Laws in Quantitative Linguistics - User contributions [en]

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Polysemy and age

2012-01-26T13:03:25Z

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'''1. Problem and history'''

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

'''2. Hypothesis'''

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

'''3. Derivation'''

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

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

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

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

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

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

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

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

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

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

'''5. References'''

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

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

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

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

[[Category:Unfertig]]

Phoneme entropy and inventory

2012-01-26T13:02:53Z

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'''1. Problem and history'''

Entropy is a measure of disorder of the phoneme system or rather a measure of deviation from the uniformity of phoneme frequencies. If <math>-ld p_x</math> is the measure of self-information of a phoneme (px is the relative frequency of phoneme x) then

(1)<math> H = -\sum_{x=1}^K p_x ld p_x</math>

is the measure of average uncertainty or entropy. For different types of entropy see Naranan, Balasubrahmanyan (2000). Since px is estimated as fx/N (N being the sample size), (1) can be written as

(2)<math> H = ld N -\frac{1}{N}\sum_{x=1}^K f_x ld f_x</math>

Since ranked frequencies of phonemes (<math>\rightarrow</math>) abide by a specific distribution, there is the justified question whether H depends on the size of the inventory of phonemes K.
Altmann and Lehfeldt (1980) used the model of the 1-displaced right truncated geometric distribution to show that entropy depends on K, Zörnig and Altmann (1983, 1984) used the Zipf-Mandelbrot distribution to derive another formula.
Cohen, Mantegna and Havlin (1997) observed a parabolic curve for the dependence between word inventory and entropy.

'''2. Hypothesis'''

''The entropy depends on the size of the phoneme inventory''.

'''3. Derivation'''

'''3.1. From the geometric distribution''' (Altmann, Lehfeldt 1980)

Let the ranked frequencies of phonemes (<math>\rightarrow</math>) follow the 1-displaced right truncated geometric distribution defined as

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

with

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

then (1) can be written as

(4)<math> H =-\sum_{x=1}^K aq^{x-1} ld (aq^{x-1})</math>.

Summing and using (3) one obtains

(5)<math> H= -ld a -\frac{\lbrack q-K(a+q-1)\rbrack ld q}{1-q}</math>.

Since a + q -1 ≈ 0 and <math>q^K\rightarrow 0</math> for great K, (5) results in

(6)<math> H= -ld(1-q)-\frac{q ld q}{1-q}</math>.

Substituting <math> q = \frac{K-2}{K+2}</math> as the first approximation (see also Phonemes: Repeat rate)

one obtains

(7)<math> H = -ld\begin{bmatrix}\left(\frac{4}{K+2}\right)\left(\frac{K-2}{K+2}^{\frac{K-2}{4}}\right)\end{bmatrix}</math> .

(Example: see below)

'''3.2. From the Zipf-Mandelbrot distribution''' (Zörnig, Altmann 1984)

Let the ranked frequencies be distributed according to

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

where c = 1 brought already a good approximation in case of Repeat rate (<math>\rightarrow</math>), we consider

(9)<math> H =-\sum_{x=1}^K \frac{A}{B+x}ld \frac{A}{B+x}</math>,

A being the normalizing constant, i.e.

(10)<math>A^{-1} = \sum_{x=1}^K \frac{1}{B+x}</math>.

Since ld x = (ld e)(ln x) we obtain from (9)

(11)<math>H = -A ld e\sum_{x=1}^K \frac{1}{B+x}ln\frac{A}{B+x}</math>

= <math> -A ld e \begin{bmatrix}\ln A \sum_{x=1}^K \frac{1}{B+x}- \sum_{x=1}^K \frac{\ln(B+x)}{B+x}\end{bmatrix}</math> .

Inserting (10) in (11) yields

(12)<math> H = ld e \begin{bmatrix} -\ln A + A \sum_{x=1}^K \frac{\ln(B+x)}{B+x}\end{bmatrix}</math>.

Approximating A by an appropriate integral, we obtain

(13)<math> A = \frac{1}{ln\frac{B+K}{B+1}}</math>.

and in the same way

(14)<math>\sum_{x=1}^K \frac{ln(B+x)}{B+x}= \int_{1}^{K}\frac{ln(B+x)}{B+x}dx = \frac{1}{2}\ln\lbrack (B+K)(B+1)\rbrack\ln\frac{B+K}{B+1}</math>.

