Law - Revision history

imported>Anonymous at 14:02, 21 April 2009

2009-04-21T14:02:54Z

← Older revision		Revision as of 14:02, 21 April 2009
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	The third kind of law is the developmental one. Here, a property is related to time. The best known example is the Piotrowski Law, which represents the development (increase and/or decrease) of the portion of new units or forms over time. This law is a typical growth process and can be derived from a simple differential equation with the solution:		The third kind of law is the developmental one. Here, a property is related to time. The best known example is the Piotrowski Law, which represents the development (increase and/or decrease) of the portion of new units or forms over time. This law is a typical growth process and can be derived from a simple differential equation with the solution:

	<math> p(t) = {\frac{1}{1+a \~~centerdot~~ e ^{-r \~~centerdot~~ c \~~centerdot~~ t}}} </math>		<math> p(t) = {\frac{1}{1+a \cdot e ^{-r \cdot c \cdot t}}} </math>

	where p is the proportion of new forms at time t, c is the saturation value, and a and b are empirical parameters. Fig. 2 shows the increase of the forms with /u/ at the cost of the older form with /a/ in the German word ward>wurde (/vart/ > /vurde/) in the time period from 1445 to 1925.		where p is the proportion of new forms at time t, c is the saturation value, and a and b are empirical parameters. Fig. 2 shows the increase of the forms with /u/ at the cost of the older form with /a/ in the German word ward>wurde (/vart/ > /vurde/) in the time period from 1445 to 1925.

imported>Anonymous: Additing solution formula for piotrowski

2009-04-21T14:01:51Z

Additing solution formula for piotrowski

← Older revision		Revision as of 14:01, 21 April 2009
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	Other examples of functional laws are the dependence of word (or morph) frequency on word (or morph) length, the frequency of syntactic constructions on their complexity, of polysemy on length, of length on age, etc.		Other examples of functional laws are the dependence of word (or morph) frequency on word (or morph) length, the frequency of syntactic constructions on their complexity, of polysemy on length, of length on age, etc.
	The third kind of law is the developmental one. Here, a property is related to time. The best known example is the Piotrowski Law, which represents the development (increase and/or decrease) of the portion of new units or forms over time. This law is a typical growth process and can be derived from a simple differential equation with the solution:		The third kind of law is the developmental one. Here, a property is related to time. The best known example is the Piotrowski Law, which represents the development (increase and/or decrease) of the portion of new units or forms over time. This law is a typical growth process and can be derived from a simple differential equation with the solution:
	,
			<math> p(t) = {\frac{1}{1+a \centerdot e ^{-r \centerdot c \centerdot t}}} </math>

	where p is the proportion of new forms at time t, c is the saturation value, and a and b are empirical parameters. Fig. 2 shows the increase of the forms with /u/ at the cost of the older form with /a/ in the German word ward>wurde (/vart/ > /vurde/) in the time period from 1445 to 1925.		where p is the proportion of new forms at time t, c is the saturation value, and a and b are empirical parameters. Fig. 2 shows the increase of the forms with /u/ at the cost of the older form with /a/ in the German word ward>wurde (/vart/ > /vurde/) in the time period from 1445 to 1925.

imported>Anonymous at 13:57, 21 April 2009

2009-04-21T13:57:31Z

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	<math>		<math>T = aL^b</math>

	~~T = aL^b~~


	~~</math>~~

	where y is the number of types, x -- the number of tokens (= text position), and b -- an empiri-cal parameter, a text characteristic. The parameter a is equal to unity if types and tokens are measured in terms of the same unit (as in almost all cases). The law is valid whether word-forms or lemmas are counted, just with a different parameter b. This parameter is also an indi-cator of the morphological type of the language under study if word-forms are considered because morphologically rich languages display a faster increase in word-form types than isolating languages.		where y is the number of types, x -- the number of tokens (= text position), and b -- an empiri-cal parameter, a text characteristic. The parameter a is equal to unity if types and tokens are measured in terms of the same unit (as in almost all cases). The law is valid whether word-forms or lemmas are counted, just with a different parameter b. This parameter is also an indi-cator of the morphological type of the language under study if word-forms are considered because morphologically rich languages display a faster increase in word-form types than isolating languages.

imported>Anonymous at 13:56, 21 April 2009

2009-04-21T13:56:48Z

← Older revision		Revision as of 13:56, 21 April 2009
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	<math>		<math>

