Laws in Quantitative Linguistics - User contributions [en]

Vowel duration

2011-04-15T17:44:33Z

KHBest:

'''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, K.-H. Best

'''5. References'''

'''Best, Karl-Heinz''' (2009). Gesetzmäßigkeiten der Lautdauer. ''Glottotheory'' 1, 1-9.

'''Best, Karl-Heinz''' (2011). Zur Gesetzmäßigkeit der Vokalquantität im Deutschen. ''Naukovyj Visnyk Cernivec’koho Universytetu: Hermans’ka filolohija''. Vypusk 532, 3-13.

'''Geršić, S., Altmann, G.''' (1988). Ein Modell für die Variabilität der Vokaldauer. ''Glottometrika 9, 49-58''.

[[Category:Unfertig]]

User:KHBest

2010-02-21T10:52:16Z

KHBest: aktualisiert

Dr. '''[[Karl-Heinz Best]]''', [[Akademischer Oberrat]], [[Seminar]] für deutsche [[Philologie]], [[Georg-August-Universität Göttingen]], Käte-Hamburger-Weg 3, 37073 Göttingen. Mitglied der Abteilung Sprachwissenschaft bis 2008; Spezialgebiet: [[Quantitative Linguistik]] mit den Schwerpunkten Sprachwandel, Entlehnungen ([[Fremdwort|Fremdwörter]]/[[Lehnwort|Lehnwörter]]), Spracherwerb sowie Gesetzmäßigkeiten in Sprachstruktur und Sprachverwendung; [[Morphologie (Sprache)|Morphologie]].

Langfristige Aktivitäten: Mitarbeit bei den Göttingen und Halleschen Tagungen zum Thema ''Wissenstransfer (Transferwissenschaften)'', Mitherausgeber der Zeitschrift ''Glottometrics'', Mitglied im ''Editorial Board'' der Zeitschrift ''Göttinger Beiträge zur Sprachwissenschaft'', Leiter des Göttinger ''Projekt Quantitative Linguistik''. Schwerpunkt der Arbeit im Projekt ist die Überprüfung und Weiterentwicklung von Hypothesen zu Gesetzen der Sprachstruktur, -entwicklung und -verwendung sowie des Spracherwerbs. Sozusagen als „Nebenprodukt“ fallen in vielen Fällen neue Statistiken zu den behandelten Themen an. Einen weiteren Schwerpunkt bilden seit mehreren Jahren eine Reihe von Untersuchungen zur Bedeutung verschiedener Wissenschaftler für die Entwicklung der ''Quantitativen Linguistik''.

Im Ruhestand; Lehrtätigkeit im Sommersemester 2009 beendet. Weiterhin Forschungsarbeit und Prüfungen.

; Veröffentlichungen:
* Karl-Heinz Best: ''Probleme der Analogieforschung''. Hueber, München 1973. (Diss. phil., Bochum 1971.)
* Karl-Heinz Best & Jörg Kohlhase (Hrsg.): ''Exakte Sprachwandelforschung''. Edition Herodot, Göttingen 1983. ISBN 3-88694-024-1.
* Karl-Heinz Best (Hrsg.): ''Glottometrika 16. The Distribution of Word and Sentence Length''. Wissenschaftlicher Verlag Trier, Trier 1997. ISBN 3-88476-276-1.
* Karl-Heinz Best (Hrsg.): ''Häufigkeitsverteilungen in Texten''. Peust & Gutschmidt, Göttingen 2001. ISBN 3-933043-08-5.
* Karl-Heinz Best: ''Quantitative Linguistik: Eine Annäherung.'' 3., stark überarbeitete und ergänzte Aufl. Göttingen: Peust & Gutschmidt 2006. (1. Aufl. – 2001, 2. Aufl. – 2003) ISBN 3-933043-17-4.
* Karl-Heinz Best: ''LinK – Linguistik in Kürze mit einem Ausblick auf die Quantitative Linguistik. Skript.'' 5., durchgesehene Ausgabe. RAM-Verlag, Lüdenscheid 2008. (1. Auflage – 2002, 2. – 2003, 3. – 2005, 4. – 2007) Adresse: [http://www.ram-verlag.de www.ram-verlag.de – „Web-Journals“].

Außerdem bisher: seit 1971 über 100 Buchanzeigen in der Zeitschrift ''Germanistik'', einige Rezensionen und anderes, über 160 Aufsätze, darunter Mitarbeit an

* Reinhard Köhler, Gabriel Altmann, Rajmund G. Piotrowski (Hrsg.): ''Quantitative Linguistik - Quantitative Linguistics. Ein internationales Handbuch''. de Gruyter, Berlin/ New York 2005, ISBN 3-11-015578-8 (siehe eigene Beiträge, Namensregister).

