'''1. Problem and history'''

The problem is to find a function or a distribution for the phoneme frequencies 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 1846, 1852; Meyer 1869; Bourdon 1892) and developed quickly on practical grounds: stenographers, printers, constructors of typewriters, decoders, etc. needed urgently the frequency of letters for their own purposes. Förstemann and Meyer pursued comparative aims, e.g. the problem of the relation between consonants and vowels in the examined languages (Old Indian, Greek, Latin and Gothic) and its impact for the development of languages.

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, namely the the geometric (and the right truncated geometric) distribution, was proposed by Sigurd (1968). Good (1969) brought a partial-sums distribution (Whitworth distribution) whose modelling was revived in word length (<math>\rightarrow</math>) research. Tuldava (1971/1995) considered different possibilities, 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 (1998, 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. The modelling has been performed in two ways: (i) a continuous curve has been fitted to the proportions of phonemes, (ii) a discrete distribution has been fitted. It can be shown that continuous curves have their analogues in discrete distributions.

'''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 approximated 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>

telling that the change of frequency (y) is inversely proportional to the rank (x) and yielding

(2)<math>y = a + b \ln x\quad</math> ,

where b is negative. This curve is frequently used in other domains, too (cf. also Martindale et al. 1996; Laherrère, Sornette 1998).

'''3.2. Derivations related to the unified theory (→) are'''

'''(a) Zipf´s law''' (zeta function)

When formula (2) of the unified theory (<math>\rightarrow</math>) is used with <math>a_0 = a_2 = a_3 = ... = 0, a_1 = -b,</math> this yields

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

telling that the relative rate of change of frequency is proportional to the relative rate of change of rank, resulting in

(4)<math>y = Ax^{-b}\quad</math>.

This is, perhaps, the most disseminated formula in linguistics representing the power law.

'''(b) Yule´s species/genera function (1924)'''

When formula (2) of the unified theory is used with <math>a_0 = c_', a_1 = b, a_2 = a_3 = ... = =,</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 x^{-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,</math>, this yields

(7)<math> \frac{dy}{y} = \left( -\frac{a_1}{x} + \frac{a_2}{x^2} \right)dx</math>

resulting in

(8)<math> y= Ce^{-a_2/x}x^{-a_1}</math>,

derived by the authors in a different way.

'''(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, a_1 - b_1 = b</math>, this results in

(11)<math> y_x = \frac{\begin{pmatrix} b + x \\ x - 1 \end{pmatrix}}{\begin{pmatrix} a + x \\ x - 1 \end{pmatrix}}y_1\quad</math>, x = 1,2,3,...

This proved to be a very good model for letter distribution in English and German (Best 2005).

All these formulas can be transformed in distributions by appropriate normalizing. Several distributions have been derived directly, namely

'''(e) Geometric distribution'''

Sigurd (1968) used simply the 1-displaced geometric distribution. It can be obtained from formula (9) setting <math>a_i = 0 (i = 1,2,3,...)</math>, which yields

(12)<math>y_{x+1}= (1+a_0)y_x\quad</math>.

For <math>-1 < a_0 < 0, 1+a_0 = q, 1-q = p, y_x = P_x</math> one obtains 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>

'''(f) Negative hypergeometric distribution'''

A systematic analysis of Slavic languages and German (Grzybek & Kelih 2003, 2003a,b, 2005, 2006b; Grzybek, Kelih, & Altmann 2004, 2006a,b; Best 2005a,b) showed that the most stable distribution for letter frequencies follows from the unified theory by setting <math>a_1 = (K+n-1)(-K+M+1)(-K+M-n), a_2 = (M-1)(K-M+n), a_0 = b_2 = 0, b_1 = -K+M-n,</math>yielding

(15)<math> P_x = \frac{(M+x-1)(K-M+n-x)}{x(n-x+1)}P_{x-1}</math>

from which

(16)<math> P_x = \frac{\begin{pmatrix} M+x-1 \\ x \end{pmatrix}\begin{pmatrix} K-M+n-x-1 \\ n-x \end{pmatrix}}{\begin{pmatrix} K+n-1 \\ n \end{pmatrix}} = \frac{\begin{pmatrix} -M \\ x \end{pmatrix}\begin{pmatrix} -K+M \\ n-x \end{pmatrix}}{\begin{pmatrix} -K \\ n \end{pmatrix}}\quad x= 0,1,...,n</math>

which is usually displaced by 1 step to the right.

