Laws in Quantitative Linguistics - User contributions [en]

Frequency and polytextuality

2007-06-22T08:23:14Z

Rkoehler:

'''1. Problem and history'''

Polytextuality measures the degree of independence of the usability of a word (in general of a linguistic unit) of its co-text or context. Linguistic units such as phonemes, syllables, morae, morphemes, words etc. differ in their usability with respect to different environments. The environment of a syllable, mora or morphem consists of the words in which they occur, the environment of a word consists of phrases, sentences, or texts. The number of different environments is often called the number of types. The frequency of a given entity in all its environments in, say, a corpus, is considered as the number of tokens. It can be shown that there exists a lawful relationship between the number of types (environments) and the number of tokens (frequency) of units on the given level.
The degree of independence of the unsability of a unit from its context (or, the variability of contextes with respect to a given unit), can be measured in several ways. Word polytextuality is often measured in terms of the number of different texts in a text corpus which contain at least one token of the given word. Polytextuality of morphemes or syllables are usually measured with reference to an inventory such as a dictionary.

The relationship between frequency and polytextuality has been postulated and investigated by R. Köhler (1986) as a complement to other quantitative properties, in order to integrate it into his synergetic control cycle. As a consequence of an erroneous identification with another “type-token” problem (<math>\rightarrow</math>), one can find this relationship also under the name “(morphological) productivity” (cf. Baayen 2001), which in turn represents a slightly different aspect (cf. Wimmer, Altmann 1995). The relationship was studied in different works on language synergetics (cf. e.g. Gieseking 2002), Köhler (2005) reformulated the pertinent part of his control cycle and Tamaoka, Altmann (2005) showed by means of Japanese morae that the unified theory (<math>\rightarrow</math>) leads to an identical result.

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

'''2. Hypothesis'''

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

'''3. Derivation'''

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

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

where x is polytexty, y is frequency and c represents some additional factors. The resulting solution,

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

was used to model polytexty and frequency of Japanese morae in a Japanese corpus.

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

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

i.e.

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

from which it follows that

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

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

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

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

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

Thus Köhler´s model explains also the additional factors.

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

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

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

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

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

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

'''Example 2'''. Polytextuality of German words as a function of their frequencies

Köhler (1986) published his result on data of a German text corpus (LIMAS), respresented here by the following graph:

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

'''5. References'''

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

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

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

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

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

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

[[Category:Quantitative properties]]

Frequency and polytextuality

2007-06-22T08:16:58Z

Rkoehler:

'''1. Problem and history'''

Polytextuality measures the degree of independence of the usability of a word (in general of a linguistic unit) of its co-text or context. Linguistic units such as phonemes, syllables, morae, morphemes, words etc. differ in their usability with respect to different environments. The environment of a syllable, mora or morphem consists of the words in which they occur, the environment of a word consists of phrases, sentences, or texts. The number of different environments is often called the number of types. The frequency of a given entity in all its environments in, say, a corpus, is considered as the number of tokens. It can be shown that there exists a lawful relationship between the number of types (environments) and the number of tokens (frequency) of units on the given level.
The degree of independence of the unsability of a unit from its context (or, the variability of contextes with respect to a given unit), can be measured in several ways. Word polytextuality is often measured in terms of the number of different texts in a text corpus which contain at least one token of the given word. Polytextuality of morphemes or syllables are usually measured with reference to an inventory such as a dictionary.

The relationship between frequency and polytextuality has been postulated and investigated by R. Köhler (1986) as a complement to other quantitative properties, in order to integrate it into his synergetic control cycle. As a consequence of an erroneous identification with another “type-token” problem (<math>\rightarrow</math>), one can find this relationship also under the name “(morphological) productivity” (cf. Baayen 2001), which in turn represents a slightly different aspect (cf. Wimmer, Altmann 1995). The relationship was studied in different works on language synergetics (cf. e.g. Gieseking 2002), Köhler (2005) reformulated the pertinent part of his control cycle and Tamaoka, Altmann (2005) showed by means of Japanese morae that the unified theory (<math>\rightarrow</math>) leads to an identical result.

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

'''2. Hypothesis'''

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

'''3. Derivation'''

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

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

where x is polytexty, y is frequency and c represents some additional factors. The resulting solution,

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

was used to model polytexty and frequency of Japanese morae in a Japanese corpus.

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

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

i.e.

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

from which it follows that

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

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

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

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

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

Thus Köhler´s model explains also the additional factors.

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

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

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

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

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

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

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

'''5. References'''

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

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

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

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

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

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

[[Category:Frequency]]

Frequency and polytextuality

2007-06-22T08:16:01Z

Rkoehler:

'''1. Problem and history'''

Polytextuality measures the degree of independence of the usability of a word (in general of a linguistic unit) of its co-text or context. Linguistic units such as phonemes, syllables, morae, morphemes, words etc. differ in their usability with respect to different environments. The environment of a syllable, mora or morphem consists of the words in which they occur, the environment of a word consists of phrases, sentences, or texts. The number of different environments is often called the number of types. The frequency of a given entity in all its environments in, say, a corpus, is considered as the number of tokens. It can be shown that there exists a lawful relationship between the number of types (environments) and the number of tokens (frequency) of units on the given level.
The degree of independence of the unsability of a unit from its context (or, the variability of contextes with respect to a given unit), can be measured in several ways. Word polytextuality is often measured in terms of the number of different texts in a text corpus which contain at least one token of the given word. Polytextuality of morphemes or syllables are usually measured with reference to an inventory such as a dictionary.

The relationship between frequency and polytextuality has been postulated and investigated by R. Köhler (1986) as a complement to other quantitative properties, in order to integrate it into his synergetic control cycle. As a consequence of an erroneous identification with another “type-token” problem (<math>\rightarrow</math>), one can find this relationship also under the name “(morphological) productivity” (cf. Baayen 2001), which in turn represents a slightly different aspect (cf. Wimmer, Altmann 1995). The relationship was studied in different works on language synergetics (cf. e.g. Gieseking 2002), Köhler (2005) reformulated the pertinent part of his control cycle and Tamaoka, Altmann (2005) showed by means of Japanese morae that the unified theory (<math>\rightarrow</math>) leads to an identical result.

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

'''2. Hypothesis'''

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

'''3. Derivation'''

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

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

where x is polytexty, y is frequency and c represents some additional factors. The resulting solution,

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

was used to model polytexty and frequency of Japanese morae in a Japanese corpus.

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

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

i.e.

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

from which it follows that

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

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

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

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

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

Thus Köhler´s model explains also the additional factors.

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

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

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

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

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

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

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

'''5. References'''

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

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

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

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

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

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

[[Category:law]]

Frequency and polytextuality

2007-06-22T08:12:46Z

Rkoehler:

'''1. Problem and history'''

Polytextuality measures the degree of independence of the usability of a word (in general of a linguistic unit) of its co-text or context. Linguistic units such as phonemes, syllables, morae, morphemes, words etc. differ in their usability with respect to different environments. The environment of a syllable, mora or morphem consists of the words in which they occur, the environment of a word consists of phrases, sentences, or texts. The number of different environments is often called the number of types. The frequency of a given entity in all its environments in, say, a corpus, is considered as the number of tokens. It can be shown that there exists a lawful relationship between the number of types (environments) and the number of tokens (frequency) of units on the given level.
The degree of independence of the unsability of a unit from its context (or, the variability of contextes with respect to a given unit), can be measured in several ways. Word polytextuality is often measured in terms of the number of different texts in a text corpus which contain at least one token of the given word. Polytextuality of morphemes or syllables are usually measured with reference to an inventory such as a dictionary.

The relationship between frequency and polytextuality has been postulated and investigated by R. Köhler (1986) as a complement to other quantitative properties, in order to integrate it into his synergetic control cycle. As a consequence of an erroneous identification with another “type-token” problem (<math>\rightarrow</math>), one can find this relationship also under the name “(morphological) productivity” (cf. Baayen 2001), which in turn represents a slightly different aspect (cf. Wimmer, Altmann 1995). The relationship was studied in different works on language synergetics (cf. e.g. Gieseking 2002), Köhler (2005) reformulated the pertinent part of his control cycle and Tamaoka, Altmann (2005) showed by means of Japanese morae that the unified theory (<math>\rightarrow</math>) leads to an identical result.

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

'''2. Hypothesis'''

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

'''3. Derivation'''

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

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

where x is polytexty, y is frequency and c represents some additional factors. The resulting solution,

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

was used to model polytexty and frequency of Japanese morae in a Japanese corpus.

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

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

i.e.

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

from which it follows that

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

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

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

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

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

Thus Köhler´s model explains also the additional factors.

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

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

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

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

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

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

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

'''5. References'''

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

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

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

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

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

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

[[Category:Unfertig]]

Frequency and polytextuality

2007-06-22T08:12:31Z

Rkoehler:

'''1. Problem and history'''

Polytextuality measures the degree of independence of the usability of a word (in general of a linguistic unit) of its co-text or context. Linguistic units such as phonemes, syllables, morae, morphemes, words etc. differ in their usability with respect to different environments. The environment of a syllable, mora or morphem consists of the words in which they occur, the environment of a word consists of phrases, sentences, or texts. The number of different environments is often called the number of types. The frequency of a given entity in all its environments in, say, a corpus, is considered as the number of tokens. It can be shown that there exists a lawful relationship between the number of types (environments) and the number of tokens (frequency) of units on the given level.
The degree of independence of the unsability of a unit from its context (or, the variability of contextes with respect to a given unit), can be measured in several ways. Word polytextuality is often measured in terms of the number of different texts in a text corpus which contain at least one token of the given word. Polytextuality of morphemes or syllables are usually measured with reference to an inventory such as a dictionary.