Inserting (13) and (14) in (12) and ordering one obtains at last

(15)<math> H = ld e \ln\lbrack \sqrt{(B+K)(B+1)}\ln\frac{B+K}{B+1}\rbrack</math>.

'''Example'''. Entropy of phoneme systems for 63 languages.
Altmann, Lehfeldt (1980) and Zörnig, Altmann (1984) fitted the above curves (7) and (15) to the empirical entropies in 63 languages and obtained the results shown in Table 1. Zörnig and Rothe (1990) added further 8 data from French and German. The best value of B was iteratively established at B = 0.61 and B = 0.27. However, B is not yet interpreted. The languages are the same as in the tables in Repeat rate (<math>\rightarrow</math>). Here the coefficient of determination has been ascertained in such a way that the values of H for the same K were averaged.

<div align="center">[[Image:Tabelle1_PEaI.jpg]]</div>

Evidently, the fitting is slightly better using Zipf-Mandelbrot´s distribution. Again, even if the fitting is satisfactory, further research must be done.

<div align="center">[[Image:Grafik1_PEaI_Kopie.jpg]]</div>
<div align="center">Fig. 1. Fitting curves (7) –––– and (15) ------ to the entropies of 63 languages</div>

Note. Using the geometric distribution for the derivation of Repeat rate (R) and Entropy (H) the following relationship between them follows

(16)<math>R = \frac{1- \begin{bmatrix}\frac{2^{-H}(K+2)}{4}\end{bmatrix}^{\frac{4}{K-2}}}{1+\begin{bmatrix}\frac{2^{-H}(K+2)}{4}\end{bmatrix}^{\frac{4}{K-2}}}</math>

and

(17)<math> H = -ld \begin{bmatrix}\begin{pmatrix}\frac{2R}{R+1}\end{pmatrix}\begin{pmatrix}\frac{1-R}{+R}\end{pmatrix}^{\frac{1-R}{2R}}\end{bmatrix}</math> .

Using the Zipf-Mandebrot distribution we obtain

(18)<math> H = \frac{1}{\ln 4}\ln \frac{K-1}{R}</math>
and

(19)<math> R = \frac{K-1}{4^H}</math>.

so that it is sufficient to compute one of these values.

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

'''5. References''' (contains also references in which merely entropy has been computed)

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

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

'''Cohen, A., Mantegna, R.N., Havlin, S'''. (1997). Numerical analysis of word frequencies in artificial and natural language texts? ''Fractals 5(1), 93-104''.

'''Feng, Zh.''' (1984). Hanzi de shang (Entropy of Chinese characters). In: Wenzi gaige 4, ….[Reprint in Chen, Y. (1989)(ed.), ''Xiandai Hanyu dingliang fenxi: 267-278 (Quantitative analysis of modern Chinese).'' Shanghai: Shanghai Jiaoyu chubanse.

'''Jakopin, F.''' (2002). ''Entropija v slovenskih leposlovnih besedilih''. Ljubljana: ZRC SAZU.

'''Kučera, K'''. (2001). The development of entropy and redundancy in Czech from the 13th to the 20th century: Is there a linguistic arrow of time. In: Uhlířova, L., Wimmer, G., Altmann, G., Köhler, R. (Eds.), ''Text as a linguistic paradigm: levels, constituents, constructs. Festschrift in honour of Ludek Hřebíček: 153-162''. Trier: WVT.

'''Lin, L'''. (2001). Guabyu Hanzi tongji tezheng de ji ge wenti. In: Su, P. (ed.), ''Hiandai Hanzixue cankao ziliao:''….. (Reference data to modern sinographics.) Peking: beijing Daxue chubanshe.

'''Lua, K.T.''' (1994). Frequency-rank curves and entropy for Chinese characters and words. ''Computer Processing of Chinese and Oriental Languages 8,(1), 37-52''.

'''Naranan, S., Balasubrahmanyan, V.K.''' (2000). Information theory and algorithmic complexity: Applications to linguistic discourses and DNA sequences as complex systems. Part I: Efficiency of the genetic code of DNA. ''J. of Quantitative Linguistics 7, 129-151''; Part II: Conmplexity of DNA sequences, analogy with linguistic discourses. ''J. of Quantitative Linguistics 7, 153-183''.

