''Busch, A.''' (2002). ''Zur Entwicklung der Satzlängen in deutscher Fachsprache''. Staatsexamensarbeit, Göttingen.

'''Graudina, L.V.''' (1964). Razvitie nulevoj formy roditel´nogo množestvennogo u suščestvitel´nych – edinic izmerenija. In: Razvitie ''grammatiki i leksiki sovremennogo russkogo jazyka: 210-221.'' Moskva: Nauka.

'''Greimas, A.J.''' (1966). Les aspects quantitatifs en linguistique diachronique. In: ''Statistique et analyse linguistique. Colloque de Strasbourg 20-22 avril 1964: 113-120.'' Paris.

'''Imsiepen, U.''' (1983). Die e-Epithese bei starken Verben im Deutschen. In: Best, Kohlhase (eds.) 1983: 119-114.

'''Klein, S.''' (1964). ''Dynamic simulation of historical change in language using Monte Carlo techniques.'' Santa Monica: System Development Corporation.

'''Klein, S.''' (1965). Some components of a program for dynamic modelling of historical change in language. In: [1.] ''International Conference on Computational Linguistics. New York, 19.-21.5.1965.''

'''Kohlhase, J.''' (1983). Die Entwicklung von ''ward'' zu ''wurde'' beim Nürnberger Chronisten Heinrich Deichsler. In: Best, Kohlhase (eds.) 1983: 103-106.

'''Körner, H.''' (2001). Der Zuwachs der Wörter auf -ion im Deutschen. ''Glottometrics 2, 82-86''.

'''Körner, H.''' (2003). ''Wortschatzentwicklung im Deutschen''. Untersuchung zur Überprüfung des Piotrowski-Gesetzes. Magisterarbeit, Göttingen.

'''Körner, H.''' (2004). Zur Entwicklung des deutschen (Lehn)Wortschatzes. ''Glottometrics 7, 25-49.''

'''Kroch, A.S.''' (1989a). Function and grammar in the history of English: periphrastic do. In: Fasold, R.W., Schiffrin, D. (eds.), Language change and variation: 133-172. Amsterdam: Benjamins.

'''Kroch, A.S.''' (1989b). Reflexes of grammar in patterns of language change. In: Fasold, R.W., Schiffrin, D. (eds.), ''Language change and variation: 199-244''. Amsterdam: Benjamins.

'''Kroch, A.S.''' (2001). Syntactic change. In: Baltin, M, Collins, C. (eds.), ''The handbook of contemporary syntactic theory: 699-729.'' Oxford: Blackwell.

'''Lazard, G.''' (1965). Les empruntes arabes dans la prose persane du X-e au XII-e siècle : aperçu statistique. ''Revue de l´Ecole National des language orientales 2, 53-67''.

'''Lehfeldt, W., Altmann, G.''' (2003). The process of fall of the reduced vowels in Old Russian in the lingth of the Piotrovskij law. ''Russian Linguistics 27, 141-149.''

'''Leopold, E.''' (1998). ''Stochastische Modellierung lexikalischer Evolutionsprozesse''. Hamburg: Kovač.

'''Leopold, E.''' (2005). Das Piotrowski-Gesetz. In: Köhler, R., Altmann, G., Piotrowski, R. (Hg.), ''Quantitative Linguistik – Quantitative Linguistics. Ein internationales Handbuch: 627-633.'' Berlin/ N.Y.: de Gruyter.

'''Müller-Hasemann, W.''' (1983). Das Eindringen englischer Wörter ins Deutsche ab 1945. In: Best, Kohlhase (eds.) 1983: 143-160.

'''Ogura, M.''' (1993). The development of periphrastic do in Enbglish: A case of lexical diffusion in syntax. ''Diachronica 10, 51-85''.

'''Ommen, E.''' (2003). ''Quantitative Untersuchungen zur Syntax des Deutschen.'' Staatsexamensarbeit, Göttingen.

'''Osgood, Ch.E., Sebeok, Th.''' (1965). ''Psycholinguistics''. Bloomington: Indiana University Press.

'''Piotrovskaja, A.A. Piotrovskij, R.G.''' (1974). Matematičeskie modeli v diachronii i tekstoobrazovanii. In: ''Statistika reči i avtomatičeskij analiz teksta: 361-400''. Leningrad: Nauka.

'''Piotrowski, R.G.''' (1960). ''Formirovanie artiklja v romanskich jazykach''. Moskva-Leningrad: Nauka.

'''Piotrovski, R.G., Bektaev, K.B., Piotrovskaja, A.A.''' (1997). ''Matematičeskaja lingvistika''. Moskva: Nauka.

'''Piotrowski, R.G., Bektaev, K.B., Piotrovskaja, A.A.''' (1985). ''Mathematische Linguistik''. Bochum, Brockmeyer.

'''Prün, C.''' (1995). ''Die linguistischen Hypothesen von G.K. Zipf aus systemtheoretischer Sicht''. Trier: Magisterarbeit.

'''Richter, E.''' (1930). Zipf, George Kingsley: Relative frequency as a determinant of phonetic change. ''Archiv für das Studium der neueren Sprachen 157, 291-296 [Review].''

'''Rozwadowski, J.''' (1909). Ein quantitatives Gesetz der Sprachentwicklung. ''Indogermanische Forschungen 25, 38-50.''

'''Rozwadowski, J.''' (1960). O pewnym prawie ilościowym rozwoju języka. In: Rozwadowski, J. (ed.), ''Wybór pism. 3: 96-105''. Warszawa.

'''Sankoff, D.''' (1969). ''Historical linguistics as stochastic process.'' Montreal. Diss., McGill Univ., Montreal.

'''Sankoff, D.''' (1971). Stochastic models for glottochronology. In: Hodson, F.R. et al. (eds.), ''Anglo-Romanian Conference on Mathematics in the Archaeological and Historical Sciences. Mamaia, Romania, 1970.'' Edinburgh: Edinburgh Univ. Press, 1971. S.381-386.

'''Sankoff, D.''' (1972). Lexical replacement processes. ''Computer Studies in the Humanities and Verbal Behaviour 4, 208-212.''

'''Tuldava, J.''' (1998). ''Probleme und Methoden der quantitativ-systemischen Lexikologie''. Trier: WVT.

'''Vulanović, R.''' (2003). Fitting periphrastic do in affirmative declaratives. QUALICO 2003, Atlanta.

'''Weinreich, U., Labov, W., Herzog, M.E.''' (1968). Empirical foundations for a theory of language change. In: Lehmann, W.P., Malkiel, Y. (eds.), ''Directions for historical linguistics: 95-188.'' Austin: University of Texas Press.

'''Winter, W.''' (1971). Formal frequency and linguistic change. Some preliminary comments. ''Folia Linguistica 5, 55-61''.

'''Zipf, G.K.''' (1946). Cultural-chronological strata in speech. ''Journal of abnormal and social psychology 41, 351-355.''

'''Zipf, G.K.''' (1947). Prehistoric `cultural strata´ in the evolution of Germanic: The case of Gothic. ''Modern language notes 62, 522-530''.

'''Zipf, G.K.''' (1949). ''Human behavior and the principle of least effort''. Cambridge, Mass: Addison-Wesley.

Sound change

2006-07-24T12:58:10Z

Altmann:

'''1. Problem and history'''

The approach is based on Zipf´s ''Principle of least effort'' according to which the increasing production effort of a sound increases the probability of its change (Zipf 1935, 1949).
Boretzky (1977: 119) says it explicitely: “Die Sprecher sind bestrebt, die Laute beizubehal-ten, zu deren Artikulation wenig Anstrengung erforderlich ist, und die Laute als erste zu verändern oder ganz wegfallen zu lassen, deren Hervorbringung besonders viel Energie erfordert.“
The first quantification goes back to Job (1983) who derived a formula for the probability of change depending on articulation effort. Since articulation effort is merely one side of Zipf´s principle, the effort of the hearer in perceiving and decoding the sound must be taken into account, too. The first extended model was derived by Job and Altmann (1985) whose further extension was performed by Prün (1995).

'''2. Hypothesis'''

''The probability of change of a sound is a function of its articulation and perception effort''. (Job, Altmann 1985)

“Articulation effort” is the energy necessary to utter the sound. It can be measured e.g. by muscle participation in articulation.
“Perception effort” could be measured as a function of the proportion of confusions of the given sound with other sounds (cf. Benzecri 1970; Clarke 1957; Conrad 1963; Conrad, Hull 1964; Gersic, Naumann, Altmann 1985; Klatt 1968; Miller, Nicely 1976; Nakatani 1972; Singh 1966, 1971; Thürmann 1974; Tversky 1977; Wickelgren 1965a,b,c, 1966a,b,c; Wilson 1963).

'''3. Derivation'''

3.1. The “effort” model

Let A be the articulation effort of the speaker, <math>A \epsilon (0,M)</math>, then the perception effort of the hearer is its complement, i.e. M-A. Let further <math>A/M = x \epsilon (0,1)</math> be the normed articulation effort and w the need for change.
Then it holds for this need that

(1)<math> \frac{dw}{dx} = \frac{aw}{1-x} - \frac{bw}{x}</math>

One assumes that the greater x (= normed effort), the greater the need for change with the speaker (= aw) and the smaller with the hearer (= bw).
The solution of (1) yields

(2)<math> w = K(1-x)^{-a}x^{-b}, \quad 0<x<1</math>

presented graphically in Fig. 1.

<div align="center">[[Image:Grafik11_SC.jpg]]</div>
<div align="center">Fig. 1. Function (2)</div>

If we consider w as the probability of the need for change, then K is the norming constant which can be obtained as the usual beta function, i.e. we obtain finally the probability function

(3)<math> f_w(x) = \frac{1}{B(1-a, 1-b)}(1-x)^{-a}x^{-b}, \quad 0<x<1</math>

The minimum of this curve is x = b/(a+b).
In order to estimate the probability of change, we use the distribution function of (3) and compute the probability of a change of a sound as

(a)If the effort of the speaker is x > b/(a+b) then we compute

(4)<math> P_w(x) = \frac{1}{B(1-a, 1-b)}\int_{b/(a+b)}^{x}(1-x)^{-a}x^{-b}dx, \quad 0<x<1</math>

i.e. the plane under the curve from its minimum to x.

(b) If the effort of the speaker is x < b/(a+b), then we compute

(5)<math> P_w(x) = \frac{1}{B(1-a, 1-b)}\int_x^{b/(a+b)}(1-x)^{-a}x^{-b}dx, \quad 0<x<1</math>

No testing has been performed as yet and the parameters a and b have not been interpreted physically.

'''3.2. Prün´s extension'''

Since in sense of Zipf a change depends also on the frequency of the sound, Prün (1995) ex-tended the above formula inserting in it the relative frequency f of the sound and obtained

(6)<math> f(x) = K(1-x)^{-af}x^{-b(1-f)}</math>

The differential equation leading to this result is

(7)<math> \left( \frac{dy}{dx} + \frac{dy}{df}\right)\frac{1}{y} = af - b (1-f)+b \ln x - a \ln (1-x)</math>

but it is not yet interpreted. A graphical example can be seen in Fig. 2.

<div align="center">[[Image:Grafik2_SC.jpg]]</div>
<div align="center">Fig. 2. Function (6) with special values (Prün 1995)</div>

'''4. Authors''': U. Strauss, G. Altmann

'''5. References'''

'''Job, U'''. (1983). Bemerkungen zu einer theoreti¬schen Konzeption von Sprachwandel. In: K.-H. Best und J. Kohlhase (eds.), ''Exakte Sprachwandelforschung: 5 – 19''. Göttingen: Edition Herodot.

'''Job, U., Altmann, G.''' (1985). Ein Modell der anstrengungsbedingten Lautveränderung. ''Folia Linguistica Historica 6, 401-407''.

'''Lehfeldt, W'''. (2005). The fall of jers in the light of Menzerath´s law. In Grzybek, P. (ed.), ''Word length studies and related issues: 215-218''. Boston/Dordrecht: Kluwer.

'''Lehfeldt, W., Altmann, G'''. (2000). Padenie reducirovannych v svete zakona P. Mencerata. ''Russian Linguistics 26, 327-344''.

'''Lehfeldt, W., Altmann, G'''. (2003). The process of the fall of the reduced vowels in Old Russian in the light of the Piotrovskii law. ''Russian Lingusitics 27, 141-149''.

'''Prün, C'''. (1995). ''Die linguistischen Hypothesen von G.K. Zipf aus systemtheoretischer Sicht''. Trier: Magisterarbeit.

Speech act length

2006-07-24T12:56:29Z

Altmann:

'''1. Problem and history'''

Illocutive speech acts have different length in conversation. Their length is measured in terms of the number of different acts of a participant in uninterrupted sequence. The length of illocutive speech acts is part of the dynamics of discourse.
The only approach has been made by Rothe, Altmann and Wagner (1992) analyzing a German polylog of two children and two adults. The following example illustrates the analysis of the text (Gabi is child, Marion is adult):

'''Gabi'''

/Zeig.Hinweis/

'''Marion'''

/Anfrag/

/Anred/

'''Gabi '''
/Feststell/

/Appellier (Zustim)/

'''Marion'''

/Zustimm/

'''Gabi'''

/Behaupt.Vergleich/

'''Marion'''

/Feststell.Widersprech/

'''Gabi'''

/Bejah/Recht-geb/

/Ueberleg/

/Behaupt.Ueberleg/

/Behaupt.Vergleich/

'''2. Hypothesis'''

''Speech acts abide by the usual length distribution following from the unified theory'' (<math>\rightarrow</math>).

'''3. Derivation'''

In the usual “length”-approach, <math>P_x = g(x)P_{x-1}</math> , one inserts g(x) = (a+bx)/(cx) and solving for x = 1,2,… one obtains the positive negative binomial distribution

(1)<math> P_x = {k+x-1 \choose x}\frac{p^k q^x}{1-p^k}, \quad x= 1, 2, ...</math>

'''Example''': Illocution chains in a German polylogue

Rothe, Altmann and Wagner (1992) fitted the negative binomial distribution to the length of illocution chains of two children (Gabi, Klaus) and two adults (Marion, Nora) from the Dortmunder Korpus of spontaneous children speech (s. Steinsträter 1988) and obtained the results in Table 1.
As can be seen, all fittings are satisfactory. The parameter p symbolizes probably the hierarchy of speakers: the greater p, the more peripheral is the role of the speaker in the polylogue.

<div align="center">[[Image:Tabelle11_SAL.jpg]]</div>

'''4. Authors''': U. Strauss, G. Altmann

'''5. References'''

'''Rothe, U., Altmann, G., Wagner, K.''' (1992). Verteilung der Länge von Sprechakten in der Kindersprache. In: Wagner, K.R. (Hrsg.), ''Kindesprachstatistik: 47-56.'' Essen: Die blaue Eule.

'''Steinsträter, Ch'''. (1988). ''Zur kindlichen Sprachhandlungsfähigkeit: Analyse eines Korpus spontaner Sprechsprache einer Fünfjährigen''. Dortmund: Diss.

'''Wagner, K.R., Steinsträter, Ch'''. (1987). Wörterbuch der illokutiven Typen zum Korpus Teresa. In: Wagner, K.R. (Hrsg.), ''Wortschatz-Erwerb: 59-81''. Bern: Lang.

Syllable length

2006-07-24T12:55:19Z

Altmann:

'''1. Problem and history'''

The problem is to find the distribution of syllable length measured in terms of letter, phone or phoneme numbers. It is connected with the general problem of length (<math>\rightarrow</math>). Another problem is that of syllable duration which was considered merely in connection with hierarchic relations (<math>\rightarrow</math>).
The research is very young, it began in the Göttingen Project with works by Zuse (1998) and Cassier (1998), up to now further tests have been performed merely by Best (2001b), Rottmann (1999, 2002) and Schneemann (2001).

'''2. Hypothesis'''

''Syllable length abides by a regular distribution following from the unified theory (<math>\rightarrow</math>).''

'''3. Derivation'''

Since there are no zero-phoneme syllables, one solves the formulas of the “length approach” for the displaced case, namely

(1)<math> P_{x+1} = g(x)P_x, \quad x=1, 2, 3, ...</math>.

Up to now only three different cases have been observed:

'''3.1. The 1-displaced hyperbinomial distribution''' (Rottmann 1999)

Substituting in (1)

<math>g(x) = \frac{q(n-x+1)}{m+x-1}</math>

one obtains the 1-displaced hyperbinomial distribution

(2) <math>P_x = \frac{{n \choose x-1}}{{m+x-2 \choose x-1}}q^{x-1}P_1, x= 1, 2, 3, ..., n; m>0, n \epsilon N, 0<q<1</math>

P1 being the norming constant <math>P_1^{-1} = _2 F_1 (-n,1:m:-q)</math> (see Appendix). It is a special case of the unified theory (11) when <math>1+ a_0 = -q, a_1 = q(m+n), a_2 = 0, b_1= m-1, c_1 = 1\quad</math>.

'''Example''': Syllable length in East Slavonic

Rottmann (1999) examined the syllable length in East Slavonic texts. One of his results is presented in Table 1 and Fig. 1.

<div align="center">[[Image:Tabelle11_SL.jpg]]</div>

<div align="center">[[Image:Grafik1_SL.jpg]]</div>
<div align="center">Fig. 1. Fitting the 1-displaced hyperbinomial distribution to the data in Table 1</div>

'''3.2. The 1-displaced Conway-Maxwell-Poisson distribution (Cassier 1998, Best 2001b)'''

Substituting in (1)

<math>g(x) = \frac{a}{x^b}</math>

one obtains the 1-displaced Conway-Maxwell-Poisson distribution

(3)<math> P_x = \frac{Ca^{x-1}}{(x-1)!^b}, \quad x= 1, 2, 3, ...; a, b >0</math>,

wehere C is the normalizing constant <math> C^{-1} = \sum_{j=0}^\infty \frac{a^j}{j^b}</math> . It is a special case of the unified theory, formula (11) when <math>a_0 = -1, a_1 = a, b_1 = a_2 = a_3 = ... = 0, c_1 = b</math>.

'''Example''': Syllable length in German

Best (2001b) examined syllable length in 21 German journalistic texts. In text 5 he obtained the results presented in Table 2 and Fig. 2.

<div align="center">[[Image:Tabelle22_SL.jpg]]</div>

<div align="center">[[Image:Grafik2_SL.jpg]]</div>
<div align="center">Fig. 2. Fitting the 1-displaced Conway-Maxwell-Poisson distribution to the data in Table 2</div>

'''3.3. The positive Cohen-Poisson distribution (Zuse 1998, Rottmann 1999)'''

If the first class is modified such that

<math> P_1 = (1-\alpha)C\quad</math>

<math> P_2 = \frac{a}{2}C</math>

<math> P_x = \frac{a}{x}P_{x-1}, \quad x=2, 3, 4, ...</math>

then one obtains the positive Cohen-Poisson distribution

(4)<math>P_x = f(n)=\begin{cases} \frac{(1-a\alpha}{e^{-a}-1-a\alpha}, & x=1 \\ \frac{a^x}{x!(e^{-a}-1-a\alpha}, & x=2,3,4... \end{cases}</math>

'''Example''': Syllable length in East Slavonic

Rottmann (1999) fitted the positive Cohen-Poisson distribution to East Slavonic texts and in three cases where the hyperbinomial distribution was not adequate he obtained satisfactory results. Table 3 and Fig. 3. Show one of the fittings.

<div align="center">[[Image:Tabelle33_SL.jpg]]</div>

<div align="center">[[Image:Grafik3_SL.jpg]]</div>
<div align="center">Fig. 3. Fitting the positive Cohen-Poisson distribution to the data in Table 3.</div>

'''4. Authors''': U. Strauss, G. Altmann, K.-H. Best

'''5. References'''

'''Best, K.-H'''. (2001b). Silbenlängen in Meldungen der Tagespresse. In: Best, K.H. (ed.), ''Häufigkeitsverteilungen in Texten: 15-32''. Göttingen: Peust & Gutschmidt.

'''Cassier, F.U.''' (1998). ''Silbenlängen in Meldungen der deutschen Tagespresse''. Göttingen: Staatsexamensarbeit.

'''Cassier, F.U.''' (2001). Silbenlängen in Meldungen der deutschen Tagespresse. In: Best, K.H. (ed.), ''Häufigkeitsverteilungen in Texten: 33-42''. Göttingen: Peust & Gutschmidt.

'''Gorot´, E.I'''. (1990). Izomorfnye i otličitel´nye čerty morfemy i sloga v raspredelenii dliny. In: ''Kvantitativnaja lingvistika i avtomatičeskij analiz tekstov: 32-36.'' Tartu.

'''Kučera, H.''' (1963). Entropy, redundancy and functional load in Russian and Czech. In: ''American Contributions to the Fifth International Cobgress of Slavists: 191-219''. The Hague: Mouton.

'''Ludvíková, M.''' (1977). On some statistical differences in two spoken texts on the syllabic level. ''Prague Studies in Matgematical Linguistics 5, 91-104''.

'''Malécot, A., Johnston, R., Kizziar, R.A.''' (1972). Syllabic rate and utterance length in French. ''Phonetica 26, 35-261''.

'''Rottmann, O. A'''. (1999). Word and syllable lengths in East Slavonic. ''J. of Quantitative Linguistics 6: 235-238.''

'''Rottmann, O. A.''' (2002) Syllable length in Russian, Bulgarian, Old Church Slavonic and Slovene. ''Glottometrics 2: 87-94''.

'''Rykov, V.''' (1994). Menzerath law for printed speech. In: ''2nd International Conference on Quantitative Linguistics, September 20-24, 1994, Moscow: 199-200''. Moscow: Lomonosov Moscow State University.

'''Schneemann, O. F'''. (2001). ''Sprachstatistische Untersuchungen zu Wort- und Silbenlängen in deutschen Musikzeitschriften''. Staatsexamensarbeit, Göttingen.

'''Zuse, M'''. (1998). ''Silbenlängen in deutschen und englischen Pressetexten der Gegenwart''. Göttingen: Staatsexamensarbeit.

Syllable structure

2006-07-24T12:54:19Z

Altmann:

'''1. Problem and history'''

If syllables are coded in the canonical form, V, CV, CVC, etc. where the vowel (V) is the syllable nucleus and the consonant (C) is the periphery, we obtain a set of structural types whose number follows a specific two-dimensional distribution. The types can be presented in tabular form

<div align="center">[[Image:Tabelle1_SS.jpg]]</div>

Where the frequency of the type CVVCC is given as n12, i symbolizing the number of consonants before and j that behind the nucleus. There are at most r onsets and s codas in the syllable. Let <math>P_{ij}*</math> be the corresponding probability.

'''2. Hypothesis'''

''The distribution of canonical syllable types abides by the (modified) bivariate Conway-Maxwell-Poisson distribution''.

'''3. Derivation'''

Starting from the unified theory we assume that the probabilities of syllable types are joined with the following proportionality relation

(1)<math> P_{i,j}* \propto P_{i, j-1}*</math>

<math> P_{i,j}* \propto P_{i-1, j}*</math>

i.e. a class is proportional to its left or upper neighbour. In the first step Zörnig and Altmann (1993) used the simple Menzerathian relatioship (see Hierarchic relations) as proprotionality function and obtained

(2)<math> P_{i,j}* = \frac{b}{j^m}P_{i, j-1}*</math>
<math> P_{i,j}* = \frac{a}{i^k}P_{i-1, j}*, \quad m,k \epsilon \mathfrak{R}</math>

The solution of (2) yields

(3)<math> P_{i,j}* = \frac{a^1 b^j}{(i!)^k (j!)^m}P_{0,0}*, \quad i= 0,1,...,r, \quad j=0,1,...,s</math>

where the norming constant is <math> P_{0,0}^{-1} = \sum_{i=0}^r \sum_{j=0}^s\frac{a^i b^j}{(i!)^k(j!)^m}</math> .

Since the particular case examined (Indonesian) has a strong preference for the CVC type ( = <math>n_11</math>) the distribution must be modified and renormed. Weighting <math>P_11*</math> by β the other probabilities must be weighted so that <math>\beta P_11* + \alpha(1-P_11*) = 1</math>. As a result we obtain

(4)<math> P_{i,j}= \begin{cases} \beta P_{i,j}*, & i=j=1 \\ \alpha P_{i,j}*, & i= 0,1,...,r,\quad j=0,1,...,s,\quad i \ne 1 or j \ne 1\end{cases}</math>

Zörnig and Altmann (1993) show also the estimators won from frequency classes as

(5)<math>\hat a = n_{10} / n_{00}</math>

<math>\hat b = n{01}/n_{00}</math>

<math>\hat k \ln(\hat a n_{10}/n_{20})</math>

<math>\hat m \ln(\hat b n_{01}/n_{02})</math>

<math>\hat \alpha = 1+ \frac{n_{10}\hat b - n_{11}}{N}</math>

<math>\hat \beta = \hat \alpha n_{11} /(n_{10}\hat b)</math>

'''Example'''. Structure of Indonesian syllables

In about 15000 Indonesian word forms from different texts Zörnig and Altmann (1993) found 610 syllable types shown in Table 1.

<div align="center">[[Image:Tabelle11_SS.jpg]]</div>

Using (3), (4) and (5) they obtained

<math>P_{i,j}= \begin{cases} 0.713P*_{i,j}, & i\ne 1 or j\ne 1 \\ 1.291P*_{i,j}, & i=j=1 \end{cases}</math>

with

<math> P*_{i,j}=\frac{(o.014)6^{i+j}}{(i!)^{4.585}(j!)^{4.948}}, \quad 0\le i, j<4</math>.

The estimated parameters are: a = b = 6, k = 4.585, m = 4.948, α = 0.713, β = 1.291, <math>P*_{00}</math> = 0.014, N = 610 and the fitting is presented in Table 2. For example <math>NP_{12}</math> is computed as

<math> NP_{1,2}= 610\frac{0.713(0.014)6^{1+2}}{1!^{4.585} 2!^{4.948}}=610(0.06985) = 42.61</math> .

<div align="center">[[Image:Tabelle22_SS.jpg]]</div>

The result is acceptable without testing.

'''4. Authors''': U. Strauss, G. Altmann

'''5. References'''

'''Lee, Sank-Oak''' (1986). An explanation of syllable structure change. ''Korean Language Research 22, 195-213.''

'''Vennemann, T.''' (1982) (ed.). Zur Silbenstruktur der deutschen Standardsprache. ''Silben, Segmente, Akzente: 261-305''. Tübingen: Narr.

'''Zörnig, P., Altmann, G'''. (1993). A model for the distribution of syllable types. ''Glottometrika 14, 190-196''.

Synonymy

2006-07-24T12:53:15Z

Altmann:

'''1. Problem and history'''

Two words are synonymous if they have the same meaning. Even if absolute synonymy is seldom, approximate synonymy is usual in specific contexts. Synonymy arises through (→) diversification of the meaning of words and can be eliminated by unification (cf. Altmann 1986).

From this point of view the problem has been analyzed merely by Wimmer and Altmann (2001a) who studied the synonymy in Italian and Ziegler (2001a) who studied it in German and English.

'''2. Hypothesis'''

''The rise and elimination of synonyms is controled by a birth-and-death process''.

'''3. Derivation'''

'''3.1. Wimmer-Altmann´s model'''

The dynamics of synonymy is a Poissonian birth-and-death process with constant birth (λ) and death rates (μ) respectively. Inserting them in the balancing equation we obtain

(1)<math>\lambda P_0 = \mu P_1

<math> (\lambda + \mu) P_x = \lambda P_{x+1}, \quad x=1,2,3,...</math>

Stepwise solution and setting q = λ/μ, (0 < q < 1) yields the geometric distribution

(2)<math>P_x = pq^{x-1}, \quad x=0,1,2,...</math>

Since words with 0 synonyms are not contained in the usual dictionary of synonyms, one either uses the 1-displaced geometric distribution which can be arrived at by a displacement in (1), i.e.

(3)<math> P_x= pq^{x-1}, \quad x=1,2,3</math>

or one takes these words with 0 synonymy from a usual but incommensurable dictionary. In that case it is advisable to modify (2) and use

(4)<math> P_x = \begin{cases} 1-\alpha, & x=0 \\ \alpha pq^{x-1}, & x=1,2,3,... \end{cases}</math>

'''Example''': Synonymy in Italian

Wimmer and Altmann (2001a) took a random sample (15 pages) from the Italian dictionary of P. Stopelli, Sinonimi e Contrari con generici, Spezifici, Analoghi, Inversi. (Garzanti 1998) and obtained the distribution as presented in Table 1. Separartely they used the dictionary of P. Giovanelli, W. Frenzel, Langenscheidts Handwörterbuch Italienisch (Berlin, Langenscheid 1998) from wich thay sampled words with 0 synonyms (i.e. those not occurring in the dictionary of synonyms but laying between the synonyms on the given page). The results of fitting of (3) and (4) to these data are shown in Table 1 and Fig. 1.

<div align="center">[[Image:Tabelle11_S.jpg]]</div>

<div align="center">[[Image:Grafik1_S.jpg]]</div>
<div align="center">Fig. 1. Fitting the modified geometric distribution to the data in Table 1</div>

The fitting in both cases is very satisfactory.

'''3.2. Ziegler´s model'''

Ziegler (2001) ascertained that for German and English data the assumptions in 3.1 are not sufficient. Using the same process he postulated

<math> \lambda = a + bx\quad</math>

<math>\mu = x\quad</math>

and obtained the balancing equations in form

(5)<math> aP_0 = P_1\quad</math>

<math> (a+bx+x)P_x = [a+b(x-1)]P_{x-1}+(x+1)P_{x+1}, \quad x=1,2,3,...</math>

leading by stepwise addition of lines to

(6)<math> P_x= \frac{a+b(x-1)}{x}P_{x-1}</math>

Setting a/b = k and b = q one obtains

(7)<math> P_x = \frac{k+x-1}{x}qP_{x-1}</math>

resulting in the negative binomial distribution. Displacing it conventionally one step to the right one gets

(8)<math> P_x= {k+x-2 \choose x-1}p^k q^{x-1}, \quad x=1,2,3,...</math>

'''Example'''. Synonymy in German dictionary

Ziegler analyzed German data using Der Duden Bd. 8 (1997, second edition) and obtained results presented in Table 2 and Fig. 2. Since for x = 34 to x = 71 there are no observed cases, the only case in x = 72 was pooled with that of x = 33. The present results are slightly different from those in Ziegler (2001)

<div align="center">[[Image:Tabelle22_S.jpg]]</div>

<div align="center">[[Image:Grafik2_S.jpg]]</div>
<div align="center">Fig. 2. Fitting the negative binomial distribution to the data in Table 2</div>

Note. For English neither (4) nor (8) were sufficient, perhaps because of the extensive language mixture. Ziegler proposed the mixed negative binomial distribution.

'''4. Authors:''' U. Strauss, G. Altmann

'''5. References'''

'''Der Duden Bd. 8''' (<math>1997^2</math>). ''Duden. Die sinn- und sachverwandte Wörter. Synonymwörterbuch der deutschen Sprache''. Mannheim: Dudenverlag.

'''Giovanelli, P., Frenzel, W.''' (1998). Langenscheidts Handwörterbuch Italienisch. Berlin: Langenscheid.