; ad personam:
* Wilfried Kürschner (Hrsg.): ''Linguisten-Handbuch. Band 1: A - L. Biographische und bibliographische Daten deutschsprachiger Sprachwissenschaftlerinnen und Sprachwissenschaftler der Gegenwart.'' Narr, Tübingen 1994, Seite 69. ISBN 3-8233-5000-5.

== Weblinks ==
{{PND|108146065}}
* [http://wwwuser.gwdg.de/~kbest Homepage von Karl-Heinz Best]
* [http://wwwuser.gwdg.de/~kbest/publist.htm Verzeichnis der Publikationen und Vorträge]

Übernommen von: http://de.wikipedia.org/wiki/Benutzer:Dr._Karl-Heinz_Best

Vowel duration

2010-02-21T10:34:09Z

KHBest: lit erg

'''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, K.-H. Best

'''5. References'''

'''Best, Karl-Heinz''' (2009). Gesetzmäßigkeiten der Lautdauer. ''Glottotheory 1, 1-9.

'''Geršić, S., Altmann, G.''' (1988). Ein Modell für die Variabilität der Vokaldauer. ''Glottometrika 9, 49-58''.

[[Category:Unfertig]]

Diversification

2009-03-05T17:33:56Z

KHBest: lit erg

'''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

'''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.

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'''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''.

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'''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''

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'''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

'''Best, K.-H.''' (2008). Zur Diversifikation lateinischer und griechischer Hexameter. ''Glottometrics 17: 43-50.

'''Best, K.-H.''' (2009). Diversifikation des Phonems /r/ im Deutschen. ''Glottometrics 18: 26-31.

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'''Hammerl, R., Sambor, J.''' (1993a). ''O statystycznych prawach jezykowych. Warszawa'': Polskie Towarzystwo Semiotyczne.

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'''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.

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'''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.

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</div>

Rhythmic units

2008-06-02T09:20:01Z

KHBest: ergänzt

'''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:Tabelle3333_RU.jpg]]</div>

<div align="center">[[Image:Grafik33_RU.jpg]]</div>

'''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.''' (2006). Rhythmische Einheiten im Altgriechischen. ''Göttinger Beiträge zur Sprachwissenschaft 13, 73-76.''

'''Best, K.-H.''' (2007). Lorenzo Bianchi (1889-1960). ''Glottometrics 14, 72-74.''

'''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.

'''Eom, J.''' (2006). ''Rhythmus im Akzent. Zur Modellierung der Akzentverteilung als einer Grundlage des Sprachrhythmus im Russischen.'' München: Sagner. (Diss.phil., Göttingen, 2006)

'''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.

'''Knaus, M.''' (2008). Zur Verteilung rhythmischer Einheiten in russischer Prosa. ''Glottometrics 16, 57-62.''

'''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]]

Change in language

2008-06-02T09:12:31Z

KHBest: ergänzt

'''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 concerned. 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 Beöthy, Altmann (1982), Altmann, von Buttlar, Rott, Strauß (1983) combining Piotrowski´s findings 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-Hasemann (1983), Best, Altmann (1986), Kroch (1989a,b, 2001), Best, Beöthy, Altmann (1990), Tuldava (1998), Best (2001), Bresnan, Dingare, Manning (2001), Vulanović (2003), Best (2006). 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 Lehnwortschatz. ''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? ''Naukovyj Visnyk Černivec´koho Universytetu 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.''' (2006a). Zum Computerwortschatz im Deutschen. ''Naukovyj Visnyk Cernivec’koho Universytetu: Hermans’ka filolohija. Vypusk 289, 10-24.''

'''Best, K.-H.'''(2006b). Italianismen im Deutschen. ''Göttinger Beiträge zur Sprachwissenschaft 13, 77-86.''

'''Best, K.-H.'''(2006c). ''Quantitative Linguistik - Eine Annäherung.'' 3., stark überarbeitete und ergänzte Auflage. Göttingen: Peust & Gutschmidt.

'''Best, K.-H.'''(2006d). Quantitative Untersuchungen zu den Jiddismen im Deutschen. ''Jiddistik-Mitteilungen 36, 1-14.''

'''Best, K.-H.'''(2006e). Wortlängen im Deutschen. ''Göttinger Beiträge zur Sprachwissenschaft 13, 23-49.''

'''Best, K.-H.''' (2007a). Zur Entwicklung des Wortschatzes der Elektrotechnik, Informationstechnik und Elektrophysik im Deutschen. ''Glottometrics 15, 24-27.''

'''Best, K.-H.''' (2007b). 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, S. 45-62.