'''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 (à). The Good distribution has the form

(17)<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:Tabelle11_PF.jpg]]</div>

Except for the geometric series, all of them yield in this case a good – approximately equal – fitting. In Fig. 1, only fitting of (11) is shown.

<div align="center">[[Image:Grafik11_PF.jpg]]</div>
<div align="center">Fig. 1. Fitting function (11) to Hawaiian phoneme frequencies</div>

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

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File:Tabelle111 Div.jpg

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Ahans:

Diversification

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Ahans:

'''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:Tabelle111_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

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'''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.''

'''Altmann, G.''' (1996). Diversification processes of the word. ''Glottometrika 15, 102-111.''

'''Altmann, G.''' (2005). Diversification processes. In: Köhler, R., Altmann, G., Piotrowski, R.G. (eds.), ''Handbook of Quantitative Linguistics: 646-658''. Berlin: de Gryuter.

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

'''Becker, H.''' (1995). ''Die Wirtschaft in der deutschsprachigen Presse''. Frankfurt: Lang.

'''Beöthy, E., Altmann, G.''' (1984a). The diversification of meaning of Hungarian verbal prefixes. II. ki-. ''Finnisch-Ugrische Mitteilungen 8, 29-37.''

'''Beöthy, E., Altmann, G.''' (1984b). Semantic diversification of Hungarian verbal prefixes. III.föl-, el-, be-. ''Glottometrika 7, 73-100''.

'''Beöthy, E., Altmann, G.''' (1991). The diversification of meaning of Hungarian verbal prefixes. I.''meg-.'' In: Rothe, U. (ed) ''1991: 60-66''.

'''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''.

'''Best, K.-H.''' (1997). Zur Wortartenhäufigkeit in Texten deutscher Kurzprosa. ''Glottometrika 16, 276-285''.

'''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,.

'''Köhler, R.''' (1991). ''Diversification of coding methods in grammar''. In: Rothe, U. (ed.), Diversification processes in language: Grammar: 47-55. Hagen: Rottman.

'''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''.

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

'''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.

'''Pawlowski, A.''' (1999). The quantitative approach in cultural anthropology: Application of linguistic corpora in the analysis of basic colour terms. ''J. of Quantitative Linguistics 6, 222 234''.

'''Raether, A., Rothe, U.''' (1991). Diversifikation der deutschen Komposita. In: Rothe, U. (ed.) ''1991: 85-91''.

'''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''.

'''Rothe, U.''' (1990a). Semantische Motivation der Genuszuweisung. ''Glottometrika 11, 95-106''.

'''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.

'''Rothe, U.''' (1991c). Distribution of spelling errors by Japanese English-users. In: Rothe, U. (ed.), ''Diversification processes in language: grammar'': 168-171. Hagen: Rottmann.

'''Saukkonen, P., Haipus, M., Niemikorpi, A., Sulkala, H.''' (1979). ''Suomen kielen taajuussa nasto. A frequency dictionary of Finnish''. Porvoo-Helsinki: Juva.

'''Schweers, A., Zhu, J.''' (1991). Wortartenklassifikation im Lateinischen, Deutschen und Chinesischen. In: Rothe U. 1991: 157-167.

'''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.''

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

'''Wimmer, G., Altmann, G.''' (1999). ''Thesaurus of univariate discrete probability distributions''. Essen: Stamm.

'''Ziegler, A.''' (1998b). 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, con-structs. Festschrift in honour of Ludek Hřebíček: 295-312.'' Trier: WVT

'''Ziegler, A., Best, K.-H., Altmann, G.''' (2001). A contribution to text spectra. ''Glottometrics 1, 97-108''.

'''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>

File:Grafik11 PF.jpg

2006-07-18T15:46:07Z

Ahans:

2006-07-11T13:42:59Z

Ahans:

Length of syntactic constructions

2006-07-11T13:36:35Z

Ahans:

'''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''': G. Altmann

'''5. References'''

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

Length of syntactic constructions

2006-07-11T13:36:17Z

2006-07-11T12:54:07Z

Ahans:

Vowel duration

2006-07-11T12:53:49Z

Ahans:

Text-blocks

2006-07-11T12:48:22Z

Ahans:

'''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''': 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.)