'''Stopelli, P'''. (1998). ''Sinonimi e contrari con generici, spezifici, analoghi, inversi''. Milano: Garzanti.

'''The Nuttall''' (1<math>976^21</math>). ''The Nuttall dictionary of English synonyms and antonyms''. Ed. by G.E. Christ. Ner York-London: Warne.

'''Wimmer, G., Altmann, G.''' (2001a). Two hypotheses on synonymy. In: Ondrejovič, S., Považaj, M. (eds.), ''Lexicographica ´99: 218-225''. Bratislava: Veda.

'''Ziegler, A.''' (2001a). Zum Gesetz der Synonymie. Modellanpassungen im Deutschen und Englischen. In: Ondrejovič, S., Považaj, M. (eds.), ''Lexicographica ´99: 230-236'' . Bratislava: Veda

Text-blocks

2006-07-24T12:52:00Z

Altmann:

'''1. Problem and history'''

Let the text be divided in (not necessarily equal) passages of words, e.g. 100, 200, pagewise, sentencewise, etc. and the distribution of a chosen linguistic entity in these passages is sought. The passages can contain this entity zero times, once, twice,… . The variable x is thus the number of the given entity in that passage, and fx is the number of passages containing this entity x times.
The origin of the reseach goes back to E. Zwirner and K. Zwirner (1935, 1938) who considered the distribution of different sounds in text-blocks and assumed the “law of small numbers” as the generating mechanism. Frumkina (1962) who probably did not know the work of Zwirners considered word occurrence as a “rare event” and applied automatically the Poisson distribution, Mosteller and Wallace (1964) derived the negative binomial distribution, Brainerd added the mixed Poisson distribution (1972a), Piotrowski, Bektaev, Piotrowskaja (1985) used the binomial distribution, some Russian authors used the normal distribution and Altmann, Burdinski (1982) who baptized this mechanism as '''Frumkina´s law''' derived the negative hypergeometric distribution which will be presented here. Leopold (1998) gives hints to other possible distributions. Köhler (2001) examined the distribution of syntactic constructions in text blocks. Best (2005) brought a general survey of results up to now.
Piotrowski (1984) mentions the following applications of the text-block law:

(1) It can help to ascertain mechanically the membership of a word to a word class.

(2) It can help to identify terminologically or semantically dominant text units.

(3) It enables us to measure and ascertain the stylistic individuality of the text.

(4) It enables us to diagnostify the foci of some psychic deseases (cf. Paškovskij, Srebrjanskaja 1971).

(5) It helps to construct learning automata.

'''2. Hypothesis'''

''The distribution of individual entities in text passages abides by the negative hypergeometric distribution''.

'''3. Derivation (Altmann, Burdinski 1982)'''

Let the probability of a word A in language be p. This is Herdan´s (1956) assumption but as a matter of fact, fixed probabilities of language units are illusory. Nevertheless, this assumption can be used because p will be randomized. In a text passage in which A can occur maximally n times, the probability that it will occur exactly x times is given by the binomial distribution

(1)<math> P(X=x|p) = f(x|p) = {n \choose x}p^x (1-p)^{n-x}, \quad x = 0,1,2,...,n</math>.

However, p is not constant since its value depends on the kind of text, on the length of the passage and especially on the environment (e.g. it cannot occur three times one behind the other). Thus the probability of its occurrence in an individual position of the passage is a variable with its own distribution.

Altmann and Burdinski (1982) assumed that p has a beta distribution given as

(2)<math> f(p) = \frac{1}{b(M,K-M}p^{M-1}(1-p)^{K-M-1}, \quad 0<p<1</math>,

where B(.) is the beta function (<math>\rightarrow</math> Appendix).
The common distribution of x and p is now

(3)<math> f(x,p) f(x|p)f(p) = {n \choose x}p^x (1-p) ^{n-x} \frac{1}{B(M,K-M)} p^{M-1} (1-p) ^{K-M-1}</math>

which can be solved for x by integrating (3) according to p.
As a result we obtain

(4)<math> f(x) = P_x = {n \choose x}\frac{B(M-x, K-m + n-x)}{B(M,K-M)} = \frac{{-M \choose x}{-K+M \choose n-x}}{{-K \choose n}}, \quad x= 0,1,...,n</math>

i.e. the negative hypergeometric distribution. Here M, N, n are parameters.
It can easily be shown that the other distributions mentioned above are limiting cases of the negative hypergeometric:

(i) when <math> K\rightarrow \infty</math>, <math> M\rightarrow \infty</math>, <math> M/K\rightarrow p</math> then the negative hypergeometric distribution converges to the binomial distribution (Piotrowski et al. version) (see (1));

(ii) when <math> K\rightarrow \infty</math>, <math> M\rightarrow \infty</math>,<math> n\rightarrow \infty</math>, <math> Mn/K\rightarrow a</math> then the negative hypergeometric distribution converges to the Poisson distribution (Brainerd´s version):

<math> P_x = \frac{a^x e^{-a}}{x!}, \quad x=0,1,2,...:\quad a>0</math>

(iii) when <math> K\rightarrow \infty</math>, <math> n\rightarrow \infty</math>, <math> K/(K+n)\rightarrow p</math> then the negative hypergeometric distribution converges to the negative binomial distribution (Mosteller-Wallace´ version):

<math> P_X = {k+x-1 \choose x}p^k q^x, \quad x= 0,1,2,...;\quad k<0;0<p<1;q=1-p</math>

Thus each of the above models has its partial justification. The normal distribution is not taken into account since it is continuous but the convergence to it can easily be shown.

'''Example''': Distribution of nouns in text blocks

Piotrowski, Bektaev, Piotrowskaja (1985) examined the distribution of nouns in passages in Auezov´s novel “Put´ Abaja” and found the frequencies given in Table 1 to which they fitted the binomial distribution. In the last column the negative hypergeometric distribution is shown.
Since there are no passages without nouns, both theoretical distributions are 1-displaced.

<div align="center">[[Image:Tabelle11_TB.jpg]]</div>

In both cases the fitting is satisfactory, the negative hypergeometric is somewhat better.

<div align="center">[[Image:Grafik1_TB.jpg]]</div>
<div align="center">Fig 1. Fitting the binomial distribution to data of Piotrowski et al. (1985)</div>

<div align="center">[[Image:Grafik2_TB.jpg]]</div>
<div align="center">Fig. 2. Fitting the negative hypergeometric distribution to data of Piotrowski et al. (1985)</div>

'''Example''': Distribution of the Russian preposition “bez” in text blocks

Frumkina (1962) examined the occurrence of the Russian preposition “bez” in 110 passages consisting of 1000 words each from texts by Pushkin and fitted the Poisson distribution.
In the last column of Table 2 the negative hypergeometric distribution is shown, too.

<div align="center">[[Image:Tabelle22_TB.jpg]]</div>

<div align="center">[[Image:Grafik3_TB.jpg]]</div>
<div align="center">Fig. 3. Fitting the Poisson distribution to Frumkina´s data</div>

<div align="center">[[Image:Grafik4_TB.jpg]]</div>
<div align="center">Fig. 4. Fitting the negative hypergeometric distribution to Frumkina´s data</div>

'''Example:''' Distribution of the article “das” in German text blocks

Altmann and Burdinski (1982) examined the occurrence of the German article “das” in nominative in passages from S. Lenz “Deutschstunde”. They fitted the negative hypergeo-metric distribution changing stepwise n and showed the gradual convergence to the negative binomial distribution (cf. Table 3)

<div align="center">[[Image:Tabelle3_TB.jpg]]</div>

As can be seen in Table 3, all fittings are good and improve with increasing n and K. This is a sign of convergence to the negative binomial distribution which, as a matter of fact, shows the best result.

<div align="center">[[Image:Grafik5_TB.jpg]]</div>
<div align="center">Fig. 5. Fitting the negative binomial distribution to data of Lenz</div>

'''Example.''' Distribution of the indirect object (Köhler 2001)

Köhler (2001) analyzed syntactic constructions in text blocks e.g. participle clauses, relative clauses, infinitival clauses, prepositional objects, indirect objects, logical direct objects and stated that all follow the negative binomial distribution. The fitting of this distribution to the Susanne Corpus (Sampson 1995) to the number of blocks with x occurrences of indirect object is shown in Table 4 and Fig. 6.

<div align="center">[[Image:Tabelle4_TB.jpg]]</div>

Further investigations have been carried out on letters (Schulte 2002, Suhren 2002), grammatical and lexical words (Best 2001, ²2003; Billmeier 1968, Muller 1972; Suhren 2002), semantic groups of words (Muller 1972) and groups consisting of 3 words (Piotrowski, Bektaev, Piotrowskaja 1985). They all abide by the law of text blocks, too (Best 2005). Knauer (1955: 146) yields the proportion of vowels in text-blocks of 100 phones in French and Italian.

'''4. Authors''': U. Strauss, G. Altmann, K.-H. Best

'''5. References'''

'''Altmann, G.''' (1988a). ''Wiederholungen in Texten''. Bochum, Brockmeyer.

'''Altmann, G., Burdinski, V'''. (1982). Towards a law of word repetitions in text-blocks. ''Glottometrika 4, 147-167''.

'''Bektaev, K.B., Lukjanenkov''' (1971). O zakonach raspredelenija edinic pis'mennoj reči. In: Piotrowski, R.G. (ed.), ''Statistika reči i avtomatičeskij analiz teksta: 47-112''. Leningrad: Nauka.

'''Best, K.-H.''' (2001; ²2003). ''Quantitative Linguistik. Eine Annäherung''. 2., überarbeitete und erweiterte Auflage. Göttingen: Peust & Gutschmidt.

'''Best, K.-H.''' (2005). Sprachliche Einheiten in Textblöcken. ''Glottometrics 9, 1-12.''

'''Billmeier, G'''. (1968). Über die Signifikanz von Auswahltexten. Untersuchung auf der Grundlage von Zeitungstexten. In: Moser, Hugo u.a. (Hrsg.), ''Forschungsberichte des Instituts für deutsche Sprache 2, 126-171''.

'''Brainerd, B.''' (1972a). Article use as an indirect indicator of style among English-language authors. In: Jäger, S. (ed.), ''Linguistik und Statistik: 11-32''. Braunschweig, Vieweg.

'''Frumkina, R.M.''' (1962). O zakonach raspredelenija slov i klassov slov. In: Mološnaja, T.N. (ed.), ''Strukturno-tipologičeskie issledovanija: 124-133''. Moskva: ANSSSR.

'''Herdan, G.''' (1956). Language as Choice and Chance. Groningen: Nordhoff.

'''Knauer, K.''' (1955). Grundfragen einer mathematischen Stilistik. ''Forschungen und Fortschritte 29, 140-149''.

'''Köhler, R.''' (2001). The distribution of some syntactic construction types in text blocks. In Uhlířova, L., Wimmer, G., Altmann, G., Köhler, R. (Eds.), ''Text as a linguistic paradigm: levels, constituents, constructs. Festschrift in honour of Ludek Hřebíček: 136-148.'' Trier: WVT.

'''Leopold, E.''' (1998). ''Stochastische Modellierung lexikalischer Evolutionsprozesse''. Hamburg: Kovač.

'''Maškina, L.E.''' (1968). ''O statističeskich metodach issledovanija leksiko-grammatičeskoj distribucii.'' Minsk, Diss.

'''Morton, A.Q., Levison, M.''' (1966). Some indicators of authorship in Greek prose. In: Leed, J. (ed.), ''The computer and literary style: 141-179''. Kent, Ohio: Kent State UP.

'''Mosteller, F., Wallace, D.L.''' (1964). ''Inference and disputed authorship: The Federalist''. Reading, Mass, Addison-Wesley.

'''Muller, Ch.''' (1972). ''Einführung in die Sprachstatistik''. München: Hueber.

'''Paškovskij, V.E., Srebrjanskaja, I.I.''' (1971). Statističeskie ocenki pis'mennoj reči bol'nych šizofreniej. In: ''Inženernaja lingvistika''. Leningrad.

'''Piotrowski, R.G.''' (1984). ''Text – Computer – Mensch''. Bochum: Brockmeyer.

'''Piotrowski, R.G., Bektaev, K.B., Piotrowskaja, A.A.''' (1985). ''Mathematische Linguistik''. Bochum, Brockmeyer.

'''Suhren, S.''' (2002). ''Untersuchung zum Gesetz von Zwirner, Zwirner und Frumkina am Beispiel des niederdeutschen „De lütte Prinz“.'' Staatsexamensarbeit, Göttingen.

'''Zwirner, E., Ezawa, K.''' (Hrsg.) (1966, 1968, 1969). ''Phonometrie, Erster-Dritter Teil''. Basel/ New York: Karger.

'''Zwirner, E., Zwirner, K.''' (1935). Lauthäufigkeit und Zufallsgesetz. ''Forschungen und Fortschritte 11, Nr. 4: 43-45''. (Also in: Zwirner & Ezawa (Hrsg.), Dritter Teil: 55-59.)

'''Zwirner, E., Zwirner, K'''. (1938). Lauthäufigkeit und Sprachvergleichung. ''Monatsschrift für höhere Schulen 37: 246-253''. (Also in: Zwirner & Ezawa (Hrsg.), Dritter Teil, 68-74.)

Vowel duration

2006-07-24T12:51:00Z

Altmann:

'''1. Problem and history'''

In speech, vowel duration is not constant but varies at every occasion of pronouncing the given vowel. Phoneticians found a number of conditions under which a vowel is pronounced longer or shorter, e.g. position in the word, accent, length of the word (<math>\rightarrow</math> Hierarchy), phonological length etc. Geršiċ and Altmann (1988) tried to show that the distribution of durations for an individual speaker is lawlike, i.e. that there is a mechanism controlling the durations. Their derivation is shown below.

2. Hypothesis

The length of vowels in speech is regularly distributed in dependence on different local, global and momentaneuos factors.

3. Derivation

The duration of vowels is influenced by three different kinds of forces:

(a) Forces responsible for innovation, emotionality, need for expression etc. leading to fluctuation, i.e. deviation from norms and self-organization. This is the class of Bühler´s expression function (Bühler 1934), Zipf´s diversification (Zipf 1949) etc.

(b) Forces fixing or modifying locally or globally the vowel duration, like phonological norms (short, long), suprasegmentals (tone, accent) and combinatorial factors (neighbour-hood).

(c) Forces restricting the speaker, damping the fluctuation, caring for equilibria like Bühler´s representation function, Zipf´s unification, pressure or control of the community.

Let

x = duration
f(x) = the probability fucntion of duration
S = expressive force of the speaker
B = accentuation
F = phonological length
K = quantity of the following consonant
H = the controlling force of the community

then

(1)<math> \frac{f'(x)}{f(x)}= \frac{Sx + Bx + F + K/x}{Hx}</math>.

Setting (S+B)/H = a, F/(S+B) = b, K/(S+B) = c one obtains

(2)<math> \frac{d[f(x)]}{f(x)}= \left( a + \frac{ab}{x} + \frac{ac}{x^2} \right)dx</math>

resulting in

(3)<math> f(x) = Nx^{ab}e^{ax-ac/x}, \quad 0 < x < R</math>

which can be simply written as (A = ab, B = a, C = -ac)

(4)<math> f(x) = Nx^A e^{Bx+C/x}, \quad 0 < x < R</math>

where N is the normalizing constant. It can easily be seen that this is just a special case of formula (2) and (3) of the unified theory (<math>\rightarrow</math>).

'''Example.''' Vowel duration in Batschka-German

Geršiċ and Altmann (1988) measured the duration of five categories of vowels (short non-ac-centuated, short semi-accentuated, short accentuated, long semi-accentuated, long accentuat-ed) with three speakers of Batschka-German. The duration was measured in intervals of 20 milliseconds, i.e. 20-39, 40-59, 60-79,… and transformed in X = X´/20 so that they obtained intervals 1,2), 2,3),… The probability in the interval was computed by numerical integration and optimization. Table 1 and Fig. 1 show the fitting of (4) to the distribution of the duration of short accentuated vowels with Speaker I.

<div align="center">[[Image:Tabelle1_VD.jpg]]</div>

The fitting is very good but not all results were satisfactory.

'''4. Authors:''' U.Strauss, G. Altmann

'''5. References'''

'''Geršić, S., Altmann, G.''' (1988). Ein Modell für die Variabilität der Vokaldauer. ''Glottometrika 9, 49-58''.
.
(Graph)

[[Category:Unfertig]]

Word length and phoneme inventory

2006-07-24T12:49:05Z

Altmann:

'''1. The problem and its history'''

If there are many phonemes in the inventory, then the given language does not need to build long words. The allowed combinations yield a suffcient number of words (lexicon) with suffcient redundancy. If the number of phonemes in the inventory is smaller, then either the mean word length is greater or there are suprasegmental means furnishing sufficient differentiation.

The first theoretical results were brought by Köhler (1986, 1987). The relationship shown below follows from his synergetic self-regulation cycle. Concrete evidence from some African languages was presented by Nettle (1995, 1997, 1998). As usual, one can consider some qualities as constant, in order to obtain a simple testable hypothesis. Nettle holds frequency, lexicon size and redundancy constant.

'''2. Hypothesis'''

''The greater the size of the phoneme inventory, the smaller is the mean word length''.

'''3. Derivation'''

In terms of synergetic linguistics the hypothesis can be translated in a simple differential equation

<math> \frac{dy}{y}= \frac{b}{x}dx</math>

yielding

<math> y=ax^b\quad</math>

with negative b, where y is the mean word length and x is the size of the inventory.

'''Example'''. Some African languages (Nettle 1998)

<div align="center">[[Image:Tabelle11_WLaPI.jpg]]</div>

<div align="center">[[Image:Grafik1_WLaPI.jpg]]</div>

Inspite of a very good F-test the result is not satisfactory. Further testing using other languages with smaller inventories is necessary, and as soon as possible the consideration of the other factors which here were considered constant.

'''4. Authors:''' R. Köhler, G. Altmann

'''5. References'''

'''Köhler, R.''' (1986). ''Zur linguistischen Synergetik'': Struktur und Dynamik der Lexik. Bochum: Brockmeyer.

'''Köhler, R.''' (1987). Systems theoretical linguistics. ''Theoretical Linguistics 14, 241-257.''

'''Nettle, D.''' (1995). Segmental inventory size, word length, and communicative efficiency. ''Linguistics 33, 359-368''.

'''Nettle, D.''' (1998). Coevolution of phonology and the lexicon in twelve languages of West Africa.'' J. of Quantitative Linguistics 5(3), 240-245''.

[[Category:Unfertig]]

Length of syntactic constructions

2006-07-24T12:46:18Z

Altmann:

'''1. Problem and history'''

The length of a syntactic construction is defined as the number of terminal nodes belonging to it, while complexity (<math>\rightarrow</math>) is he number of its immediate constituents. These two properties are interrelated.

The first distribution models were proposed by Köhler and Altmann (2000), no further development is known. As can be seen, the result is a special case of length (<math>\rightarrow</math>) distributions.

'''2. Hypothesis'''

''The length of syntactic constructions abides by the positive negative binomial distribution''.

'''3. Derivation'''

The quantities necessary for the derivation are shown in the chapter “Syntactic structures: Complexity” (<math>\rightarrow</math>). Here the requirement minX = 0, because length depends on complexity and minX is given implicitely. Using the approach proposed for modeling complexity we obtain

(1)<math> P_x = \frac{max H + x}{x}\frac{E}{I(K)}P_{x-1}</math>.

Setting again maxH = k-1, E/I(K) = q (0 < q < 1) and solving (1) we obtain

(2)<math> P_x= {k+x-1 \choose x}\frac{p^k q^x}{1-p^k}, \quad x=1,2,3,... </math>

If maxH tends to -1, i.e. k <math>\rightarrow</math> 0, we obtain the logarithmic distribution

(3)<math> P_x= \frac{q^x}{-x \ln(1-q)}, \quad x=1,2,3,...</math>

but in particular cases it is necessary to modify the probability in x = 1 and one obtains the extended variants of (2) and (3), namely

(4)<math> P_x = \begin{cases} 1-\alpha, & x=1 \\ \alpha {k+x-2 \choose x-1}\frac{p^k q^x}{1-p^k}, & x=2,3,... \end{cases}</math>

(5)<math>P_x = \begin{cases} 1-\alpha, & x=1 \\ \frac{\alpha q^{x-1}}{-(x-1)\ln (1-q)}, & x=2,3,4,... \end{cases}</math>

Example: The length of syntactic constructions in the Susanne corpus
The result of fitting of the extended logarithmic distribution to the data in Susanne corpus are shown in Table 1 and Fig. 1 (Köhler, Altmann 2000).

<div align="center">[[Image:Tabelle111_SCL.jpg]]</div>

Though the chi-square value is very high, the fit is satisfactory as shown by the value of C. The greatest divergence is in the middle range where one can observe strong fluctuation.

'''4. Authors''': U. Strauss, G. Altmann

'''5. References'''

'''Köhler, R., Altmann, G.''' (2000). Probability distributions of syntactic units and properties. ''J. of Quantitative Linguistics 7, 189-200''.

Position of syntactic structures

2006-07-24T12:44:17Z

Altmann:

'''1. Problem and history'''

According to Köhler (1999) the number of allowed constituent types and functions decreases with increasing position in the construct because the necessary storage space in the memory increases from position to position. Thus the number of constituents in a corpus located at individual positions x (x = 1,2,3,…) must follow a specific regularity.
The problem was tackled for the first time by Köhler (1999), the probability distribu-tion was derived by Köhler and Altmann (2000).

'''2. Hypothesis'''

''The number of constituents at given positions follows the binomial distribution''.

'''3. Derivation'''

The quantities necessary for the derivation are the same as with complexity ( Complexity of syntactic structures). However, here complexity is implicit (the complexity of a constituent is its maximum position), thus minX = 0. The compactness maxH is constant and can be written as n+1. Since increasing the position decreases the probability, x in the numerator has negative sign. Setting p/q for E/I(K) and taking the displacement into account (there is no position 0) we obtain

(1)<math> P_{x+1}= \frac{n-x+1}{x}\frac{p}{q}p_x, \quad x=1,2,...,n</math>

resulting in the 1-displaced binomial distribution

(2)<math> P_x = {n \choose x-1}p^{x-1}q^{n-x+1},\quad x=1,2,...,n+1</math>.

Since there are also data displaying bimodal course (e.g. the Negra-Korpus), formula (2) can be modified as follows (P’x = parent 1-displaced binomial distribution)

(3)<math> P_X = \begin{cases} P'_1 + \alpha P'_2, & x=1 \\ P'_2(1-\alpha), & x=2 \\P'_x, & x=3,4,5,... \end{cases}</math>

resulting in the Cohen-binomial distribution

(4)<math> P_X = \begin{cases} q^n (1+\alpha np/q), & x=1 \\ npq^{n-1}(1-\alpha), & x=2 \\ {n \choose x-1}p^{x-1}q^{n-x+1}, & x= 3,4,...,n+1 \end{cases}</math>

'''Example''': Distribution of the position in the Susanne corpus
Köhler and Altmann futted to these data the 1-displaced binomial distribution and obtained the results presented in Table 1 and Fig. 1.

<div align="center">[[Image:Tabelle11_PoSS.jpg]]</div>

<div align="center">[[Image:Grafik1_PoSS.jpg]]</div>
<div align="center">Fig. 1. Fitting the 1-displaced binomial to the data in Table 1</div>

Example: Distribution of the position in the Negra-Korpus
Köhler and Altmann fitted the Cohen-binomial distribution to the Negra-Korpus and obtained the results in Table 2 and Fig. 2.

<div align="center">[[Image:Tabelle22_PoSS.jpg]]</div>

<div align="center">[[Image:Grafik2_PoSS.jpg]]</div>
<div align="center">Fig. 2. Fitting the Cohen-binomial distribution to the data in Table 2</div>

'''4. Authors:''' U. Strauss, G. Altmann

'''5. References'''

'''Köhler, R.''' (1999). Syntactic structures: Properties and interrelations. ''J. of Quantitative Linguistics 6, 46-57''.

'''Köhler, R., Altmann, G'''. (2000). Probability distributions of syntactic units and properties. ''J. of Quantitative Linguistics 7, 189-200''.

Polysemy

2006-07-20T08:40:50Z

Altmann:

'''1. Problem and history'''

The simplest problem connected with polysemy is its frequency distribution in the dictionary. One can easily see that in a monolingual dictionary lexical items have different number of explications. Sometimes the explications are numbered and indicate meaning difference or nuance. This is the only source of approaching the problem. The number of meanings of a word is sometimes called semantic size or semantic volume (cf. Tuldava 1998: 119). Some reserchers stated that the mean semantic volume of different word classes differs significantly (cf. Višnjakova 1976; Tuldava 1979; Ufimceva 1968: 89)
Historically, G.K. Zipf (1935, 1945a, 1949) was probably the first who has shown that meaning is associated with frequency, with length and other properties analyzed in synergetic linguistics (see Vol. 2). Nevertheless, the distribution of the semantic volume can be studied separately (see Introduction, Chapter 2), since it represents a loop in the system. Another question is the frequency of polysemic words in text.
The first trials to find a theoretical distribution can be found in Krylov, Jakubovskaja (1977), Krylov (1982) who derived the geometric distribution, Tuldava (1979) who used on empirical grounds a kind of exponential function and Levickij, Drebet, Kiiko (1999) who used rather the mixed geometric distribution, assuming that the words in the dictionary are of a mixed character. A good survey of problems can be found in Hoffmann (2001).

'''2. Hypothesis'''

''The number of words (<math>f_x</math>) with x meanings is a function of x.''

'''3. Derivation'''

'''3.1. Waring distribution'''

Wimmer and Altmann (1999a) considered the building of polysemy as a simple Poissonian birth-and-death process and set up the equations

(1)<math>\lambda_0 P_0 = \mu_1 P_1\quad</math>

<math>(\lambda _x + \mu_x)P_x = \lambda_{x-1}P_{x-1} + \mu_{x+1}P_{x+1}, \quad x= 1, 2,...</math>

Inserting <math>\lambda_x = a+x, \mu_x = a+b+x</math> and displacing the distribution one step to the right – because zero-semic words are not taken into account - they obtained the Waring distribution

(2)<math> P_x = \frac{ba^{(x-1)}}{(a+b)(a+b+1)^{(x-1)}},\quad x=1, 2, ...</math>

'''3.2. Bissinger-geometric distribution'''

Since the Waring distribution can be presented by a backward recurrence formula, Wimmer and Altmann (1999a) consider also the possibility of forward dependence using a parent distribution {Px* }x≥0 and building partial sums in the form

(3)<math> P_0= f_0 (P*_1 + P*_2 + P*_3 + P*_4 + ...)</math>

<math> P_1 = f_1 (P*_2 + P*_3 + P*_4 +...)</math>

<math> P_2 = f_2 (P*_3 + P*_4 +...)</math>

From the great number of possibilities (cf. Wimmer, Altmann 2000) they chose the simple form

(4)<math> f_1 = (P*_{i+1} + P*_{i+2} + ...) = c \sum_{j=i+1}^\infty\frac{P*_1}{j}, \quad i=0, 1, ...</math>

yielding <math> c = \frac{1}{1-P*_0}</math> as a normalizing constant.. Using the geometric distribution as parent distribution and displacing the result one step to the right they obtain

(5)<math>P_x = \frac{p}{q}\sum_{j=x}^\infty\frac{q^j}{j}, \quad x= 1, 2, 3, ...</math>

known as Bissinger-geometric distribution (cf. Wimmer, Altmann 1999).

'''Example:''' Polysemy in Maori (New Zealand)

Using the data ascertained by V. Krupa, Wimmer and Altmann (1999a) fitted the Waring and the Bissinger-geometric distributions to Maori dictionary data as shown in Table 1.

<div align="center">[[Image:Tabelle1_P.jpg]]</div>

[[[Please complete:

'''3.3. Tuldava´s version'''
<math>y_x = a \exp (-bx^c), \quad y_x </math> = number of words with x meanings

'''3.4. Krylov-Jakubovskaja´s version''' (

<math> y_x = e^{-bx}</math>]]]

'''4. Authors:''' U. Straus, G. Altmann

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'''Wimmer, G., Altmann, G'''. (2000). On the generalization of the STER distribution applied to generalized hypergeometric parents. ''Acta Universitatis Palackiensis Olomouciensis, Facultats rerum naturalium Mathematica 39, 215-247''.

'''Wolff, D'''. (1972). Bedeutungshäufigkeit und ihr statistisches Verhalten. ''Beiträge zur Linguistik und Informationsverarbeitung 22, 33-44''.

'''Zipf, G.K.''' (1935).''The psycho-biology of language''. Boston: Houghton Mifflin.

'''Zipf, G.K'''. (1945a). The meaning-frequency relationship of words. ''J. of General Psychology 33, 251-255''.

'''Zipf, G.K.''' (1949). ''Human behavior and the principle of least effort''. Cambridge: Addison-Wesley.

Polysemy

2006-07-20T07:53:19Z

Altmann:

'''1. Problem and history'''

The simplest problem connected with polysemy is its frequency distribution in the dictionary. One can easily see that in a monolingual dictionary lexical items have different number of explications. Sometimes the explications are numbered and indicate meaning difference or nuance. This is the only source of approaching the problem. The number of meanings of a word is sometimes called semantic size or semantic volume (cf. Tuldava 1998: 119). Some reserchers stated that the mean semantic volume of different word classes differs significantly (cf. Višnjakova 1976; Tuldava 1979; Ufimceva 1968: 89)
Historically, G.K. Zipf (1935, 1945a, 1949) was probably the first who has shown that meaning is associated with frequency, with length and other properties analyzed in synergetic linguistics (see Vol. 2). Nevertheless, the distribution of the semantic volume can be studied separately (see Introduction, Chapter 2), since it represents a loop in the system. Another question is the frequency of polysemic words in text.
The first trials to find a theoretical distribution can be found in Krylov, Jakubovskaja (1977), Krylov (1982) who derived the geometric distribution, Tuldava (1979) who used on empirical grounds a kind of exponential function and Levickij, Drebet, Kiiko (1999) who used rather the mixed geometric distribution, assuming that the words in the dictionary are of a mixed character. A good survey of problems can be found in Hoffmann (2001).

'''2. Hypothesis'''

''The number of words (<math>f_x</math>) with x meanings is a function of x.''