'''Best, K.-H., Altmann, G.''' (1986). Untersuchungen zur Gesetzmäßigkeit von Entlehnungsprozessen 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.

Text-blocks

2007-04-19T19:27:06Z

KHBest: aktualis.

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

'''Best, K.-H.''' (2006). Quantitative Untersuchungen zum Niederdeutschen und Niederländischen. ''Göttinger Beiträge zur Sprachwissenschaft 13, 51-71.''

'''Best, K.-H.''' (2006). ''Quantitative Linguistik. Eine Annäherung''. 3., stark überarbeitete und ergänzte Auflage. Göttingen: Peust & Gutschmidt.

'''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.)

Change in language

2007-04-19T19:23:31Z

KHBest:

'''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 concerned. 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 Beöthy, Altmann (1982), Altmann, von Buttlar, Rott, Strauß (1983) combining Piotrowski´s findings 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-Hasemann (1983), Best, Altmann (1986), Kroch (1989a,b, 2001), Best, Beöthy, Altmann (1990), Tuldava (1998), Best (2001), Bresnan, Dingare, Manning (2001), Vulanović (2003), Best (2006). 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 Lehnwortschatz. ''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? ''Naukovyj Visnyk Černivec´koho Universytetu 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.'''(2006). Wortlängen im Deutschen. ''Göttinger Beiträge zur Sprachwissenschaft 13, 23-49.''

'''Best, K.-H.'''(2006). Quantitative Untersuchungen zu den Jiddismen im Deutschen. ''Jiddistik-Mitteilungen 36, 1-14.''

'''Best, K.-H.'''(2006). ''Quantitative Linguistik - Eine Annäherung.'' 3., stark überarbeitete und ergänzte Auflage. Göttingen: Peust & Gutschmidt.

'''Best, K.-H.'''(2006). Italianismen im Deutschen. ''Göttinger Beiträge zur Sprachwissenschaft 13, 77-86.''

'''Best, K.-H.''' (?). 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 Entlehnungsprozessen 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.

Rhythmic units

2007-04-19T19:20:19Z

KHBest:

'''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:Tabelle3333_RU.jpg]]</div>

<div align="center">[[Image:Grafik33_RU.jpg]]</div>

'''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.''' (2006). Rhythmische Einheiten im Altgriechischen. ''Göttinger Beiträge zur Sprachwissenschaft 13, 73-76.''

'''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]]

Change in language

2007-04-19T19:16:34Z

KHBest: erg

'''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 concerned. 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 Beöthy, Altmann (1982), Altmann, von Buttlar, Rott, Strauß (1983) combining Piotrowski´s findings 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-Hasemann (1983), Best, Altmann (1986), Kroch (1989a,b, 2001), Best, Beöthy, Altmann (1990), Tuldava (1998), Best (2001), Bresnan, Dingare, Manning (2001), Vulanović (2003), Best (2006). 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.

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Hierarchic relations

2007-03-16T16:57:18Z

KHBest: lit erg

'''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

2007-03-16T16:50:23Z

KHBest:

'''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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Phoneme frequency

2007-03-16T08:33:48Z

KHBest: Lit. ergänzt

'''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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'''Veenker, W.''' (1982a). Konfrontierende Darstellung zur phonologischen Statistik der unga-rischen und finnischen Schriftsprache. Nyelvtudományi közlemények 84, 305-348a.

'''Veenker, W.''' (1982b). Zur phonologischen Statistik der syrjänischen Sprache. Etudes Finno-Ougriennes 15, 435-445.

'''Verglas, A.''' (1962). Remarques sur la relation entre rang et fréquence des lettres français. Bulletin d´information du laboratoire d´analyse lexicographique 6, 29-40.

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'''Vértes, E'''. (1970). Beiträge zu den typologischen Fragen des Ostjakischen. In: Dezsö, L., Hajdú, P (eds.), ''Theoretical problems of typology and the Northern Eurasian languages: 135-144''. Amsterdam: Grüner.

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'''Weidert, A'''. (1972). Die Vokalphoneme des Khasi, III. Teil. ''Zeitschrift für Phonetik, Sprach-wissenschaft und Kommunikationsforschung 25, 506-521''.

'''Weiss, M'''. (1962). Über die relative Häufigkeit der Phoneme des Schwedischen. ''Statistical methods in Linguistics 1, 41-55''.

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'''Wimmer, G., Altmann, G'''. (1999). s. Essen: Stamm.

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'''Wioland, F'''. (1972). Estimation de la „fréquence” des phonèmes en français parlé. ''Travaux de l´Institut phonétique de Strasbourg 4, 177-204''.