The relationship between frequency and polytextuality has been postulated and investigated by R. Köhler (1986) as a complement to other quantitative properties, in order to integrate it into his synergetic control cycle. AS a consequence of an erroneous identification with another “type-token” problem (<math>\rightarrow</math>), one can find this relationship also under the name “(morphological) productivity” (cf. Baayen 2001), which in turn represents a slightly different aspect (cf. Wimmer, Altmann 1995). The relationship was studied in different works on language synergetics (cf. e.g. Gieseking 2002), Köhler (2005) reformulated the pertinent part of his control cycle and Tamaoka, Altmann (2005) showed by means of Japanese morae that the unified theory (<math>\rightarrow</math>) leads to an identical result.

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

'''2. Hypothesis'''

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

'''3. Derivation'''

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

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

where x is polytexty, y is frequency and c represents some additional factors. The resulting solution,

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

was used to model polytexty and frequency of Japanese morae in a Japanese corpus.

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

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

i.e.

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

from which it follows that

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

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

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

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

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

Thus Köhler´s model explains also the additional factors.

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

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

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

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

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

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

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

'''5. References'''

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

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

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

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

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

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

[[Category:Unfertig]]

Frequency and polytextuality

2007-06-22T08:11:45Z

Rkoehler:

'''1. Problem and history'''

Polytextuality measures the degree of independence of the usability of a word (in general of a linguistic unit) of its co-text or context. Linguistic units such as phonemes, syllables, morae, morphemes, words etc. differ in their usability with respect to different environments. The environment of a syllable, mora or morphem consists of the words in which they occur, the environment of a word consists of phrases, sentences, or texts. The number of different environments is often called the number of types. The frequency of a given entity in all its environments in, say, a corpus, is considered as the number of tokens. It can be shown that there exists a lawful relationship between the number of types (environments) and the number of tokens (frequency) of units on the given level.
The degree of independence of the unsability of a unit from its context (or, the variability of contextes with respect to a given unit), can be measured in several ways. Word polytextuality is often measured in terms of the number of different texts in a text corpus which contain at least one token of the given word. Polytextuality of morphemes or syllables are usually measured with reference to an inventory such as a dictionary.

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

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

'''2. Hypothesis'''

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

'''3. Derivation'''

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

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

where x is polytexty, y is frequency and c represents some additional factors. The resulting solution,

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

was used to model polytexty and frequency of Japanese morae in a Japanese corpus.

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

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

i.e.

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

from which it follows that

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

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

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

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

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

Thus Köhler´s model explains also the additional factors.

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

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

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

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

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

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

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

'''5. References'''

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

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

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

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

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

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

[[Category:Unfertig]]

Frequency and polytextuality

2007-06-22T08:10:07Z

Rkoehler:

'''1. Problem and history'''

Polytextuality measures the degree of independence of the usability of a word (in general of a linguistic unit) of its co-text or context. Linguistic units such as phonemes, syllables, morae, morphemes, words etc. differ in their usability with respect to different environments. The environment of a syllable, mora or morphem consists of the words in which they occur, the environment of a word consists of phrases, sentences, or texts. The number of different environments is often called the number of types. The frequency of a given entity in all its environments in, say, a corpus, is considered as the number of tokens. It can be shown that there exists a lawful relationship between the number of types (environments) and the number of tokens (frequency) of units on the given level.
The degree of independence of the unsability of a unit from its context (or, the variability of contextes with respect to a given unit), can be measured in several ways. Word polytextuality is often measured in terms of the number of different texts in a text corpus which contain at least one token of the given word. Polytextuality of morphemes or syllables are usually measured with reference to an inventory such as a dictionary.

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

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

'''2. Hypothesis'''

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

'''3. Derivation'''

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

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

where x is polytexty, y is frequency and c represents some additional factors. The resulting solution,

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

was used to model polytexty and frequency of Japanese morae in a Japanese corpus.

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

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

i.e.

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

from which it follows that

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

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

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

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

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

Thus Köhler´s model explains also the additional factors.

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

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

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

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

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

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

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

'''5. References'''

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

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

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

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

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

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

[[Category:Unfertig]]

Frequency and polytextuality

2007-06-22T08:08:07Z

Rkoehler:

'''1. Problem and history'''

Polytextuality measures the degree of independence of the usability of a word (in general of a linguistic unit) of its co-text or context. Linguistic units such as phonemes, syllables, morae, morphemes, words etc. differ in their usability with respect to different environments. The environment of a syllable, mora or morphem consists of the words in which they occur, the environment of a word consists of phrases, sentences, or texts. The number of different environments is often called the number of types. The frequency of a given entity in all its environments in, say, a corpus, is considered as the number of tokens. It can be shown that there exists a lawful relationship between the number of types (environments) and the number of tokens (frequency) of units on the given level.
The degree of independence of the unsability of a unit from its context (or, the variability of contextes with respect to a given unit), can be measured in several ways. Word polytextuality is often measured in terms of the number of different texts in a text corpus which contain at least one token of the given word.

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

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

'''2. Hypothesis'''

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

'''3. Derivation'''

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

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

where x is polytexty, y is frequency and c represents some additional factors. The resulting solution,

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

was used to model polytexty and frequency of Japanese morae in a Japanese corpus.

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

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

i.e.

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

from which it follows that

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

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

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

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

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

Thus Köhler´s model explains also the additional factors.

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

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

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

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

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

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

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

'''5. References'''

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

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

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

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

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

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

[[Category:Unfertig]]

Frequency and polytextuality

2007-06-22T08:06:44Z

Rkoehler:

'''1. Problem and history'''

Polytextuality measures the degree of independence of the usability of a word (in general of a linguistic unit) of its co-text or context. Linguistic units such as phonemes, syllables, morae, morphemes, words etc. differ in their usability with respect to different environments. The environment of a syllable, mora or morphem consists of the words in which they occur, the environment of a word consists of phrases, sentences, or texts. The number of different environments is often called the number of types. The frequency of a given entity in all its environments in, say, a corpus, is considered as the number of tokens. It can be shown that a certain, there exists a lawful relationship between the number of types (environments) and the number of tokens (frequency) of units on the given level.
The degree of independence of the unsability of a unit from its context (or, the variability of contextes with respect to a given unit), can be measured in several ways. Word polytextuality is often measured in terms of the number of different texts in a text corpus which contain at least one token of the given word.

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

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

'''2. Hypothesis'''

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

'''3. Derivation'''

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

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

where x is polytexty, y is frequency and c represents some additional factors. The resulting solution,

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

was used to model polytexty and frequency of Japanese morae in a Japanese corpus.

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

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

i.e.

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

from which it follows that

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

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

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

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

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

Thus Köhler´s model explains also the additional factors.

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

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

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

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

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

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

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

'''5. References'''

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

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

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

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

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

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

[[Category:Unfertig]]

Frequency and polytextuality

2007-06-22T07:58:42Z

Rkoehler:

'''1. Problem and history'''

Polytextuality measures the degree of independence of the usability of a word (in general of a linguistic unit) of its co-text or context. Linguistic units such as syllable, mora, morphem, word etc. differ in their usability with respect to different environments. The environment of a syllable, mora or morphem consists of the words in which they occur, the environment of a word consists of phrases, sentences, or texts. The number of different environments is often called the number of types. The frequency of a given entity in all its environments in, say, a corpus, is considered as the number of tokens. It can be shown that a certain, there exists a lawful relationship between the number of types (environments) and the number of tokens (frequency) of units on the given level.
The degree of independence of the unsability of a unit from its context (or, the variability of contextes with respect to a given unit), can be measured in several ways. Word polytextuality is often measured in terms of the number of different texts in a text corpus which contain at least one token of the given word.

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

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

'''2. Hypothesis'''

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

'''3. Derivation'''

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

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

where x is polytexty, y is frequency and c represents some additional factors. The resulting solution,

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

was used to model polytexty and frequency of Japanese morae in a Japanese corpus.

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

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

i.e.

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

from which it follows that

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

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

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

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

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

Thus Köhler´s model explains also the additional factors.

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

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

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

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

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

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

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

'''5. References'''

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

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

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

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

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

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

[[Category:Unfertig]]

Complexity of syntactic constructions

2006-12-09T18:05:25Z

Rkoehler:

'''1. Problem and history'''

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

'''2. Hypothesis'''

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

'''3. Derivation'''

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

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

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

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

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

Assumptions: The number of constructions with complexity x+1 is proportional to that with complexity x. maxH increases the probability of a higher complexity, minX decreases it. The greater E the more complexity is needed to code the individual messages. On the other hand, the greater the inventory I(K) the less complexity is needed. Within a given period of time, the relation E/I(K) can be considered as a constant, say q.
With these assumptions, we obtain

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

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

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

resulting in

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

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

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

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

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

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

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

'''5. References'''

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

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

Complexity of syntactic constructions

2006-12-04T18:26:14Z

Rkoehler:

'''1. Problem and history'''

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

'''2. Hypothesis'''

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

'''3. Derivation'''

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

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

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

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

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

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

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

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

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

resulting in

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

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

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

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

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

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

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

'''5. References'''

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

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

Frequency and polytextuality

2006-08-10T08:33:30Z

Rkoehler:

'''1. Problem and history'''

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

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

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

'''2. Hypothesis'''

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

'''3. Derivation'''

Since in most cases linguistic properties are related by their relative rates of change, Tamaoka and Altmann (2005), taking into account some ceteris paribus factors, and leaning against the unfied theory (<math>\rightarrow</math>) set up the equation

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

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

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

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

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

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

from which

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

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

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

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

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

Thus Köhler´s model explains also the additional factors.