'''3. Derivation'''

'''3.1. Waring distribution'''

Wimmer and Altmann (1999a) considered the building of polysemy as a simple Poissonian birth-and-death process and set up the equations

(1)<math>\lambda_0 P_0 = \mu_1 P_1\quad</math>

<math>(\lambda _x + \mu_x)P_x = \lambda_{x-1}P_{x-1} + \mu_{x+1}P_{x+1}, \quad x= 1, 2,...</math>

Inserting <math>\lambda_x = a+x, \mu_x = a+b+x</math> and displacing the distribution one step to the right – because zero-semic words are not taken into account - they obtained the Waring distribution

(2)<math> P_x = \frac{ba^{(x-1)}}{(a+b)(a+b+1)^{(x-1)}},\quad x=1, 2, ...</math>

'''3.2. Bissinger-geometric distribution'''

Since the Waring distribution can be presented by a backward recurrence formula, Wimmer and Altmann (1999a) consider also the possibility of forward dependence using a parent distribution {Px* }x≥0 and building partial sums in the form

(3)<math> P_0= f_0 (P*_1 + P*_2 + P*_3 + P*_4 + ...)</math>

<math> P_1 = f_1 (P*_2 + P*_3 + P*_4 +...)</math>

<math> P_2 = f_2 (P*_3 + P*_4 +...)</math>

From the great number of possibilities (cf. Wimmer, Altmann 2000) they chose the simple form

(4)<math> f_1 = (P*_{i+1} + P*_{i+2} + ...) = c \sum_{j=i+1}^\infty\frac{P*_1}{j}, \quad i=0, 1, ...</math>

yielding <math> c = \frac{1}{1-P*_0}</math> as a normalizing constant.. Using the geometric distribution as parent distribution and displacing the result one step to the right they obtain

(5)<math>P_x = \frac{p}{q}\sum_{j=x}^\infty\frac{q^j}{j}, \quad x= 1, 2, 3, ...</math>

known as Bissinger-geometric distribution (cf. Wimmer, Altmann 1999).

'''Example:''' Polysemy in Maori (New Zealand)

Using the data ascertained by V. Krupa, Wimmer and Altmann (1999a) fitted the Waring and the Bissinger-geometric distributions to Maori dictionary data as shown in Table 1.

<div align="center">[[Image:Tabelle1_P.jpg]]</div>

[[[Please complete:

'''3.3. Tuldava´s version'''
<math>y_x = a \exp (-bx^c), \quad y_x </math> = number of words with x meanings

'''3.4. Krylov-Jakubovskaja´s version''' (

<math> y_x = e^{-bx}</math>]]]

'''4. Authors:''' U. Strauus, G. Altmann

'''5. References'''

'''Altmann, G.''' (1985). Semantische Diversifikation. ''Folia Linguistica 19, 177-200''.
Altmann, G., Bagheri, D., Goebl, H., Köhler, R., Prün, C. (2002). ''Einführung in die quantitative Lexikologie''. Götingen: Peust & Gutschmidt.

'''Altmann, G., Best, K.-H., Kind, B.''' (1987). Eine Verallgemeinerung des Gesetzes der semantischen Diversifikation. ''Glottometrika 8, 130-139''.

'''Hammerl, R'''. (1991). ''Untersuchungen zur Struktur der Lexik: Aufbau eines lexikalischen Basismodells.'' Trier, WVT.

'''Hoffmann, Ch'''. (2001). Polylexie lexikalischer Einheiten in Texten. In: Uhlířova, L., Wimmer, G., Altmann, G., Köhler, R. (Eds.), ''Text as a linguistic paradigm: levels, constituents, constructs. Festschrift in honour of Ludek Hřebíček: 76-97''. Trier: WVT.

'''Krott, A.''' (2002). Ein funktionalanalytisches Modell der Wortbildung. In: Köhler, R. (ed.), ''Korpuslinguistische Untersuchungen in die quantitative und systemtheoretische Linguistik: 75-126''. http://ubt.opus.hbz-nrw.de/volltexte/2004/279/

'''Krylov, Ju.K'''. (1982a). Ob odnoj paradigme lingvostatističeskich raspredelenij. ''Acta et Commentationens Universitatis Tartuensis 628, 80-102''.

'''Krylov, Ju.K'''. (1982b). Eine Untersuchung statistischer Gesetzmäßigkeiten auf der paradigmatischen Ebene der Lexik natürlicher Sprachen. In: Guiter, H., Arapov, M.V. (eds.), ''Studies on Zipf´s law: 234-262''. Bochum: Brockmeyer.

Krylov, Jakubovskaja 1977;

'''Levickij, V'''. (2005). Polysemie. In: Köhler, R., Altmann, G., Piotrowski, R.G. (eds.), ''Handbook of Quantitative Linguistics: 458-464.'' Berlin: de gruyter.

'''Levickij, V.V., Drebet, V.V., Kiiko, S.V.''' (1999). Some quantitative characteristics of polysemy of verbs, nouns and adjectives in the German language. ''J. of Quantitative Linguistics 6, 172-187''.

'''Polikarpov, A.A.''' (1987). Polisemija. Sistemno-kvantitativnye aspekty. In: Tuldava, J.A. (ed.), ''Kvantitativnaja lingvistika i avtomatičeskij analiz tekstov: 135-154''. Tartu: Učenye zapiski Tartuskogo gosudarstvennogo universiteta 774.

'''Rothe, U.''' (1983). Wortlänge und Bedeutungsmenge. Eine Untersuchung zum Menzerathschen Gesetz an drei romanischen Sprachen. ''Glottometrika 5, 101-112''.

Tuldava 1979,

'''Tuldava, J.''' (1998). ''Probleme und Methoden der quantitativ-systemischen Lexikologie''. Trier: WVT.

Ufimceva 1968;

Višnjakova 1976;

'''Wimmer, G., Altmann, G'''. (1999a). Rozdelenie polysémie v maorijčine. In: Ondrejovič, S., Genzor, J. (eds.), ''Pange lingua. Zborník na počest´ Viktora Krupu: 17-24''. Bratislava: Veda.

'''Wimmer, G., Altmann, G'''. (2000). On the generalization of the STER distribution applied to generalized hypergeometric parents. ''Acta Universitatis Palackiensis Olomouciensis, Facultats rerum naturalium Mathematica 39, 215-247''.

*'''Wolff, D'''. (1972). Bedeutungshäufigkeit und ihr statistisches Verhalten. ''Beiträge zur Linguistik und Informationsverarbeitung 22, 33-44''.

'''Zipf, G.K.''' (1935).''The psycho-biology of language''. Boston: Houghton Mifflin.

'''Zipf, G.K'''. (1945a). The meaning-frequency relationship of words. ''J. of General Psychology 33, 251-255''.

'''Zipf, G.K.''' (1949). ''Human behavior and the principle of least effort''. Cambridge: Addison-Wesley.

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Phoneme entropy and inventory

2006-07-20T07:50:14Z

Altmann:

'''1. Problem and history'''

Entropy is a measure of disorder of the phoneme system or rather a measure of deviation from the uniformity of phoneme frequencies. If <math>-ld p_x</math> is the measure of self-information of a phoneme (px is the relative frequency of phoneme x) then

(1)<math> H = -\sum_{x=1}^K p_x ld p_x</math>

is the measure of average uncertainty or entropy. For different types of entropy see Naranan, Balasubrahmanyan (2000). Since px is estimated as fx/N (N being the sample size), (1) can be written as

(2)<math> H = ld N -\frac{1}{N}\sum_{x=1}^K f_x ld f_x</math>

Since ranked frequencies of phonemes (<math>\rightarrow</math>) abide by a specific distribution, there is the justified question whether H depends on the size of the inventory of phonemes K.
Altmann and Lehfeldt (1980) used the model of the 1-displaced right truncated geometric distribution to show that entropy depends on K, Zörnig and Altmann (1983, 1984) used the Zipf-Mandelbrot distribution to derive another formula.
Cohen, Mantegna and Havlin (1997) observed a parabolic curve for the dependence between word inventory and entropy.

'''2. Hypothesis'''

''The entropy depends on the size of the phoneme inventory''.

'''3. Derivation'''

'''3.1. From the geometric distribution''' (Altmann, Lehfeldt 1980)

Let the ranked frequencies of phonemes (<math>\rightarrow</math>) follow the 1-displaced right truncated geometric distribution defined as

(3)<math> P_x = aq^{x-1}, \quad x= 1, 2, ..., K</math>

with

(3)<math> a= \frac{p}{1-q^K}, \quad p=1-q</math>

then (1) can be written as

(4)<math> H =-\sum_{x=1}^K aq^{x-1} ld (aq^{x-1})</math>.

Summing and using (3) one obtains

(5)<math> H= -ld a -\frac{\lbrack q-K(a+q-1)\rbrack ld q}{1-q}</math>.

Since a + q -1 ≈ 0 and <math>q^K\rightarrow 0</math> for great K, (5) results in

(6)<math> H= -ld(1-q)-\frac{q ld q}{1-q}</math>.

Substituting <math> q = \frac{K-2}{K+2}</math> as the first approximation (see also Phonemes: Repeat rate)

one obtains

(7)<math> H = -ld\begin{bmatrix}\left(\frac{4}{K+2}\right)\left(\frac{K-2}{K+2}^{\frac{K-2}{4}}\right)\end{bmatrix}</math> .

(Example: see below)

'''3.2. From the Zipf-Mandelbrot distribution''' (Zörnig, Altmann 1984)

Let the ranked frequencies be distributed according to

(8)<math> P_x = \frac{A}{(B+x)^c}, \quad x = 1, 2, ..., K</math>

where c = 1 brought already a good approximation in case of Repeat rate (<math>\rightarrow</math>), we consider

(9)<math> H =-\sum_{x=1}^K \frac{A}{B+x}ld \frac{A}{B+x}</math>,

A being the normalizing constant, i.e.

(10)<math>A^{-1} = \sum_{x=1}^K \frac{1}{B+x}</math>.

Since ld x = (ld e)(ln x) we obtain from (9)

(11)<math>H = -A ld e\sum_{x=1}^K \frac{1}{B+x}ln\frac{A}{B+x}</math>

= <math> -A ld e \begin{bmatrix}\ln A \sum_{x=1}^K \frac{1}{B+x}- \sum_{x=1}^K \frac{\ln(B+x)}{B+x}\end{bmatrix}</math> .

Inserting (10) in (11) yields

(12)<math> H = ld e \begin{bmatrix} -\ln A + A \sum_{x=1}^K \frac{\ln(B+x)}{B+x}\end{bmatrix}</math>.

Approximating A by an appropriate integral, we obtain

(13)<math> A = \frac{1}{ln\frac{B+K}{B+1}}</math>.

and in the same way

(14)<math>\sum_{x=1}^K \frac{ln(B+x)}{B+x}= \int_{1}^{K}\frac{ln(B+x)}{B+x}dx = \frac{1}{2}\ln\lbrack (B+K)(B+1)\rbrack\ln\frac{B+K}{B+1}</math>.

Inserting (13) and (14) in (12) and ordering one obtains at last

(15)<math> H = ld e \ln\lbrack \sqrt{(B+K)(B+1)}\ln\frac{B+K}{B+1}\rbrack</math>.

'''Example'''. Entropy of phoneme systems for 63 languages.
Altmann, Lehfeldt (1980) and Zörnig, Altmann (1984) fitted the above curves (7) and (15) to the empirical entropies in 63 languages and obtained the results shown in Table 1. Zörnig and Rothe (1990) added further 8 data from French and German. The best value of B was iteratively established at B = 0.61 and B = 0.27. However, B is not yet interpreted. The languages are the same as in the tables in Repeat rate (<math>\rightarrow</math>). Here the coefficient of determination has been ascertained in such a way that the values of H for the same K were averaged.

<div align="center">[[Image:Tabelle1_PEaI.jpg]]</div>

Evidently, the fitting is slightly better using Zipf-Mandelbrot´s distribution. Again, even if the fitting is satisfactory, further research must be done.

<div align="center">[[Image:Grafik1_PEaI_Kopie.jpg]]</div>
<div align="center">Fig. 1. Fitting curves (7) –––– and (15) ------ to the entropies of 63 languages</div>

Note. Using the geometric distribution for the derivation of Repeat rate (R) and Entropy (H) the following relationship between them follows

(16)<math>R = \frac{1- \begin{bmatrix}\frac{2^{-H}(K+2)}{4}\end{bmatrix}^{\frac{4}{K-2}}}{1+\begin{bmatrix}\frac{2^{-H}(K+2)}{4}\end{bmatrix}^{\frac{4}{K-2}}}</math>

and

(17)<math> H = -ld \begin{bmatrix}\begin{pmatrix}\frac{2R}{R+1}\end{pmatrix}\begin{pmatrix}\frac{1-R}{+R}\end{pmatrix}^{\frac{1-R}{2R}}\end{bmatrix}</math> .

Using the Zipf-Mandebrot distribution we obtain

(18)<math> H = \frac{1}{\ln 4}\ln \frac{K-1}{R}</math>
and

(19)<math> R = \frac{K-1}{4^H}</math>.

so that it is sufficient to compute one of these values.

'''4. Authors''': U. Strauss, G. Altmann

'''5. References''' (contains also references in which merely entropy has been computed)

'''Altmann, G., Bagheri, D., Goebl, H., Köhler, R., Prün, C.''' (2002). ''Einführung in die quantitative Lexikologie''. Götingen: Peust & Gutschmidt.

'''Altmann, G., Lehfeldt, W.''' (1980). ''Einführung in die quantitative Phonologie''. Buchum: Brockmeyer.

'''Cohen, A., Mantegna, R.N., Havlin, S'''. (1997). Numerical analysis of word frequencies in artificial and natural language texts? ''Fractals 5(1), 93-104''.

'''Feng, Zh.''' (1984). Hanzi de shang (Entropy of Chinese characters). In: Wenzi gaige 4, ….[Reprint in Chen, Y. (1989)(ed.), ''Xiandai Hanyu dingliang fenxi: 267-278 (Quantitative analysis of modern Chinese).'' Shanghai: Shanghai Jiaoyu chubanse.

'''Jakopin, F.''' (2002). ''Entropija v slovenskih leposlovnih besedilih''. Ljubljana: ZRC SAZU.

'''Kučera, K'''. (2001). The development of entropy and redundancy in Czech from the 13th to the 20th century: Is there a linguistic arrow of time. In: Uhlířova, L., Wimmer, G., Altmann, G., Köhler, R. (Eds.), ''Text as a linguistic paradigm: levels, constituents, constructs. Festschrift in honour of Ludek Hřebíček: 153-162''. Trier: WVT.

'''Lin, L'''. (2001). Guabyu Hanzi tongji tezheng de ji ge wenti. In: Su, P. (ed.), ''Hiandai Hanzixue cankao ziliao:''….. (Reference data to modern sinographics.) Peking: beijing Daxue chubanshe.

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Hierarchic relations

2006-07-18T14:39:14Z

Altmann:

'''1. Problem and history'''

In different domains of language one observed the fact that the length of a construct influences the length of its consitutents. Usually the constituents get smaller with increasing length of the construct but not in all cases. The problem is to find a theoretical model encompassing all dependencies of this kind. The constructs whose length is the independent variable are: hreb, sentence, rhythmic unit, word, syllable; the constituents whose length or duration is the dependent variable are: sentence, clause, word, syllable, morph, sound. There is also the possibility to consider two independent variables, e.g. word (measured in number of syllables) and syllable (measured in number of sounds) , while the dependent variable is the syllable duration.
Up to now the following particular cases have been examined (see Table 1)

<div align="center">[[Image:Tabelle11_HR.jpg]]</div>

The origin of the problem can be found in 19th century, in phonetics, probably for the first time with Sievers (1876, 1901) who measured the syllable duration in rhythmic units (Sprechakte). A number of phoneticians tested different hypotheses, a part of which corroborated, another part falsified it. The isochrony hypothesis in English is a special case of this problem. The problem was generalized by Menzerath who stated that ''the greater the whole the smaller its parts'' (1954: 101). Different researchers proposed some empirical formulas (Fónagy, Magdics 1960; Nooteboom 1972, 1973; Landblom, Rapp 1972), Altmann (1980) set up the pertinent differential equation and called the result Menzerath´s law. Hřebíček (1992, 1995, 1997) showed that the whole hierarchy of textual levels is based on this dependence and called it ''Menzerath-Altmann´s law''.

There is a great number of individual examinations in different domains of languge. In phonetics the most exhaustive is Weber (1998), in textology the works by Hřebíček (see above) and a mixture of problems including biology and sociology can be found in Altmann, Schwibbe (1989). Bohn (1998) analyzed the relationship between Chines characters and the complexity of composing graphemes, length of words and simplicity of characters, clause length and word length, sentence length and clause length. (cf. also Menzel 2005).
The law has a strong corroboration not only within linguistics but displays analogies to other sciences. Thus the simple form of Menzerath´s law is identical with the allometric law in biology and with power laws current in different sciences. Its correspondences can be found in (i) molecular biology, (ii) sociology of baboons, (iii) in the domain of self-organized criticality, (iv) in chaos research, (v) in the theory of fractals, (vi) in information theory.
It has been observed that construct length is not always the only cause of shortening of the constituents. Also accent, vowel quality, syllable structure, frequency etc. can intervene (cf. Weber 1998). In that case more complex formulas must be used.
The law has several consequences, all of which must still be tested (cf. Altmann, Schwibbe 1989: 8-14):
1. In longer words more phonetic changes occur than in shorter ones.

2. In languages with greater average word length more phonetic/phonemic changes occur than within the same time interval in languages with smaller average word length

3. The adding of an affix to a word evokes the tendency to reduce the inventory of consonants of the word.

4. The shortening of average syllables length in Hypothesis 3 can also be achieved by inserting epenthetic vowels between the stem and affix (or compounding stem).

5. Partial reduplication is more frequent in natural languages than full reduplication.

6. Short roots/morphemes/stems build more compounds or derived words than long ones.

7. The more elements there are in a compound the shorter they are (see the hypotheses on compounds <math>\leftarrow</math>)

8. Fenk-Fenk (to be inserted)

There are different interpretations of the law:

1. In general, long constructs contain more redundancy than short ones. In order to prevent excessive growth of redundancy one can reduce the size of the constituents. The size, the place and the time of this reduction is not known and cannot be predicted. The analogy to self-organized criticality of sand-piles is evident (cf. Bak 1996).

2. Köhler (1989) shows that mechanism of shortening is a consequence of restrictions of the memory: the longer the construct, the more place must be reserved for the structural information between the constituents, thus the size of the constituents must be reduced.

'''2. Hypothesis'''

''The size of the components is a function of the construct size''.

'''3. Derivation'''

The average size of constituents changes with the increase of the size of the construct. It is assumed that the relative rate of change of the size of components is proportional to the rate of change of the size of constructs, the proportionality function being

<math> g(x) = a_0 + \frac{a_1}{x} + \frac{a_2}{x^2}</math>.

Thus in case that all other variables other than construct size are subsumed under the ceteris paribus condition one obtains

(1)<math> \frac{dy}{y-d}= \left(a_0 \frac{a_1}{x}+ \frac{a_2}{x^2}\right)</math>.

where d is the minimal value y can attain.
In case that there is another independent variable, z, one starts from

(2)<math>\frac{dy}{dx}\frac{1}{y-d}=a_0 + \frac{a_1}{x}+\frac{a_2}{x^2}\quad</math> and <math>\frac{dy}{dz}\frac{1}{y-d}=b_0 + \frac{b_1}{z}+\frac{b_2}{z^2}</math>

which can be extended to any number of variables.
The solution of (1) yields

(3)<math> y= Cx^{a_1}e^{a_0 x-a_2/x} + d</math>

the combining (2) the solution is

(4)<math> y= Cx^{a_1}z^{b_1}e^{a_0 x + b_0 z-a_2 /x-b_2 /z} + d</math>

In different applications the following special cases of (3) have been used (d > 0)

(1a) <math> y=ax^{-b}(+d), \quad x= 1, 2, 3, ...; \quad a, b > 0</math>

(1b)<math> y=ax^b e^{cx}(+d), \quad x= 1, 2, 3, ...; \quad a > 0</math>

(1c)<math> y=ae^{-cx}(+d), \quad x= 1, 2, 3, ...; \quad a, c > 0</math>

(1d) <math> y=ae^{c/x}(+d), \quad x= 1, 2, 3, ...; \quad a, c > 0</math>

Solution (4) is still very seldom. Both approaches are special cases of the unified theory (<math>\leftarrow</math>).

'''Examples''':

'''4. Authors: U. Strauss, G. Altmann'''

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[[Category:Unfertig]]

Hierarchic relations

2006-07-18T14:23:06Z

Altmann:

'''1. Problem and history'''

In different domains of language one observed the fact that the length of a construct influences the length of its consitutents. Usually the constituents get smaller with increasing length of the construct but not in all cases. The problem is to find a theoretical model encompassing all dependencies of this kind. The constructs whose length is the independent variable are: hreb, sentence, rhythmic unit, word, syllable; the constituents whose length or duration is the dependent variable are: sentence, clause, word, syllable, morph, sound. There is also the possibility to consider two independent variables, e.g. word (measured in number of syllables) and syllable (measured in number of sounds) , while the dependent variable is the syllable duration.
Up to now the following particular cases have been examined (see Table 1)

<div align="center">[[Image:Tabelle11_HR.jpg]]</div>

The origin of the problem can be found in 19th century, in phonetics, probably for the first time with Sievers (1876, 1901) who measured the syllable duration in rhythmic units (Sprechakte). A number of phoneticians tested different hypotheses, a part of which corroborated, another part falsified it. The isochrony hypothesis in English is a special case of this problem. The problem was generalized by Menzerath who stated that ''the greater the whole the smaller its parts'' (1954: 101). Different researchers proposed some empirical formulas (Fónagy, Magdics 1960; Nooteboom 1972, 1973; Landblom, Rapp 1972), Altmann (1980) set up the pertinent differential equation and called the result Menzerath´s law. Hřebíček (1992, 1995, 1997) showed that the whole hierarchy of textual levels is based on this dependence and called it ''Menzerath-Altmann´s law''.

There is a great number of individual examinations in different domains of languge. In phonetics the most exhaustive is Weber (1998), in textology the works by Hřebíček (see above) and a mixture of problems including biology and sociology can be found in Altmann, Schwibbe (1989). Bohn (1998) analyzed the relationship between Chines characters and the complexity of composing graphemes, length of words and simplicity of characters, clause length and word length, sentence length and clause length. (cf. also Menzel 2005).
The law has a strong corroboration not only within linguistics but displays analogies to other sciences. Thus the simple form of Menzerath´s law is identical with the allometric law in biology and with power laws current in different sciences. Its correspondences can be found in (i) molecular biology, (ii) sociology of baboons, (iii) in the domain of self-organized criticality, (iv) in chaos research, (v) in the theory of fractals, (vi) in information theory.
It has been observed that construct length is not always the only cause of shortening of the constituents. Also accent, vowel quality, syllable structure, frequency etc. can intervene (cf. Weber 1998). In that case more complex formulas must be used.
The law has several consequences, all of which must still be tested (cf. Altmann, Schwibbe 1989: 8-14):
1. In longer words more phonetic changes occur than in shorter ones.

2. In languages with greater average word length more phonetic/phonemic changes occur than within the same time interval in languages with smaller average word length

3. The adding of an affix to a word evokes the tendency to reduce the inventory of consonants of the word.

4. The shortening of average syllables length in Hypothesis 3 can also be achieved by inserting epenthetic vowels between the stem and affix (or compounding stem).

5. Partial reduplication is more frequent in natural languages than full reduplication.

6. Short roots/morphemes/stems build more compounds or derived words than long ones.

7. The more elements there are in a compound the shorter they are (see the hypotheses on compounds <math>\leftarrow</math>)

8. Fenk-Fenk (to be inserted)

There are different interpretations of the law:

1. In general, long constructs contain more redundancy than short ones. In order to prevent excessive growth of redundancy one can reduce the size of the constituents. The size, the place and the time of this reduction is not known and cannot be predicted. The analogy to self-organized criticality of sand-piles is evident (cf. Bak 1996).

2. Köhler (1989) shows that mechanism of shortening is a consequence of restrictions of the memory: the longer the construct, the more place must be reserved for the structural information between the constituents, thus the size of the constituents must be reduced.

'''2. Hypothesis'''

''The size of the components is a function of the construct size''.

'''3. Derivation'''

The average size of constituents changes with the increase of the size of the construct. It is assumed that the relative rate of change of the size of components is proportional to the rate of change of the size of constructs, the proportionality function being

<math> g(x) = a_0 + \frac{a_1}{x} + \frac{a_2}{x^2}</math>.

Thus in case that all other variables other than construct size are subsumed under the ceteris paribus condition one obtains

(1)<math> \frac{dy}{y-d}= \left(a_0 \frac{a_1}{x}+ \frac{a_2}{x^2}\right)</math>.

where d is the minimal value y can attain.
In case that there is another independent variable, z, one starts from

(2)<math>\frac{dy}{dx}\frac{1}{y-d}=a_0 + \frac{a_1}{x}+\frac{a_2}{x^2}\quad</math> and <math>\frac{dy}{dz}\frac{1}{y-d}=b_0 + \frac{b_1}{z}+\frac{b_2}{z^2}</math>

which can be extended to any number of variables.
The solution of (1) yields

(3)<math> y= Cx^{a_1}e^{a_0 x-a_2/x} + d</math>

the combining (2) the solution is

(4)<math> y= Cx^{a_1}z^{b_1}e^{a_0 x + b_0 z-a_2 /x-b_2 /z} + d</math>

In different applications the following special cases of (3) have been used (d > 0)

(1a) <math> y=ax^{-b}(+d), \quad x= 1, 2, 3, ...; \quad a, b > 0</math>

(1b)<math> y=ax^b e^{cx}(+d), \quad x= 1, 2, 3, ...; \quad a > 0</math>

(1c)<math> y=ae^{-cx}(+d), \quad x= 1, 2, 3, ...; \quad a, c > 0</math>

(1d) <math> y=ae^{c/x}(+d), \quad x= 1, 2, 3, ...; \quad a, c > 0</math>

Solution (4) is still very seldom. Both approaches are special cases of the unified theory (<math>\leftarrow</math>).

'''Examples''':

'''4. Authors: U. Strauss, G. Altmann'''

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[[Category:Unfertig]]

Phoneme frequency

2006-07-16T08:42:28Z

Altmann:

'''1. Problem and history'''

The problem is to find a rank-frequency distribution for the phoneme of a text or of a corpus. Sometimes letters or even sounds are counted, which is fully justified. In the same way one could count e.g. the syllables of the Japanese katakana or hiragana. The number of examinations is enormous, some of them give the absolute frequencies other ones merely the proportions.

The counting began in the 19th century (Förstemann 1852; Bourdon 1892) and developed quickly on practical grounds since stenographers, printers, constructors of typewriters, decoders etc., needed urgently the frequency of letters for their own purposes.

The first who considered phonemes from the frequency point of view and set up hypotheses was G.K. Zipf (1929, 1935, 1949). Afterwards a great number of works appeared using phoneme frequencies for finding other interrelations. The first empirical model for a distribution, namely the geometric (and the right truncated geometric) distribution, was proposed by Sigurd (1968). Good (1969) brought a partial-sums distribution whose modelling was revived in word length (<math>\rightarrow</math>) research. Altmann (1993) used the synergetic way of modelling and derived a special function for this purpose. Martindale, Gusein-Zade, Mckenzie and Borodovsky (1996) compared several curves (functions) and many data in order to find the “best” model. Altmann and Lehfeldt (1980) and Zörnig, Altmann (1983, 1984) developed hypotheses on the entropy (<math>\rightarrow</math>) and the repeat rate (<math>\rightarrow</math>) of phonemes, Kubáček (1994) derived the formula for the necessary size of the phoneme count in order to attain confident counts. Naranan and Balasubrahmanyan (2000) developed a theory from which different curves for phoneme frequencies are derivable.

Not all arguments holding for word frequencies are valid in this domain.

'''2. Hypothesis'''

''The ranked frequencies of phonemes follow a regular probability function or a regular monotone decreasing function''.

The result depends on whether one considers the ranked frequencies as a discrete distribution (normalized) or merely a regular series approached by a continuous function (not normalized).

'''3. Derivation'''

The formulas used up to now can be derived from different approaches.

'''3.1. Tuldava´s approach (1988)'''

This approach can be represented by the simple differential equation

(1) <math> y' = \frac{b}{x}</math>

to obtain

(2)<math>A y = a + b \ln x \quad</math>

This curve is frequently used in other domains, too (cf. also Martindale et al. 1996).

'''3.2. Derivations related to the unified theory (→) are'''

'''(a) Zipf´s law (zeta distribution)'''

When formula (2) of the unified theory is used with
<math> a_0 = a_2 = a_3 = ... = 0, a_1 = -b \quad</math>, this yields

(3) <math>\frac{dy}{y} = -\frac{b}{x}dx</math>

resulting in

(4)<math> y = Ax^{-b}\quad</math>.

This is, perhaps, the most disseminated formula in linguistics.

'''(b) Yule´s species/genera function'''

When formula (2) of the unified theory is used with <math>a_0 = c, a_1 = b, a_2 = a_3 = ... = 0\quad</math>, this yields

(5)<math> \frac{dy}{y}= \left(c- \frac{b}{x} \right)dx</math>

resulting in

(6)<math> y= ae^{cx}x^{-b}= ad^x b^{-b}\quad</math> .

'''(c) Naranan and Balasubrahmanyan´s (1992a,b, 2000) function'''

When formula (2) of the unified theory is used with <math>a_0 = 0, a_3 = a_4 = ... = 0\quad</math>, this yields

(7)<math> \frac{dy}{y}\left(- \frac{a_1}{x}{a_2}{x^2} \right)dx</math>

resulting in

(8)<math> y= Ce^{-a_2/x}x^{-a_1}</math>.

'''(d) Altmann´s ranking function (1993)'''

Using formula (11) of the unified theory, which can be written as

(9)<math> y_x = \left(1-a_0 + \frac{a_1}{(x-b_1)^{c_1}} + \frac{a_2}{(x-b_2)^{c_2}} \right)y_{x-1}</math> ,

and reparametrizing <math>a_i = 0 (i = 0,2,3,...), c_1 = 1</math>, yields

(10)<math> y_x = \left(1+ \frac{a_1}{x-b_1} \right)y_{x-1}</math>.

Upon setting <math>b_1 = -a</math>, <math>a_1 - b_1 = b\quad</math>, this results in

(11)<math>y_x = \frac{\begin{pmatrix} b+x \\ x-1 v \end{pmatrix}}{\begin{pmatrix} a+x \\ x-1 \end{pmatrix}}y_1 \quad, x = 1,2,3,...</math>

All these formulas can be transformed in distributions by appropriate normalizing.

'''(e) Geometric distribution'''

Sigurd (1968) used simply the geometric distribution. It can be obtained from formula (10) of the unified theory setting <math>a_i = 0 (i = 1,2,3,...)\quad</math>, which yields

(12)<math> y_{x+1}= (1+a_0)y_x\quad</math>.

For<math>- < a_ < 0, 1+a_ = q, 1-q = p, y_ = Px\quad</math>tains the usual (1-displaced) geometric distribution

(13)<math> P_x = pq^{x-1}, \quad x = 1,2,3,...</math>

The same result was proposed also by Orlov, Boroda, Nadarejšvili (1982). Treating directly the relative frequencies one can write (13) as

(14)<math> y_x = y_1 q^{x-1}, \quad x=1,2,3,...</math>

'''3.3. Partial-sums distributions (Good 1969)'''

Good (1969) introduced a new distribution, mentioned in Martindale et al. (1996). It is a so-called partial-sums distribution, namely a “sterred” discrete uniform distribution (cf. Wimmer, Altmann 1999). Their provenience is shown in the chapter on Word frequency (<math> \rightarrow</math>)has the form

(15)<math>P_x = \frac{1}{n}\sum_{i=x}^n \frac{1}{i},\quad x=1,2,...,n</math>.