'''Rothe, U., Zörnig, P.''' (1989). The entropy of phoneme frequencies. German and French. ''Glottometrika 11, 199-205''.

'''Yannakoudakis, E.J., Tsomokos, I., Hutton, P.J'''. (1990). n-Grams and their implication to natural language understanding. Pattern Recognition 23,(5), 509-528.

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

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

Morph length

2012-01-26T11:05:22Z

LQL Admin:

'''1. Problem and history'''

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

'''2. Hypothesis'''

''2.1. Morph length in texts abides by a regular probability distribution derived form the unified theory, namely the 1-displaced Hyperpoisson distribution.''

''2.2. If morphs have specific forms, the distribution is multimodal and must be modeled by an appropriate approach''.

'''3. Derivation'''

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

'''5. References'''

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

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

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

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

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

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

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

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

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

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

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

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

[[Category:Unfertig]]

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2005-12-16T10:54:42Z

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Welcome to the '''Encyclopedia of Linguistic Laws and the LQL''' server. The quest
for laws of language and text during the last decades has resulted in a
wealth of law hypotheses and accepted laws. It has become difficult to
achieve a systematic overview of the relevant studies and findings.
Therefore, we have launched a project with the aim to collect as completely
and systematically as possible the corresponding literature and to form a
handbook on this basis. As this endeavor will take some time it seems
advantageous to set up a wiki at the same time, which will serve as a
growing online-encyclopedia and as the basis for book publications. We hope
that many users will support us with this work.

We have as resources for this project not more than the time and the effort
which we can devote to it ourselves. Therefore, to avoid the effort to find
and remove unwished (because misleading or irritating) contributions to this
wiki, LQL can be changed or added by authorised editors only. Everyone who
wants to help us is invited to send us contributions via [mailto:koehler@uni-trier.de email]. Thank you
for your support.

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This wiki has been set up by Götz Burger. We would like to express our
thanks for his valuable support of the project.

Main Page

2005-12-16T10:52:26Z

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Welcome to the '''Encyclopedia of Linguistic Laws and the LQL''' server. The quest
for laws of language and text during the last decades has resulted in a
wealth of law hypotheses and accepted laws. It has become difficult to
achieve a systematic overview of the relevant studies and findings.
Therefore, we have launched a project with the aim to collect as completely
and systematically as possible the corresponding literature and to form a
handbook on this basis. As this endeavor will take some time it seems
advantageous to set up a wiki at the same time, which will serve as a
growing online-encyclopedia and as the basis for book publications. We hope
that many users will support us with this work.

We have as resources for this project not more than the time and the effort
which we can devote to it ourselves. Therefore, to avoid the effort to find
and remove unwished (because misleading or irritating) contributions to this
wiki, LQL can be changed or added by authorised editors only. Everyone who
wants to help us is invited to send us contributions via email. Thank you
for your support.

1. January 2006,
Gabriel Altmann, Reinhard Köhler, Relja Vulanovic

This wiki has been set up by Götz Burger. We would like to express our
thanks for his valuable support of the project.

Main Page

2005-12-16T10:51:48Z

LQL Admin:

Welcome to the '''Encyclopedia of Linguistic Laws and the LQL''' server. The quest
for laws of language and text during the last decades has resulted in a
wealth of law hypotheses and accepted laws. It has become difficult to
achieve a systematic overview of the relevant studies and findings.
Therefore, we have launched a project with the aim to collect as completely
and systematically as possible the corresponding literature and to form a
handbook on this basis. As this endeavor will take some time it seems
advantageous to set up a wiki at the same time, which will serve as a
growing online-encyclopedia and as the basis for book publications. We hope
that many users will support us with this work.
We have as resources for this project not more than the time and the effort
which we can devote to it ourselves. Therefore, to avoid the effort to find
and remove unwished (because misleading or irritating) contributions to this
wiki, LQL can be changed or added by authorised editors only. Everyone who
wants to help us is invited to send us contributions via email. Thank you
for your support.
1. January 2006,
Gabriel Altmann, Reinhard Köhler, Relja Vulanovic

This wiki has been set up by Götz Burger. We would like to express our
thanks for his valuable support of the project.

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Herzlich Willkommen....Herr Köhler..... :)

Schön, dass Sie vorbeischauen....

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