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Text-blocks

2007-03-16T08:31:00Z

KHBest: Lit. ergänzt

'''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.''

'''Best, K.-H.''' (2006). Quantitative Untersuchungen zum Niederdeutschen und Niederländischen. ''Göttinger Beiträge zur Sprachwissenschaft 13'', 51-71.

'''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.)

Sentence and clause length

2007-03-16T08:28:55Z

KHBest: Ergänzung Lit.

'''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.

'''Altmann, G'''. (1988b). Verteilungen von Satzlängen. Glottometrika 9, 147-170.

'''Altmann, G.''' (1992). Sherman´s laws of sentence length distribution. In: Saukkonen, P. (ed.), ''What is Language Synergetics?: 38-39''. Oulu: University of Oulu.

'''Bartkowiakowa, A.''' (1963). O rozkładzie i kolejności zdań współrzędnych i podrzędnych w utworach powieściowych żeromskiego i Sienkiewicza. ''Zastosowania matematyki, Tom VII, 133-154''.

'''Best, K.-H.''' (2001a). Wie viele Wörter enthalten Sätze im Deutschen? Ein Beitrag zu den Shermann-Altmann-Gesetzen. In: Best, K.-H. (ed.), ''Häufigkeitsverteilungen in Texten: 167-201''. Göttingen: Peust & Gutschmidt.

'''Best, K.-H'''. (2001b). Satzlängen im Deutschen: Verteilungen, Mittelwerte, Sprachwandel. ''Göttinger Beiträge zur Sprachwissenschaft 7, 7-31''.

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

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

'''Best, K.-H'''. (2003). ''Quantitative Linguistik. Eine Annäherung''. 2., überarb. u. erw. Aufl. Göttingen: Peust & Gutschmidt. (3. Aufl. in Vorbereitung)

'''Best, K.-H'''. (2005). Satzlänge. In: Altmann, G., Köhler, R., Piotrowski, R. (eds.), ''Quantitative Linguistik - Quantitative Linguistics. Ein internationales Handbuch: 298-304''. Berlin/ N.Y.: de Gruyter.

'''Best, K.-H.''' (2006). Quantitative Untersuchungen zum Niederdeutschen und Niederländischen. ''Göttinger Beiträge zur Sprachwissenschaft 13'', 51-71.

'''Best, K.-H.''' (2006). Verteilung von Phrasen- und Subsatzlängen in deutscher Fachsprache. ''Naukovyj Visnyk Černivec’koho Universytetu: Hermans’ka filolohija.'' Vypusk 319-320, 113-120.

'''Bohn, H. (1998).''' ''Quantitative Untersuchungen der modernen chinesischen Sprache und Schrift''. Hamburg: Kovač.

'''Buch, K.R.''' (1969). A note on sentence-length as random variable. In: Doležel, L., Bailey, R.W. (eds.), ''Statistics and Style: 76-79''. New York: Elsevier.

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

'''Clayman, D.L.''' (1981). Sentence length in Greek hexameter poetry. In: Grotjahn, R. (ed.), ''Hexameter Studies: 107-136''. Bochum: Brockmeyer.

'''Dshurjuk, T.V., Levickij, V.V.''' (2003). Satztypen und Satzlängen im Funktional- und Autorenstil. ''Glottometrics 6, 40-51''.

'''Eggers, Hans''' (1962). Zur Syntax der deutschen Sprache der Gegenwart. Studium Generale 15, 49-59.

'''Eggers, Hans''' (1967). Beobachtungen zur Häufigkeit deutscher Wortformen. ''Wirkendes Wort 17, 93-105''.

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User:KHBest

2007-03-15T16:17:43Z

KHBest:

Dr. Karl-Heinz Best, Seminar für deutsche Philologie, Georg-August-Universität Göttingen, Käte-Hamburger-Weg 3, 37073 Göttingen. Mitglied der Abteilung Sprachwissenschaft; Spezialgebiet: Quantitative Linguistik mit den Schwerpunkten Sprachwandel, Entlehnungen (Fremdwörter/Lehnwörter), Spracherwerb sowie Gesetzmäßigkeiten in Sprachstruktur und Sprachverwendung; Morphologie. Langfristige Aktivitäten: Mitarbeit bei den Göttingen - Halleschen Tagungen zum Thema Wissenstransfer (Transferwissenschaften), Mitherausgeber der Zeitschrift Glottometrics, Mitglied im editorial board der Zeitschrift Göttinger Beiträge zur Sprachwissenschaft, Leiter des Göttinger Projekt Quantitative Linguistik. Im Internet: http://wwwuser.gwdg.de/~kbest

Von „http://de.wikipedia.org/wiki/Benutzer:Dr._Karl-Heinz_Best“