'''Example:''' Frequency of phonemes in Hawaiian

In Table 1 and Fig. 1 one can find the fitting of the above formulas to the relative frequencies of Hawaiian phonemes. If functions are used, normalizing is not necessary.

<div align="center">[[Image:Tabelle1_PF.jpg]]</div>

Except for the geometric distribution, all of them yield in this case a good – approximately equal – fitting. In Fig. 1, only fitting (11) is shown.

<div align="center">[[Image:Grafik1_PF.jpg]]</div>
<div align="center">Fig. 1. Fitting function (11) to Hawaiian phoneme frequencies</div>

'''4. Authors: G. Altmann'''

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Hierarchic relations

2006-07-16T08:25:52Z

Altmann:

'''1. Problem and history'''

In different domains of language one observed the fact that the length of a construct influences the length of its consitutents. Usually the constituents get smaller with increasing length of the construct but not in all cases. The problem is to find a theoretical model encompassing all dependencies of this kind. The constructs whose length is the independent variable are: hreb, sentence, rhythmic unit, word, syllable; the constituents whose length or duration is the dependent variable are: sentence, clause, word, syllable, morph, sound. There is also the possibility to consider two independent variables, e.g. word (measured in number of syllables) and syllable (measured in number of sounds) , while the dependent variable is the syllable duration.
Up to now the following particular cases have been examined (see Table 1)

<div align="center">[[Image:Tabelle11_HR.jpg]]</div>

The origin of the problem can be found in 19th century, in phonetics, probably for the first time with Sievers (1876, 1901) who measured the syllable duration in rhythmic units (Sprechakte). A number of phoneticians tested different hypotheses, a part of which corroborated, another part falsified it. The isochrony hypothesis in English is a special case of this problem. The problem was generalized by Menzerath who stated that ''the greater the whole the smaller its parts'' (1954: 101). Different researchers proposed some empirical formulas (Fónagy, Magdics 1960; Nooteboom 1972, 1973; Landblom, Rapp 1972), Altmann (1980) set up the pertinent differential equation and called the result Menzerath´s law. Hřebíček (1992, 1995, 1997) showed that the whole hierarchy of textual levels is based on this dependence and called it ''Menzerath-Altmann´s law''.

There is a great number of individual examinations in different domains of languge. In phonetics the most exhaustive is Weber (1998), in textology the works by Hřebíček (see above) and a mixture of problems including biology and sociology can be found in Altmann, Schwibbe (1989). Bohn (1998) analyzed the relationship between Chines characters and the complexity of composing graphemes, length of words and simplicity of characters, clause length and word length, sentence length and clause length. (cf. also Menzel 2005).
The law has a strong corroboration not only within linguistics but displays analogies to other sciences. Thus the simple form of Menzerath´s law is identical with the allometric law in biology and with power laws current in different sciences. Its correspondences can be found in (i) molecular biology, (ii) sociology of baboons, (iii) in the domain of self-organized criticality, (iv) in chaos research, (v) in the theory of fractals, (vi) in information theory.
It has been observed that construct length is not always the only cause of shortening of the constituents. Also accent, vowel quality, syllable structure, frequency etc. can intervene (cf. Weber 1998). In that case more complex formulas must be used.
The law has several consequences, all of which must still be tested (cf. Altmann, Schwibbe 1989: 8-14):
1. In longer words more phonetic changes occur than in shorter ones.

2. In languages with greater average word length more phonetic/phonemic changes occur than within the same time interval in languages with smaller average word length

3. The adding of an affix to a word evokes the tendency to reduce the inventory of consonants of the word.

4. The shortening of average syllables length in Hypothesis 3 can also be achieved by inserting epenthetic vowels between the stem and affix (or compounding stem).

5. Partial reduplication is more frequent in natural languages than full reduplication.

6. Short roots/morphemes/stems build more compounds or derived words than long ones.

7. The more elements there are in a compound the shorter they are (see the hypotheses on compounds <math>\leftarrow</math>)

8. Fenk-Fenk (to be inserted)

There are different interpretations of the law:

1. In general, long constructs contain more redundancy than short ones. In order to prevent excessive growth of redundancy one can reduce the size of the constituents. The size, the place and the time of this reduction is not known and cannot be predicted. The analogy to self-organized criticality of sand-piles is evident (cf. Bak 1996).

2. Köhler (1989) shows that mechanism of shortening is a consequence of restrictions of the memory: the longer the construct, the more place must be reserved for the structural information between the constituents, thus the size of the constituents must be reduced.

'''2. Hypothesis'''

''The size of the components is a function of the construct size''.

'''3. Derivation'''

The average size of constituents changes with the increase of the size of the construct. It is assumed that the relative rate of change of the size of components is proportional to the rate of change of the size of constructs, the proportionality function being

<math> g(x) = a_0 + \frac{a_1}{x} + \frac{a_2}{x^2}</math>.

Thus in case that all other variables other than construct size are subsumed under the ceteris paribus condition one obtains

(1)<math> \frac{dy}{y-d}= \left(a_0 \frac{a_1}{x}+ \frac{a_2}{x^2}\right)</math>.

where d is the minimal value y can attain.
In case that there is another independent variable, z, one starts from

(2)<math>\frac{dy}{dx}\frac{1}{y-d}=a_0 + \frac{a_1}{x}+\frac{a_2}{x^2}\quad</math> and <math>\frac{dy}{dz}\frac{1}{y-d}=b_0 + \frac{b_1}{z}+\frac{b_2}{z^2}</math>

which can be extended to any number of variables.
The solution of (1) yields

(3)<math> y= Cx^{a_1}e^{a_0 x-a_2/x} + d</math>

the combining (2) the solution is

(4)<math> y= Cx^{a_1}z^{b_1}e^{a_0 x + b_0 z-a_2 /x-b_2 /z} + d</math>

In different applications the following special cases of (3) have been used (d > 0)

(1a) <math> y=ax^{-b}(+d), \quad x= 1, 2, 3, ...; \quad a, b > 0</math>

(1b)<math> y=ax^b e^{cx}(+d), \quad x= 1, 2, 3, ...; \quad a > 0</math>

(1c)<math> y=ae^{-cx}(+d), \quad x= 1, 2, 3, ...; \quad a, c > 0</math>

(1d) <math> y=ae^{c/x}(+d), \quad x= 1, 2, 3, ...; \quad a, c > 0</math>

Solution (4) is still very seldom. Both approaches are special cases of the unified theory (<math>\leftarrow</math>).

'''Examples''':

'''4. Authors: U. Strauss, G. Altmann'''

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[[Category:Unfertig]]

Sentence and clause length

2006-07-16T07:54:10Z

Altmann:

'''1. Problem and history'''

The problem is to find the distribution of sentence and/or clause length in texts. To this end two previous problems must be solved or operationalized:

(i) The determination of sentence boundaries (cf. esp. Niehaus 2001) which is extremely difficult in speech but feasible in written texts. Usually it is achieved with the aid of numerous criteria which can differ from language to language.

(ii) The determination of the measurement unit. One obtains different models for different measurement units, e.g. clauses or words or syllables or phonemes. Today merely the first two are used, sometimes in modified form. Several hundreds of tests in different languages performed in the framework of Göttingen Project corroborate this kind of modelling (Altmann 1988b; Best 2001a,b, 2003, 2006; Grzybek 1995, 1999, 2001; Jing 2001; Kaßel, Livesey 2001; Niehaus 1997, 2001; Rheinländer 2000; Rottmann 2001; Roukk 2001, 2001a; Strehlow 1997; Uhlířová 2001; Wittek 2001).

Historically, the first who tackled the problem was Sherman (1888) using sentence length for characterization purposes. Altmann (1988b) baptized these laws to Sherman´s laws. Older empirical data were collected especially in Classical Greek, in English, Russian and Slovak (Marckworth, Bell 1967; Martynenko 1965; Mistrík 1967; Morton 1965; Morton, Levison 1966; Morton, McLeman 1966; Clayman 1981). Modelling goes back to Yule (1939, 1944), Williams (1940, 1970), Lesskis (1962), Martynenko (1965), Fucks (1955; 1970/71), Buch (1969), Sichel (1974); today it is part of the unified theory (<math>\rightarrow</math>), the first models in this direction were set up by Altmann (1988b).

'''2. Hypothesis'''

''Sentence and clause lengths in texts abide by regular probability distributions derived from the unified theory (<math>\rightarrow</math>).''

'''3. Derivation''' (Altmann 1988b)

The speaker (writer) tends to prolong the actual length of the sentence x adding further clauses, affecting it linearly in the form a + bx. The consideration for the hearer (reader) brakes this trend with force cx. Thus, if sentence length is measured in the number of clauses, one obtains

(1)<math> g(x)=\frac{a+bx}{cx}</math>.

Inserting g(x) as a proportionality function in (10) esp. Example 6 of Unified Theory (à) and reparametrizing, one obtains the negative binomial distribution

(2)<math> P_x={k+x-1 \choose x}p^k q^x \quad, x=0,1,2,...;\quad 0<p<1;\quad q=1-p;\quad k>0</math>.

Some authors define sentence as having at least one clause even if there is no finite verb in it. In that case, one solves the pertinent difference equation for x = 1, 2,... and obtains the positive negative binomial distribution

(3)<math> P_x={k+x-1 \choose x}\frac{p^k q^x}{1-p^k} \quad, x=1,2,3,...;\quad 0<p<1;\quad q=1-p;\quad k>0</math>. .

Example: Sentence length in clauses

Roukk (2001) measured the sentence length (in clauses) in Čechov´s stories and fitted the positive negative binomial distribution as given in Table 1 and Fig. 1.

<div align="center">[[Image:Tabelle1_SaCL.jpg]]</div>

<div align="center">[[Image:Grafik1_SaCL.jpg]]</div>
<div align="center">Fig. 1. Fitting the positive negative binomial distribution to Roukk´s data</div>

'''Example''': Length of clauses in Bulgarian

Uhlířová (2001) measured the length of clauses in a collection of letters of a Bulgarian native speaker and obtained the results in Table 2, Fig. 2.

<div align="center">[[Image:Tabelle2_SaCL.jpg]]</div>

<div align="center">[[Image:Grafik2_SaCL.jpg]]</div>
<div align="center">Figure 2. Fitting the negative binomial distribution to clause length in Bulgarian (Uhlířová 2001)</div>

If the measurement unit is '''word''', then there is an intermediate level exerting a constant effect d added to cx, i.e. one obtains

(4)<math>P_x= \frac{a+bx}{d+cx}P_{x-1}</math> ,

yielding, after reparametrization, the hyperpascal distribution

(5)<math> P_x= \frac{{k+x-1 \choose x}}{{m+x-1 \choose x}}q^x C, \quad x=0,1,...;\quad k,m>0,\quad0<q<1;</math>

C being the normalizing constant,<math> C^{-1}= _2 F_1 (k,1;m;q)\quad</math>.

'''Example'''. Sentence length in Herodot´s Book 1

Altmann (1988) has shown that under this condition the sentence length follows the hyperpascal distribution using 244 texts. The fitting of (5) to Herodot´s Book 1 is shown in Table 3 and Fig. 3.

<div align="center">[[Image:Tabelle3_SaCL.jpg]]</div>

<div align="center">[[Image:Grafik3_SaCL.jpg]]</div>
<div align="center">Figure 3. Fitting the hyperpascal distribution to sentence length in Herodot´s Book 1</div>

Under favourable boundary conditions, one can use all special or limiting cases of these distributions (cf. Wimmer, Altmann 1999).

Example: Sentence length in Old Church Slavonic using the positive Poisson distribution
For Old Church Slavonic texts, Rottmann (2001) and for German texts Wittek (2001) use the positive Poisson distribution which is the limiting case of the positive negative binomial distribution

(6)<math> P_x=\frac{a^x}{x!(e^a -1)},\quad x=1,2,3\quad a>0</math>.

<div align="center">[[Image:Tabelle4_SaCL.jpg]]</div>

<div align="center">[[Image:Grafik4_SaCL.jpg]]</div>
<div align="center">Figure 4. Fitting the positive Poisson distribution to Old Church Slavonic data.</div>

'''Example''': Modified positive Poisson distribution for German texts

Wittek (2001) obtained good results for 80 German texts using the positive Poisson distribution. However, in 4 cases he was forced to modify the first two classes of the positive Poisson distribution and obtained

(7)<math> P-x=\begin{cases} \frac{(1-\alpha)a}{e^a -1}, & x=1 \\ \frac{a}{e^a -1}\left( \frac{a}{2} + \alpha \right), & x=2,\quad a>0;\quad 0<\alpha <1 \\ \frac{a^x}{x! (e^a -1)}, & x=3,4,... \end{cases}</math>

The results of fitting are shown in Table 5.

<div align="center">[[Image:Tabelle5_SaCL.jpg]]</div>

Uhlířová (2001) used for Bulgarian clauses the mixed negative binomial distribution, but it can be shown that the usual negative binomial distribution is sufficient.

'''4. Authors''': U. Strauss, G. Altmann, K.-H. Best

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Hierarchic relations

2006-07-16T07:50:20Z

Altmann:

'''1. Problem and history'''

In different domains of language one observed the fact that the length of a construct influences the length of its consitutents. Usually the constituents get smaller with increasing length of the construct but not in all cases. The problem is to find a theoretical model encompassing all dependencies of this kind. The constructs whose length is the independent variable are: hreb, sentence, rhythmic unit, word, syllable; the constituents whose length or duration is the dependent variable are: sentence, clause, word, syllable, morph, sound. There is also the possibility to consider two independent variables, e.g. word (measured in number of syllables) and syllable (measured in number of sounds) , while the dependent variable is the syllable duration.
Up to now the following particular cases have been examined (see Table 1)

<div align="center">[[Image:Tabelle11_HR.jpg]]</div>

The origin of the problem can be found in 19th century, in phonetics, probably for the first time with Sievers (1876, 1901) who measured the syllable duration in rhythmic units (Sprechakte). A number of phoneticians tested different hypotheses, a part of which corroborated, another part falsified it. The isochrony hypothesis in English is a special case of this problem. The problem was generalized by Menzerath who stated that ''the greater the whole the smaller its parts'' (1954: 101). Different researchers proposed some empirical formulas (Fónagy, Magdics 1960; Nooteboom 1972, 1973; Landblom, Rapp 1972), Altmann (1980) set up the pertinent differential equation and called the result Menzerath´s law. Hřebíček (1992, 1995, 1997) showed that the whole hierarchy of textual levels is based on this dependence and called it ''Menzerath-Altmann´s law''.

There is a great number of individual examinations in different domains of languge. In phonetics the most exhaustive is Weber (1998), in textology the works by Hřebíček (see above) and a mixture of problems including biology and sociology can be found in Altmann, Schwibbe (1989). Bohn (1998) analyzed the relationship between Chines characters and the complexity of composing graphemes, length of words and simplicity of characters, clause length and word length, sentence length and clause length. (cf. also Menzel 2005).
The law has a strong corroboration not only within linguistics but displays analogies to other sciences. Thus the simple form of Menzerath´s law is identical with the allometric law in biology and with power laws current in different sciences. Its correspondences can be found in (i) molecular biology, (ii) sociology of baboons, (iii) in the domain of self-organized criticality, (iv) in chaos research, (v) in the theory of fractals, (vi) in information theory.
It has been observed that construct length is not always the only cause of shortening of the constituents. Also accent, vowel quality, syllable structure, frequency etc. can intervene (cf. Weber 1998). In that case more complex formulas must be used.
The law has several consequences, all of which must still be tested (cf. Altmann, Schwibbe 1989: 8-14):
1. In longer words more phonetic changes occur than in shorter ones.

2. In languages with greater average word length more phonetic/phonemic changes occur than within the same time interval in languages with smaller average word length

3. The adding of an affix to a word evokes the tendency to reduce the inventory of consonants of the word.

4. The shortening of average syllables length in Hypothesis 3 can also be achieved by inserting epenthetic vowels between the stem and affix (or compounding stem).

5. Partial reduplication is more frequent in natural languages than full reduplication.

6. Short roots/morphemes/stems build more compounds or derived words than long ones.

7. The more elements there are in a compound the shorter they are (see the hypotheses on compounds <math>\leftarrow</math>)

8. Fenk-Fenk (to be inserted)

There are different interpretations of the law:

1. In general, long constructs contain more redundancy than short ones. In order to prevent excessive growth of redundancy one can reduce the size of the constituents. The size, the place and the time of this reduction is not known and cannot be predicted. The analogy to self-organized criticality of sand-piles is evident (cf. Bak 1996).

2. Köhler (1989) shows that mechanism of shortening is a consequence of restrictions of the memory: the longer the construct, the more place must be reserved for the structural information between the constituents, thus the size of the constituents must be reduced.

'''2. Hypothesis'''

''The size of the components is a function of the construct size''.

'''3. Derivation'''

The average size of constituents changes with the increase of the size of the construct. It is assumed that the relative rate of change of the size of components is proportional to the rate of change of the size of constructs, the proportionality function being

<math> g(x) = a_0 + \frac{a_1}{x} + \frac{a_2}{x^2}</math>.

Thus in case that all other variables other than construct size are subsumed under the ceteris paribus condition one obtains

(1)<math> \frac{dy}{y-d}= \left(a_0 \frac{a_1}{x}+ \frac{a_2}{x^2}\right)</math>.

where d is the minimal value y can attain.
In case that there is another independent variable, z, one starts from

(2)<math>\frac{dy}{dx}\frac{1}{y-d}=a_0 + \frac{a_1}{x}+\frac{a_2}{x^2}\quad</math> and <math>\frac{dy}{dz}\frac{1}{y-d}=b_0 + \frac{b_1}{z}+\frac{b_2}{z^2}</math>

which can be extended to any number of variables.
The solution of (1) yields

(3)<math> y= Cx^{a_1}e^{a_0 x-a_2/x} + d</math>

the combining (2) the solution is

(4)<math> y= Cx^{a_1}z^{b_1}e^{a_0 x + b_0 z-a_2 /x-b_2 /z} + d</math>

In different applications the following special cases of (3) have been used (d > 0)

(1a) <math> y=ax^{-b}(+d), \quad x= 1, 2, 3, ...; \quad a, b > 0</math>

(1b)<math> y=ax^b e^{cx}(+d), \quad x= 1, 2, 3, ...; \quad a > 0</math>

(1c)<math> y=ae^{-cx}(+d), \quad x= 1, 2, 3, ...; \quad a, c > 0</math>

(1d) <math> y=ae^{c/x}(+d), \quad x= 1, 2, 3, ...; \quad a, c > 0</math>

Solution (4) is still very seldom. Both approaches are special cases of the unified theory (<math>\leftarrow</math>).

'''Examples''':

'''4. Authors: U. Strauss, G. Altmann'''

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[[Category:Unfertig]]

Diversification

2006-07-16T07:10:28Z

Altmann:

'''1. Problem and history'''

Diversification is a process of enlarging the number of forms or meanings of any linguistic entity. It can be ''paradigmatic'', e.g. the rise of cases, numbers, tenses, etc., ''syntactic'', e.g. the rise of allophones, allomorphs etc., ''geographical'', e.g. the increase of different expressions of a concept, ''social'', e.g. the rise different words or meanings of a word or different pronunciations, ''idiolectal'' within a community, ''semantic'', e.g. the increase of synonymy and polysemy, ''contextual'', e.g. the increase of using a unit in different contexts. It comprises a number of phenomena dispersed in this volume.

For the sake of illustration let us show some concrete examples:

(1) The word can enlarge its class membership without any change, e.g. through conversion: “the hand”, “to hand”.

(2) The stem enlarges its class membership through derivation, e.g. German "Bild", "bilden", "bildhaft", or vocalization in Semitic languages, etc.

(3) The stem can enlarge its applicability within one class through derivation e.g. German "Blut", "Blutung", "Bluter", or through vocalization, etc.

(4) The stem can enlarge its applicability within one class through compounding e.g. "Blut", "Blutdruck", "Blutdurst", etc.

(5) If a language abandons the isolating morphology, then morphemes diversify into several morphs because of agglutination or inflection (sequential or syntactic dependence).

(6) The word can enlarge its applicability in the sentence by acquiring several functions, i.e. it enlarges its dispositional properties, which are different from the constant grammatical properties, e.g. practically every word can become the subject of a sentence.

(7) Verbs can enlarge their valence, i.e. their combinability with different cases.

(8) The word can enlarge its cotextuality (cf. Köhler 1986), i.e. its ability to occur in several contexts (where "context" can be defined in several ways). The reverse of this kind of diversification process is a part of style formation, where a "position" diversifies, i.e., a position in a given context can be filled with different units (words, sentences, etc.).

(9) A concept can be expressed by different forms, giving rise to dialects, sociolects, idiolects, or to synonymy.

(10) A word can acquire different meaning (polysemy).

(11) Every word can acquire different associations (connotations).

Diversified entities abide by a ranking law, i.e. if the members of the diversified entity are ordered according to their frequency, then the frequencies are “lawfully” connected.
The factors of diversification can be as follows (Altmann 2005):

(a) ''Random fluctuation'' which is omnipresent in any language phenomena.

(b) ''Environmentally conditioned variation'' forcing an element to acquire different forms or meaning nuances in different environments.

(c) ''Conscious change'' through conscious creation, borrowing, emotionality etc.

(d) ''Self-organisatory triggering'' of a process to a limit, causing changes in other levels, too.

(e) ''System modification'' joined with local or global modifications in a subsystem,

(f) ''Köhlerian requirements'' (Köhler 1986, 1987, 1989, 1990, 1991) forcing to take into account collateral pressures form different sides. They are as follows: (i) ''The trend for minimal coding and deciding effort'', (ii) ''sufficient redundancy'', (iii) ''the coding requirement in general'', (iv) ''context economy vs. context specificity'', and (v) ''invariance vs. flexibility of relation between expression and meaning.''
The concepts of diversification and unification go back to G.K. Zipf (1935, 1949). Together they are called “Zipfian processes”. The scope of the phenomena is enormous. Semantic phenomena have been examined by Beöthy and Altmann (1984a,b, 1991), Altmann (1985a), Altmann, Best, Kind (1987); grammatical phenomena were analyzed in the omnibus volume edited by Rothe (1991), where also a study on spelling errors in Japanese can be found, and dialectal diversification was studied by Altmann (1985b)
The laws hold usually for ranked nominal classes of limited size.

'''2. Hypothesis'''

''Every linguistic entity diversifies, i.e. it generates variants and secondary forms and acquires membership in different classes. The ranked frequencies of individual entities abide by a rank-frequency distribution (or a rank-frequency series).''

A “rank-frequency distribution” (series) is a function expressing the decrease of frequencies ranked according to their magnitude. There are, ''eo ipso'', no bell-shaped rank-frequency distributions.

“Variants” are all free or conditional “non-standard” forms of the entity, e.g. allophones, allomorphs, dialectal or sociolectal expressions of a concept, etc.

“Secondary forms” are in some way derived from the primary form, e.g. secondary meanings (polysemy), cases, times, moods, aspects, etc.

“Classes” are built by a class-building criterion, e.g. derivates, compounds, declination classes, word classes (Wortarten), even semantic classes, etc.

'''Corollary''': ''If the above hypothesis holds, then the frequencies of elements of a linguistic class are not distributed uniformly''.

In a “uniform distribution” all frequencies are equal.
The corollary is rather a well corroborated inductive generalization. Some theoretical rank-frequency distributions can result in the discrete uniform distribution for special values of parameters but they are not actual in linguistics.

'''3. Derivation'''

'''3.1. Altmann´s approach A (1991).'''

Since the entities are ranked and the corollary holds, it is true that for the probabilities of classes it holds that

<math>P_x\le P_{x-1}</math>

Since <math>P_x</math> and <math>P_{x-1}</math> (x = 2,3,…) are joined in a law-like manner, we can write

(1) <math>P_x=g(x)P_{x-1}\quad</math>, where <math>g(x)\le 1\quad</math>.

Furthermore, g(x) can be written as

<math>g(x)=\frac{f(x)}{h(x)}</math>,

where f(x) is a function composed of a language constant a and the diversifying effect of the speaker bx, i.e. f(x) = a+bx, while h(x) contains the controlling, regulating effect of the hearer (community) cx, i.e.

<math>g(x)=\frac{a+bx}{cx}</math>, <math>\quad a+bx\le cx</math> (a, b, and c are assumed positive),

so that

(2)<math>P_x=\frac{a+bx}{cx}p_{x-1}</math>.

In order to obtain a known distribution, one can reparametrize (2) by writing a/b = k-1 and b/c = q, and solving (2) for Px. One obtains

(3)<math>P_x=\begin{pmatrix}k&+&x&-&1\\&&x\end{pmatrix}\frac{p^kq^x}{1-p^k}, \quad x=1,2,3,...</math>

yielding the zero-truncated (positive) negative binomial distribution. The condition <math>g(x)\le 1</math> is fulfilled if <math>kq\le 1</math>.
Using (1) Altmann (1991) showed a number of other possibilities of obtaining a diversification distribution.

'''3.2. Alternative derivation (Altmann 1985b)'''

For the purposes of dialectal variation captured in terms of numbers of lexeme variants on maps of a dialect atlas, Altmann (1985) used the birth-and-death process based on the following assumptions:

(a) In a time interval Δt the birth of a new variant is proportional to the length of the interval, i.e. aΔt.

(b) The assertion of a variant against x rivals is propotional to the number of rivals and the length of the interval, i.e. bxΔt.

(c) The death of a variant is proportional to the number of variants and the length of the interval, i.e. cxΔt.

(d) No change (birth, death or assertion) in Δt is given as the complement to the above changes: 1 – [a+(b+c)x]Δt ignoring intervals smaller then Δt.

(e) The events are independent and the probability of more then one event in the interval is zero.

Thus the probability that there are x-1 variants and a new variant arises or asserts itself against x-1 rivals is

<math>a\triangle tP_{x-1}(t) + b(x-1)\triangle tP_{x-1}(t)</math>;

the probability that there are x+1 variants and one dies is

<math>c(x+1)\triangle tP_{x+1}(t)</math>;

the probability that nothing happens in Δt is

<math>{{1-[a+(b+c)x]\triangle t}}P_x(t)</math>.

Putting these probabilities together we obtain the probability that in the interval (t, t+Δt) there will be exactly x variants as

<math>P_x(t+\triangle t) = [a+b(x-1)]\triangle tP_{x-1}(t) + c(x+1)\triangle tP_{x+1}(t) + {1-[a+(b+c)x]\triangle t}P_x(t)</math>.

Substracting Px from both sides and dividing them by Δt, we obtain

<math>\frac{P_x(t+\triangle t)-P_x(t)}{\triangle t}= [a+b(x-1)P_{x-1}(t)+c(x+1)P_{x+1}(t)-[a+(b+c)x]P_x(t)]</math>.

Letting <math>\triangle t\rightarrow 0</math> we finally obtain

<math>\frac{dP_x(t)}{dt}=[a+b(x-1)P_{x-1}(t)+c(x+1)P_{x+1}(t)-[a+(b+c)x]P_x(t)]</math>

<math>\frac{dP_0(t)}{dt}=cP_1(t)-aP_0(t)</math>

Solving the balancing equations holding for the steady state

<math>-aP_0+cP_1=0,\quad</math>

<math>-[a+(b+c)x]P_x+[a+b(x-1)]P_{x-1}+c(x+1)P_{x+1}=0, \quad x\ge 1,</math>

and setting b/c = q and a/b = k results again in the negative binomial distribution

(4)<math>P_x=\begin{pmatrix}k&+&x&-&1\\&&x\end{pmatrix}p^kq^x, \quad x=0,1,2,...</math>

For dialect maps, (4) is to be understood as the probability that the basic lexeme has x variants, i.e. if on a map there is only one unique form, then x = 0.

'''Example: Goebl´s law (dialectal diversification)'''

Goebl (1984) studied the dialect maps of North West France and Italy and brought the distribution of the numbers of variants in the atlases. Since dialectal variants of a concept arise by a birth-and-death process, the number of maps containing x variants must follow the negative binomial distribution. One of these distributions is shown in Table 1 (Fig. 1).

<div align="center">[[Image:Tabelle11_Div.jpg]]</div>

<div align="center">[[Image:DivFig1.JPG]]</div>
<div align="center">Fig. 1.Fitting the negative binomial distribution to Goebl´s data</div>

'''Example: Beöthy´s law (semantic diversification)'''

According to this law ''the ranked frequencies of the elements of a semantic class are distributed according'' to (3) or (5) (see below). Rothe (1991c) brings a survey of semantic classes abiding by these laws. Testing has been perfomed for meanings of different Hungarian verbal prefixes (Beöthy, Altmann 1984a,b, 1991), Slovak verbal prefixes (Nemcová 1991), the Japanese postposition ni (Roos 1991), German compounds (Raether, Rothe 1991), the German particle ''von'' (Best 1991), the German preposition ''auf'' (Fuchs 1991), the English preposition ''in'' (Hennern 1991), the Polish preposition ''w'' (Hammerl, Sambor 1991), Russian conjunctions ''a'' and ''no'' (Kuße 1991), the French conjunction ''et'' (Rothe 1986), the German genitive (Rothe 1991b), word class distribution in Latin, German and Chinese (Schweers, Zhu 1991), in German (Best 1994, 1997b, 2000b, 2001e; Hammerl 1989; Judt 1995), in Arabic (Altmann 1991a), in Portuguese (Ziegler 1998, 2001), in French (Judt 1995), spelling errors by Japanese English-users (Rothe 1991), word building patterns in Early High German (Best 1990).
In the example (Table 2, Fig. 2) one finds the ranked distribution of German neologisms of the type “Noun + Noun” categorized in 13 groups from Raether, Rothe (1991).

<div align="center">[[Image:Tabelle2_Div.jpg ]]</div>

The result shows that nominal classifications of language entities abide by this type of diversification law.

<div align="center">[[Image:Grafik_2_Div.jpg]]</div>
<div align="center">Fig. 2. Fitting the positive negative binomial distribution (3) to Raether-Rothe data</div>

'''3.3. Hřebíček ´s approach (1996)'''

Hřebíček used two assumptions:
(i) The logarithm of the ratio of the probabilities <math>P_1</math> and <math>P_x</math> is proportional to the logarithm of the classe size, i.e

<math>\ln(P_1/P_x)\propto\ln x\quad</math>

(ii) the proportionality function is given by the logarithm of Menzerath´s law (<math>\rightarrow</math> Hierarchy), i.e.

<math>\ln(P_1/P_x)=\ln(AX^b)\ln x\quad</math>,

yielding the solution

(5)<math>P_x=P_1x^{-(a+b\ln x)}, \quad x=1,2,3,...</math>.