2006-07-11T12:43:54Z

Ahans:

2006-07-11T12:10:33Z

Ahans:

File:Tabelle11 Div.jpg

2006-07-11T12:10:21Z

Ahans:

Gap formation

2006-07-11T09:57:23Z

Ahans:

'''1. Problem and history'''

The ''distance'' (gap) between two identical entities in text can be measured in two ways: (i) in terms of the number of other intervening entities and (ii) as the number of steps from the previous occurrence of the entity to the next one. In the sequence

1 0 0 0 1

method (i) results in a gap of length 3, method (ii) yields 4. However, some entities cannot occur in direct neighbourhood, e.g. the same preposition.

The ''entities'' can be of any kind: word classes, lengths, structural types, clause types, phonemes, individual words occurring x-times, types of verse, etc.

The investigation was initiated by G.K. Zipf, who found different aspects of distances, or intervals, or gaps, between identical entities in text (Zipf 1935, 1937a,b, 1945, 1946, 1949). The first models were set up by Spang-Hanssen (1956), Yngve (1956) and Uhlířová (1967). Herdan (1966: 127-130) and Králík (1977) considered the gap as the time between two consecutive Poisson events and obtained the exponential distribution. Brainerd (1976) considered the sequence of entities as a two-state Markov chain and derived models of different order. Strauß, Sappok, Diller, and Altmann (1984) considered identical entities as an urn and derived the negative binomial distribution using the Poisson pure birth model. Zörnig (1984a,b) derived the model for the random distribution of distances. Hrebicek (2000), leaning against his general text theory, found that even distances abide by Menzerath´s law.

'''2. Hypothesis'''

According to a generalized Skinner hypothesis ''the probability of a small distance (gap) between identical entities in text is greater than the probability of greater distances''. The hypothesis is based on the reinforcement of a stimulus which dies away.

Corollary: If Skinner´s hypothesis does not hold, then the gaps are distributed randomly and follow the Zörnig model (see below).

'''3. Derivations'''

'''3.1. The geometric model'''

Spang-Hanssen (1956), Yngve (1956) and Uhlířová (1967) assumed that if the probability of an entity A is p and that of non-A 1 - p = q, then the probability of a distance of size x is given simply by the geometric distribution

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

which is adequate in many cases.

'''3.2. Markov chain model'''

Since the geometric distribution represents merely a Markov chain of zeroth order, not taking sequential dependencies into account, Brainerd (1976) considered higher chain orders and operated with transitions between elements A (= 1) and non-A (= 0). For the first order chain, he obtained the probability of no distance (x = 0) from the transition 11 as P(1|1). For all other distances, we consider 100…01, which means that there is a transition from 1 to 0 in the first step, then x-1 transitions between zeroes, P(0|0), and finally the transition from 0 to 1, yielding <math>P(0|1)P(1|0)P(0|0)^{x-1}</math>. In a similar way one can obtain dependencies of higher order. For the first three orders he obtained the following distributions:

(2) <math>P_x = \begin{cases} P(1|1), & \quad x=0
\\ P(0|1)P(1|0)P(0|0)^{x-1},& \quad x=1,2,...\end{cases}</math>

(3)<math>P_x= \begin{cases} P(1|1), & \quad x=0\\P(0|1)P(1|0),& \quad x=1\\P(0|1)P(0|10)P(1|00)P(0|00)^{x-2}, & \quad x=2,3,...\end{cases}</math>

(4)<math>P_x= \begin{cases} P(1|1), & \quad x=0\\P(01|1),& \quad x=1\\P(00|1)P(1|100), & \quad x=2\\P(00|1)P(0|100)P(1|000)P(0|000)^{x-3}, & \quad x=3,4,...\end{cases}</math>

One sees that the higher the order of the chain, the more extensive is the modification of the simple geometric distribution. For example (4) can be simply written as

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

The parameters could express some properties of the given entity but there are no further exmanations in this direction.
There are two problems with this approach: (a) Markov chains do not consider forward dependencies which are usual in text, (b) stepwise modification would capture any empirical distribution but at costs of explanatory power. A simple description of these chains can be found in Altmann (1988a) and a survey of modified distribution in Wimmer, Witkovský, Altmann (1999).