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

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

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

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

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

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

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

'''5. References'''

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

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

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

'''Tamaoka, K., Altmann, G.''' (2005). On the relation between types and tokens of Japanese morae………….

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

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

[[Category:Unfertig]]

Frequency and polytextuality

2006-08-10T08:32:41Z

Rkoehler:

'''1. Problem and history'''

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

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

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

'''2. Hypothesis'''

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

'''3. Derivation'''

Since in most cases linguistic properties are related by their relative rates of change, Tamaoka and Altmann (2005), taking into account some ceteris paribus factors, and leaning against the unfied theory (<math>\rightarrow</math>) set up the equation

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

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

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

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

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

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

from which

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

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

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

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

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

Thus Köhler´s model explains also the additional factors.

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

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

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

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

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

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

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

'''5. References'''

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

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

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

'''Tamaoka, K., Altmann, G.''' (2005). On the relation between types and tokens of Japanese morae………….

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

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

[[Category:Unfertig]]

Hierarchic relations

2006-06-29T21:30:13Z

Rkoehler:

'''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: G. Altmann'''

'''5. References'''

'''Abercrombie, D'''. (1967). ''Elements of general phonetics''. Edinburgh: University Press. (p.97)

'''Äima, F.''' (1918). Phonetik und Lautlehre des Inarilappischen. ''Mémoires de la Societé Finno-Ougrienne 43, 1-249.'' (p.204)

'''Altmann, G.''' (1980). Prolegomena to Menzerath´s law. ''Glottometrika 2, 1-10''.

'''Altmann, G'''. (1983). H. Arens´ „verborgene Ordnung“ und das Menzerathsche Gesetz. In: Faust, M., Harweg, R., Lehfeldt, W. (Hrsg.), ''Allgemeine Sprachwissenschaft, Sprachtypologie und Textlinguistik: 31-39''. Tübingen: Narr.

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

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

'''Altmann, G., Schwibbe, M'''. (1989). ''Das Menzerathsche Gesetz in informationsverarbeitenden Systemen''. Hildesheim, Olms.

'''Arens, H'''. (1965). ''Verborgene Ordnung''. Düsseldorf: Schwann.

'''Auer, P., Uhmann, S'''. (1988). Silben- und akzentzählende Sprachen. ''Zeitschrift für Sprachwissenschaft 7, 214-259.''

'''Bak, P.''' (1996) ''How nature works. The science of self-organized crit''icality. New York: Copernicus-Springer.

'''Bennett, S.G'''. (1935). Vergleichende experimentalphonetische Untersuchung der ungespannten Plosive im Englischen, Deutschen und Französischen. Diss. (Footnote 18)

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

'''Bohn, H.''' (2002). Untersuchungen zur chinesischen Sprache und Schrift. In: Köhler, R. (ed.), ''Korpuslinguistische Untersuchungen in die quantitative und systemtheoretische Linguistik: 127-177''. http://ubt.opus.hbz-nrw.de/volltexte/2004/279/

'''Boroda, M.G., Altmann, G'''. (1991). Menzerath's law in musical texts. ''Musikometrika 3, 1-13.''

'''Changizi, M.A'''. (2001). Universal scaling laws for hierarchical complexity in langauegs, organisms, bahaviors and other combinatorial systems. ''J. of Theoretical Biology 211, 277-295.''

'''Collinder, B'''. (1964). Das Wort als phonetische Einheit. In: Collinder, B. (Hrsg.), ''Sprachwissenschaft und Wahrscheinlichkeit: 203-217. Uppsala: Almquist, Wiksells.'' (1939: 66)

'''Cramer, I.M'''. (2005). The parameters of the Altmann-Menzerath law. ''J. of Quantitative Linguistics 12(1), 41-52.''

'''Cramer, I.M'''. (2005a). Das Menzerathsceh Gesetz. In: Altmann, G., Köhhler, R., Piotrowski, R.G. (eds.), ''Handbook of Quantitative Linguistics: 659-688''. Berlin: de Gruyter.

'''Donner, K.''' (1912). Salmin murteen kvantiteettisuehtista. ''Suomi IV, 9, II''. (p. 33)

'''Elert, C.C.''' (1965). ''Phonological studies of quantity in Swedish''. Uppsala. (p. 138)

'''Evertz,''' (1929). ''Beiträge zur Phonetik der südfranzösischen Plosive''. Diss. (footnote 21)

'''Faure, G., Hirst, D.J., Chafcouloff, M.''' (1980). Rhythm in English: Isochronism, potch, and pereceived stress. In: Linda, R., Schooneveld, C.H.v. (eds.), The ''melody of language: 71-79''. Bartimore: University Park Press.

'''Fenk, A., Fenk-Oczlon, G'''. (1993). Menzerath´s law and the constant flow of linguistic information. In: Köhler, R., Rieger, B.B. (eds.), ''Contributions to quantitative linguistics: 11-31''. Dordrecht: Kluwer.

'''Fenk-Oczlon, G., Fenk, A'''. (1995). Selbstorganisation und natürliche Typologie. ''Sprachtypologie und Universalienforschung 48, 223-238''.

'''Fenk-Oczlon, G., Fenk, A'''. (1999). Cognition, quantitative linguistics, and systemic typology. Linguistic Typology 3, 151-177.

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

'''Fischer-Jörgensen, E'''. (1964). Sound duration and place of articulation. ''Zeitschrift für Phonetik 17, 175-207''.

'''Fónagy, I'''. (1959).'' A költöi nyelv hangtanából.'' Budapest.

'''Fónagy, I, Magdics, K'''. (1960). Speed of utterance in phrases of different length. ''Language and Speech 3, 179-182''.

'''Gaitenby, I.H'''. (1965). ''The elastic word''. Haskins, Status Report on speech research 1965/SR-2,3.1-3.12.

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

'''Geršić, S., Altmann, G'''. (1980). Laut – Silbe – Wort und das Menzerathsche Gesetz. ''Frankfurter Phonetische Beiträge 3, 115-123''.

'''Grégoire, A'''. (1899). Variation de la durée de la syllable français. ''La Parole 1'' (p. 161).

'''Grzybek, P'''. (1998).

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

'''Hegedüs, L'''. (1956). Sprechtempoanalysen im Ungarischen. ''Zeitschrift für Phonetik 10.''

'''Heups, G'''. (1983). Untersuchungen zum Verhältnis von Satzlänge und Clauselänge am Beispiel deutscher Texte verschiedener Textklassen. ''Glottometrika 5, 113-133''.

'''Hřebíček, L'''. (1990). Menzerath-Altmann's law on the semantic level. ''Glottometrika 11, 47-56.''

'''Hřebíček, L''' (1990a). The constants of the Menzerath-Altmann law. ''Glottometrika 12, 61-71''.

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

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[[Category:Unfertig]]

Length of syntactic constructions

2006-06-29T21:25:38Z

Rkoehler: Syntactic constructions: Length moved to Length of syntactic constructions:

'''1. Problem and history'''

The length of a syntactic construction is defined as the number of terminal nodes belonging to it, while complexity (<math>\rightarrow</math>) is he number of its immediate constituents. These two properties are interrelated.

The first distribution models were proposed by Köhler and Altmann (2000), no further development is known. As can be seen, the result is a special case of length (<math>\rightarrow</math>) distributions.

'''2. Hypothesis'''

''The length of syntactic constructions abides by the positive negative binomial distribution''.

'''3. Derivation'''

The quantities necessary for the derivation are shown in the chapter “Syntactic structures: Complexity” (<math>\rightarrow</math>). Here the requirement minX = 0, because length depends on complexity and minX is given implicitely. Using the approach proposed for modeling complexity we obtain

(1)<math> P_x = \frac{max H + x}{x}\frac{E}{I(K)}P_{x-1}</math>.

Setting again maxH = k-1, E/I(K) = q (0 < q < 1) and solving (1) we obtain

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

If maxH tends to -1, i.e. k <math>\rightarrow</math> 0, we obtain the logarithmic distribution

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

but in particular cases it is necessary to modify the probability in x = 1 and one obtains the extended variants of (2) and (3), namely

(4)<math> P_x = \begin{cases} 1-\alpha, & x=1 \\ \alpha {k+x-2 \choose x-1}\frac{p^k q^x}{1-p^k}, & x=2,3,... \end{cases}</math>

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

Example: The length of syntactic constructions in the Susanne corpus
The result of fitting of the extended logarithmic distribution to the data in Susanne corpus are shown in Table 1 and Fig. 1 (Köhler, Altmann 2000).

Table 1
Fitting the extended logarithmic distribution to the data in Susanne corpus

Though the chi-square value is very high, the fit is satisfactory as shown by the value of C. The greatest divergence is in the middle range where one can observe strong fluctuation.