If (5) is considered a probability distribution, then P1 is the norming constant, otherwise it is estimated as the size of the first class, x = 1. Since the frequency of the first class x = 1 is decisive for the form of the distribution, one usually ascribes it a special value α, modifying (5) as

(6)<math>P_x=\begin{cases}a, & x=1\\\frac{(1-a)x^{(a+b\ln x)}}{T}, & x=2,3,...,(n)\end{cases}</math>

where <math>T=\sum_{j=2}^nj^{-(a+b\ln j)}</math>, 0 < α < 1, <math>a,b\in\mathfrak{R}</math> so that <math>P_x</math> converges for <math>n\rightarrow\infty</math>. This version corroborates again the relevance of Menzerath´s law (<math>\rightarrow</math>). Distributions (5) or (6) are called ''Zipf-Alekseev distributions''. If ''n'' is finite, (6) is called ''modified right truncated Zipf-Alekseev distribution'' (see Wimmer, Altmann 1999).
Even though (3) and (5) are quite different, it can be shown that they are special cases of the Siromoney-Dirichlet distribution

(7)<math>P_x=\frac{a_xe^{-\theta b_x}}{f(\theta)}, \quad x=1,2,3,...</math> <math>f(\theta)=\sum_{j=1}^\infty a_je^{-\theta b_j}<\infty</math>

(i) If <math>a_x = k^{(x)}/x!, b_x = x, e^{-\theta} = q\quad</math>, we obtain the positive negative binomial distribution with parameters (k, p) (q = 1-p);

(ii) if <math>\theta = 1, a_x = 1, b_x = (a+b \quad\ln \quad x)\ln x</math>, we obtain the Zipf-Alekseev distribution (a,b);

(iii) the 1-displaced negative binomial distribution, which would be obtained with the conventional displacement of (4), would result if <math>a_x = k^{(x-1)}/(x-1)!, b_x = x-1, e^{-\theta} = q\quad</math>.

Formula (7) admits to the development of further theoretical approaches (see Wimmer, Altmann 1999).

'''Example''': Association law

The connotations of a word diversify because everybody can have different associations. Nevertheless, within a community of speakers, they are distributed in a very regular way suggesting a background mechanism which can be captured as a law.
In the dictionaries of word associations (see e.g. Palermo, Jenkins 1964), the responses to a stimulus word are ordered according to the number of test persons that gave the same response, i.e. they are ranked according to their frequency of occurrence. The test persons are usually classified according to age, sex, education, occupation, social status etc. Quantitative modelling began most probably in Horvath (1963) and continued in Haight (1966), Haight, Jones (1974), Lánský, Radil-Weiss (1980) who used the logarithmic, the Yule, the Borel and the Haight-zeta distributions, none of which gave satisfactory results. Dolinskij (1988, 1994) proposed the Zipf-Alekseev distribution, Altmann (1992) added the 1-displaced negative binomial and modified the Zipf-Alekseev distributions.
In Table 3 (Figure 3) one finds the fitting of the Zipf-Alekseev distribution to the rank-frequency of associations of the word “high” (4th grade, male) as given by Palermo, Jenkins (1964).

<div align="center">Table 3</div>

<div align="center">Fitting model (5) to the associations of the word “high” (4th grade, male) </div>
<div align="center">given by Palermo, Jenkins (1964)</div>

<div align="center">[[Image:Tabelle_3_Divers.jpg]]</div>

The result represents a perfect fit that has been found in all cases of associations.

<div align="center">[[Image:Grafik_3_Div.jpg]]</div>

Fig. 3. Fitting the Zipf-Alekseev distribution (5) to the word associations of “high”

<div align="left">
'''4. Author''': U. Strauss, G. Altmann

'''5. References'''

'''Alekseev, P. M.''' (1978), O nelinejnych formulirovkach zakona Cipfa. In: Piotrovskij, R.G. (ed.), ''Statistika reči i avtomatičeskij analiz teksta'': 53-65. Moskva/Leningrad: Naučnyj sovet po kompleksnoj probleme “Kibernetika” AN SSSR.

'''Altmann, G.''' (1985a). Semantische Diversifikation. ''Folia Linguistica 19, 177-200.''

'''Altmann, G.''' (1985b). Die Entstehung diatopischer Varianten. Ein stochastisches Modell. ''Zs. für Sprachwissenschaft 4, 139-155''.

'''Altmann, G.''' (1991). Modelling diversification phenomena in language. In: Rothe 1991: 33-46.

'''Altmann, G.''' (1991a). Word class diversification of Arabic verbal roots. In: Rothe 1991: 57-59.

'''Altmann, G.''' (1992). Two models for word association data. ''Glottometrika 13, 105-120.''

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'''Beöthy, E., Altmann, G.''' (1984a). The diversification of meaning of Hungarian verbal prefixes. II. ki-. ''Finnisch-Ugrische Mitteilungen 8, 29-37.''

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'''Best, K.-H.''' (1990). Die semantische Diversifikation eines Wortbildungsmusters im Frühneuhochdeutschen. ''Glottometrika 11, 107-110''.

'''Best, K.-H.''' (1991). Von: Zur Diversifikation einer Partikel des Deutschen. In: Rothe U. (ed) 1991: ''94-104''.

'''Best, K.H.''' (1993). Zur Wortartenhäufigkeit in Texten deutscher Kurzprosa der Gegenwart. ''Glottometrika 15, 1993, 1-11''.

'''Best, K.-H.''' (1994). Word class frequencies in contemporary German short prose texts. ''J. of Quantitative Linguistics 1, 144-147''.

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'''Best, K.-H.''' (2000). Verteilung der Wortarten in Anzeigen. ''Göttinger Beiträge zur Sprachwissenschaft 4, 37-51''

'''Best, K.-H.''' (2001). Zur Gesetzmäßigkeit der Wortartenverteilungen in deutschen Pressetexten. ''Glottometrics 1, 1-26''.

'''Best, K.-H.''' (2003). ''Quantitative Linguistik: eine Annäherung.'' 2nd ed. Göttingen: Peust & Gutschmidt.

'''Best, K.-H.''' (2007). Diversifikation bei Eigennamen. In: Grzybek, P., Köhler, R. (eds.), ''Exact Methods in the Study of language and Text: 21-31''. Berlin: de Gruyter

'''Brüers, N., Heeren, A.''' (2004). Plural-Allomorphe in Briefen Heinrich von Kleists. ''Glottometrics 7: 85-90.''

'''Dolinskij, V.A.''' (1988). Raspredelenie reakcij v ekseprimentach po verbal´nym associacijam. ''Acta et Commentationes Universitatis Tartuensis 827, 80-101.''

'''Dolinskij, V.A.''' (1994). Moscow Student´s word associations. In: ''2nd International Confer ence on Quantitative Linguistics, September 20-24, 1994, Moscow: 66-68.'' Moscow: Lomonosov Moscow State University.

'''Fuchs, R.''' (1991). Semantische Diversifikation der deutschen Präposition ''auf''. In: Rothe, U. (ed.) 1991: ''105-115''.

'''Goebl, H.''' (1984). ''Dialektometrische Studien I''. Tübingen: Niememyer.

'''Haight, F.A.''' (1966). Some statistical problems in connection with word association data. ''J. of Mathematical Psychology 3, 217-233''.

'''Haight, F.A., Jones, R.B'''. (1974). A probabilistic treatment of qualitative data with special reference to word association tests. ''J. of Mathematical Psychology 11, 237-244.''

'''Hammerl, R.''' (1989). Untersuchungen zur Verteilung der Wortarten im Text. ''Glottometrika 11, 142-156''.

'''Hammerl, R.''' (1991). ''Untersuchungen zur Struktur der Lexik: Aufbau eines lexikalischen Basismodells''. Trier, WVT.

'''Hammerl, R., Sambor, J.''' (1991). Untersuchungen zur Verteilung der Bedeutungen der polyfunktionalen polnischen Präposition ‘w’ im Text. In: Rothe, U. (ed.), ''1991: 127-137''.

'''Hammerl, R., Sambor, J.''' (1993a). ''O statystycznych prawach jezykowych. Warszawa'': Polskie Towarzystwo Semiotyczne.

'''Hennern, A.''' (1991). Zur semantischen Diversifikation von „in“ im Englischen. In: Rothe, U. (Hrsg.), ''Diversification processes in language: grammar: 116-126''. Hagen: Rottmann.

'''Horvath, W.J.''' (1963). A stochastic model for word association tests. ''Psychological Review 70, 361-364''.

'''Hřebíček, L.''' (1996). Word associations and text. ''Glottometrika 15, 12-17.''

'''Jakubajtis, T.A'''. (1981). ''Časti reči i tipi tekstov''. Riga: Zinatne.

'''Judt, B.''' (1995). ''Wortartenhäufigkeiten im Deutschen und Französischen''. Göttinen: Staats examensarbeit.

'''Junger, J.''' (1989). Diversification in the modern Hebrew verbal system. ''Glottometrika 10, 71 99''.

'''Kločkova, E.A.''' (1968). O raspredelenii klassov slov v nekotorych funkcional´nach stiljach russ kogo jazyka. In: ''Voprosy slavjanskogo jazykoznanija: 109-118''. Saratov.

'''Köhler, R.''' (1986), ''Zur linguistischen Synergetik. Struktur und Dynamik der Lexik.'' Bochum: Bockmeyer.

'''Köhler, R.''' (1987), Systems theoretical linguistics. ''Theoretical Linguistics 14, 241-57.''

'''Köhler, R.''' (1989). Linguistische Analyseebenen, Hierarchisierung und Erklärung im Modell der sprachlichen Selbstregulation. ''Glottometrika 11, 1-18'' (Ed. L. Hřebíček). Bochum: Brockmeyer.

'''Köhler, R.''' (1990). Elemente der synergetischen Linguistik. In: ''Glottometrika 12, 179-187''. (Ed. R.Hammerl). Bochum: Brockmeyer,.

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'''Köhler, R.''' (1991). Diversification of coding methods in grammar. In: Rothe, U. (Hrsg.), ''Diversification processes in language: grammar: 47-55''. Hagen: Rottmann.

'''Krylov, Ju.K.''' (1982a).Ob odnoj paradigme lingvostatističeskich raspredelenij. ''Acta et Commentationens Universitatis Tartuensis 628, 80-102''.

'''Krylov, Ju.K.''' (1982b). Eine Untersuchung statistischer Gesetzmäßigkeiten auf der paradigmatischen Ebene der Lexik natürlicher Sprachen. In: Guiter, H., Arapov, M.V. (eds.), ''Studies on Zipf´s law: 234-262.'' Bochum: Brockmeyer.

'''Kuße, H.''' (1991). A und no in N.M. Karamzins Pis´ma Russkogo Putesetvennika. In: Rothe, U. (ed.), ''Diversification processes in language: grammar: 173-182''. Hagen: Rottmann.

'''Lánský, P., Radil-Weiss, T.''' (1980). A generalization of the Yule-Simon model, with special reference to word association tests and neural cell assembly formation. ''J. of Mathematical Psychology 21, 53-65''.

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'''Nemcová, E.''' (1991). Semantic diversification of Slovak verbal prefixes. In: Rothe, U. (ed.), ''Diversification processes in language: grammar: 67-74''. Hagen: Rottmann.

'''Palermo, D.S., Jenkins, J.J.''' (1964). ''Word association norms. Grade School through College''. Minneapolis: University of Minnesota Press.

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'''Roos, U.''' (1991). Diversifikation der japanischen Postposition “-ni”. In: Rothe, U. (ed.), ''Diversification processes in language: grammar:'' 75-82. Hagen: Rottmann.

'''Rothe, U.''' (1986). ''Die Semantik des kontextuellen et''. Frankfurt: Lang.

'''Rothe, U.''' (1990). Verteilung der Suffixe denominaler Verben nach ihren semantischen Wortbildungsmustern. ''Glottometrika 12, 107-114''.

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'''Rothe, U.''' (1990b). Semantische Beziehungen zwischen Präfixen deutscher denominaler Verben und der motivierenden Nomina. ''Glottometrika 11, 111-121''.

'''Rothe, U.''' (ed.) (1991). ''Diversification processes in language: grammar''. Hagen: Rottmann.

'''Rothe, U.''' (1991a). Diversification processes in grammar. An introduction. In: Rothe, U. (ed.), Diversification processes in language: grammar: 3-32. Hagen: Rottmann.

'''Rothe, U.''' (1991b). Diversification of the case in German: genitive. In: Rothe, U. (ed.), ''Diversification processes in language: grammar'': 140-156. Hagen: Rottmann.

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'''Schweiger, F.''' (1987). Zu den Modellen der semantischen Diversifikation von G. Altmann. ''Folia Linguistica 21, 191-194''.

'''Tiščenko, V.''' (1970). Častota častii movi v riznich funkcional´nych stiljach sučasnoj ukrains´koj movi. In: ''Pitanija strukturnoi leksikologii. Kiiv.''

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'''Zipf, G. K.''' (1935). ''The psycho-biology of language. An introduction to dynamic philology''. Boston: Houghton Mifflin.

'''Zipf, G.K.''' (1949). ''Human behavior and the principle of least effort.'' Cambridge: Addison Wesley.

'''Zsilka, T.''' (1974). ''Stilisztika és statisztika''. Budapest.
</div>

Word length and frequency

2006-07-14T17:00:21Z

Altmann:

'''1. Problem and history'''

There are two problems that must be strictly separated:

(a) The use of words of specific length in texts. Here ''word forms'' are meant and neither the size of the lexicon nor the number of phonemes in the phoneme inventory are relevant. Other properties like polysemy, polytexty, or synthetism can be added for modelling textual word length.

(b) Frequency as a factor of ''lemma construction'' in the dictionary, where frequency is taken from a frequency dictionary, and the dictionary size, as well as the size of the phoneme inventory, are relevant.

In (b), problems arise because the size of the dictionary can be merely coarsely estimated. There are even authors considering it infinite (cf. Piotrowski, Bektaev, Piotrovskaja 1985; Kornai 2002), which is a reasonable assumption. If, however, it is considered as a fixed finite value, it can be taken into account. Further, frequency dictionaries strongly depend on the kind of the analyzed texts. In order to secure a reliable frequency dictionary even for a short time period, an astronomical number of words must be counted.

Problem (a) is easily solvable but it must be considered for individual texts or group of texts from the same author. Here we consider merely the problem of dependence of word form length on its frequency in texts. In case of rejecting this relationship in some texts, further variables can be joined, e.g. number of meanings.

Another problem is frequency of occurrence and duration (→).
G.K. Zipf (1935:25) set up two hypotheses on the relation of frequency and length:

(I) “The magnitude of words tends, on the whole, to stand in an inverse (not necessarily proportionate) relationship to the number of occurrences.”
His second hypothesis on the variety of words occurring x-times is simply that of the distribution of word frequency (→).

Zipf himself demonstrated the inverse of (I) using Kaeding´s frequency dictionary of German (Kaeding 1897), i.e. he simply demonstrated the distribution of lengths which turned out to be monotone decreasing (→ word length). Baker (1951) used the letters of a woman with psychic deseases and divided the words in frequency groups; some authors use frequency directly, other ones use the ranks as independent variable. Word length has been counted in terms of the number of phonemes (e.g. Miller, Newman, Friedman 1958) or syllables. Different empirical formulas have been proposed (Belonogov 1962, Guiraud 1954, Kalinin 1964, Guiter 1977). Köhler (1986) discovered an oscillation of lengths and Köhler, Zörnig, Brinkmöller (1990) smoothed the data by taking gliding means. Grzybek, Altmann (2002) and Strauss, Grzybek and Altmann (2005) have shown the dependence of length on frequency in ten languages using individual texts, but other researchers rather use corpora representing mixed samples. Different aspects of this relationship have been shown in Strauss, Grzybek, Altmann (2005).
Krott (1996, 2002) stated the same relationship between frequency and morpheme length.

'''2. Hypothesis'''

''The mean syllabic length (y) of word forms in texts decreases with their frequency of occurrence (x).''

Here each form is considered separately and the mean length of all forms with the same frequency is considered y. Since frequency can be considered in relative form, x can be considered continuous. Further hypotheses can be derived from the above-mentioned one, e.g. shorter forms of the word are more frequent than its longer forms (e.g. case, modus, aspect, tenses). Or, derivates and compounds of a lexeme occur more seldom than the lexeme itself.

'''3. Derivation'''

Köhler (1986), Strauss, Grzybek, Altmann (2005) start from the assumption that the relative rate of change of mean word length decreases proportionally to the relative rate of change of the frequency, as is very usual in synergetic linguistics. If zero-syllabic clitics are considered parts of the following words, then the mean length cannot take values less than 1. Thus the differential equation is

(1) <math>\frac{dL}{L-1}= -b\frac{dF}{F}</math>

from which the well-known formula

(2) <math>L=aF^b+1,\quad a>0,\quad b<0</math>

follows. Here <math>a = e^C\quad</math> (C being the integration sonstant). If one considers frequency as depending on length, inversion yields <math>F = A(L-1)^{-B}\quad</math> with <math>A =a^{1/b},\quad B =1/b</math>. (2) is a special case of of the unified derivation (see Introduction, 4.1). Other empirical formulas can be found in the references.

'''Example.''' Russian text

Strauss, Grzybek, Altmann (2005) present the result for the first chapter of Tolstoj´s Anna Karenina. For each frequency class the mean word-form length has been computed. Zero-syllabic prepositions (''k, s, v'') were considered proclitics. Frequency classes containing fewer than 10 records were pooled and the unweighted average was computed. The result is presented in Table 1 and Fig. 1. In the frist column, the frequency classes are shown, in the second the observed mean word form lengths, and in the third the theoretical mean lengths according to (2).

<div align="center">[[Image:Tabelle_11_WLFR.jpg]]</div>

<div align="center">[[Image:Figur_1_WL-FR.jpg]]</div>

<div align="center">Fig. 1. Fitting (2) to the data from Anna Karenina Ch. 1.</div>

'''4. Author:''' U. Strauss, G. Altmann

'''5. References'''

'''Baker, S.J.''' (1951). A linguistic law of constancy: II. ''The J. of General Psychology 44, 113-120.''

'''Belonogov, G.G.''' (1962). O nekotorych statističeskich zakonomernostjach ruskoj pis´mennoj reči. ''Voprosy jazykoznanija 11/1, 100-101''.

'''Breiter, M.A.''' (1994). Length of Chinese words in relation to their other systemic features. ''J. of Quantitative Linguistics 1, 224-231.''

'''Bürmann, C., Frank, H., Lorenz, L.''' (1963). Informationstheoretische Untersuchungen über Rang und Länge deutscher Wörter. ''Grundlagenstudien aus Kybernetik und Geisteswissenschaft 4 (3-4), 73-90.''

'''Fenk-Oczlon, G.''' (2001). Familiarity, information flow, and linguistic form. In: Bybee, J., Hopper, P. (eds.). ''Frequency and the emergence of linguistic structure: 431-448. Amsterdam: Benjamins''.

'''Gieseking, K.''' (2002). Untersuchungen zur Synergetik der englischen Lexik. In: Köhler, R. (ed.), ''Korpuslinguistische Untersuchungen in die quantitative und systemtheoretische Linguistik: 387-433''. http://ubt.opus.hbz-nrw.de/volltexte/2004/279/

'''Grzybek, P., Altmann, G.''' (2002). Oscillation in the frequency-length relationship. ''Glottometrics 5, 97-107''.

'''Guiraud, P.''' (1954). ''Les caractères statistiques du vocabulaire. Essai de méthodologie.'' Paris: P.U.F.

'''Guiter, H.''' (1977). Les relations /frequence-longeuer-sens/ des mots (langue romanes et anglais). ''XVI Congresso Internazionale di Linguistica e Filologia Romanza, Napoli, 15-20 Aprile 1974, 373-381.'' Napoli: Macchiaroli.

'''Hammerl, R.''' (1990). Länge - Frequenz, Länge - Rangnummer. Überprüfung von zwei lexikalischen Modellen. ''Glottometrika 12, 1-24''.

'''Hammerl, R.''' (1991). ''Untersuchungen zur Struktur der Lexik: Aufbau eines lexikalischen Basismodells''. Trier, WVT.

'''Herdan, G.''' (1966). ''The advanced theory of language as choice and chance''. Berlin, Springer.

'''Kaeding, F.W.''' (1897-98). ''Häufigkeitswörterbuch der deutschen Sprache''. Steglitz: Selbstverlag.

'''Kalinin, V.M'''. (1964a). O statistike literaturnogo teksta. ''Voprosy jazykoznanija Nr. 1, 123-127''.

'''Köhler, R.''' (1986). Zur ''linguistischen Synergetik: Struktur und Dynamik der Lexik.'' Bochum: Brockmeyer.

'''Köhler, R.'''(2005). Synergetic linguistics. In: Köhler, R., Altmann, G., Piotrowski, R.G. (eds.), ''Quantitative Linguistics. An International Handbook: 760-774''. Berlin: de Gruyter.

'''Köhler, R., Zörnig, P., Brinkmöller.''' (1990). Differential equation models for the oscillation of the word length as a function of the frequency. ''Glottometrika 12, 25-40''.

'''Kornai, A.''' (2002). How many words are there? ''Glottometrics 4, 61-86''.

'''Krott, A.''' (1996). Some remarks on the relation between word length and morpheme length. ''J. of Quantitative Linguistics 3, 29-37.''

'''Krott, A.''' (2002). Ein funktionalanalytisches Modell der Wortbildung. In: Köhler, R. (ed.), ''Korpuslinguistische Untersuchungen in die quantitative und systemtheoretische Linguistik: 75-126.'' http://ubt.opus.hbz-nrw.de/volltexte/2004/279/

'''Leopold, E.''' (1997). Frequency spectra within word length classes. In: ''Third International Conference on Quantitative Linguistics, August 26-29, 1997, Helsinki, Finland: 156''. Helsinki: Monila.

'''Leopold, E.''' (1998). ''Stochastische Modellierung lexikalischer Evolutionsprozesse.'' Hamburg: Kovač

'''Leopold, E.''' (2000a). Length-distribution of words with coinciding frequency. In: ''Proceedings of the fourth conference of the International Quantitative Linguistic Association, Prague, August 24-26: 76-77''.

'''Miller, G.A., Newman, E.B., Friedman, E.A.''' (1958). Length-frequency statistics for written English.. ''Information and Control 1, 370-389''.

'''Miyajima, T.''' (1992). Relationship in the length, age and frequency of Classical Japanese words. ''Glottometrika 13, 219-229''.

'''Piotrowski, R.G., Bektaev, K.B., Piotrovskaja, A.A.''' (1985). ''Mathematische Linguistik''. Bochum, Brockmeyer.

'''Sanada, H.''' (1999). Analysis of Japanese vocabulary by the theory of synergetic linguistics. ''J. of Quantitative Linguistics 6, 239-251''.

'''Strauss, U., Grzybek, P., Altmann, G.''' (2005). Word length and word frequency. In: Grzybek, P. (ed.), ''Word length studies and related issues: 255-272''. Boston/Dordrecht: Kluwer.

'''Tuldava, J.''' (1995). ''Methods in quantitative linguistics''. Trier: WVT.

'''Zipf, G.K.''' (1932). ''Selected studies of the principle of relative frequency in language.'' Cam-bridge, Mass.: Harvard University Press.

'''Zipf, G.K.''' (1935).''The psycho-biology of language''. Boston: Houghton Mifflin.

[[Category:word]] [[Category:length]] [[Category:frequency]]

Word length and frequency

2006-07-14T16:59:11Z

Altmann:

'''1. Problem and history'''

There are two problems that must be strictly separated:

(a) The use of words of specific length in texts. Here ''word forms'' are meant and neither the size of the lexicon nor the number of phonemes in the phoneme inventory are relevant. Other properties like polysemy, polytexty, or synthetism can be added for modelling textual word length.

(b) Frequency as a factor of ''lemma construction'' in the dictionary, where frequency is taken from a frequency dictionary, and the dictionary size, as well as the size of the phoneme inventory, are relevant.

In (b), problems arise because the size of the dictionary can be merely coarsely estimated. There are even authors considering it infinite (cf. Piotrowski, Bektaev, Piotrovskaja 1985; Kornai 2002), which is a reasonable assumption. If, however, it is considered as a fixed finite value, it can be taken into account. Further, frequency dictionaries strongly depend on the kind of the analyzed texts. In order to secure a reliable frequency dictionary even for a short time period, an astronomical number of words must be counted.

Problem (a) is easily solvable but it must be considered for individual texts or group of texts from the same author. Here we consider merely the problem of dependence of word form length on its frequency in texts. In case of rejecting this relationship in some texts, further variables can be joined, e.g. number of meanings.

Another problem is frequency of occurrence and duration (→).
G.K. Zipf (1935:25) set up two hypotheses on the relation of frequency and length:

(I) “The magnitude of words tends, on the whole, to stand in an inverse (not necessarily proportionate) relationship to the number of occurrences.”
His second hypothesis on the variety of words occurring x-times is simply that of the distribution of word frequency (→).

Zipf himself demonstrated the inverse of (I) using Kaeding´s frequency dictionary of German (Kaeding 1897), i.e. he simply demonstrated the distribution of lengths which turned out to be monotone decreasing (→ word length). Baker (1951) used the letters of a woman with psychic deseases and divided the words in frequency groups; some authors use frequency directly, other ones use the ranks as independent variable. Word length has been counted in terms of the number of phonemes (e.g. Miller, Newman, Friedman 1958) or syllables. Different empirical formulas have been proposed (Belonogov 1962, Guiraud 1954, Kalinin 1964, Guiter 1977). Köhler (1986) discovered an oscillation of lengths and Köhler, Zörnig, Brinkmöller (1990) smoothed the data by taking gliding means. Grzybek, Altmann (2002) and Strauss, Grzybek and Altmann (2005) have shown the dependence of length on frequency in ten languages using individual texts, but other researchers rather use corpora representing mixed samples. Different aspects of this relationship have been shown in Strauss, Grzybek, Altmann (2005).
Krott (1996, 2002) stated the same relationship between frequency and morpheme length.

'''2. Hypothesis'''

''The mean syllabic length (y) of word forms in texts decreases with their frequency of occurrence (x).''

Here each form is considered separately and the mean length of all forms with the same frequency is considered y. Since frequency can be considered in relative form, x can be considered continuous. Further hypotheses can be derived from the above-mentioned one, e.g. shorter forms of the word are more frequent than its longer forms (e.g. case, modus, aspect, tenses). Or, derivates and compounds of a lexeme occur more seldom than the lexeme itself.

'''3. Derivation'''

Köhler (1986), Strauss, Grzybek, Altmann (2005) start from the assumption that the relative rate of change of mean word length decreases proportionally to the relative rate of change of the frequency, as is very usual in synergetic linguistics. If zero-syllabic clitics are considered parts of the following words, then the mean length cannot take values less than 1. Thus the differential equation is

(1) <math>\frac{dL}{L-1}= -b\frac{dF}{F}</math>

from which the well-known formula

(2) <math>L=aF^b+1,\quad a>0,\quad b<0</math>

follows. Here <math>a = e^C\quad</math> (C being the integration sonstant). If one considers frequency as depending on length, inversion yields <math>F = A(L-1)^{-B}\quad</math> with <math>A =a^{1/b},\quad B =1/b</math>. (2) is a special case of of the unified derivation (see Introduction, 4.1). Other empirical formulas can be found in the references.

'''Example.''' Russian text

Strauss, Grzybek, Altmann (2005) present the result for the first chapter of Tolstoj´s Anna Karenina. For each frequency class the mean word-form length has been computed. Zero-syllabic prepositions (''k, s, v'') were considered proclitics. Frequency classes containing fewer than 10 records were pooled and the unweighted average was computed. The result is presented in Table 1 and Fig. 1. In the frist column, the frequency classes are shown, in the second the observed mean word form lengths, and in the third the theoretical mean lengths according to (2).

<div align="center">[[Image:Tabelle_11_WLFR.jpg]]</div>

<div align="center">[[Image:Figur_1_WL-FR.jpg]]</div>

<div align="center">Fig. 1. Fitting (2) to the data from Anna Karenina Ch. 1.</div>

'''4. Author:''' G. Altmann

'''5. References'''

'''Baker, S.J.''' (1951). A linguistic law of constancy: II. ''The J. of General Psychology 44, 113-120.''

'''Belonogov, G.G.''' (1962). O nekotorych statističeskich zakonomernostjach ruskoj pis´mennoj reči. ''Voprosy jazykoznanija 11/1, 100-101''.

'''Breiter, M.A.''' (1994). Length of Chinese words in relation to their other systemic features. ''J. of Quantitative Linguistics 1, 224-231.''

'''Bürmann, C., Frank, H., Lorenz, L.''' (1963). Informationstheoretische Untersuchungen über Rang und Länge deutscher Wörter. ''Grundlagenstudien aus Kybernetik und Geisteswissenschaft 4 (3-4), 73-90.''

'''Fenk-Oczlon, G.''' (2001). Familiarity, information flow, and linguistic form. In: Bybee, J., Hopper, P. (eds.). ''Frequency and the emergence of linguistic structure: 431-448. Amsterdam: Benjamins''.

'''Gieseking, K.''' (2002). Untersuchungen zur Synergetik der englischen Lexik. In: Köhler, R. (ed.), ''Korpuslinguistische Untersuchungen in die quantitative und systemtheoretische Linguistik: 387-433''. http://ubt.opus.hbz-nrw.de/volltexte/2004/279/

'''Grzybek, P., Altmann, G.''' (2002). Oscillation in the frequency-length relationship. ''Glottometrics 5, 97-107''.

'''Guiraud, P.''' (1954). ''Les caractères statistiques du vocabulaire. Essai de méthodologie.'' Paris: P.U.F.

'''Guiter, H.''' (1977). Les relations /frequence-longeuer-sens/ des mots (langue romanes et anglais). ''XVI Congresso Internazionale di Linguistica e Filologia Romanza, Napoli, 15-20 Aprile 1974, 373-381.'' Napoli: Macchiaroli.

'''Hammerl, R.''' (1990). Länge - Frequenz, Länge - Rangnummer. Überprüfung von zwei lexikalischen Modellen. ''Glottometrika 12, 1-24''.

'''Hammerl, R.''' (1991). ''Untersuchungen zur Struktur der Lexik: Aufbau eines lexikalischen Basismodells''. Trier, WVT.

'''Herdan, G.''' (1966). ''The advanced theory of language as choice and chance''. Berlin, Springer.

'''Kaeding, F.W.''' (1897-98). ''Häufigkeitswörterbuch der deutschen Sprache''. Steglitz: Selbstverlag.

'''Kalinin, V.M'''. (1964a). O statistike literaturnogo teksta. ''Voprosy jazykoznanija Nr. 1, 123-127''.

'''Köhler, R.''' (1986). Zur ''linguistischen Synergetik: Struktur und Dynamik der Lexik.'' Bochum: Brockmeyer.

'''Köhler, R.'''(2005). Synergetic linguistics. In: Köhler, R., Altmann, G., Piotrowski, R.G. (eds.), ''Quantitative Linguistics. An International Handbook: 760-774''. Berlin: de Gruyter.

'''Köhler, R., Zörnig, P., Brinkmöller.''' (1990). Differential equation models for the oscillation of the word length as a function of the frequency. ''Glottometrika 12, 25-40''.

'''Kornai, A.''' (2002). How many words are there? ''Glottometrics 4, 61-86''.

'''Krott, A.''' (1996). Some remarks on the relation between word length and morpheme length. ''J. of Quantitative Linguistics 3, 29-37.''

'''Krott, A.''' (2002). Ein funktionalanalytisches Modell der Wortbildung. In: Köhler, R. (ed.), ''Korpuslinguistische Untersuchungen in die quantitative und systemtheoretische Linguistik: 75-126.'' http://ubt.opus.hbz-nrw.de/volltexte/2004/279/

'''Leopold, E.''' (1997). Frequency spectra within word length classes. In: ''Third International Conference on Quantitative Linguistics, August 26-29, 1997, Helsinki, Finland: 156''. Helsinki: Monila.