'''3.3. Urn model'''

The derivations in 3.1 and 3.2 do not take Skinner´s hypothesis into account, they are rather of local character. Strauß, Sappok, Diller and Altmann (1984) consider two occurrences of element A as an urn which exerts influence on acceptance or rejection of new non-A elements.
Let the placement of non-A elements between two A elements be a Poisson pure birth process (see Appendix) in which new non-A elements can only be inserted but not taken away, yielding

(5)<math>P^'_0(t)=-\lambda_0P_0(t)</math>

<math>P^'_x(t)=-\lambda_xP_x(t)+\lambda_{x-1}P_{x-1}(t), \quad x=1,2,3,...</math>

If there is no trend, i.e. the “balls” fall in the urns randomly, then <math>\lambda_x= a</math> (a constant) and the process results in the Poisson distribution. If however, the urns exert influence, the result may be different. If an urn repells new balls the more, the more balls are already in it, then one can write <math>\lambda_x= n-x</math>, insert it in (5), and obtain the binomial distribution.

However, Skinner´s hypothesis says that there is a tendency to produce more small distances and enlarge the long ones. This means that an urn attracts the more new balls the more are already in it. Substituting in <math>\lambda_x = k+x</math> in (5), one obtains

(6)<math>P^'_0(t)=-kP_0(t),</math>

<math>P^'_x(t)=-(k+x)P_x(t)+(k-x-1)P_{x-1}(t), \quad x=1,2,3...</math>

Solving (6) in the usual way and setting <math>e^{-t} = q</math>, one obtains the negative binomial distribution

(7)<math>P_x = \begin{pmatrix}k+x-1\\x\end{pmatrix}p^kq^x, \quad x=0,1,2,...; \quad k>0; \quad 0<p<1; \quad q=1-p</math>

Example: Distances between equal rhythmic structures in hexameter

Strauß et al. (1984) examined the occurrence of verses with the structure DSSS (D – dactylus, S – spondeus) in 300 lines of Bridges´ “Poems in Classical prosody. Epistle II: To a Socialist in London” and recorded the distances between them. They obtained the results in the first and the second columns of Table 1.
The geometric d., the negative binomial d. and the Markov chain of first order were fitted to these data.

<div align="center">[[Image:Tabelle111_GF.jpg]]</div>

As it can be seen, the first order Markov chain yields the best fit for this type of data.

'''3.4. The Menzerathian model'''

Starting from a different philosophy of texts, Hřebíček (2000) assumes that not only hierarchical relations but also sequential ones abide by the simplest form of Menzerath´s law (for derivation see Hierarchic relations) yielding

(8)<math>y_x = ax^{-b}\quad </math>

where <math>y_x</math> is the frequency of the distance x between identical units. It can be considered either as a usual function (without norming), or as a probability function representing the zeta distribution (a being the norming constant and b > 1). Testing with good results has been performed for words of high frequency in Czech and Turkish (Hřebíček 2000). Hřebíček used method (ii) for measuring distances and still another method consisting of counting the intervening sentences.

Example: Distances between the personal name “Nihat” in a Turkish text

Hřebíček (2000: 32-34) pooled the distances in intervals and obtained the results presented in Table 2.

<div align="Center">[[Image:Tabelle2_GF.jpg ]]</div>

The fitting is satisfactory.

'''3.5. Zörnig´s model of random distribution of distances between any number of identical entities (Zörnig 1984a,b).'''