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

'''5. References'''

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

Syntactic constructions: Length

2006-06-29T21:25:38Z

Rkoehler: Syntactic constructions: Length moved to Length of syntactic constructions:

#redirect [[Length of syntactic constructions:]]

Frequency and polytextuality

2006-06-29T21:24:13Z

Rkoehler: Frequency and polytexty moved to Frequency and polytextuality

'''1. Problem and history'''

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

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

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

'''2. Hypothesis'''

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

'''3. Derivation'''

Since in most cases linguistic properties are related by their relative rates of change, Tamaoka and Altmann (2005), taking into account some ceteris paribus factors, and leaning against the unfied theory (<math>\rightarrow</math>) set up the equation

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

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

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

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

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

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

from which

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

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

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

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

Fig. 1. The relationship between polytextuality and frequency in general

Thus Köhler´s model explains also the additional factors.

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

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

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

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

<div align="center">[[Image:Tabelle1_Freq.jpg]]</div>

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

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

'''5. References'''

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

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

'''Köhler (2005)………….'''

'''Tamaoka, K., Altmann, G.''' (2005). On the relation between types and tokens of Japanese morae………….

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

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

[[Category:Unfertig]]

Frequency and polytextuality

2006-06-29T21:22:27Z

Rkoehler:

'''1. Problem and history'''

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

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

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

'''2. Hypothesis'''

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

'''3. Derivation'''

Since in most cases linguistic properties are related by their relative rates of change, Tamaoka and Altmann (2005), taking into account some ceteris paribus factors, and leaning against the unfied theory (<math>\rightarrow</math>) set up the equation

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

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

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

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

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

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

from which

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

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

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

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

Fig. 1. The relationship between polytextuality and frequency in general

Thus Köhler´s model explains also the additional factors.

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

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

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

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

<div align="center">[[Image:Tabelle1_Freq.jpg]]</div>

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

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

'''5. References'''

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

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

'''Köhler (2005)………….'''

'''Tamaoka, K., Altmann, G.''' (2005). On the relation between types and tokens of Japanese morae………….

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

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

[[Category:Unfertig]]

Word length and frequency

2006-02-22T12:22:06Z

Rkoehler:

'''1. Problem and history'''

There are two problems that must be strictly separated:

(a) The use of words of specific length in texts. Here ''word forms'' are meant and neither the size of the lexicon nor the number of phonemes in the phoneme inventory are relevant. Other properties like polysemy, polytexty, or synthetism can be added for modelling textual word length.

(b) Frequency as a factor of ''lemma construction'' in the dictionary, where frequency is taken from a frequency dictionary, and the dictionary size, as well as the size of the phoneme inventory, are relevant.

In (b), problems arise because the size of the dictionary can be merely coarsely estimated. There are even authors considering it infinite (cf. Piotrowski, Bektaev, Piotrovskaja 1985; Kornai 2002), which is a reasonable assumption. If, however, it is considered as a fixed finite value, it can be taken into account. Further, frequency dictionaries strongly depend on the kind of the analyzed texts. In order to secure a reliable frequency dictionary even for a short time period, an astronomical number of words must be counted.

Problem (a) is easily solvable but it must be considered for individual texts or group of texts from the same author. Here we consider merely the problem of dependence of word form length on its frequency in texts. In case of rejecting this relationship in some texts, further variables can be joined, e.g. number of meanings.

Another problem is frequency of occurrence and duration (→).
G.K. Zipf (1935:25) set up two hypotheses on the relation of frequency and length:

(I) “The magnitude of words tends, on the whole, to stand in an inverse (not necessarily proportionate) relationship to the number of occurrences.”
His second hypothesis on the variety of words occurring x-times is simply that of the distribution of word frequency (→).

Zipf himself demonstrated the inverse of (I) using Kaeding´s frequency dictionary of German (Kaeding 1897), i.e. he simply demonstrated the distribution of lengths which turned out to be monotone decreasing (→ word length). Baker (1951) used the letters of a woman with psychic deseases and divided the words in frequency groups; some authors use frequency directly, other ones use the ranks as independent variable. Word length has been counted in terms of the number of phonemes (e.g. Miller, Newman, Friedman 1958) or syllables. Different empirical formulas have been proposed (Belonogov 1962, Guiraud 1954, Kalinin 1964, Guiter 1977). Köhler (1986) discovered an oscillation of lengths and Köhler, Zörnig, Brinkmöller (1990) smoothed the data by taking gliding means. Grzybek, Altmann (2002) and Strauss, Grzybek and Altmann (2005) have shown the dependence of length on frequency in ten languages using individual texts, but other researchers rather use corpora representing mixed samples. Different aspects of this relationship have been shown in Strauss, Grzybek, Altmann (2005).
Krott (1996, 2002) stated the same relationship between frequency and morpheme length.

'''2. Hypothesis'''

''The mean syllabic length (y) of word forms in texts decreases with their frequency of occurrence (x).''

Here each form is considered separately and the mean length of all forms with the same frequency is considered y. Since frequency can be considered in relative form, x can be considered continuous. Further hypotheses can be derived from the above-mentioned one, e.g. shorter forms of the word are more frequent than its longer forms (e.g. case, modus, aspect, tenses). Or, derivates and compounds of a lexeme occur more seldom than the lexeme itself.

'''3. Derivation'''

Köhler (1986), Strauss, Grzybek, Altmann (2005) start from the assumption that the relative rate of change of mean word length decreases proportionally to the relative rate of change of the frequency, as is very usual in synergetic linguistics. If zero-syllabic clitics are considered parts of the following words, then the mean length cannot take values less than 1. Thus the differential equation is

(1) <math>\frac{dL}{L-1}= -b\frac{dF}{F}</math>

from which the well-known formula

(2) <math>L=aF^b+1,\quad a>0,\quad b<0</math>

follows. Here <math>a = e^C\quad</math> (C being the integration sonstant). If one considers frequency as depending on length, inversion yields <math>F = A(L-1)^{-B}\quad</math> with <math>A =a^{1/b},\quad B =1/b</math>. (2) is a special case of of the unified derivation (see Introduction, 4.1). Other empirical formulas can be found in the references.

'''Example.''' Russian text

Strauss, Grzybek, Altmann (2005) present the result for the first chapter of Tolstoj´s Anna Karenina. For each frequency class the mean word-form length has been computed. Zero-syllabic prepositions (''k, s, v'') were considered proclitics. Frequency classes containing fewer than 10 records were pooled and the unweighted average was computed. The result is presented in Table 1 and Fig. 1. In the frist column, the frequency classes are shown, in the second the observed mean word form lengths, and in the third the theoretical mean lengths according to (2).

<div align="center">[[Image:Tabelle_1_WL-FR.jpg]]</div>

<div align="center">[[Image:Figur_1_WL-FR.jpg]]</div>

<div align="center">Fig. 1. Fitting (2) to the data from Anna Karenina Ch. 1.</div>

'''4. Author:''' G. Altmann

'''5. References'''

'''Baker, S.J.''' (1951). A linguistic law of constancy: II. ''The J. of General Psychology 44, 113-120.''

'''Belonogov, G.G.''' (1962). O nekotorych statističeskich zakonomernostjach ruskoj pis´mennoj reči. ''Voprosy jazykoznanija 11/1, 100-101''.

'''Breiter, M.A.''' (1994). Length of Chinese words in relation to their other systemic features. ''J. of Quantitative Linguistics 1, 224-231.''

'''Bürmann, C., Frank, H., Lorenz, L.''' (1963). Informationstheoretische Untersuchungen über Rang und Länge deutscher Wörter. ''Grundlagenstudien aus Kybernetik und Geisteswissenschaft 4 (3-4), 73-90.''

'''Fenk-Oczlon, G.''' (2001). Familiarity, information flow, and linguistic form. In: Bybee, J., Hopper, P. (eds.). ''Frequency and the emergence of linguistic structure: 431-448. Amsterdam: Benjamins''.

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

'''Grzybek, P., Altmann, G.''' (2002). Oscillation in the frequency-length relationship. ''Glottometrics 5, 97-107''.

'''Guiraud, P.''' (1954). ''Les caractères statistiques du vocabulaire. Essai de méthodologie.'' Paris: P.U.F.

'''Guiter, H.''' (1977). Les relations /frequence-longeuer-sens/ des mots (langue romanes et anglais). XVI Congresso Internazionale di Linguistica e Filologia Romanza, Napoli, 15-20 Aprile 1974, 373-381. Napoli: Macchiaroli.

'''Hammerl, R.''' (1990). Länge - Frequenz, Länge - Rangnummer. Überprüfung von zwei lexikalischen Modellen. ''Glottometrika 12, 1-24''.

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

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

'''Kaeding, F.W.''' (1897-98). ''Häufigkeitswörterbuch der deutschen Sprache''. Steglitz: Selbstverlag.

'''Kalinin, V.M'''. (1964a). O statistike literaturnogo teksta. ''Voprosy jazykoznanija Nr. 1, 123-127''.

'''Köhler, R.''' (1986). Zur ''linguistischen Synergetik: Struktur und Dynamik der Lexik.'' Bochum: Brockmeyer.

'''Köhler, R., Zörnig, P., Brinkmöller.''' (1990). Differential equation models for the oscillation of the word length as a function of the frequency. ''Glottometrika 12, 25-40''.

'''Kornai, A.''' (2002). How many words are there? ''Glottometrics 4, 61-86''.

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

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

'''Leopold, E.''' (1997). Frequency spectra within word length classes. In: ''Third International Conference on Quantitative Linguistics, August 26-29, 1997, Helsinki, Finland: 156''. Helsinki: Monila.

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

'''Leopold, E.''' (2000a). Length-distribution of words with coinciding frequency. In: ''Proceedings of the fourth conference of the International Quantitative Linguistic Association, Prague, August 24-26: 76-77''.