'''Leopold, E.''' (1998). ''Stochastische Modellierung lexikalischer Evolutionsprozesse.'' Hamburg: Kovač

'''Leopold, E.''' (2000a). Length-distribution of words with coinciding frequency. In: ''Proceedings of the fourth conference of the International Quantitative Linguistic Association, Prague, August 24-26: 76-77''.

'''Miller, G.A., Newman, E.B., Friedman, E.A.''' (1958). Length-frequency statistics for written English.. ''Information and Control 1, 370-389''.

'''Miyajima, T.''' (1992). Relationship in the length, age and frequency of Classical Japanese words. ''Glottometrika 13, 219-229''.

'''Piotrowski, R.G., Bektaev, K.B., Piotrovskaja, A.A.''' (1985). ''Mathematische Linguistik''. Bochum, Brockmeyer.

'''Sanada, H.''' (1999). Analysis of Japanese vocabulary by the theory of synergetic linguistics. ''J. of Quantitative Linguistics 6, 239-251''.

'''Strauss, U., Grzybek, P., Altmann, G.''' (2005). Word length and word frequency. In: Grzybek, P. (ed.), ''Word length studies and related issues: 255-272''. Boston/Dordrecht: Kluwer.

'''Tuldava, J.''' (1995). ''Methods in quantitative linguistics''. Trier: WVT.

'''Zipf, G.K.''' (1932). ''Selected studies of the principle of relative frequency in language.'' Cam-bridge, Mass.: Harvard University Press.

'''Zipf, G.K.''' (1935).''The psycho-biology of language''. Boston: Houghton Mifflin.

[[Category:word]] [[Category:length]] [[Category:frequency]]

References

2006-07-13T06:43:10Z

Altmann:

'''1. Problem and history'''

“Reference” is any sign joining a sentence with any preceding sentence. References are the means for making the text a compact entity. In qualitative linguistics there is an ample literature on kinds and behavior of references. They are the basis for evaluating text cohesion. The sentences joined by a commoh reference are called ''hrebs''.
The only law, known in literature as “Hřebíček´s reference law”, originates from Hřebíček´s (1985) derivation. Altmann (1988: 81-85) proposed merely some further problems for investigation. The law was corroborated on many Turkish texts. Formula (4) supports Herdan´s version of the type-token ratio (<math>\rightarrow</math>).

'''2. Hypothesis'''

''The number of references in text depends on the number of words and the number of sentences''.

“Word” is every word-like entity (token) in the text.
“Sentence” in written texts is demarcated by orthographic signs.

'''3. Derivation'''

Let

r = number of references in text

s = number of sentences in text

n = number of word tokens in text

v = number of word types in text (vocabulary of the text)

w = vocabulary richness

a, b, c, A, B = constants

Hřebíček´s assumptions:

(i) the richer the vocabulary, the smaller is the number of references

(ii) the more sentences are in the text, the greater the number of references.

The change of the number of references relative to the change of the vocabulary richness is proportional to the number of sentences

<math>\frac{\partial r}{\partial w}=As</math>

and, at the same time, the change of the number of references relative to the change of the number of sentences is proportional to the vocabulary richness of the text

<math>\frac{\partial r}{\partial s}=Aw</math>

yielding the solution

(1)<math>r=csw\quad</math> (c = AB).

Taking the simplest interpretation of vocabulary richness w as

<math>w=\frac{v}{n}</math>

we obtain

(2)<math>r=cs\frac{v}{n}</math> .

Using Herdan´s (1966: 76) type-token ratio (<math>\rightarrow</math>) expressing the vocabulary of the text as a power function of its length

(3)<math> v=n^a,\quad 0 < a < 1 </math>

and inserting it in (2) one obtains

(4)<math> r= csn^{a-1}=csn^b, \quad a-1 = b \quad -1 < b < 0</math>

meeting assumptions (i) and (ii).

'''Example''': The course of references in a Turkish text

Hřebíček (1992) examined the course of references in several Turkish texts. One of these cases is shown in Table 1.

<div align="center">[[Image:Tabelle1_R.jpg]]</div>

'''4. Authors''': U. Strauss, G. Altmann

'''5. References'''

'''Altmann, G.''' (1988a). ''Wiederholungen in Texten''. Bochum, Brockmeyer.

'''Hřebíček, L'''. (1985). Text as a unit and co-references. In: Ballmer, Th.T. (ed.), ''Linguistic dynamics'': 190-198. New York, de Gruyter.

'''Hřebíček, L'''. (1986). Cohesion in Ottoman poetic texts. ''Archiv orientální 54,252-256''.

'''Hřebíček, L'''. (1989). A syntactic variable on the text level. Glottometrika ''10, 204-218''.

'''Hřebíček, L.''' (1992). ''Text in communication: Supra-sentence structure''. Bochum, Brockmeyer.

'''Hřebíček, L'''. (2000). ''Variation in sequences''. Prague: Oriental Institute.
Hřebíček 1985, 1986, 1989, 1992, 2000; Altmann 1988.

'''Hřebíček, L'''. (2006). Text laws. In: Köhler, R., Altmann, G., Piotrowski, R.G. (eds.), ''Quantitative Linguistics. An International Handbook: 348-361.'' Berlin: de Gruyter.

'''Mehler, A. ''' (2006). Eigenschaften der textuellen Einheiten und Systeme. In: Köhler, R., Altmann, G., Piotrowski, R.G. (eds.), ''Quantitative Linguistics. An International Handbook: 325-348.'' Berlin: de Gruyter.

[[[Das zeichnen geht schwer, da zwei unabhängige Variablen drin sind. Mit Harvard Graphics wirds]]]

[[Category:Unfertig]]

Polysemy and length

2006-07-13T06:12:38Z

Altmann:

'''1. Problem and history'''

The relation between polysemy and length of words is one of the oldest problems in quantitative linguistics introduced by Zipf (see esp. 1949). It is also part of Köhler´s self-regulation cycle (1986). The question is the direction of dependence. Since word prolongation by affixation, compounding, reduplication etc. results from semantic needs, namely specification of meaning, polysemy is the independent variable; though in a complete self-regulation cycle both directions can be tested.
Hypotheses were set up and tested by Altmann, Beöthy and Best (1982) for German, Hungarian and Slovak, by Sambor (1984) for Polish, German, Slovak and Hungarian, by Fickermann, Markner-Jäger, Rothe (1984) for German, Swedish and Indonesian and by Köhler (1999a) for Maori (New Zealand). Köhler considered the both directions (cf. also Length, frequency, polysemy)

'''2. Hypothesis'''

''Length is a function of polysemy''.

'''3. Derivation'''

The relative rate of change of length (L) is proportional to the relative rate of change of the polysemy (P) amplified by an additive constant expressed as

(1)<math> \frac{dL}{L}= ( a+\frac{b}{P} )dP</math>

resulting in

(2)<math> L= Cp^b e^{aP}\quad</math>.

The analogous relation holds if in (1) one exchanges the variables, i.e. if one considers P = f(L). The result is evidently special case of the Unified theory, see Example 1 in § 4.1.

'''Example'''. Length and polysemy in Maori

Köhler (1999) showed on Maori data that the direction P = f(L) is slightly more significant then the other way round and that in most cases a = 0 is sufficient, i.e. mathematically L = CPb would be more correct. The result of fitting is showed in Table 1 and Fig. 1

<div align="center">[[Image:Tabelle1_PaL.jpg]]</div>

<div align="center">[[Image:Grafik1_PaL.jpg]]</div>
<div align="center">Fig. 1. Length of lexemes as a function of polysemy in Maori (Köhler 1999)</div>

'''4. Authors''': U. Strauss, G. Altmann

'''5. References'''

'''Altmann, G., Bagheri, D., Goebl, H., Köhler, R., Prün, C'''. (2002). ''Einführung in die quantitative Lexikologie''. Götingen: Peust & Gutschmidt.

'''Altmann, G., Beöthy, E., Best, K.-H'''. (1982). Die Bedeutungskomplexität der Wörter und das Menzerathsche Gesetz. ''Zeitschrift für Phonetik, Sprachwissenschaft und Kommunikations-forschung 35, 537-543''.

'''Altmann, G., Schwibbe, M.''' (1989). ''Das Menzerathsche Gesetz in informationsverarbeitenden Systemen''. Hildesheim, Olms.

'''Breiter, M.A.''' (1994). Length of Chinese words in relation to their other systemic features. ''J. of Quantitative Linguistics 1(3), 224-231''.

'''Drebet, V.V., Levickij, V.V., Cherubim, D.''' (1996). Morphologische Faktoren bei der Polysemie der deutschen Adjektive. ''Naukovy Visnyk Cerniveckoho Universitetu 1, 29-32.''

'''Elts, J'''. (1995). Word length and its sematic complexity. In: Mikk, J. (ed.), ''Family and Textbooks: 115-126''. Tartu: University of Tartu.

'''Fickermann, I., Markner-Jäger, B, Rothe, U'''. (1984). Wortlänge und Bedeutungskomplexität. ''Glottometrika 6, 115-1''26.

'''Gieseking, K'''. (2002). Untersuchungen zur Synergetik der englischen Lexik. In: Köhler, R. (ed.), ''Korpuslinguistische Untersuchungen in die quantitative und systemtheoretische Linguistik: 387-433''. http://ubt.opus.hbz-nrw.de/volltexte/2004/279/

'''Guiter, H.''' (1977), Les relations /fréquence-longeur-sens/ des mots (langues romanes et anglais). ''Actes du 14ème congrès international de linguistique et philologie romanes. Napoli 15-20 Aprile 1974, Vol. 4, 1977: 373-381''. Napoli: Macchiaroli.

'''Hammerl, R.''' (1991). ''Untersuchungen zur Struktur der Lexik: Aufbau eines lexikalischen Basismodells''. Trier, WVT.

'''Hammerl, R., Sambor, J'''. (1993). ''O statystycznych prawach jezykowych''. Warszawa: Polskie Towarzystwo Semiotyczne.

'''Hoffmann, Ch.''' (2001). Polylexie lexikalischer Einheiten in Texten. In: Uhlířova, L., Wimmer, G., Altmann, G., Köhler, R. (Eds.), ''Text as a linguistic paradigm: levels, constituents, constructs. Festschrift in honour of Ludek Hřebíček: 76-97.'' Trier: WVT.

'''Köhler, R.''' (1986). ''Zur linguistischen Synergetik: Struktur und Dynbamik der Lexik''. Bochum: Brockmeyer.

'''Köhler, R.''' (1999a). Der Zusammenhang zwischen Lexemlänge und Polysemie im Maori. In: Ondrejovič, S., Genzor, J. (eds.), ''Pange lingua. Zborník na počest´Viktora Krupu: 27-33''. Bratislava: Veda.

'''Krylov, Ju.K.''' (1982). Eine Untersuchung statistischer Gesetzmäßigkeiten auf der paradig-matischen Ebene der Lexik natürlicher Sprachen. In: Guiter, H., Arapov, M.V. (Eds.), ''Studies on Zipf´s law: 234-262''. Bochum: Brockmeyer.

'''Mikk, J., Uibo, H., Elts, J.''' (2001). Word length as an indicator of semantic complexity. In Uhlířova, L., Wimmer, G., Altmann, G., Köhler, R. (Eds.), ''Text as a linguistic paradigm: levels, constituents, constructs. Festschrift in honour of Ludek Hřebíček: 187-195.'' Trier: WVT.

'''Rothe, U.''' (1983). Wortlänge und Bedeutungsmenge. Eine Untersuchung zum Menzerathschen Gesetz an drei romanischen Sprachen. Glottometrika 5, 101-112.
Sambor, J. (1984). Menzerath’s law and the polysemy of words. ''Glottometrika 6, 94-114''.

'''Zipf, G.K.''' (1935). ''The psycho-biology of language''. Boston: Houghton Mifflin.

'''Zipf, G.K.''' (1945). The meaning-frequency relationship of words. J. of General Psychology 33, 251-256.

'''Zipf, G.K.''' (1949). ''Human behavior and the principle of least effort''. Cambridge: Addison-Wesley.

Word associations

2006-07-11T15:46:46Z

Altmann:

'''1. Problem and history'''

Giving a word as a stimulus, different persons respond with different words, e.g. “music” as stimulus can evoke “violin”, “Chopin”, “love”, “melody”, etc. Asking many persons one can observe that the frequency of particular responses (associations) is not equal, on the contrary, the response words can be ranked according to their frequency. The problem is to find the adequate ranking (<math> \rightarrow</math>) distribution. Associations are thus both ranking and diversification problems.

Not considering qualitative work and compilation of frequency lists having mostly the character of voluminous books, the first attempt at modelling was probably made by Horvath (1963) who used inductively the Yule distribution. Haight (1966) compared the Borel, the Yule, the logarithmic distributions with the distribution derived by him for this purpose called now Haight-zeta distribution (cf. Wimmer, Altmann 1999) but none of them could yield adequate results. Haight and Jones (1974) as well as Lánský and Radil-Weiss (1980) tried another approach but attained good results only in about 50% of cases. Dolinskij (1994, 1988) used for this purpose the Zipf-Alekseev distribution which is a generalization of Zipf distribution. Altmann (1992) has shown that the deviations from this distribution are extremely small (P ≈ 1 in almost all cases) but the foundation of this distribution was performed by Hřebíček (1995, 1996, 1997). Altmann (1992) used the usual proportionality approach with speaker-hearer balance and obtained the negative binomial distribution.

'''2. Hypothesis'''

''The ranking of word associations abides by a regular ranking distribution''.

'''3. Derivation'''

'''3.1. The Zipf-Alekseev model''' (Hřebíček 1997: 43)

Hřebíček starts from two assumptions:

(i) The frequency of an association <math>f_x</math> at any rank x is proportional to the frequency at the first rank <math>f_1</math>.

(ii) The frequency fx at rank x is inversely proportional to the rank x.
Putting this together, we obtain

<math> f_x\approx f_1 \frac{1}{x}</math>

or, in logarithmic form

<math>\ln (f_1/f_2) \approx \ln x</math>

The proportionality is given by Menzerath´s law, i.e.

(1)<math>\ln(P_1/P_2) = \ln(Ax^b) \ln x \quad</math> .

Solving for fx yields

(2)<math> P_x = P_1 x ^{-(a+b\ln x)}\quad</math>.

Usually <math>P_1</math> is so important that one gives it a special value, i.e. one modifies the distribution obtaining

(3)<math> P_x = \begin{cases} \alpha, & x = 1 \\ \frac{(1-\alpha)x^{-(a+b \ln x)}}{T}, & x = 2,3,...,n \end{cases}</math>.

with <math> T = \sum_{j=2}^n j^{-(a+b \ln j)}, a, b \epsilon \Re \quad, n \epsilon N, \quad 0 < \alpha < 1</math>.

'''3.2. The negative binomial model''' (Altmann 1992)

Assumption: The probability <math>P_x</math> at rank x is proprotional to the probability <math>P_{x-1}</math> at rank x-1, the proportionality being g(x) = (a+bx)/(cx). If ranking begins with x = 1, one solves the equation for the displaced form, i.e.

(4)<math> P_{x+1} = \frac{a+bx}{cx}P_x</math>

and after reparametrization one obtains the 1-displaced negative binomial distribution

(5)<math> P_x = {k+x-2 \choose x-1}p^k q^{x-1}, \quad x=1,2,3,...</math>

where a/b = k-1, b/c = q.

'''Example''': Associations of the word “high”

Altmann (1992) used the associations of the word “high” (4th grade, male) from Palermo, Jenkins (1964) and obtained the result presented in Table 1 and Fig. 1.

<div align="center">[[Image:Tabelle11_WA.jpg]]</div>

<div align="center">[[Image:Grafik1_WA.jpg]]</div>

Both results are excellent.

'''4. Authors:''' U. Strauss, G. Altmann, J. Eom

'''5. References:'''

'''Altmann, G'''. (1992). Two models for word association data. ''Glottometrika 13, 105-120''.

'''Altmann, G., Bagheri, D., Goebl, H., Köhler, R., Prün, C'''. (2002). ''Einführung in die quantitative Lexikologie.'' Götingen: Peust & Gutschmidt.

'''Dolinskij, V.A'''. (1994). Moscow Student´s word associations. In: ''2nd International Conference on Quantitative Linguistics, September 20-24, 1994, Moscow: 66-68. Moscow:'' Lomonosov Moscow State University.

'''Dolinskij, V.A'''. (1988). Raspredelenie reakcij v ekseprimentach po verbal´nym associacijam. ''Acta et Commentationes Universitatis Tartuensis 827, 80-101''.

'''Haight, F.A'''. (1966). Some statistical problems in connection with word association data. ''J. of Mathematical Psychology 3, 217-233''.

'''Haight, F.A., Jones, R.B'''. (1974). A probabilistic treatment of qualitative data with special reference to word association tests. ''J. of Mathematical Psychology 11, 237-244.''

'''Horvath, W.J.''' (1963). A stochastic model for word association tests. ''Psychological Review 70, 361-364.''

'''Hřebíček, L.''' (1995). ''Text levels. Language constructs, constituents and Menzerath-Altmann law.'' Trier: WVT.

'''Hřebíček, L.''' (1996). Word associations and text. ''Glottometrika 15, 12-17''.

'''Hřebíček, L.''' (1997). ''Lectures on text theory''. Prague: Oriental Institute.

'''Lánský, P., Radil-Weiss, T'''. (1980). A generalization of the Yule-Simon model, with special reference to word association tests and neural cell assembly formation. ''J. of Mathematical Psychology 21, 53-65''.

'''Palermo, D.S., Jenkins, J.J'''. (1964): ''Word association norms''. Grade School through College. Minneapolis: University of Minnesota Press.

'''Wimmer, G., Altmann, G'''. (1999). ''Thesaurus of univariate discrete probability distributions.'' Essen: Stamm.

Sentence and clause length

2006-07-07T15:23:55Z

Altmann:

'''1. Problem and history'''

The problem is to find the distribution of sentence and/or clause length in texts. To this end two previous problems must be solved or operationalized:

(i) The determination of sentence boundaries (cf. esp. Niehaus 2001) which is extremely difficult in speech but feasible in written texts. Usually it is achieved with the aid of numerous criteria which can differ from language to language.

(ii) The determination of the measurement unit. One obtains different models for different measurement units, e.g. clauses or words or syllables or phonemes. Today merely the first two are used, sometimes in modified form. Several hundreds of tests in different languages performed in the framework of Göttingen Project corroborate this kind of modelling (Altmann 1988b; Best 2001a,b, 2003, 2006; Grzybek 1995, 1999, 2001; Jing 2001; Kaßel, Livesey 2001; Niehaus 1997, 2001; Rheinländer 2000; Rottmann 2001; Roukk 2001, 2001a; Strehlow 1997; Uhlířová 2001; Wittek 2001).

Historically, the first who tackled the problem was Sherman (1888) using sentence length for characterization purposes. Altmann (1988b) baptized these laws to Sherman´s laws. Older empirical data were collected especially in Classical Greek, in English, Russian and Slovak (Marckworth, Bell 1967; Martynenko 1965; Mistrík 1967; Morton 1965; Morton, Levison 1966; Morton, McLeman 1966; Clayman 1981). Modelling goes back to Yule (1939, 1944), Williams (1940, 1970), Lesskis (1962), Martynenko (1965), Fucks (1955; 1970/71), Buch (1969), Sichel (1974); today it is part of the unified theory (<math>\rightarrow</math>), the first models in this direction were set up by Altmann (1988b).

'''2. Hypothesis'''

''Sentence and clause lengths in texts abide by regular probability distributions derived from the unified theory (<math>\rightarrow</math>).''

'''3. Derivation''' (Altmann 1988b)

The speaker (writer) tends to prolong the actual length of the sentence x adding further clauses, affecting it linearly in the form a + bx. The consideration for the hearer (reader) brakes this trend with force cx. Thus, if sentence length is measured in the number of clauses, one obtains

(1)<math> g(x)=\frac{a+bx}{cx}</math>.

Inserting g(x) as a proportionality function in (10) esp. Example 6 of Unified Theory (à) and reparametrizing, one obtains the negative binomial distribution

(2)<math> P_x={k+x-1 \choose x}p^k q^x \quad, x=0,1,2,...;\quad 0<p<1;\quad q=1-p;\quad k>0</math>.

Some authors define sentence as having at least one clause even if there is no finite verb in it. In that case, one solves the pertinent difference equation for x = 1, 2,... and obtains the positive negative binomial distribution

(3)<math> P_x={k+x-1 \choose x}\frac{p^k q^x}{1-p^k} \quad, x=1,2,3,...;\quad 0<p<1;\quad q=1-p;\quad k>0</math>. .

Example: Sentence length in clauses

Roukk (2001) measured the sentence length (in clauses) in Čechov´s stories and fitted the positive negative binomial distribution as given in Table 1 and Fig. 1.

<div align="center">[[Image:Tabelle1_SaCL.jpg]]</div>

<div align="center">[[Image:Grafik1_SaCL.jpg]]</div>
<div align="center">Fig. 1. Fitting the positive negative binomial distribution to Roukk´s data</div>

'''Example''': Length of clauses in Bulgarian

Uhlířová (2001) measured the length of clauses in a collection of letters of a Bulgarian native speaker and obtained the results in Table 2, Fig. 2.

<div align="center">[[Image:Tabelle2_SaCL.jpg]]</div>

<div align="center">[[Image:Grafik2_SaCL.jpg]]</div>
<div align="center">Figure 2. Fitting the negative binomial distribution to clause length in Bulgarian (Uhlířová 2001)</div>

If the measurement unit is '''word''', then there is an intermediate level exerting a constant effect d added to cx, i.e. one obtains

(4)<math>P_x= \frac{a+bx}{d+cx}P_{x-1}</math> ,

yielding, after reparametrization, the hyperpascal distribution

(5)<math> P_x= \frac{{k+x-1 \choose x}}{{m+x-1 \choose x}}q^x C, \quad x=0,1,...;\quad k,m>0,\quad0<q<1;</math>

C being the normalizing constant,<math> C^{-1}= _2 F_1 (k,1;m;q)\quad</math>.

'''Example'''. Sentence length in Herodot´s Book 1

Altmann (1988) has shown that under this condition the sentence length follows the hyperpascal distribution using 244 texts. The fitting of (5) to Herodot´s Book 1 is shown in Table 3 and Fig. 3.

<div align="center">[[Image:Tabelle3_SaCL.jpg]]</div>

<div align="center">[[Image:Grafik3_SaCL.jpg]]</div>
<div align="center">Figure 3. Fitting the hyperpascal distribution to sentence length in Herodot´s Book 1</div>

Under favourable boundary conditions, one can use all special or limiting cases of these distributions (cf. Wimmer, Altmann 1999).

Example: Sentence length in Old Church Slavonic using the positive Poisson distribution
For Old Church Slavonic texts, Rottmann (2001) and for German texts Wittek (2001) use the positive Poisson distribution which is the limiting case of the positive negative binomial distribution

(6)<math> P_x=\frac{a^x}{x!(e^a -1)},\quad x=1,2,3\quad a>0</math>.

<div align="center">[[Image:Tabelle4_SaCL.jpg]]</div>

<div align="center">[[Image:Grafik4_SaCL.jpg]]</div>
<div align="center">Figure 4. Fitting the positive Poisson distribution to Old Church Slavonic data.</div>

'''Example''': Modified positive Poisson distribution for German texts

Wittek (2001) obtained good results for 80 German texts using the positive Poisson distribution. However, in 4 cases he was forced to modify the first two classes of the positive Poisson distribution and obtained

(7)<math> P-x=\begin{cases} \frac{(1-\alpha)a}{e^a -1}, & x=1 \\ \frac{a}{e^a -1}\left( \frac{a}{2} + \alpha \right), & x=2,\quad a>0;\quad 0<\alpha <1 \\ \frac{a^x}{x! (e^a -1)}, & x=3,4,... \end{cases}</math>

The results of fitting are shown in Table 5.

<div align="center">[[Image:Tabelle5_SaCL.jpg]]</div>

Uhlířová (2001) used for Bulgarian clauses the mixed negative binomial distribution, but it can be shown that the usual negative binomial distribution is sufficient.

'''4. Authors''': U. Strauss, G. Altmann, K.-H. Best

'''5. References'''

'''Admoni, Wladimir''' (1973). ''Die Entwicklungstendenzen des deutschen Satzbaus von heute''. München: Hueber.

'''Altmann, G'''. (1988a). ''Wiederholungen in Texten''. Bochum, Brockmeyer.

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Word classes

2006-07-07T04:23:46Z

Altmann:

'''1. Problem and history'''

Does the frequency of different word classes abide by a special distribution law? Evidently, word classes are nominal entities, thus they must be ranked (see Ranking →). Historically, word classes represent the diversification of an amorphous word stock which began to be partitioned by the development of grammar, thus this is a problem of diversification (→).

The first investigations have been performed by Hammerl (1990) who obtained the Zipf-Alekseev distribution, just as Schweers, Zhu (1991) did. Köhler (1991) studied the problem from the diversification point of view. A number of individual studies on texts in German (Best 1994, 1997, 2000, 2001a, b; Judt 1995), Russian (Bosselmann 2001), French (Judt 1995), Latin (Schweers, Zhu 1991) Chinese (Zhu, Best 1992, Schweers, Zhu 1991), Czech (Uhlířová 2000) and Portuguese (Ziegler 1998, 2001) has been performed and brought different results. The word classes were considered in their classical version, nobody tested other possible classifications. Statistical tests for word classes have been set up by Wimmer and Altmann (2001).
The hypothesis concerning word classes is merely a special case of a more general hypothesis encompassing any kind of classes of linguistic entities.

'''2. Hypothesis'''

''If language entities are ordered in classes, then their ranked frequencies follow a regular probability distribution or a regular ranking series''.

Seen from the opposite point of view, if the ranked frequencies are properly distributed we have a preliminary approximate corroboration of the “correctness” of the classification, i.e. we approximate some linguistic-psychological truth.

'''3. Derivation'''

'''3.1. The Zipf-Alekseev distribution'''

The derivation is shown in Word associations (<math>\rightarrow</math>), Chapter 3.1. Different derivations are shown in Hammerl (1990) and Hřebíček (2000: 14f). The formula used in right truncated form and with modified class x = 1 is

<math> P_x = \begin{cases} \alpha, & x=1 \\ \frac{(1-\alpha)x^{a+b \ln x}}{T} & 2,3,...,n \end{cases}</math>

where <math> T= \sum_{j=2}^N j^{a+b \ln x}</math>.

Example: Word classes in Portuguese (Ziegler 2001).

<div align="center">[[Image:Tabelle11_WCL.jpg]]</div>

<div align="center">[[Image:Grafik1_WCL.jpg]]</div>
<div align="center">Figure 1. Fitting of modified right truncated Zipf-Alekseev distribution to Portuguese data</div>

'''3.2. The negative hypergeometric distribution'''

The frequency of ranked word classes abides by the usual proportionality relation between neighboring classes. Using the proportionality function

<math> g(x)= \frac{(M+x-1)(K-M+n-x)}{x(n-x+1)}</math>

and solving with displacement

(1)<math> P_{x+1}=\frac{(M+x-1)(K-M+n-x)}{x(n-x+1)}P_x</math>

one obtains the 1-displaced negative hypergeometric distribution

(2)<math> P_x = \frac{{-M \choose x-1}{-K+M \choose n-x+1}}{{-K \choose n}}, \quad x=1,2,3,...,n,\quad K\geq M\geq 0,\quad n \epsilon N</math>

Example: Word classes in Latin (Schweers, Zhu 1991)
Schweers and Zhu examined word classes in Latin, German and Chinese and found satisfactory fits for Latin and German. For Latin, they took Caesar´s “Bellum Gallicum”, Book 1, Chapters 1-8, § 2, and obtained the results in Table 2. The majority of researchers used this distribution.

<div align="center">[[Image:Tabelle222_WCL.jpg]]</div>

<div align="center">[[Image:Grafik2_WCL.jpg]]</div>
<div align="center">Figure 2. Fitting the negative hypergeometric distribution to Latin data</div>

'''3.3. Altmann´s series'''

Altmann (1993) did not strive for a distribution but simply for a series capturing the decreasing frequencies (absolute or relative) of ranked classes. Consider the Zipf-Mandelbrot law as a (not normalized) continuous function. Its differential equation is

(2)<math> \frac{dy}{y}= -\frac{c}{a+x}dx</math> ,

i.e. a special case of the unified theory (→). Since ranking proceeds in unit steps, dx = 1 and dy = yx+1 – yx, we obtain

(3)<math> \frac{y_{x+1}-y_x}{y_x}= -\frac{c}{a+x}</math>.

Reordering, setting a-c = b, and solving (3), results in

(4)<math> y_x = \frac{{b+x \choose x-1}}{{a+x \choose x-1}}y_1, \quad x=1,2,3,...</math>

The parameters fulfill one of the conditions (i) a > b 0, (ii) a > 0, -1 < b < 0, (iii) b < a < 0 when a and b are not integers.

Example: Word class distribution in a German text (Best 1997)

Altmann used (4) only for phoneme ranking, but Best (1994, 1997) used it also for word classes with very satisfactory results. The fitting of (4) to the relative frequencies of word classes in a German text (Bichsel, P., Der Mann, der nichts mehr wissen wollte) yielded results presented in Table 3 and Fig. 3.

Here x are the ranked word classes, yx are their relative frequencies. The fit is very satisfactory.

<div align="center">[[Image:Grafik3_WCL.jpg]]</div>

'''4. Authors:''' U. Strauss, G. Altmann, K.-H. Best

'''5. References'''

'''Altmann, G.''' (1991). Word class diversification of Arabic verbal roots. In: Rothe, U. (ed.), ''Diversification Processes in Language: Grammar: 57-59''. Hagen: Rottmann.

'''Altmann, G'''. (1993). Phoneme counts. ''Glottometrika 14, 55-70''.

'''Best, K.-H.''' (1994). Word class frequencies in contemporary German short prose texts. ''J. of Quantitative Linguistics 1, 144-147''.

'''Best, K.-H'''. (1997). Zur Wortartenhäufigkeit in Texten deutscher Kurzprosa der Gegenwart. ''Glottometrika 16, 276-285''.

'''Best, K-H.''' (1998). Zur Interaktion der Wortarten in Texten. ''Papiere zur Linguistik 58: 83-95''.

'''Best, K.-H.''' (2000). Verteilung der Wortarten in Anzeigen. ''Göttinger Beiträge zur Sprachwissenschaft 4, 37-51''.

'''Best, K.-H.''' (2001a). Zur Gesetzmäßigkeit der Wortartenverteilungen in deutschen Pressetexten. ''Glottometrics 1, 1-26''.

'''Best, K.H.''' (2001b). ''Quantitative Linguistik. Eine Annäherung.'' Göttingen: Peust & Gutschmidt.

'''Bosselmann, A.''' (2001). ''Wortartenverteilungen in russischen Texten''. Msc.

'''Hammerl, R.''' (1990). Untersuchungen zur Verteilung der Wortarten im Text. ''Glottometrika 11, 142-156''.