In the above models the entities of the text were always divided dichotomically to elements A and non-A. One can also add all distances of the same size or examine the distances for each element separately.
If the distances between identical entities are random, then they follow the distribution

(9)<math>P_x = \frac {(n-x-1)!}{n!(n-m)}\sum_{i=1}^m k_i(k_i-1)(n-k_i)_{(x)}</math>,

or, if we are interested in fequencies, we have, with N = n-m,

(10)<math>NP_x = \frac{(n-x-1)!}{n!}\sum_{i=1}^m k_i (k_i-1)(n-k_i)_{(x)}</math>

where

n = number of elements in the sequence

m = number of different element types

<math>k_i</math> = frequency of occurrence of elements of type i (i = 1,2,...,m)

<math>r_{(x)} = r(r-1)(r-2)...(r-x+1)</math>

'''Example''': Distances in an artificial case

Let us consider the following sequence:

A B A C D B C A D D B

Here

''n = 11''

''m = 4 (A,B,C,D)''

<math>k_1 = k_A = 3</math>

<math>k_2 = k_B = 3</math>

<math>k_3 = k_C = 2</math>

<math>k_4 = k_D = 3</math>

We find here following distances:

Between the A´s 1 and 4

Between the B´s 3 and 4

Between the C´s 2

Between the D´s 3 and 0

Using (2) we compute the theoretical frequency of distance 2:

<math>NP_2=\frac{(11-2-1)!}{11!}(3(3-1)(11-3)(11-2)+3(3-1)(11-3)(11-2)+2(2-1)(11-2)(11-1)+3(3-1)(11-3)(11-2))=1.3553.</math>

In the same way one can compute the other distances and compare them with the real ones.

'''4. Author''': U. Strauss, G. Altmann, L. Hřebíček

'''5. References'''

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

'''Brainerd, B'''. (1976). On the Markov nature of text. ''Linguistics 176, 5-30.''

'''Chen, Y.-S.''' (1988). An exponential recurrence distribution in the Simon-Yule model of text. ''Cybernetics and Systems: An International Journal 19, 521-545.''

'''Chen, Y.-S., Chong, P.P., Kim, J.-S.''' (1992). A self-adaptive statistical language model for speech recognition. Cybernetica 35(2), 103-127.

'''Herdan, G.''' (1966). ''The advanced theory of language as choice and chance.'' Berlin, Springer (p. 127-130).

'''Hřebíček, L.''' (2000). ''Variation in sequences''. Prague: Oriental Institute

'''Králík, J.''' (1977). An application of exponential distribution law in quantitative linguistics. ''Prague Studies in Mathematical Linguistics 5, 223-235.''

'''Prün, C.''' (1997). A text linguistic hypothesis of G.K. Zipf. ''J. of Quantitative Linguistics 4, 244-251.''

'''Spang-Hanssen, H.''' (1956). The study of gaps between repetitions. In: Halle, M. (Ed.), ''For Roman Jakobson: 497-502''. The Hague: Mouton.

'''Strauß, U., Sappok, Ch., Diller, H.J., Altmann, G.''' (1984). Zur Theorie der Klumpung von Textentitäten. ''Glottometrika 7, 73-100''.

'''Uhlířová, L.''' (1967). Statistics of word order of direct object in Czech. ''Prague Studies in Mathematical Linguistics 2, 37-49''.

'''Wimmer, G., Witkovský, V., Altmann, G.''' (1999). Modification of probability distributions applied to word length research.'' J. of Quantitative Linguistics 6, 257-268.''

'''Yngve, V. (1956).''' Gap analysis and syntax. ''IRE Transactions PGIT-2, 106-112.''

'''Zipf, G.K.''' (1935). ''The psycho-biology of language: an introduction to dynamic phlology.'' Boston: Houghton Mifflin.

'''Zipf, G.K.''' (1937a). Observations on the possible effect of mental age upon the frequency-distribution of words from the viewpoint of dynamic philology. ''Journal of Psychology 4, 239-244.''

'''Zipf, G.K.''' (1937b). Statistical methods in dynamic philology (Reply to M. Joos). Language 132, 60-70.

'''Zipf, G.K.''' (1945). The repetition of words, time-perspective and semantic balance. ''The J. of General Psychology 32, 127-148.''

'''Zipf, G.K.''' (1946). The psychology of language. In: Hariman, P.L. (ed.), Encyclopedia of Psychology: 332-341. New York: Philosophical Library.

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

'''Zörnig, P'''. (1984a). The distribution of the distance between like elements in a sequence I. ''Glottometrika 6, 1-15.''

'''Zörnig, P'''. (1984b). The distribution of the distance between like elements in a sequence II. ''Glottometrika 7, 1-14.''

'''Zörnig, P'''. (1987). A theory of distances between like elements in a sequence. ''Glottometrika 8, 1-22.''