'''Miller, G.A., Newman, E.B., Friedman, E.A.''' (1958). Length-frequency statistics for written English.. ''Information and Control 1, 370-389''.

'''Miyajima, T.''' (1992). Relationship in the length, age and frequency of Classical Japanese words. ''Glottometrika 13, 219-229''.

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

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

'''Strauss, U., Grzybek, P., Altmann, G.''' (2005). Word length and word frequency. In: Grzybek, P. (ed.), ''Word length studies and related issues: 255-272''. Boston/Dordrecht: Kluwer.

'''Tuldava, J.''' (1995). ''Methods in quantitative linguistics''. Trier: WVT.

'''Zipf, G.K.''' (1932). ''Selected studies of the principle of relative frequency in language.'' Cam-bridge, Mass.: Harvard University Press.

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

[[Category:word]] [[Category:length]] [[Category:frequency]]

Gap formation

2006-02-22T12:21:44Z

Rkoehler:

'''1. Problem and history'''

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

1 0 0 0 1

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

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

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

'''2. Hypothesis'''

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

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

'''3. Derivations'''

'''3.1. The geometric model'''

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

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

which is adequate in many cases.

'''3.2. Markov chain model'''

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

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

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

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

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

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

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

'''3.3. Urn model'''

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

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

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

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

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

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

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

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

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

Example: Distances between equal rhythmic structures in hexameter

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

<div align="center">Table 1</div>
<div align="center">Fitting of distance models to the data of Strauß et al. (1984)</div>

<div align="Center">[[Image:Tabelle_3_Div.jpg]]</div>

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

'''3.4. The Menzerathian model'''

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

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

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

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

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

<div align="center">Table 2</div>
<div align="center">Distances between the name “Nihat” in a Turkish text (Hřebíček 2000)</div>

<div align="Center">[[Image:Tabelle_2_GF.jpg ]]</div>

The fitting is satisfactory.

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

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

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

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

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

where

n = number of elements in the sequence

m = number of different element types

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

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

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

Let us consider the following sequence:

A B A C D B C A D D B

Here

''n = 11''

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

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

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

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

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

We find here following distances:

Between the A´s 1 and 4

Between the B´s 3 and 4

Between the C´s 2

Between the D´s 3 and 0

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

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

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

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

'''5. References'''

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Diversification

2006-02-22T12:21:19Z

Rkoehler:

'''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).

[[Image:Tabelle1_Div.jpg]]

[[Image:DivFig1.JPG]]

Fig. 1.Fitting the negative binomial distribution to Goebl´s data

'''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">Table 2</div>

<div align="center">Rank-frequency distribution of German neologisms “Noun+Noun” of different grammatical-semantical categories according to Raether, Rothe (1991)</div>

<div align="center">[[Image:Tabelle_2_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>

Fig. 2. Fitting the positive negative binomial distribution (3) to Raether-Rothe data

'''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)\sim\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''': G. Altmann

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

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

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

'''Zsilka, T.''' (1974). ''Stilisztika és statisztika''. Budapest.
</div>

Compounds and polysemy

2006-02-22T12:20:22Z

Rkoehler:

'''1. Problem and history'''

The building of compounds from a stem (word, lexeme, morpheme) depends on different factors, one of them being polysemy and other ones frequency, length, polytexty, word class, age of the stem, etc. Some hypotheses have been collected by Altmann (1989, <math>\rightarrow</math> Compounds: further hypotheses) but derivations and testing is not yet advanced. The most complex problem is the operational definition of the compound word which can be different for different languages. The law could, perhaps, help us to decide for a special definition.
One of the hypotheses considered several times is the relation between polysemy and compounding. The start was made by Rothe (1988) who considered it special case of Menzerath´s law and especially by Steiner (1995, 2002) who embedded the problem in the framework of synergetic linguistics (see also Krott 2002). Steiner speaks rather about “composition activity”. Rothe took a sample and Steiner considered the complete German dictionary.
Below we show a simple dependence, but the problem should be treated as a multidimensional one, i.e. there are always more factors affecting the properties of the compound.

'''2. Hypothesis'''

''The greater the polysemy/polylexy of a stem/lexeme, the more compounds it forms.''

The wording is different with Rothe and Steiner but the meaning of the hypothesis is the same. This is the same relation as between polysemy and word length since words are prolonged by affixation, compounding, reduplication etc.

'''3. Derivation'''

The relative rate of change of the number of compounds (dy/y) built from a stem is proportional to the relative rate of change of its polysemy (x), i.e.

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

resulting in

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

'''Example.''' Polysemy and compounds in German (Rothe 1988)

Rothe took a sample of 1858 lexemes form “Wahrig: Deutsches Wörterbuch” (1985) and tested the hypothesis separately on nouns, verbs, adjectives/adverbs and on the complete sample. Since the independent variable is the number of meanings, the number of compounds has been averaged. For the nouns, Rothe obtained the results shown in Table 1 and Fig. 1.

<div align="center">[[Image:Figur1 CaP.jpg]]</div>

<div align="center">[[Image:Grafik1_CaPneu.jpg]]</div>
<div align="center">Fig. 1. Dependence of the number of compounds on polysemy</div>

Here the fitting is good but some data display greater dispersion. A multidimensional approach would probably improve the results.

'''4. Author:''' G. Altmann

'''5. References'''

'''Altmann, G.''' (1989). Hypotheses about compounds. ''Glottometrika 10, 100-107.''

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

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

'''Rothe, U.''' (1988). Polylexy and compounding. ''Glottometrika 9, 121-134.''

'''Sambor, J.''' (1984). Menzerath's law and the polysemy of words. In: J. Boy & R. Köhler (Hg.), ''Glottometrika 6, 94-114.'' Bochum: Brockmeyer.

'''Sambor, J.''' (1989). Polnische Version des Projekts "Sprachliche Synergetik. Teil I. Quantitative Lexikologie". In: R. Hammerl (ed.), ''Glottometrika 10, 171-197''. Bochum: Brockmeyer.

'''Steiner, P.''' (1995). Effects of polylexy on compounding. ''J. of Quantitative Linguistics 2, 133-140.''

'''Steiner, P.''' (2002). Polylexie und Kompositionsaktivität in Text und Lexik. In: Köhler, R. (ed.), ''Korpuslinguistische Untersuchungen in die quantitative und systemtheoretische Linguistik: 209-251.'' http://ubt.opus.hbz-nrw.de/volltexte/2004/279/

Compounds: further hypotheses

2006-02-22T12:20:01Z

Rkoehler:

'''1. Problem and history'''

Altmann (1989) set up a number of hypotheses on compounds resulting from the synergetic view of language. Up to now, only one of them has been corroborated both theoretically and empirically (<math>\rightarrow</math> Compounds and polysemy). They are presented according to the variable with which compound formation is joined. The problems concerning mathematical modelling, measurement, and testing are here not taken into account; this is preliminarily a mere collect¬ion of hypotheses.

'''2. Hypotheses'''

'''2.1. Hypotheses concerning meaning'''

'''Hypothesis 1'''

''The number of compounds in a language (having compounds) decreases proportionally to the measure of semantic correspondence of the components with the compound; or, inversely, the greater the semantic correspondence of the compound with its components, the greater the number of compounds in a language.''

'''Hypothesis 2'''

''The greater the polylexy of a word, the more compounds there are of which it is a component (<math>\rightarrow</math> Compounds and polysemy).''

'''Hypothesis 3'''

''The longer the compound, the greater its semantic correspondence with its components.''

'''Hypothesis 4'''

''The longer a compound, the fewer meanings it has (on average).''

'''2.2. Hypotheses concerning length'''

'''Hypothesis 1'''

''The shorter a word, the more frequently it occurs in compounds.''

'''Hypothesis 2'''

''The longer the compound, the shorter its components.
This is a simple consequence of Menzerath-Altmann´s law (<math>\rightarrow</math> Hierarchy).''

'''Hypothesis 3'''

''The number of compounds decreases with their increasing length.''

'''2.3. Hypotheses concerning frequency'''

'''Hypothesis 1'''

''The more frequent a word, the more compounds it produces.''

'''2.4. Hypotheses concerning cotextuality (polytexty)'''

'''Hypothesis 1'''

''The greater the cotextuality of a word, the more compounds it produces.''

'''Hypothesis 2'''

''The longer a compound, the smaller its cotextuality.''

'''2.5. Hypothesis concerning age'''

'''Hypothesis 1'''

''The older a word, the more compounds it produces.
Here word classes must be treated separately.''

'''3. Derivations'''

''The derivation of individual hypotheses, if it exists, can be found in the pertinent chapter (see the references marked with →).''

'''4. Author: G. Altmann'''

'''5. References'''

'''Altmann, G.''' (1989). Hypotheses about compounds. ''Glottometrika 10, 100-107.''

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

Change in language

2006-02-22T12:19:33Z

Rkoehler:

'''1. Problem and history'''

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

Qualitative change

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

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

Volume change

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

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

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

'''2. Hypothesis'''

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

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

'''3. Derivation'''

The interaction can be presented in the form

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

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

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

C = limit of change

t > 0, time

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

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

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

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

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

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

<div align="center">[[Image:Figur1_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:Figur2_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:Grafik3_CiL.jpg]]</div>

The fitting is very good.

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

'''4. Author:''' G. Altmann

'''5. References'''

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

'''Best, K.-H.''' (2004). Kürzungstendenzen im Deutschen aus der Sicht der Quantitativen Linguistik. In: Bär, J. A., Roelcke, T., & Steinhauer, A. (Hrsg.), ''Sprachliche Kürze. Konzeptuelle, strukturelle und pragmatische Aspekte.'' Berlin/ New York: de Gruyter. (Im Druck)

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

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

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

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

'''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''.