'''Hřebíček, L.''' (2000). Variation in sequences. Prague: Oriental Institute.

'''Hudson, R.''' (1994). About 37 % of word-tokens are nouns. ''Language 70, 331-339''.

'''Judt, B.''' (1995). ''Wortartenhäufigkeiten im Deutschen und Französischen''. Göttingen: Staatsexamensarbeit.

'''Köhler, R.''' (1991). Diversification of coding methods in grammar. In: Rothe, U. (ed.), ''Diversification Processes in Language: Grammar: 47-55''. Hagen: Rottmann.

'''Lauter, J.''' (1966). ''Untersuchungen zur Sprache von Kants “Kritik der reinen Vernunft”.'' Köln: Westdeutscher Verlag.

'''Mizutani, S.''' (1989). Ohno's lexical law: its data adjustment by linear regression. In: Mizutani, S. (ed.), ''Japanese Quantitative Linguistics''. Bochum: Brockmeyer. 1-13.

'''Schindelin, C.''' (2005). Die quantitative Erforschung der chinesischen Sprache und Schrift. In: ''Köhler, R., Altmann, G., Piotrowski, R.G. (eds.), ''Quantitative Linguistics - An Inernational handbook: 947-970.''' Berlin: de Gruyter.

'''Schweers, A., Zhu, J'''. (1991). Wortartenklassifikation im Lateinischen, Deutschen und Chinesischen. In: Rothe, U. (ed.), ''Diversification Processes in Language: Grammar: 157-167''. Hagen: Rottmann.

'''Uhlířová, L.''' (2000). On language modelling in automatic speech recognition. ''J. of Quantitative Linguistics 7, 209-216''.

'''Wimmer, G., Altmann, G.''' (2001). Some statistical investigations concerning word classes. ''Glottometrics 1, 109-123''.

'''Zhu, J, Best, K.-H.''' (1992). Zum Wort im modernen Chinesisch. ''Oriens Extremus 35, 45-60.''

'''Ziegler, A.''' (1998). Word class frequencies in Brazilian-Portuguese texts. ''J. of Quantitative Linguistics 5, 269-280.''

'''Ziegler, A.''' (2001). Word class frequencies in Portuguese press texts. In: Uhlířová, L., Wimmer, G., Altmann, G., Köhler, R. (eds.), ''Text as a Linguistic Paradigm: Levels, Constituents, Constructs. Festschrift in Honour of Ludek Hřebíček: 295-312''. Trier: WVT.

'''Ziegler, A., Best, K.-H., Altmann, G'''. (2002). Nominalstil. ''ETC 2, 72-85''.

'''Ziegler, A., Best, K.-H., Altmann, G.''' (2001). A contribution to text spectra. ''Glottometrics 1, 97-108''.

[[Category:Unfertig]]

Hierarchic relations

2006-07-07T03:37:43Z

Altmann:

'''1. Problem and history'''

In different domains of language one observed the fact that the length of a construct influences the length of its consitutents. Usually the constituents get smaller with increasing length of the construct but not in all cases. The problem is to find a theoretical model encompassing all dependencies of this kind. The constructs whose length is the independent variable are: hreb, sentence, rhythmic unit, word, syllable; the constituents whose length or duration is the dependent variable are: sentence, clause, word, syllable, morph, sound. There is also the possibility to consider two independent variables, e.g. word (measured in number of syllables) and syllable (measured in number of sounds) , while the dependent variable is the syllable duration.
Up to now the following particular cases have been examined (see Table 1)

<div align="center">[[Image:Tabelle11_HR.jpg]]</div>

The origin of the problem can be found in 19th century, in phonetics, probably for the first time with Sievers (1876, 1901) who measured the syllable duration in rhythmic units (Sprechakte). A number of phoneticians tested different hypotheses, a part of which corroborated, another part falsified it. The isochrony hypothesis in English is a special case of this problem. The problem was generalized by Menzerath who stated that ''the greater the whole the smaller its parts'' (1954: 101). Different researchers proposed some empirical formulas (Fónagy, Magdics 1960; Nooteboom 1972, 1973; Landblom, Rapp 1972), Altmann (1980) set up the pertinent differential equation and called the result Menzerath´s law. Hřebíček (1992, 1995, 1997) showed that the whole hierarchy of textual levels is based on this dependence and called it ''Menzerath-Altmann´s law''.

There is a great number of individual examinations in different domains of languge. In phonetics the most exhaustive is Weber (1998), in textology the works by Hřebíček (see above) and a mixture of problems including biology and sociology can be found in Altmann, Schwibbe (1989). Bohn (1998) analyzed the relationship between Chines characters and the complexity of composing graphemes, length of words and simplicity of characters, clause length and word length, sentence length and clause length. (cf. also Menzel 2005).
The law has a strong corroboration not only within linguistics but displays analogies to other sciences. Thus the simple form of Menzerath´s law is identical with the allometric law in biology and with power laws current in different sciences. Its correspondences can be found in (i) molecular biology, (ii) sociology of baboons, (iii) in the domain of self-organized criticality, (iv) in chaos research, (v) in the theory of fractals, (vi) in information theory.
It has been observed that construct length is not always the only cause of shortening of the constituents. Also accent, vowel quality, syllable structure, frequency etc. can intervene (cf. Weber 1998). In that case more complex formulas must be used.
The law has several consequences, all of which must still be tested (cf. Altmann, Schwibbe 1989: 8-14):
1. In longer words more phonetic changes occur than in shorter ones.

2. In languages with greater average word length more phonetic/phonemic changes occur than within the same time interval in languages with smaller average word length

3. The adding of an affix to a word evokes the tendency to reduce the inventory of consonants of the word.

4. The shortening of average syllables length in Hypothesis 3 can also be achieved by inserting epenthetic vowels between the stem and affix (or compounding stem).

5. Partial reduplication is more frequent in natural languages than full reduplication.

6. Short roots/morphemes/stems build more compounds or derived words than long ones.

7. The more elements there are in a compound the shorter they are (see the hypotheses on compounds <math>\leftarrow</math>)

8. Fenk-Fenk (to be inserted)

There are different interpretations of the law:

1. In general, long constructs contain more redundancy than short ones. In order to prevent excessive growth of redundancy one can reduce the size of the constituents. The size, the place and the time of this reduction is not known and cannot be predicted. The analogy to self-organized criticality of sand-piles is evident (cf. Bak 1996).

2. Köhler (1989) shows that mechanism of shortening is a consequence of restrictions of the memory: the longer the construct, the more place must be reserved for the structural information between the constituents, thus the size of the constituents must be reduced.

'''2. Hypothesis'''

''The size of the components is a function of the construct size''.

'''3. Derivation'''

The average size of constituents changes with the increase of the size of the construct. It is assumed that the relative rate of change of the size of components is proportional to the rate of change of the size of constructs, the proportionality function being

<math> g(x) = a_0 + \frac{a_1}{x} + \frac{a_2}{x^2}</math>.

Thus in case that all other variables other than construct size are subsumed under the ceteris paribus condition one obtains

(1)<math> \frac{dy}{y-d}= \left(a_0 \frac{a_1}{x}+ \frac{a_2}{x^2}\right)</math>.

where d is the minimal value y can attain.
In case that there is another independent variable, z, one starts from

(2)<math>\frac{dy}{dx}\frac{1}{y-d}=a_0 + \frac{a_1}{x}+\frac{a_2}{x^2}\quad</math> and <math>\frac{dy}{dz}\frac{1}{y-d}=b_0 + \frac{b_1}{z}+\frac{b_2}{z^2}</math>

which can be extended to any number of variables.
The solution of (1) yields

(3)<math> y= Cx^{a_1}e^{a_0 x-a_2/x} + d</math>

the combining (2) the solution is

(4)<math> y= Cx^{a_1}z^{b_1}e^{a_0 x + b_0 z-a_2 /x-b_2 /z} + d</math>

In different applications the following special cases of (3) have been used (d > 0)

(1a) <math> y=ax^{-b}(+d), \quad x= 1, 2, 3, ...; \quad a, b > 0</math>

(1b)<math> y=ax^b e^{cx}(+d), \quad x= 1, 2, 3, ...; \quad a > 0</math>

(1c)<math> y=ae^{-cx}(+d), \quad x= 1, 2, 3, ...; \quad a, c > 0</math>

(1d) <math> y=ae^{c/x}(+d), \quad x= 1, 2, 3, ...; \quad a, c > 0</math>

Solution (4) is still very seldom. Both approaches are special cases of the unified theory (<math>\leftarrow</math>).

'''Examples''':

'''4. Authors: U. Strauss, G. Altmann'''

'''5. References'''

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[[Category:Unfertig]]

Hierarchic relations

2006-07-06T14:39:33Z

Altmann:

'''1. Problem and history'''

In different domains of language one observed the fact that the length of a construct influences the length of its consitutents. Usually the constituents get smaller with increasing length of the construct but not in all cases. The problem is to find a theoretical model encompassing all dependencies of this kind. The constructs whose length is the independent variable are: hreb, sentence, rhythmic unit, word, syllable; the constituents whose length or duration is the dependent variable are: sentence, clause, word, syllable, morph, sound. There is also the possibility to consider two independent variables, e.g. word (measured in number of syllables) and syllable (measured in number of sounds) , while the dependent variable is the syllable duration.
Up to now the following particular cases have been examined (see Table 1)

<div align="center">[[Image:Tabelle11_HR.jpg]]</div>

The origin of the problem can be found in 19th century, in phonetics, probably for the first time with Sievers (1876, 1901) who measured the syllable duration in rhythmic units (Sprechakte). A number of phoneticians tested different hypotheses, a part of which corroborated, another part falsified it. The isochrony hypothesis in English is a special case of this problem. The problem was generalized by Menzerath who stated that ''the greater the whole the smaller its parts'' (1954: 101). Different researchers proposed some empirical formulas (Fónagy, Magdics 1960; Nooteboom 1972, 1973; Landblom, Rapp 1972), Altmann (1980) set up the pertinent differential equation and called the result Menzerath´s law. Hřebíček (1992, 1995, 1997) showed that the whole hierarchy of textual levels is based on this dependence and called it ''Menzerath-Altmann´s law''.

There is a great number of individual examinations in different domains of languge. In phonetics the most exhaustive is Weber (1998), in textology the works by Hřebíček (see above) and a mixture of problems including biology and sociology can be found in Altmann, Schwibbe (1989). Bohn (1998) analyzed the relationship between Chines characters and the complexity of composing graphemes, length of words and simplicity of characters, clause length and word length, sentence length and clause length. (cf. also Menzel 2005).
The law has a strong corroboration not only within linguistics but displays analogies to other sciences. Thus the simple form of Menzerath´s law is identical with the allometric law in biology and with power laws current in different sciences. Its correspondences can be found in (i) molecular biology, (ii) sociology of baboons, (iii) in the domain of self-organized criticality, (iv) in chaos research, (v) in the theory of fractals, (vi) in information theory.
It has been observed that construct length is not always the only cause of shortening of the constituents. Also accent, vowel quality, syllable structure, frequency etc. can intervene (cf. Weber 1998). In that case more complex formulas must be used.
The law has several consequences, all of which must still be tested (cf. Altmann, Schwibbe 1989: 8-14):
1. In longer words more phonetic changes occur than in shorter ones.

2. In languages with greater average word length more phonetic/phonemic changes occur than within the same time interval in languages with smaller average word length

3. The adding of an affix to a word evokes the tendency to reduce the inventory of consonants of the word.

4. The shortening of average syllables length in Hypothesis 3 can also be achieved by inserting epenthetic vowels between the stem and affix (or compounding stem).

5. Partial reduplication is more frequent in natural languages than full reduplication.

6. Short roots/morphemes/stems build more compounds or derived words than long ones.

7. The more elements there are in a compound the shorter they are (see the hypotheses on compounds <math>\leftarrow</math>)

8. Fenk-Fenk (to be inserted)

There are different interpretations of the law:

1. In general, long constructs contain more redundancy than short ones. In order to prevent excessive growth of redundancy one can reduce the size of the constituents. The size, the place and the time of this reduction is not known and cannot be predicted. The analogy to self-organized criticality of sand-piles is evident (cf. Bak 1996).

2. Köhler (1989) shows that mechanism of shortening is a consequence of restrictions of the memory: the longer the construct, the more place must be reserved for the structural information between the constituents, thus the size of the constituents must be reduced.

'''2. Hypothesis'''

''The size of the components is a function of the construct size''.

'''3. Derivation'''

The average size of constituents changes with the increase of the size of the construct. It is assumed that the relative rate of change of the size of components is proportional to the rate of change of the size of constructs, the proportionality function being

<math> g(x) = a_0 + \frac{a_1}{x} + \frac{a_2}{x^2}</math>.

Thus in case that all other variables other than construct size are subsumed under the ceteris paribus condition one obtains

(1)<math> \frac{dy}{y-d}= \left(a_0 \frac{a_1}{x}+ \frac{a_2}{x^2}\right)</math>.

where d is the minimal value y can attain.
In case that there is another independent variable, z, one starts from

(2)<math>\frac{dy}{dx}\frac{1}{y-d}=a_0 + \frac{a_1}{x}+\frac{a_2}{x^2}\quad</math> and <math>\frac{dy}{dz}\frac{1}{y-d}=b_0 + \frac{b_1}{z}+\frac{b_2}{z^2}</math>

which can be extended to any number of variables.
The solution of (1) yields

(3)<math> y= Cx^{a_1}e^{a_0 x-a_2/x} + d</math>

the combining (2) the solution is

(4)<math> y= Cx^{a_1}z^{b_1}e^{a_0 x + b_0 z-a_2 /x-b_2 /z} + d</math>

In different applications the following special cases of (3) have been used (d > 0)

(1a) <math> y=ax^{-b}(+d), \quad x= 1, 2, 3, ...; \quad a, b > 0</math>

(1b)<math> y=ax^b e^{cx}(+d), \quad x= 1, 2, 3, ...; \quad a > 0</math>

(1c)<math> y=ae^{-cx}(+d), \quad x= 1, 2, 3, ...; \quad a, c > 0</math>

(1d) <math> y=ae^{c/x}(+d), \quad x= 1, 2, 3, ...; \quad a, c > 0</math>

Solution (4) is still very seldom. Both approaches are special cases of the unified theory (<math>\leftarrow</math>).

'''Examples''':

'''4. Authors: U. Strauss, G. Altmann'''

'''5. References'''

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[[Category:Unfertig]]

Hierarchic relations

2006-07-06T14:03:31Z

Altmann:

'''1. Problem and history'''

In different domains of language one observed the fact that the length of a construct influences the length of its consitutents. Usually the constituents get smaller with increasing length of the construct but not in all cases. The problem is to find a theoretical model encompassing all dependencies of this kind. The constructs whose length is the independent variable are: hreb, sentence, rhythmic unit, word, syllable; the constituents whose length or duration is the dependent variable are: sentence, clause, word, syllable, morph, sound. There is also the possibility to consider two independent variables, e.g. word (measured in number of syllables) and syllable (measured in number of sounds) , while the dependent variable is the syllable duration.
Up to now the following particular cases have been examined (see Table 1)

<div align="center">[[Image:Tabelle11_HR.jpg]]</div>

The origin of the problem can be found in 19th century, in phonetics, probably for the first time with Sievers (1876, 1901) who measured the syllable duration in rhythmic units (Sprechakte). A number of phoneticians tested different hypotheses, a part of which corroborated, another part falsified it. The isochrony hypothesis in English is a special case of this problem. The problem was generalized by Menzerath who stated that ''the greater the whole the smaller its parts'' (1954: 101). Different researchers proposed some empirical formulas (Fónagy, Magdics 1960; Nooteboom 1972, 1973; Landblom, Rapp 1972), Altmann (1980) set up the pertinent differential equation and called the result Menzerath´s law. Hřebíček (1992, 1995, 1997) showed that the whole hierarchy of textual levels is based on this dependence and called it ''Menzerath-Altmann´s law''.

There is a great number of individual examinations in different domains of languge. In phonetics the most exhaustive is Weber (1998), in textology the works by Hřebíček (see above) and a mixture of problems including biology and sociology can be found in Altmann, Schwibbe (1989). Bohn (1998) analyzed the relationship between Chines characters and the complexity of composing graphemes, length of words and simplicity of characters, clause length and word length, sentence length and clause length. (cf. also Menzel 2005).
The law has a strong corroboration not only within linguistics but displays analogies to other sciences. Thus the simple form of Menzerath´s law is identical with the allometric law in biology and with power laws current in different sciences. Its correspondences can be found in (i) molecular biology, (ii) sociology of baboons, (iii) in the domain of self-organized criticality, (iv) in chaos research, (v) in the theory of fractals, (vi) in information theory.
It has been observed that construct length is not always the only cause of shortening of the constituents. Also accent, vowel quality, syllable structure, frequency etc. can intervene (cf. Weber 1998). In that case more complex formulas must be used.
The law has several consequences, all of which must still be tested (cf. Altmann, Schwibbe 1989: 8-14):
1. In longer words more phonetic changes occur than in shorter ones.

2. In languages with greater average word length more phonetic/phonemic changes occur than within the same time interval in languages with smaller average word length

3. The adding of an affix to a word evokes the tendency to reduce the inventory of consonants of the word.

4. The shortening of average syllables length in Hypothesis 3 can also be achieved by inserting epenthetic vowels between the stem and affix (or compounding stem).

5. Partial reduplication is more frequent in natural languages than full reduplication.

6. Short roots/morphemes/stems build more compounds or derived words than long ones.

7. The more elements there are in a compound the shorter they are (see the hypotheses on compounds <math>\leftarrow</math>)

8. Fenk-Fenk (to be inserted)

There are different interpretations of the law:

1. In general, long constructs contain more redundancy than short ones. In order to prevent excessive growth of redundancy one can reduce the size of the constituents. The size, the place and the time of this reduction is not known and cannot be predicted. The analogy to self-organized criticality of sand-piles is evident (cf. Bak 1996).

2. Köhler (1989) shows that mechanism of shortening is a consequence of restrictions of the memory: the longer the construct, the more place must be reserved for the structural information between the constituents, thus the size of the constituents must be reduced.

'''2. Hypothesis'''

''The size of the components is a function of the construct size''.

'''3. Derivation'''

The average size of constituents changes with the increase of the size of the construct. It is assumed that the relative rate of change of the size of components is proportional to the rate of change of the size of constructs, the proportionality function being

<math> g(x) = a_0 + \frac{a_1}{x} + \frac{a_2}{x^2}</math>.

Thus in case that all other variables other than construct size are subsumed under the ceteris paribus condition one obtains

(1)<math> \frac{dy}{y-d}= \left(a_0 \frac{a_1}{x}+ \frac{a_2}{x^2}\right)</math>.

where d is the minimal value y can attain.
In case that there is another independent variable, z, one starts from

(2)<math>\frac{dy}{dx}\frac{1}{y-d}=a_0 + \frac{a_1}{x}+\frac{a_2}{x^2}\quad</math> and <math>\frac{dy}{dz}\frac{1}{y-d}=b_0 + \frac{b_1}{z}+\frac{b_2}{z^2}</math>

which can be extended to any number of variables.
The solution of (1) yields

(3)<math> y= Cx^{a_1}e^{a_0 x-a_2/x} + d</math>

the combining (2) the solution is

(4)<math> y= Cx^{a_1}z^{b_1}e^{a_0 x + b_0 z-a_2 /x-b_2 /z} + d</math>

In different applications the following special cases of (3) have been used (d > 0)

(1a) <math> y=ax^{-b}(+d), \quad x= 1, 2, 3, ...; \quad a, b > 0</math>

(1b)<math> y=ax^b e^{cx}(+d), \quad x= 1, 2, 3, ...; \quad a > 0</math>

(1c)<math> y=ae^{-cx}(+d), \quad x= 1, 2, 3, ...; \quad a, c > 0</math>

(1d) <math> y=ae^{c/x}(+d), \quad x= 1, 2, 3, ...; \quad a, c > 0</math>

Solution (4) is still very seldom. Both approaches are special cases of the unified theory (<math>\leftarrow</math>).

'''Examples''':

'''4. Authors: U. Strauss, G. Altmann'''

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[[Category:Unfertig]]

Definition chains

2006-07-06T13:37:32Z

Altmann:

'''1. Problem and history'''

Take a noun from a monolingual dictionary and seek in its definitions (explications) the noun which is its genus proximum, i.e. only the relation of inclusion is relevant. For example ''cat'' is an ''animal''; then look at ''animal'' to find that it is a living ''creature'', which in turn is an ''organism'', etc. In this way one obtains a ''definition chain''. Thus a definition chain is a sequence of nouns in which the ith noun is member of the class encompassed by the (i+1)st noun, or in other words, the (i+1)st noun is the genus proximum of the ith noun. There are, of course, sometimes two or more genera proxima giving rise to lexeme nets.
Definition chains have different length. Taking a sample from a dictionary one sees that the number of different nouns on each higher level is smaller than that on lower levels.
The first empirical observations on chains have been made by R. Martin (1974), the first model which held only for Martin´s data was presented by Altmann and Kind (1983), who called this phenomenon '''Martin´s law'''. Hammerl´s model (1987a) presented below is widely applicable, however there are many alternative ways of establishing definition chains (cf. Sambor, Hammerl 1991, Schierholz 1989) and their final forms depends not only on criteria but also on the size of the dictionary.
If one considers all meanings of a noun, one obtains lexeme nets which can be treated as graphs, having different properties. Up to now no laws have been found for the construction of lexeme nets.

'''2. Hypothesis'''

''The number of nouns in a set of definition chains is the smaller the higher the abstraction (generality) level of the nouns.''

“Abstraction level” or “generality” is the position of a noun in the chain, e.g. in ''cat – animal – creature – organism – system'' the first noun ''cat'' has position (level, degree) 1, ''animal'' has degree 2, ''creature'' 3 etc. (cf. Kisro-Völker 1984).

The hypothesis involves that the resulting distribution is monotone decreasing.

Critical points:
(1) The lowest level can begin with 0 or 1, it is merely a convention. We use here 1.

(2) As <math>1^{st}</math>-level nouns either all nouns are admitted or only those which are not genera proxima of another nouns.

(3) There are three kinds of counting the number of nouns on a given level: (i) one counts all lexemes, (ii) one counts only different lexemes, (iii) one counts a given lexeme only on its highest level. It depends on the problem examined.
(4) In some languages compound nouns can strongly prolong the chains. One can take them into account or not.

(5) The choice of the sampling dictionary is of great improtance. Great dictionaries prolong the chains.

(6) The existence of different genera proxima for a noun leads to the problem of lexeme nets but it can be solved in that one takes only one of them.

(7) In circular definitions the last noun should be eliminated, e.g. ''Stall – Gebäude – Bauwerk – Gebäude''.

(8) Since dictionaries do not care for unambiguity of chains, one can apply one owns competence in completing or changing them. Such a procedure strongly depends on the erudition of the researcher.

(9) Not nominal lexemes should be left out, e.g. ''something that''.

(10) In German every verb can be converted to noun (''arbeiten - das Arbeiten''). In these cases the converted noun can be replaced by the equivalent noun (''Arbeit'').

(11) It is not known whether the above laws hold also for other word classes or other relations between lexemes.

'''3. Derivation'''

'''3.1. Hammerl´s version A (1987a)'''

Assumptions:

(i) The number of lexemes on level x+1 is proportional to the power of that on level x, i.e. <math>y_{x+1}\approx y_x^b</math>.

(ii) The number of lexemes on level x+1 is inversely proportional to the power of the level itself, i.e <math>y_{x+1}\approx 1/(x+1)^a</math>.

Putting these assumptions together one obtains

(1)<math>y_{x+1}\frac{c}{(x+1)^a}y_x^b</math>.

where c is the proportionality constant. In practice, formula (1) is used since the solution

(2) <math>y_x m= y_1^{b^{x-1}}\prod_{i=0}^{x-2}\begin{bmatrix} c \\ (x-i)^a\end{bmatrix}^{b'}, \quad x = 1, 2, 3, ...</math>.

is neither illuminating nor practical. If one considers it a probability distribution, it must be normed. The estimation of the parameters a,b,c can be performed using the first four observed values. Let

(3)<math>\hat a = \frac{(\ln y_3-\ln y_2)(\ln y_1-\ln y_3)-(\ln y_2-\ln y_4)(\ln y_2-\ln y_1)}{(\ln 2-\ln 4)\ln y_2+(\ln 4-\ln 3)\ln y_1+(\ln 3-\ln 2)\ln y_3}</math>

<math>\hat b = \frac{\ln y2 - a(\ln 2 - \ln 4)-y_4}{\ln y_1 - \ln y_3}</math>

<math>\hat c = exp{\mathcal f\ln y_2+a\ln 2-b\ln y_1\mathcal g}</math>

'''Example''': Definition chains for Polish nouns

Hammerl (1987b) constructed definition chains for 1000 Polish nouns using two methods:
(a) A lexeme can be repeated, i.e. the same lexeme can occur on different levels but it is counted only once on each particular level.
(b) Lexemes cannot be repeated, i.e. if a lexeme occurs on different levels (in different chains), it is counted only on its highest level.
The results are shown in Table 1.

<div align="center">[[Image:Tabelle1_def.jpg]]</div>

The fitting is so good that the empirical and the theoretical points in the graphs cannot be distinguished.

<div align="center">[[Image:Figur2_Def.jpg]]</div>
<div align="center">Fig. 1. Fitting (2) to Polish nouns (with repetition)</div>
<div align="center">[[Image:Figur3_Def.jpg]]</div>
<div align="center">Fig. 2. Fitting (2) to Polish data (without repetition)</div>

'''3.2. Hammerl´s version B (1989a''')

In Hammerl´s version A the lexemes were numbered beginning from the most concrete level upwards, i.e. (DC = definition chain)

<div align="center">[[Image:VersionA_Def.jpg]]</div>

In version B the most general noun of every chain is assigned level 1, the next one level 2 , etc., i.e. the chains begin “from above”

<div align="center">[[Image:VersionB_Def.jpg]]</div>

Hammerl starts from the assumption that for the speaker a very general way of expressing himself is sufficient, thus he has the tendency to shorten the chains which is in agreement with his least effort; the hearer, on the contrary, requires very concrete designations, in order to reduce his own decoding effort. Thus definition chains develop in language dynamically.
Consider a a constant depending on dictionary size, -bx the negative influence of the speaker upon the given level and cx the effort of the hearer to brake the intention of the speaker, one obtains the classical steady state situation of the diversification process (à) (with displacement) given as

(5)<math> \frac{\Delta y_x}{y_x} = \frac{a - bx}{cx}</math>

whose solution after reparametrisation [(c-b)/c = q, a/(a-b) = k-1 and yx = Px] yields the 1-displaced negative binomial distribution

(6)<math> P_x = {k + x - 2 \choose x - 1}p^k q^{x - 1}, \quad x = 1, 2, 3, ...</math>

If one counts a noun on each level only once, the distribution need not be monotone decreasing.

'''Example:''' Definition chains for Polish and German nouns

Hammerl (1989a: 134) showed the distribution of Polish and German nouns in definition chains built according to his version B as given in Table 2.

<div align="center">Tabelle 2</div>
<div align="center">[[Image:Tabelle2_Def.jpg]]</div>

The results of fitting are very good in both cases.

<div align="center">[[Image:Figur4_Def.jpg]]</div>

<div align="center">Fig 3. Fitting the 1-displaced negative binomial d. (6) to Polish data</div>

<div align="center">[[Image:Figur5_Def.jpg]]</div>

<div align="center">Fig. 4. Fitting the 1-displaced negative binomial d. (6) to German data</div>

'''3.3. Hammerl´s law'''

Another aspect of definition chains is the number (frequency) of lexemes occurring x-times in all examined chains. Here the ordering on different levels is irrelevant, the point is the frequency of occurrence. Hammerl (1991b) excludes two extreme possibilities:
(a) All lowest level nouns have their own individual chains, not intersecting with other ones, i.e. no lexeme of higher level is common to two chains.
(b) The nouns are semantically in such a close relationship that they all have one common “ancestor”. Since the selection of nouns from the dictionary must be random, these cases are for a sample very improbable. For the whole dictionary they are impossible.
Hammerl (1991b) used a continuos approach leading to Menzerath´s law (zeta distribution). Unfortunately, the fitting could be made satisfactory only if one modified the distribution a posteriori.
In discrete terms Hammerl´s approach can be presented in the form

(7)<math> P_x = \left( 1-\frac{1}{x} \right)^a P_{x-1}</math>

which means that the probability of lexemes occurring in all examined chains x-times is proportional to those occurring x-1 times, the proportionality function being g(x) = (1-1/x)a. Here we assume the same mechanism but define the proportionality as

(8) <math> g(x) = \left( 1-\frac{2}{a+x}\right)</math>

i.e.
(9)<math> P_{(x)} = \left( 1-\frac{2}{a+x}\right)P_{x-1}, x = 2, 3, 4, ...</math>

resulting in the Johnson-Kotz distribution (called sometimes Yule-Simon d.)

(10)<math> P_{(x)} = \left(\frac{a}{(a+x-1)(a+x)}\right), x = 1, 2, 3, ...</math>

which is a special case of the unified theory (<math>\rightarrow</math>) formula (10) substituting <math> a_0 = a_2 = ... = 0, a_1 = -2, b_1 = a</math>.

'''Example''': German definition chains

Hammerl (1991b) and Hammerl, Sambor (1997) showed data from 1000 German definition chains as given in Table 3. The frequencies greater than or equal to x = 34 were pooled in the last class. The results of fitting (10) are in the third column of Table 3 (Fig. 5). The classes were pooled so that each theoretical frequency was greater than 1.

<div align="center">[[Image:Tabelle3_Def.jpg]]</div>

The result of fitting is very good.

<div align="center">[[Image:Figur7 Def.jpg]]</div>
<div align="center">Fig. 5. Fitting the Johnson-Kotz distribution (10) to Hammerl´s frequency data</div>

'''4. Authors:''' U. Strauss, G. Altmann

'''5. References:'''

'''Altmann, G.''' (1991b). Definitionsfolgen und Lexemnetze. In: Sambor, J., Hammerl, R. (eds.) 1991: 188-198.

'''Altmann, G., Bagheri, D., Goebl, H., Köhler, R., Prün, C.''' (2002). ''Einführung in die quantitative Lexikologie''. Götingen: Peust & Gutschmidt.

'''Altmann, G., Kind, B. (1983).''' Ein semantisches Gesetz. ''Glottometrika 5'', 1-13.

'''Bagheri, D.''' (1996). Review of Hammerl, R., Sambor, J. (eds.) 1991. ''Journal of Quantitative Linguistics 3'', 152-168.

'''Bagheri, D.''' (2002). Definitionsfolgen und Lexemnetze. In: Altmann, G., Bagheri, D., Goebl, H., Köhler, R., Prün, C., Einführung ''in die quantitative Lexikologie: 94-133.'' Göttingen: Peust & Gutschmidt.

'''Gołębiewska, E.''' (1989). ''Gniazda pojęcziowe v Małym Slowniku Jzyka Polskiego''. Warszawa: Diss.

'''Gołębiewska, E.''' (1991). Statistische Strutkur der Lexemnetze in der polnischen und deutschen Sprache – eine vergleichende Analyse. In: Sambor, J., Hammerl, R. (eds.) ''1991: 120-128''.

'''Hammerl, R'''. (1987). Prawa językowe we współczesnej kwantytatywnej lingwistyce modelowej (na przykladie tzw. Prawa Martina). ''Poradnik Językowy 6, 414-428.''

'''Hammerl, R.''' (1987a). Untersuchungen zur mathematischen Beschreibung des Martingesetzes der Abstraktionsebenen. ''Glottometrika 8, 113-129''.

'''Hammerl, R'''. (1987b). Untersuchungen zur mathematischen Beschreibung des Martingesetzes der Abstraktionsebenen. ''Glottometrika 9, 105-120''.

'''Hammerl, R.''' (1988). Neue theoretische Untersuchungen im Zusammenhang mit dem Martingesetz der Abstraktionsebenen. ''Glottometrika 9, 105-120''.

'''Hammerl, R.''' (1989a). Neue Perspektiven der sprachlichen Synergetik: Begriffsstrukturen – kognitive Gesetze. ''Glottometrika 10, 129-140''.

'''Hammerl, R.''' (1989b). Untersuchung struktureller Eigenschaften von Begriffsnetzen. ''Glottometrika 10, 141-154''.

'''Hammerl, R.''' (1990a). A contribution to the examination of semantic relations between lexemes. In: Bock, H.H., Ihm, P. (eds.), ''Classification, data analysis, and knowledge organization. Models and methods of application''s: 149-155. Berlin-Heidelberg: Springer.

'''Hammerl, R'''. (1990e). Gniazda pojęcowie. Przegląd problemów. ''Poradnik Językowy 9-10, 680-693''.

'''Hammerl, R.''' (1991a). Definition von Grundbegriffen für die Untersuchung von Definitionsfolgen und Lexemnetzen. In: Sambor, J., Hammerl, R. (eds.) ''1991: 2-12''.

'''Hammerl, R'''. (1991b). Messung der Stärke von semantischen Relationen zwischen Lexemen. In: Sambor, J., Hammerl, R. (eds.) ''1991: 75-96''.

'''Hammerl, R'''. (1991c). Methodologische und methodische Probleme der Erstellung von Definitionsfolgen und Lexemnetzen. In: Sambor, J., Hammerl, R. (eds.) ''1991: 13-37''.

'''Hammerl, R'''. (1991d). Überprüfung des Martingesetzes I an deutschem Sprachmaterial. In: Sambor, J., Hammerl, R. (eds.) ''1991: 50-64''.

'''Hammerl, R'''. (1991e). Untersuchungen von Definitionsfolgen und Lexemnetzen: eine einführende Bibliographie. In: Sambor, J., Hammerl, R. (eds.) ''1991: 200-204''.

'''Hammerl, R'''. (1991f). Zur Untersuchung der semantischen Struktur der Lexik. Vergleich zweier Modelle. In: Sambor, J., Hammerl, R. (eds.) ''1991: 174-186''.

'''Hammerl, R., Rogalińska, A'''. (1991). Erstellung und Analyse von Lexemnetzen mit dem PC-Programm NETZE. In: Sambor, J., Hammerl, R. (eds.) ''1991: 157-173''.

'''Hammerl, R., Sambor, J'''. (1993a). ''O statystycznych prawach jezykowych''. Warszawa: Polskie Towarzystwo Semiotyczne.

'''Hammerl, R., Sambor,''' J. (1997). Miary stopnia bliskości semantycznej leksemów. In: J.Sambor (ed.), ''Z zagadnień kwantytatywnej semantyki kogntiywnej: 217-238''. Warszawa: Polskie Towarzystwo Semiotyczne.

'''Hammerl, R., Schulz, K.P.''' (1988). Untersuchungen von Strukturen sprachlicher Begriffe am Beispiel von Abstraktionsstrukturen. ''Referat auf der Jahrestagung der Gesellschaft für Klassifikation''. Darmstadt

'''Jankiewicz, M'''. (1991). Vergleichende Analyse von Definitionsfolgen aus zwei polnischen Wörterbüchern. In: Sambor, J., Hammerl, R. (eds.) ''1991: 39-''49.

'''Januszkiewicz, N.A'''. (1997). Angielskie gniazda leksykalne. Wybrane aspekty analizy semantycznej pojęć końcowych. In: J.Sambor (ed.), ''Z zagadnień kwantytatywnej semantyki kogntiywnej: 175-213''. Warszawa: Polskie Towarzystwo Semiotyczne.

'''Kisro-Völker, S'''. (1984). On the measurement of abstractness in lexicon. ''Glottometrika 6, 138-151''.

'''Komuda, J'''. (1986). ''Budowa ciągów definicyjnych na podstawie 6-tomowego „Rečnika srpskohrvatskogo književnogo jezyka''. Warszawa: Diss.

'''Lewczuk, J'''. (1991). Homonyme und polyseme Lexeme in Lexemnetzen. In: Sambor, J., Hammerl, R. (eds.) ''1991: 129-152''.

'''Martin, R'''. (1974). Syntaxe de la définition lexicographique: étude quantitative des définissants dans le « Dictionnaire fondamental de la langue francaise. In David, J., Martin, R. (eds.), ''Statistique et linguistique'': 61-71. Paris: Klincksieck.

'''Sambor, J'''. (1983). O budowie tzw. ciągów definicyjnych (na material definicji leksykalnych) ''Biuletyn PTJ 40, 151-165''.

'''Sambor, J'''. (1991). Definitionsfolgen und Suche nach semantischen Indefinibilien. In: Sambor, J., Hammerl, R. (eds.) ''1991: 65-73''.

'''Sambor, J'''. (1997a). Ciągi definicyjne i gniazda leksykalne jako nowy temat semantyki leksykalnej. In: J.Sambor (ed.), ''Z zagadnień kwantytatywnej semantyki kognitywnej: 9-23''. Warszawa: Polskie Towarrzystwo Semiotyczne.

'''Sambor, J. (ed.)'''. (1997b).''Z zagadnień kwantytatywnej semantyki kognitywnej''. Warszawa: Polskie Towarzystwo Semiotyczne.

'''Sambor, J.'''(1997c). Ciągi definicyjne slownikowe i psychologiczne. Zarys ekxperimentu. In: J.Sambor (ed.), ''Z zagadnień kwantytatywnej semantyki kognitywnej: 27-31''. Warszawa: Polskie Towarrzystwo Semiotyczne.

'''Sambor, J.'''(1997d). Gniazda leksykalne i pojęciowe w dwóch slownikach języka polskiego. In: J.Sambor (ed.), ''Z zagadnień kwantytatywnej semantyki kognitywnej: 109-145''. Warszawa: Polskie Towarzystwo Semiotyczne.

'''Sambor, J., Hammerl, R.''' (1997). Gniazda leksykalnne i pojęciowe w słowniku języka niemieckiego (na materiale słownika ''Handwörterbuch der deutschen Gegenwartssprache). In: J.Sambor (ed.), Z zagadnień kwantytatywnej semantyki kognitywnej: 147-173''. Warszawa: Polskie Towarzystwo Semiotyczne.

'''Sambor, J., Hammerl, R. (eds.)''' (1991). ''Definitionsfolgen und Lexemnetze 1.'' Lüdenscheid: RAM.

'''Schierholz, St'''. (1989). Kritische Aspekte zum Martinschen Gesetz. ''Glottometrika 10, 108-128''.

'''Schulz-Otto, K.-P., Hammerl, R.''' (1988). Untersuchungen sprachlicher Begriffe – am Beispiel von Abtsraktheitsstrukturen. In. Wille, R. (ed.), ''Klassifikation und Ordnung. Studien zur Klassifikation 19, 221-224''. Frankfurt: Indeks Verlag.

'''Szczekocka-Augustyn, A.''' (1997). Ciagi definicyjne w swiadomosci osób mówiacych (na materiale naczyn i mebli. In: Sambor, J. (ed.), ''Z zagadnień kwantytatywnej semantyki kogniywnej:69-105''. Warszawa: Polskie Towarrzystwo Semiotyczne.

'''Wereszczyńska, B'''. (1997). Ciągi definicyjne w świadomości osób mówiących (na materiale nazw zwierząt). In: J.Sambor (ed.), ''Z zagadnień kwantytatywnej semantyki kogniywnej: 47-67''. Warszawa: Polskie Towarrzystwo Semiotyczne.

'''Zagrodska, T.''' (1997). Ciągi definicyjne w świadomości osób mówiących (na materiale nazw roślin. In: J. Sambor (ed.), ''Z zagadnień kwantytatywnej semantyki kognitywnej: 33-46''. War-szawa: Polskie Towarzystwo Semiotyczne.

Rhythmic units

2006-07-05T13:39:14Z

Altmann:

'''1. Problem and history'''

A rhythmic unit is according to Marbe (1904) the number of non-stressed syllables between two stressed ones. Some researchers consider it as a whole consisting of a stressed and the following non-stressed syllables. The problem is to ascertain whether the length of rhythmic units abides by a special distribution.
The first numerical examinations have been performed by Marbe and Roetteken (1904). Best (2001c) assigns this problem to the “length” problems using the appropriate way of modelling. Brainerd – in another context – considered it a Markov chain. Lehfeldt and Altmann (2003) derive the model from an urn approach considering the Poissonian gap filling between accentuated syllables as a pure birth process with repulsion.
Rhythmic units were used in the study of language acquisition by children (Deußig 1927/1969).

'''2. Hypothesis'''

''The distribution of rhythmic units follows a regular probability distribution''.

'''3. Derivation'''

'''3.1. Best´s approach'''

Best starts from the usual “length approach” considering the proportionality between frequency classes, i.e.

(1)<math>P_x = g(x)P_{x-1}\quad</math> .

Setting g(x) = a/(b+x) and the necessary displacement which is conventional he obtains

(2)<math>P_{x+1} = \frac{a}{b+x}p_x, \quad x = 1, 2, 3, ...</math>

whose solution yields the 1-displaced hyper-Poisson distribution

(3)<math> P_X = \frac{a^{x-1}}{b^{x-1}_1 F_1(1; b; a)}, \quad x = 1, 2, 3,...</math>

where <math>b^(x) = b(b+1)(b+2)...(b+x-1)\quad and \quad_1 F_1 (1; b; a)\quad</math>

is the confluent hypergeometric function. The hyper-Poisson is a special case of the unified theory (<math>\rightarrow</math>) when <math>a_0 = -1, a_1 = a, b_1 = b, a_2 = 0</math>.

'''Example:''' Rhythmic units in German

Best (2001c) examined 8 texts processed by Marbe and Roetteken and obtained in 6 cases a corroboration of the model. One of the results can be seen in Table 1 and Fig.1

<div align="center">[[Image:Tabelle11_RU.jpg]]</div>

<div align="center">[[Image:Grafik1_RU.jpg]]</div>
<div align="center">Fig. 1. Fitting the hyper-Poisson to the data in Table 1</div>

'''3.2. The birth process with repulsion''' (Lehfeldt 2003)

Let the gaps between accentuated syllables are considered as urns and the non-accentuated as balls. In time interval h (time is merely an auxiliary variable) either 1 or none ball is inserted in an urn or before the first and behind the last one. Let the assumptions of the Poisson process pure birth process are fulfilled. The urns are not passive but exert influence on the acception of balls, namely the more balls are in the urn, the more the urn repulses new balls: a balanced rhythm requires a restricted number of non-accentuated syllables. Let the birth rate be <math>\lambda_x = n-x</math>. Then one obtains

(4)<math>P'_0(t) = -nP_0(t)\quad</math>

<math>P'_x(t) = (n-x+1)P_{x-1}(t)-(n-x)P_x(t), \quad x = 1, 2, ..., n</math>.

Solving (4) with boundary conditions <math> P_0(0)=1, P_x(0) = 0, x=1, 2, ..., n</math> and substituting at last <math>e^{-t} = q, p = 1-q </math>we obtain the binomial distribution

(5)<math> P_x ={n \choose x}p^x q^{n-x}, \quad x=0,1,2,...,n</math>

Brainerd (1976) considered sequences of this kind as Markov chains and obtained for the distances chains of different order represented by the modified geometric distribution (see Gap formation). Since in all models of order higher than zero the first class is modified, Lehfeldt (2003) modified it a posteriori, too, and obtained the extended positive binomial distribution

(6)<math> P_X =\begin{cases} 1-\alpha & x = 0 \\ \frac{\alpha {n\choose x}p^x q^{n-x}}{1-q^n}& x=1, 2, ..., n \end{cases}</math>

Evidently, (5) and (6) are identical when α = 1-qn.
'''
Example''': Intervalls of non-accentuated syllables in Russian

Lehfeldt (2003) examined the intervals of non-accentuated syllables in Russian and obtained the results in Table 2 and Fig. 2. Here the rhythmic unit is merely the number of non-accentuated syllables.

<div align="center">[[Image:Tabelle22_RU.jpg]]</div>

The result of fitting is satisfactory. Additional pooling had brought still better results.

<div align="center">[[Image:Grafik2_RU.jpg]]</div>
<div align="center">Fig. 2. Fitting the binomial d. to the data in Table 2</div>

'''Example''': Rhythmic units in Puschkins Werk “Vystrel”

Lehfeldt (2003) fitted the extended positive binomial distribution to the rhythmic units in Puschkins work “Vystrel” and obtained the results presented in Table 3 and Fig. 3.

<div align="center">[[Image:Tabelle33_RU.jpg]]</div>

[[[Auch EPPO, Hype, Hpas, EpBin, Bin]] Wird eingetragen! Please complete

'''4. Authors: U. Strauss, G. Altmann, K.-H. Best'''

'''5. References'''

'''Best, K.-H'''. (2001a). Zur Verteilung rhythmischer Einheiten in deutscher Prosa. In: Best, K.H. (ed.), ''Häufigkeitsverteilungen in Texten: 162-166''. Göttingen: Peust & Gutschmidt.

'''Best, K.-H'''. (2001b). Probability distributions of language entities. ''J. of Quantitative Linguistics 8, 1-11''.

'''Best, K-H.''' (2002). The distribution of rhythmic units in German short prose. ''Glottometrics 3, 136-142''.

'''Best, K.-H'''. (2005). Längen rhythmischer Einheiten. In: Altmann, G., Köhler, R., Piotrowski, R. (Hg.), ''Quantitative Linguistik - Quantitative Linguistics. Ein internationales Handbuch: 208-214''. Berlin/ N.Y.: de Gruyter.

'''Best, K.-H'''. (2005). Karl Marbe (1869-1953). ''Glottometrics 9, 74-76".

'''Best, K.-H., Kotrasch, B.''' (2005). Albert Thumb (1965-1915). ''Glottometrics 9, 82-84''.

'''Brainerd, B'''. (1976). On the Markov nature of text. ''Linguistics 176, 5-30.''

'''Deußing, H.''' (1927/1969). Der sprachliche Ausdruck des Schulkindes. In: Helmers, H. (ed.), ''Zur Sprache des Kindes: 60-131''. Darmstadt: Wissenschaftliche Buchgesellschaft.

'''Gropp, F.''' (1915). ''Zur Ästhetik und Statistik des Prosarhythmus''. Würzburg, diss.phil.

'''Kaßel, A.''' (2002). ''Zur Verteilung rhythmischer Einheiten in deutschen und englischen Texten''. Staatsexamensarbeit; Göttingen.

'''Lehfeldt, W.''' (2003). ''Akzent und Betonung im Russischen''. München: Sagner.

'''Marbe, K.''' (1904). ''Über den Rhythmus der Prosa''. Giessen: J.Ricker´sche Verlagsbuchhandlung.

'''Thumb, A.''' (1913). Satzrhythmus und Satzmelodie in der altgrichischen Prosa. In: Marbe, K. unter Mitwirkung von W. Peters, ''Fortschritte der Psychologie und ihrer Anwendungen I, 3, 139-168''. Leipzig/Berlin: Teubner.

[[Category:Unfertig]]

Word classes

2006-07-05T10:53:54Z

Altmann:

'''1. Problem and history'''

Does the frequency of different word classes abide by a special distribution law? Evidently, word classes are nominal entities, thus they must be ranked (see Ranking →). Historically, word classes represent the diversification of an amorphous word stock which began to be partitioned by the development of grammar, thus this is a problem of diversification (→).

The first investigations have been performed by Hammerl (1990) who obtained the Zipf-Alekseev distribution, just as Schweers, Zhu (1991) did. Köhler (1991) studied the problem from the diversification point of view. A number of individual studies on texts in German (Best 1994, 1997, 2000, 2001a, b; Judt 1995), Russian (Bosselmann 2001), French (Judt 1995), Latin (Schweers, Zhu 1991) Chinese (Zhu, Best 1992, Schweers, Zhu 1991), Czech (Uhlířová 2000) and Portuguese (Ziegler 1998, 2001) has been performed and brought different results. The word classes were considered in their classical version, nobody tested other possible classifications. Statistical tests for word classes have been set up by Wimmer and Altmann (2001).
The hypothesis concerning word classes is merely a special case of a more general hypothesis encompassing any kind of classes of linguistic entities.

'''2. Hypothesis'''

''If language entities are ordered in classes, then their ranked frequencies follow a regular probability distribution or a regular ranking series''.

Seen from the opposite point of view, if the ranked frequencies are properly distributed we have a preliminary approximate corroboration of the “correctness” of the classification, i.e. we approximate some linguistic-psychological truth.

'''3. Derivation'''

'''3.1. The Zipf-Alekseev distribution'''

The derivation is shown in Word associations (<math>\rightarrow</math>), Chapter 3.1. Different derivations are shown in Hammerl (1990) and Hřebíček (2000: 14f). The formula used in right truncated form and with modified class x = 1 is

<math> P_x = \begin{cases} \alpha, & x=1 \\ \frac{(1-\alpha)x^{a+b \ln x}}{T} & 2,3,...,n \end{cases}</math>

where <math> T= \sum_{j=2}^N j^{a+b \ln x}</math>.

Example: Word classes in Portuguese (Ziegler 2001).

<div align="center">[[Image:Tabelle11_WCL.jpg]]</div>

<div align="center">[[Image:Grafik1_WCL.jpg]]</div>
<div align="center">Figure 1. Fitting of modified right truncated Zipf-Alekseev distribution to Portuguese data</div>

'''3.2. The negative hypergeometric distribution'''

The frequency of ranked word classes abides by the usual proportionality relation between neighboring classes. Using the proportionality function

<math> g(x)= \frac{(M+x-1)(K-M+n-x)}{x(n-x+1)}</math>

and solving with displacement

(1)<math> P_{x+1}=\frac{(M+x-1)(K-M+n-x)}{x(n-x+1)}P_x</math>

one obtains the 1-displaced negative hypergeometric distribution

(2)<math> P_x = \frac{{-M \choose x-1}{-K+M \choose n-x+1}}{{-K \choose n}}, \quad x=1,2,3,...,n,\quad K\geq M\geq 0,\quad n \epsilon N</math>

Example: Word classes in Latin (Schweers, Zhu 1991)
Schweers and Zhu examined word classes in Latin, German and Chinese and found satisfactory fits for Latin and German. For Latin, they took Caesar´s “Bellum Gallicum”, Book 1, Chapters 1-8, § 2, and obtained the results in Table 2. The majority of researchers used this distribution.

<div align="center">[[Image:Tabelle222_WCL.jpg]]</div>

<div align="center">[[Image:Grafik2_WCL.jpg]]</div>
<div align="center">Figure 2. Fitting the negative hypergeometric distribution to Latin data</div>

'''3.3. Altmann´s series'''

Altmann (1993) did not strive for a distribution but simply for a series capturing the decreasing frequencies (absolute or relative) of ranked classes. Consider the Zipf-Mandelbrot law as a (not normalized) continuous function. Its differential equation is

(2)<math> \frac{dy}{y}= -\frac{c}{a+x}dx</math> ,

i.e. a special case of the unified theory (→). Since ranking proceeds in unit steps, dx = 1 and dy = yx+1 – yx, we obtain

(3)<math> \frac{y_{x+1}-y_x}{y_x}= -\frac{c}{a+x}</math>.

Reordering, setting a-c = b, and solving (3), results in

(4)<math> y_x = \frac{{b+x \choose x-1}}{{a+x \choose x-1}}y_1, \quad x=1,2,3,...</math>

The parameters fulfill one of the conditions (i) a > b 0, (ii) a > 0, -1 < b < 0, (iii) b < a < 0 when a and b are not integers.

Example: Word class distribution in a German text (Best 1997)

Altmann used (4) only for phoneme ranking, but Best (1994, 1997) used it also for word classes with very satisfactory results. The fitting of (4) to the relative frequencies of word classes in a German text (Bichsel, P., Der Mann, der nichts mehr wissen wollte) yielded results presented in Table 3 and Fig. 3.

Here x are the ranked word classes, yx are their relative frequencies. The fit is very satisfactory.

<div align="center">[[Image:Grafik3_WCL.jpg]]</div>

'''4. Authors:''' U. Strauss, G. Altmann, K.-H. Best

'''5. References'''

'''Altmann, G.''' (1991). Word class diversification of Arabic verbal roots. In: Rothe, U. (ed.), ''Diversification Processes in Language: Grammar: 57-59''. Hagen: Rottmann.

'''Altmann, G'''. (1993). Phoneme counts. ''Glottometrika 14, 55-70''.

'''Best, K.-H.''' (1994). Word class frequencies in contemporary German short prose texts. ''J. of Quantitative Linguistics 1, 144-147''.

'''Best, K.-H'''. (1997). Zur Wortartenhäufigkeit in Texten deutscher Kurzprosa der Gegenwart. ''Glottometrika 16, 276-285''.

'''Best, K-H.''' (1998). Zur Interaktion der Wortarten in Texten. ''Papiere zur Linguistik 58: 83-95''.

'''Best, K.-H.''' (2000). Verteilung der Wortarten in Anzeigen. ''Göttinger Beiträge zur Sprachwissenschaft 4, 37-51''.

'''Best, K.-H.''' (2001a). Zur Gesetzmäßigkeit der Wortartenverteilungen in deutschen Pressetexten. ''Glottometrics 1, 1-26''.

'''Best, K.H.''' (2001b). ''Quantitative Linguistik. Eine Annäherung.'' Göttingen: Peust & Gutschmidt.

'''Bosselmann, A.''' (2001). ''Wortartenverteilungen in russischen Texten''. Msc.

'''Hammerl, R.''' (1990). Untersuchungen zur Verteilung der Wortarten im Text. ''Glottometrika 11, 142-156''.

'''Hřebíček, L.''' (2000). Variation in sequences. Prague: Oriental Institute.

'''Hudson, R.''' (1994). About 37 % of word-tokens are nouns. ''Language 70, 331-339''.

'''Judt, B.''' (1995). ''Wortartenhäufigkeiten im Deutschen und Französischen''. Göttingen: Staatsexamensarbeit.

'''Köhler, R.''' (1991). Diversification of coding methods in grammar. In: Rothe, U. (ed.), ''Diversification Processes in Language: Grammar: 47-55''. Hagen: Rottmann.

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'''Mizutani, S.''' (1989). Ohno's lexical law: its data adjustment by linear regression. In: Mizutani, S. (ed.), ''Japanese Quantitative Linguistics''. Bochum: Brockmeyer. 1-13.

'''Schweers, A., Zhu, J'''. (1991). Wortartenklassifikation im Lateinischen, Deutschen und Chinesischen. In: Rothe, U. (ed.), ''Diversification Processes in Language: Grammar: 157-167''. Hagen: Rottmann.

'''Uhlířová, L.''' (2000). On language modelling in automatic speech recognition. ''J. of Quantitative Linguistics 7, 209-216''.

'''Wimmer, G., Altmann, G.''' (2001). Some statistical investigations concerning word classes. ''Glottometrics 1, 109-123''.

'''Zhu, J, Best, K.-H.''' (1992). Zum Wort im modernen Chinesisch. ''Oriens Extremus 35, 45-60.''

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[[Category:Unfertig]]

Rhythmic units

2006-07-04T12:00:08Z

Altmann:

'''1. Problem and history'''

A rhythmic unit is according to Marbe (1904) the number of non-stressed syllables between two stressed ones. Some researchers consider it as a whole consisting of a stressed and the following non-stressed syllables. The problem is to ascertain whether the length of rhythmic units abides by a special distribution.
The first numerical examinations have been performed by Marbe and Roetteken (1904). Best (2001c) assigns this problem to the “length” problems using the appropriate way of modelling. Brainerd – in another context – considered it a Markov chain. Lehfeldt and Altmann (2003) derive the model from an urn approach considering the Poissonian gap filling between accentuated syllables as a pure birth process with repulsion.
Rhythmic units were used in the study of language acquisition by children (Deußig 1927/1969).

'''2. Hypothesis'''

''The distribution of rhythmic units follows a regular probability distribution''.

'''3. Derivation'''

'''3.1. Best´s approach'''

Best starts from the usual “length approach” considering the proportionality between frequency classes, i.e.

(1)<math>P_x = g(x)P_{x-1}\quad</math> .

Setting g(x) = a/(b+x) and the necessary displacement which is conventional he obtains

(2)<math>P_{x+1} = \frac{a}{b+x}p_x, \quad x = 1, 2, 3, ...</math>

whose solution yields the 1-displaced hyper-Poisson distribution

(3)<math> P_X = \frac{a^{x-1}}{b^{x-1}_1 F_1(1; b; a)}, \quad x = 1, 2, 3,...</math>

where <math>b^(x) = b(b+1)(b+2)...(b+x-1)\quad and \quad_1 F_1 (1; b; a)\quad</math>

is the confluent hypergeometric function. The hyper-Poisson is a special case of the unified theory (<math>\rightarrow</math>) when <math>a_0 = -1, a_1 = a, b_1 = b, a_2 = 0</math>.

'''Example:''' Rhythmic units in German

Best (2001c) examined 8 texts processed by Marbe and Roetteke and obtained in 6 cases a corroboration of the model. One of the results can be seen in Table 1 and Fig.1

<div align="center">[[Image:Tabelle11_RU.jpg]]</div>

<div align="center">[[Image:Grafik1_RU.jpg]]</div>
<div align="center">Fig. 1. Fitting the hyper-Poisson to the data in Table 1</div>

'''3.2. The birth process with repulsion''' (Lehfeldt 2003)

Let the gaps between accentuated syllables are considered as urns and the non-accentuated as balls. In time interval h (time is merely an auxiliary variable) either 1 or none ball is inserted in an urn or before the first and behind the last one. Let the assumptions of the Poisson process pure birth process are fulfilled. The urns are not passive but exert influence on the acception of balls, namely the more balls are in the urn, the more the urn repulses new balls: a balanced rhythm requires a restricted number of non-accentuated syllables. Let the birth rate be <math>\lambda_x = n-x</math>. Then one obtains

(4)<math>P'_0(t) = -nP_0(t)\quad</math>

<math>P'_x(t) = (n-x+1)P_{x-1}(t)-(n-x)P_x(t), \quad x = 1, 2, ..., n</math>.

Solving (4) with boundary conditions <math> P_0(0)=1, P_x(0) = 0, x=1, 2, ..., n</math> and substituting at last <math>e^{-t} = q, p = 1-q </math>we obtain the binomial distribution

(5)<math> P_x ={n \choose x}p^x q^{n-x}, \quad x=0,1,2,...,n</math>

Brainerd (1976) considered sequences of this kind as Markov chains and obtained for the distances chains of different order represented by the modified geometric distribution (see Gap formation). Since in all models of order higher than zero the first class is modified, Lehfeldt and Altmann (2003) modified it a posteriori, too, and obtained the extended positive binomial distribution

(6)<math> P_X =\begin{cases} 1-\alpha & x = 0 \\ \frac{\alpha {n\choose x}p^x q^{n-x}}{1-q^n}& x=1, 2, ..., n \end{cases}</math>

Evidently, (5) and (6) are identical when α = 1-qn.
'''
Example''': Intervalls of non-accentuated syllables in Russian

Lehfeldt and Altmann (2003) examined the intervals of non-accentuated syllables in Russian and obtained the results in Table 2 and Fig. 2. Here the rhythmic unit is merely the number of non-accentuated syllables.

<div align="center">[[Image:Tabelle22_RU.jpg]]</div>

The result of fitting is satisfactory. Additional pooling had brought still better results.

<div align="center">[[Image:Grafik2_RU.jpg]]</div>
<div align="center">Fig. 2. Fitting the binomial d. to the data in Table 2</div>

'''Example''': Rhythmic units in Puschkins Werk “Vystrel”

Lehfeldt and Altmann (2003) fitted the extended positive binomial distribution to the rhythmic units in Puschkins work “Vystrel” and obtained the results presented in Table 3 and Fig. 3.

<div align="center">[[Image:Tabelle33_RU.jpg]]</div>

[[[Auch EPPO, Hype, Hpas, EpBin, Bin]] Wird eingetragen! Please complete

'''4. Authors: U. Strauss, G. Altmann, K.-H. Best'''

'''5. References'''

'''Best, K.-H'''. (2001a). Zur Verteilung rhythmischer Einheiten in deutscher Prosa. In: Best, K.H. (ed.), ''Häufigkeitsverteilungen in Texten: 162-166''. Göttingen: Peust & Gutschmidt.

'''Best, K.-H'''. (2001b). Probability distributions of language entities. ''J. of Quantitative Linguistics 8, 1-11''.

'''Best, K-H.''' (2002). The distribution of rhythmic units in German short prose. ''Glottometrics 3, 136-142''.

'''Best, K.-H'''. (2005). Längen rhythmischer Einheiten. In: Altmann, G., Köhler, R., Piotrowski, R. (Hg.), ''Quantitative Linguistik - Quantitative Linguistics. Ein internationales Handbuch: 208-214''. Berlin/ N.Y.: de Gruyter.

'''Best, K.-H'''. (2005). Karl Marbe (1869-1953). ''Glottometrics 9, 74-76".

'''Best, K.-H., Kotrasch, B.''' (2005). Albert Thumb (1965-1915). ''Glottometrics 9, 82-84''.

'''Brainerd, B'''. (1976). On the Markov nature of text. ''Linguistics 176, 5-30.''

'''Deußing, H.''' (1927/1969). Der sprachliche Ausdruck des Schulkindes. In: Helmers, H. (ed.), ''Zur Sprache des Kindes: 60-131''. Darmstadt: Wissenschaftliche Buchgesellschaft.

'''Gropp, F.''' (1915). ''Zur Ästhetik und Statistik des Prosarhythmus''. Würzburg, diss.phil.

'''Kaßel, A.''' (2002). ''Zur Verteilung rhythmischer Einheiten in deutschen und englischen Texten''. Staatsexamensarbeit; Göttingen.

'''Lehfeldt, W.''' (2003). ''Akzent und Betonung im Russischen''. München: Sagner.

'''Marbe, K.''' (1904). ''Über den Rhythmus der Prosa''. Giessen: J.Ricker´sche Verlagsbuchhandlung.

'''Thumb, A.''' (1913). Satzrhythmus und Satzmelodie in der altgrichischen Prosa. In: Marbe, K. unter Mitwirkung von W. Peters, ''Fortschritte der Psychologie und ihrer Anwendungen I, 3, 139-168''. Leipzig/Berlin: Teubner.

[[Category:Unfertig]]