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

'''Leopold, E.''' (2005). Das Piotrowski-Gesetz. In: Altmann, G., Köhler, R., 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.

Gap formation

2006-02-10T16:54:29Z

Rkoehler:

'''1. Problem and history'''

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

1 0 0 0 1

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

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

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

'''2. Hypothesis'''

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

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

'''3. Derivations'''

'''3.1. The geometric model'''

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

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

which is adequate in many cases.

'''3.2. Markov chain model'''

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

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

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

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

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

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

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

'''3.3. Urn model'''

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

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

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

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

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

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

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

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

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

Example: Distances between equal rhythmic structures in hexameter

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

<div align="center">Table 1</div>
<div align="center">Fitting of distance models to the data of Strauß et al. (1984)</div>

<div align="Center">[[Image:Tabelle_3_Div.jpg]]</div>

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

'''3.4. The Menzerathian model'''

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

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

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

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

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

<div align="center">Table 2</div>
<div align="center">Distances between the name “Nihat” in a Turkish text (Hřebíček 2000)</div>

<div align="Center">[[Image:Tabelle_2_GF.jpg ]]</div>

The fitting is satisfactory.

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

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

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

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

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

where

n = number of elements in the sequence

m = number of different element types

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

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

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

Let us consider the following sequence:

A B A C D B C A D D B

Here

''n = 11''

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

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

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

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

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

We find here following distances:

Between the A´s 1 and 4

Between the B´s 3 and 4

Between the C´s 2

Between the D´s 3 and 0

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

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

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

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

'''5. References'''

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Gap formation

2006-01-06T11:15:25Z

Rkoehler:

'''1. Problem and history'''

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

1 0 0 0 1

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

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

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

'''2. Hypothesis'''

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

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

'''3. Derivations'''

'''3.1. The geometric model'''

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

(1) <math>P_x = pq^x</math>, x = 0,1,2,…

which is adequate in many cases.

'''3.2. Markov chain model'''

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

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

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

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

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

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

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

'''3.3. Urn model'''

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

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

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

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

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

(6)<math>P^'_0(t)=-kP_0(t),</math>
<math>P^'_x(t)=-(k+x)P_x(t)+(k-x-1)P_{x-1}(t), \quad x=1,2,3...</math>

Solving (6) in the usual way and setting e-t = q, one obtains the negative binomial distribution

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

Example: Distances between equal rhythmic structures in hexameter

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

<div align="center">Table 1
Fitting of distance models to the data of Strauß et al. (1984)</div>

All parameters were optimized and the classes were pooled where necessary

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

'''3.4. The Menzerathian model'''

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

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

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

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

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

<div align="center">Table 2
Distances between the name “Nihat” in a Turkish text (Hřebíček 2000)</div>

The fitting is satisfactory.

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

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

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

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

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

where

n = number of elements in the sequence

m = number of different element types

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

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

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

Let us consider the following sequence:

A B A C D B C A D D B

Here

''n = 11''

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

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

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

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

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

We find here following distances:

Between the A´s 1 and 4

Between the B´s 3 and 4

Between the C´s 2

Between the D´s 3 and 0

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

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

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

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

'''5. References'''

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Gap formation

2006-01-06T11:13:37Z

Rkoehler:

'''1. Problem and history'''

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

1 0 0 0 1

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

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

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

'''2. Hypothesis'''

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

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

'''3. Derivations'''

'''3.1. The geometric model'''

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

(1) <math>P_x = pq^x</math>, x = 0,1,2,…

which is adequate in many cases.

'''3.2. Markov chain model'''

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

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

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

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

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

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

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

'''3.3. Urn model'''

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

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

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

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

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

(6)<math>P^'_0(t)=-kP_0(t),</math>
<math>P^'_x(t)=-(k+x)P_x(t)+(k-x-1)P_{x-1}(t), \quad x=1,2,3...</math>

Solving (6) in the usual way and setting e-t = q, one obtains the negative binomial distribution

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

Example: Distances between equal rhythmic structures in hexameter

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

<div align="center">Table 1
Fitting of distance models to the data of Strauß et al. (1984)</div>

All parameters were optimized and the classes were pooled where necessary

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

'''3.4. The Menzerathian model'''

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

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

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

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

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

<div align="center">Table 2
Distances between the name “Nihat” in a Turkish text (Hřebíček 2000)</div>

The fitting is satisfactory.

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

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

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

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

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

where

n = number of elements in the sequence

m = number of different element types

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

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

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

Let us consider the following sequence:

A B A C D B C A D D B

Here

''n = 11''

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

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

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

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

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

We find here following distances:

Between the A´s 1 and 4

Between the B´s 3 and 4

Between the C´s 2

Between the D´s 3 and 0

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

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

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

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

'''5. References'''

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Category:Frequency

2006-01-06T11:09:44Z

Rkoehler:

(first draft)

Frequency is one of the most prominent quantitative properties of linguistic units among others, such as length, comlexity, polysemy, age, polytextuality, and homonymy.

Laws and hypotheses concerning frequency are based on

(1) distributional analyses (in form of rank-frequency distributions, cf. the well-known Zipf (Zipf-Mandelbrot) law, or in the spectral form, which represents the number of units with a given frequency;

(2) functional interrelations such as the dependence of the length of many types of units on their frequency or the dependence of frequency on polytextuality;

(3) the development of the frequency of a given unit (type) over the time.

There are several linguistic units which can be investigated according to their frequency of occurrence: sounds or phonemes, letters, syllables, morph(em)s, words, word classes such as part-of-speech, and even higher units such as syntactic constructions

'''Author: R. Köhler'''

--[[User:Rkoehler|Rkoehler]] 12:09, 6 January 2006 (CET)

[[category: quantitative properties]]

Category:Length

2006-01-06T11:08:56Z

Rkoehler:

first draft)

Length is one of the most prominent quantitative properties of linguistic units among others, such as complexity, frequency, polysemy, age, polytextuality, and homonymy.

Laws and hypotheses concerning length are based on

(1) distributional analyses in the spectral form, which represents the number of units with a given frequency. Length distributions can be analysed on data from tests or from dictionaries (with different results because in dictionaries, the units occur only once;

(2) functional interrelations such as the dependence of the length of many types of units on their frequency or the dependence of polysemy on length;

There are several linguistic units which can be investigated according to their length: sounds or phonemes (duration), syllables, morph(em)s, words, and even higher units such as syntactic constructions.

Author: R. Köhler

--[[User:Rkoehler|Rkoehler]] 12:08, 6 January 2006 (CET)

[[category:quantitative properties]]

Category:Word

2006-01-06T11:06:24Z

Rkoehler:

(preliminary)

Among the linguistic units, the word is probably the most prominent object with respect to the number of quantitative studies which have been published - despite of the difficulties of a satisfactory definition. One of the most common ways to operationalise the word as unit of investigation is the typographical one: Any string of letters delimited by spaces or interpunctation is counted as a word. This definition is, of course, the easiest but also the least meaningful one from a linguistic point of view. For every kind of study, the researches has to decide which of the possible definitions of word will serve the given purpose best. The decision will also depend on factors such as which properties of a word is to be measured. If, e.g. word length lies in the focus of interest, another definition may be appropriate than if part-of-speech distribution or polysemy is investigated.

The following properties of words (as well as their distributions and interrelations between each other) have been studied in quantitative linguistics so far: frequency, length, polysemy, homonymy, age, polytextuality, level of semantic generality/specifity, motivation (compositionality of meaning), origin, ...

'''Author: R. Köhler'''

--[[User:Rkoehler|Rkoehler]] 12:05, 6 January 2006 (CET)

[[category: linguistic units]]

Category:Word

2006-01-06T11:05:38Z

Rkoehler:

(preliminary)

Among the linguistic units, the word is probably the most prominent object with respect to the number of quantitative studies which have been published - despite of the difficulties of a satisfactory definition. One of the most common ways to operationalise the word as unit of investigation is the typographical one: Any string of letters delimited by spaces or interpunctation is counted as a word. This definition is, of course, the easiest but also the least meaningful one from a linguistic point of view. For every kind of study, the researches has to decide which of the possible definitions of word will serve the given purpose best. The decision will also depend on factors such as which properties of a word is to be measured. If, e.g. word length lies in the focus of interest, another definition may be appropriate than if part-of-speech distribution or polysemy is investigated.

The following properties of words (as well as their distributions and interrelations between each other) have been studied in quantitative linguistics so far: frequency, length, polysemy, homonymy, age, polytextuality, level of semantic generality/specifity, motivation (compositionality of meaning), origin, ...

--[[User:Rkoehler|Rkoehler]] 12:05, 6 January 2006 (CET)

[[category: linguistic units]]

Main Page

2006-01-05T22:02:50Z

Rkoehler:

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.

1. January 2006,
Gabriel Altmann, [mailto:koehler@uni-trier.de Reinhard Köhler], Relja Vulanović

The wiki which is used for LQL has been set up by [mailto:burger@ldv.uni-trier.de Götz Burger]. We would like to express our
thanks for his valuable support to this project.

Category:Length

2006-01-05T21:46:56Z

Rkoehler:

first draft)

Length is one of the most prominent quantitative properties of linguistic units among others, such as frequency, polysemy, age, polytextuality, and homonymy.

Laws and hypotheses concerning length are based on

(1) distributional analyses in the spectral form, which represents the number of units with a given frequency. Length distributions can be analysed on data from tests or from dictionaries (with different results because in dictionaries, the units occur only once;

(2) functional interrelations such as the dependence of the length of many types of units on their frequency or the dependence of polysemy on length;

There are several linguistic units which can be investigated according to their length: sounds or phonemes (duration), syllables, morph(em)s, words, and even higher units such as syntactic constructions.

Author: R. Köhler

[[category:quantitative properties]]

Category:Quantitative properties

2006-01-05T21:39:40Z

Rkoehler:

(first draft)

Quantitative properties, as opposed to qualitative ones, are defined in a way which makes possible to measure them and to operate with the results, i.e. to use mathematical methods.
Qualitative properties are defined on a so-called nominal, or categorial scales, which means that the only possible operation is to compare them with respect to identity or non-identity. Every day color terms are an example of qualitative values, they constitute the quanlitative property ''color''. An object can be compared to other objects with respect to color only by qualitative means, i.e. by determining whether both objects share their color or not.

Quantitative properties can be defined on three different scales. The first one is the ordinal or comparative scale. An exaple of such a property is the way school-boys may determine which of them is the strongest one, the second strongest etc. The values of such a scale allows to arrange objects according to the corresponding order but they do not allow to give more precise information. They allow us to determine that an object possesses more or less of a given property than another one, or the same amount of it – formally: P(A) > P(B), P(A) = P(B) or P(A) < P(B). Applying this kind of concept yields a higher degree of order, viz. a ranking of the objects with respect to a given property. A linguistic example is the grammatical acceptability of sentences. The highest degree of order is achieved with the help of metrical concepts, which are needed if the difference between the amounts of a given property object a and b possess plays a role. In this case, the values of the property are mapped to the elements of an appropriate set of numbers, i.e. a set of numbers in which the relations between these numbers correspond to the relations between the values of the properties of the objects. In this way, specific operations such as subtraction correspond to specific differences or distances in the properties between the objects – formally: P(A) – P(B) = d. This enables the researcher to establish an arbitrarily fine conceptual grid within his field of study. Concepts which allow determining distances or similarities between objects are called interval-scale concepts. If another feature is added, viz. a fixed point of reference (e.g. an absolute zero) ratio-scaled concepts are obtained, which allow the operation of multiplication and division, formally: P(A) = aP(B) + d. Only the latter scale enables to formulate how many times object A has more than B of a property.

Often, quantitative concepts are introduced indirectly. Quantification can start from established (or potential) quantitative concepts and adding the needed features. One has to make sure that the conceptual scale is chosen properly, i.e. the concepts must be formed according to the mathematical operations which correspond to the properties and relations of the objects. The polysemy of words may serve as a linguistic example of an indirectly introduced quantitative concept. Polysemy is originally a qualitative concept in traditional linguistics which identifies or differentiates words with respect to lexical ambiguity. Taking this as a starting point, a quantitative variant of this concept can easily be created: It may be defined as the number of meanings of a linguistic expression; the values admitted are cardinal numbers in the interval [1,∞), i.e. the smallest possible value is 1 whereas an upper limit cannot be specified. This is a well-defined ratio-scale concept: using basic mathematical operations, differences in polysemy between words can be expressed (e.g. word x has three meanings more than word y) and even the ratio between the polysemy of two words values can be specified (word x has twice as many meanings as word y), since we have a fixed reference point: the minimum polysemy 1.
Only by means of concepts on higher scales, i.e. quantitative ones is it possible to pose deeper-reaching questions and even to make corresponding observations. Thus, without our quantitative concept of polysemy no-one could even notice that there is a lawful relation be-tween the number of meanings of a word and its length).

'''Author: R. Köhler'''

Category:Frequency

2006-01-05T19:47:11Z

Rkoehler:

(first draft)

Frequency is one of the most prominent quantitative properties of linguistic units among others, such as length, polysemy, age, polytextuality, and homonymy.

Laws and hypotheses concerning frequency are based on

(1) distributional analyses (in form of rank-frequency distributions, cf. the well-known Zipf (Zipf-Mandelbrot) law, or in the spectral form, which represents the number of units with a given frequency;

(2) functional interrelations such as the dependence of the length of many types of units on their frequency or the dependence of frequency on polytextuality;

(3) the development of the frequency of a given unit (type) over the time.

There are several linguistic units which can be investigated according to their frequency of occurrence: sounds or phonemes, letters, syllables, morph(em)s, words, word classes such as part-of-speech, and even higher units such as syntactic constructions

'''Author: R. Köhler'''

[[category: quantitative properties]]

Category:Frequency

2006-01-05T19:46:52Z

Rkoehler:

(first draft)

Frequency is one of the most prominent quantitative properties of linguistic units among others, such as length, polysemy, age, polytextuality, and homonymy.

Laws and hypotheses concerning frequency are based on

(1) distributional analyses (in form of rank-frequency distributions, cf. the well-known Zipf (Zipf-Mandelbrot) law, or in the spectral form, which represents the number of units with a given frequency.

(2) functional interrelations such as the dependence of the length of many types of units on their frequency or the dependence of frequency on polytextuality.

(3) the development of the frequency of a given unit (type) over the time.

There are several linguistic units which can be investigated according to their frequency of occurrence: sounds or phonemes, letters, syllables, morph(em)s, words, word classes such as part-of-speech, and even higher units such as syntactic constructions

'''Author: R. Köhler'''

[[category: quantitative properties]]

Category:Frequency

2006-01-05T19:43:20Z

Rkoehler:

(first draft)

Frequency is one of the most prominent quantitative properties of linguistic units among others, such as length, polysemy, age, polytextuality, and homonymy.

Laws and hypotheses concerning frequency are based on

(1) distributional analyses (in form of rank-frequency distributions, cf. the well-known Zipf (Zipf-Mandelbrot) law, or in the spectral form, which represents the number of units with a given frequency.

(2) functional interrelations such as the dependence of the length of many types of units on their frequency or the dependence of frequency on polytextuality.

There are several linguistic units which can be investigated according to their frequency of occurrence: sounds or phonemes, letters, syllables, morph(em)s, words, word classes such as part-of-speech, and even higher units such as syntactic constructions

'''Author: R. Köhler'''

[[category: quantitative properties]]

Category:Frequency

2006-01-05T19:36:49Z

Rkoehler:

= Frequency =
(first draft)
Frequency is one of the most prominent quantitative properties of linguistic units among others, such as length, polysemy, age, polytextuality, and homonymy.

Laws and hypotheses concerning frequency are based either on distributional analyses (in form of rank-frequency distributions, cf. the well-known Zipf (Zipf-Mandelbrot) law, or in the spectral form, which represents the number of units with a given frequency.

'''Author: R. Köhler'''

[[category: quantitative properties]]

Word length and frequency

2006-01-05T18:40:18Z

Rkoehler:

'''1. Problem and history'''

There are two problems that must be strictly separated:

(a) The use of words of specific length in texts. Here ''word forms'' are meant and neither the size of the lexicon nor the number of phonemes in the phoneme inventory are relevant. Other properties like polysemy, polytexty, or synthetism can be added for modelling textual word length.

(b) Frequency as a factor of ''lemma construction'' in the dictionary, where frequency is taken from a frequency dictionary, and the dictionary size, as well as the size of the phoneme inventory, are relevant.

In (b), problems arise because the size of the dictionary can be merely coarsely estimated. There are even authors considering it infinite (cf. Piotrowski, Bektaev, Piotrovskaja 1985; Kornai 2002), which is a reasonable assumption. If, however, it is considered as a fixed finite value, it can be taken into account. Further, frequency dictionaries strongly depend on the kind of the analyzed texts. In order to secure a reliable frequency dictionary even for a short time period, an astronomical number of words must be counted.

Problem (a) is easily solvable but it must be considered for individual texts or group of texts from the same author. Here we consider merely the problem of dependence of word form length on its frequency in texts. In case of rejecting this relationship in some texts, further variables can be joined, e.g. number of meanings.

Another problem is frequency of occurrence and duration (→).
G.K. Zipf (1935:25) set up two hypotheses on the relation of frequency and length:

(I) “The magnitude of words tends, on the whole, to stand in an inverse (not necessarily proportionate) relationship to the number of occurrences.”
His second hypothesis on the variety of words occurring x-times is simply that of the distribution of word frequency (→).

Zipf himself demonstrated the inverse of (I) using Kaeding´s frequency dictionary of German (Kaeding 1897), i.e. he simply demonstrated the distribution of lengths which turned out to be monotone decreasing (→ word length). Baker (1951) used the letters of a woman with psychic deseases and divided the words in frequency groups; some authors use frequency directly, other ones use the ranks as independent variable. Word length has been counted in terms of the number of phonemes (e.g. Miller, Newman, Friedman 1958) or syllables. Different empirical formulas have been proposed (Belonogov 1962, Guiraud 1954, Kalinin 1964, Guiter 1977). Köhler (1986) discovered an oscillation of lengths and Köhler, Zörnig, Brinkmöller (1990) smoothed the data by taking gliding means. Grzybek, Altmann (2002) and Strauss, Grzybek and Altmann (2005) have shown the dependence of length on frequency in ten languages using individual texts, but other researchers rather use corpora representing mixed samples. Different aspects of this relationship have been shown in Strauss, Grzybek, Altmann (2005).
Krott (1996, 2002) stated the same relationship between frequency and morpheme length.

'''2. Hypothesis'''

''The mean syllabic length (y) of word forms in texts decreases with their frequency of occurrence (x).''

Here each form is considered separately and the mean length of all forms with the same frequency is considered y. Since frequency can be considered in relative form, x can be considered continuous. Further hypotheses can be derived from the above-mentioned one, e.g. shorter forms of the word are more frequent than its longer forms (e.g. case, modus, aspect, tenses). Or, derivates and compounds of a lexeme occur more seldom than the lexeme itself.

'''3. Derivation'''

Köhler (1986), Strauss, Grzybek, Altmann (2005) start from the assumption that the relative rate of change of mean word length decreases proportionally to the relative rate of change of the frequency, as is very usual in synergetic linguistics. If zero-syllabic clitics are considered parts of the following words, then the mean length cannot take values less than 1. Thus the differential equation is

(1) <math>\frac{dL}{L-1}= -b\frac{dF}{F}</math>

from which the well-known formula

(2) <math>L=aF^b+1,\quad a>0,\quad b<0</math>

follows. Here <math>a = e^C</math> (C being the integration sonstant). If one considers frequency as depending on length, inversion yields <math>F = A(L-1)^{-B}</math> with <math>A =a^\frac{1}{b},B =\frac{1}{b}</math>. (2) is a special case of of the unified derivation (see Introduction, 4.1). Other empirical formulas can be found in the references.

'''Example.''' Russian text

Strauss, Grzybek, Altmann (2005) present the result for the first chapter of Tolstoj´s Anna Karenina. For each frequency class the mean word-form length has been computed. Zero-syllabic prepositions (''k, s, v'') were considered proclitics. Frequency classes containing fewer than 10 records were pooled and the unweighted average was computed. The result is presented in Table 1 and Fig. 1. In the frist column, the frequency classes are shown, in the second the observed mean word form lengths, and in the third the theoretical mean lengths according to (2).

<div align="center">Table 1
Fitting (2) to the data in Tolstoj´s Anna Karenina Chapter 1
(Strauss, Grzybek, Altmann (2005)</div>

<div align="center">[[Image:Tabelle_1_WL-FR_Kopie.jpg]]</div>

<div align="center">[[Image:Figur_1_WL-FR.jpg]]</div>

<div align="center">Fig. 1. Fitting (2) to the data from Anna Karenina Ch. 1.</div>

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

'''5. References'''

'''Baker, S.J.''' (1951). A linguistic law of constancy: II. ''The J. of General Psychology 44, 113-120.''
'''Belonogov, G.G.''' (1962). O nekotorych statističeskich zakonomernostjach ruskoj pis´mennoj reči. ''Voprosy jazykoznanija 11/1, 100-101''.

'''Breiter, M.A.''' (1994). Length of Chinese words in relation to their other systemic features. ''J. of Quantitative Linguistics 1, 224-231.''

'''Bürmann, C., Frank, H., Lorenz, L.''' (1963). Informationstheoretische Untersuchungen über Rang und Länge deutscher Wörter. ''Grundlagenstudien aus Kybernetik und Geisteswissenschaft 4 (3-4), 73-90.''

'''Fenk-Oczlon, G.''' (2001). Familiarity, information flow, and linguistic form. In: Bybee, J., Hopper, P. (eds.). ''Frequency and the emergence of linguistic structure: 431-448. Amsterdam: Benjamins''.

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

'''Grzybek, P., Altmann, G.''' (2002). Oscillation in the frequency-length relationship. ''Glottometrics 5, 97-107''.

'''Guiraud, P.''' (1954). ''Les caractères statistiques du vocabulaire. Essai de méthodologie.'' Paris: P.U.F.

'''Guiter, H.''' (1977). Les relations /frequence-longeuer-sens/ des mots (langue romanes et anglais). XVI Congresso Internazionale di Linguistica e Filologia Romanza, Napoli, 15-20 Aprile 1974, 373-381. Napoli: Macchiaroli.

'''Hammerl, R.''' (1990). Länge - Frequenz, Länge - Rangnummer. Überprüfung von zwei lexikalischen Modellen. ''Glottometrika 12, 1-24''.

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

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

'''Kaeding, F.W.''' (1897-98). ''Häufigkeitswörterbuch der deutschen Sprache''. Steglitz: Selbstverlag.

'''Kalinin, V.M'''. (1964a). O statistike literaturnogo teksta. ''Voprosy jazykoznanija Nr. 1, 123-127''.

'''Köhler, R.''' (1986). Zur ''linguistischen Synergetik: Struktur und Dynamik der Lexik.'' Bochum: Brockmeyer.

'''Köhler, R., Zörnig, P., Brinkmöller.''' (1990). Differential equation models for the oscillation of the word length as a function of the frequency. ''Glottometrika 12, 25-40''.

'''Kornai, A.''' (2002). How many words are there? ''Glottometrics 4, 61-86''.

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

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

'''Leopold, E.''' (1997). Frequency spectra within word length classes. In: ''Third International Conference on Quantitative Linguistics, August 26-29, 1997, Helsinki, Finland: 156''. Helsinki: Monila.

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

'''Leopold, E.''' (2000a). Length-distribution of words with coinciding frequency. In: ''Proceedings of the fourth conference of the International Quantitative Linguistic Association, Prague, August 24-26: 76-77''.

'''Miller, G.A., Newman, E.B., Friedman, E.A.''' (1958). Length-frequency statistics for written English.. ''Information and Control 1, 370-389''.

'''Miyajima, T.''' (1992). Relationship in the length, age and frequency of Classical Japanese words. ''Glottometrika 13, 219-229''.

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

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

'''Strauss, U., Grzybek, P., Altmann, G.''' (2005). Word length and word frequency. In: Grzybek, P. (ed.), ''Word length studies and related issues: 255-272''. Boston/Dordrecht: Kluwer.

'''Tuldava, J.''' (1995). ''Methods in quantitative linguistics''. Trier: WVT.

'''Zipf, G.K.''' (1932). ''Selected studies of the principle of relative frequency in language.'' Cam-bridge, Mass.: Harvard University Press.

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

[[Category:word]] [[Category:length]] [[Category:frequency]]

Main Page

2006-01-05T18:31:34Z

Rkoehler:

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.

1. January 2006,
Gabriel Altmann, [mailto:koehler@uni-trier.de Reinhard Köhler], Relja Vulanović

This wiki has been set up by [mailto:burger@ldv.uni-trier.de Götz Burger]. We would like to express our
thanks for his valuable support of the project.

Main Page

2005-12-20T18:13:19Z

Rkoehler:

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.

1. January 2006,
Gabriel Altmann, [mailto:koehler@uni-trier.de Reinhard Köhler], Relja Vulanovic

This wiki has been set up by [mailto:burger@ldv.uni-trier.de Götz Burger]. We would like to express our
thanks for his valuable support of the project.

Main Page

2005-12-20T18:12:27Z

Rkoehler:

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.

1. January 2006,
Gabriel Altmann, [mailto:koehler@uni-trier.de Reinhard Köhler], Relja Vulanovi&accentc

This wiki has been set up by [mailto:burger@ldv.uni-trier.de Götz Burger]. We would like to express our
thanks for his valuable support of the project.

Category:Probieren

2005-12-20T17:51:11Z

Rkoehler:

Studieren geht über Probieren.

Main Page

2005-12-19T12:19:55Z

Rkoehler:

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.

1. January 2006,
Gabriel Altmann, [mailto:koehler@uni-trier.de Reinhard Köhler], Relja Vulanovic

This wiki has been set up by [mailto:burger@ldv.uni-trier.de Götz Burger]. We would like to express our
thanks for his valuable support of the project.

Main Page

2005-12-19T12:19:00Z

Rkoehler:

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.

1. January 2006,
Gabriel Altmann, [mailto:koehler@uni-trier.de Reinhard Köhler], Relja Vulanovi<math>\mathrm{\acute c}</math>

This wiki has been set up by [mailto:burger@ldv.uni-trier.de Götz Burger]. We would like to express our
thanks for his valuable support of the project.

Main Page

2005-12-19T12:18:01Z

Rkoehler:

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.

1. January 2006,
Gabriel Altmann, [mailto:koehler@uni-trier.de Reinhard Köhler], Relja Vulanovi<math>\acute c</math>

This wiki has been set up by [mailto:burger@ldv.uni-trier.de Götz Burger]. We would like to express our
thanks for his valuable support of the project.

Main Page

2005-12-19T12:17:42Z

Rkoehler:

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.

1. January 2006,
Gabriel Altmann, [mailto:koehler@uni-trier.de Reinhard Köhler], Relja Vulanovi<math>\acutec</math>

This wiki has been set up by [mailto:burger@ldv.uni-trier.de Götz Burger]. We would like to express our
thanks for his valuable support of the project.

Main Page

2005-12-19T12:16:46Z

Rkoehler:

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.

1. January 2006,
Gabriel Altmann, [mailto:koehler@uni-trier.de Reinhard Köhler], Relja Vulanovi&acutec

This wiki has been set up by [mailto:burger@ldv.uni-trier.de Götz Burger]. We would like to express our
thanks for his valuable support of the project.

Main Page

2005-12-19T12:16:04Z

Rkoehler:

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.

1. January 2006,
Gabriel Altmann, [mailto:koehler@uni-trier.de Reinhard Köhler], Relja Vulanovi\&acutec

This wiki has been set up by [mailto:burger@ldv.uni-trier.de Götz Burger]. We would like to express our
thanks for his valuable support of the project.