	T = aL^~~be^{cL}~~		T = aL^b

imported>Anonymous at 13:56, 21 April 2009

2009-04-21T13:56:03Z

← Older revision		Revision as of 13:56, 21 April 2009
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	A variant of this third kind of law is based on (discrete) ‘linguistic’ instead of (continuous) physical time. The simplest way to operationalize linguistic time is the reference to text posi-tion. In oral texts, there is a direct correspondence of the sequence of linguistic units to physi-cal time intervals while written texts map this correspondence in a slightly more indirect way.		A variant of this third kind of law is based on (discrete) ‘linguistic’ instead of (continuous) physical time. The simplest way to operationalize linguistic time is the reference to text posi-tion. In oral texts, there is a direct correspondence of the sequence of linguistic units to physi-cal time intervals while written texts map this correspondence in a slightly more indirect way.
	A typical example of this variant is the type-token ratio (TTR), which was, in the beginning, a single number (the quotient of the number of different words, the types, and the number of running words, the tokens), used to characterize the vocabulary richness of a text. Later, it became apparent that this value is inappropriate for several reasons. Instead, at each text position, the number of types occurred so far is counted, which yields a monotonously increasing curve, because the number of words used before a given text position cannot decrease in the course of the rest of the text. A straightforward theoretical derivation of a corresponding law was given by Gustav Herdan (Herdan, 1966), represented by the simple formula:		A typical example of this variant is the type-token ratio (TTR), which was, in the beginning, a single number (the quotient of the number of different words, the types, and the number of running words, the tokens), used to characterize the vocabulary richness of a text. Later, it became apparent that this value is inappropriate for several reasons. Instead, at each text position, the number of types occurred so far is counted, which yields a monotonously increasing curve, because the number of words used before a given text position cannot decrease in the course of the rest of the text. A straightforward theoretical derivation of a corresponding law was given by Gustav Herdan (Herdan, 1966), represented by the simple formula:
	,

			<math>

			T = aL^be^{cL}


			</math>

	where y is the number of types, x -- the number of tokens (= text position), and b -- an empiri-cal parameter, a text characteristic. The parameter a is equal to unity if types and tokens are measured in terms of the same unit (as in almost all cases). The law is valid whether word-forms or lemmas are counted, just with a different parameter b. This parameter is also an indi-cator of the morphological type of the language under study if word-forms are considered because morphologically rich languages display a faster increase in word-form types than isolating languages.		where y is the number of types, x -- the number of tokens (= text position), and b -- an empiri-cal parameter, a text characteristic. The parameter a is equal to unity if types and tokens are measured in terms of the same unit (as in almost all cases). The law is valid whether word-forms or lemmas are counted, just with a different parameter b. This parameter is also an indi-cator of the morphological type of the language under study if word-forms are considered because morphologically rich languages display a faster increase in word-form types than isolating languages.
	A problem of the TTR is that it is not independent of the overall text length. Therefore, more complicated formulae are used to take this influence into account or quite different models (cf. Popescu, Altmann 2006, 2007) are applied.		A problem of the TTR is that it is not independent of the overall text length. Therefore, more complicated formulae are used to take this influence into account or quite different models (cf. Popescu, Altmann 2006, 2007) are applied.

imported>Anonymous at 13:53, 21 April 2009

2009-04-21T13:53:40Z

← Older revision		Revision as of 13:53, 21 April 2009
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	\quad \mbox{with} \ c_i \not= c_j, i \not= j.		\quad \mbox{with} \ c_i \not= c_j, i \not= j.
	\quad (\mbox{~~We note that~~ for}\quad k_s = 0 : \sum_{i=1}^{k_5}		\quad (\mbox{for}\quad k_s = 0 : \sum_{i=1}^{k_5}
	\frac{a_{ji}}{(x-b_{ji})^{c_s}})		\frac{a_{ji}}{(x-b_{ji})^{c_s}})

imported>Anonymous at 13:47, 21 April 2009

2009-04-21T13:47:44Z

← Older revision		Revision as of 13:47, 21 April 2009
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	and for the discrete case		and for the discrete case

			<math>

			\frac{\Delta{P_{x-1}}}{P_{x-1}}
			=
			a_0+\sum_{i=1}^{k_1}
			\frac{a_{1i}}{(x-b_{1i})^{c_1}}+\sum_{i=1}^{k_2}\frac{a_{2i}}{(x-b_{2i})^{c_2}}+
			... \quad .

			</math>

imported>Anonymous: Additing discrete case for formula one

2009-04-21T13:41:47Z

Additing discrete case for formula one

← Older revision		Revision as of 13:41, 21 April 2009
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	\quad (\mbox{We note that for}\quad k_s = 0 : \sum_{i=1}^{k_5}		\quad (\mbox{We note that for}\quad k_s = 0 : \sum_{i=1}^{k_5}
	\frac{a_{ji}}{(x-b_{ji})^{c_s}})		\frac{a_{ji}}{(x-b_{ji})^{c_s}})



	</math>		</math>






	and for the discrete case		and for the discrete case

	or

	Both are well interpretable linguistically and yield the same results as the synergetic ap-proach. The great majority of laws known up to now can be derived from the above equations (e.g. Menzerath´s law, Zipf-Mandelbrot law, Frumkina´s law, all laws of length, diversifica-tion laws, TTR, synonymy, polysemy, polytextuality laws, morphological productivity, vo-cabulary growth, Krylov´s law, the law of change, etc.). The discrete and continuous ap-proaches can be transformed into one another (cf. Mačutek, Altmann 2007) and yield all discrete probability distributions used in linguistics. The parameters are interpreted as specific language forces as known from synergetic linguistics.		Both are well interpretable linguistically and yield the same results as the synergetic ap-proach. The great majority of laws known up to now can be derived from the above equations (e.g. Menzerath´s law, Zipf-Mandelbrot law, Frumkina´s law, all laws of length, diversifica-tion laws, TTR, synonymy, polysemy, polytextuality laws, morphological productivity, vo-cabulary growth, Krylov´s law, the law of change, etc.). The discrete and continuous ap-proaches can be transformed into one another (cf. Mačutek, Altmann 2007) and yield all discrete probability distributions used in linguistics. The parameters are interpreted as specific language forces as known from synergetic linguistics.

imported>Anonymous at 13:35, 21 April 2009

2009-04-21T13:35:24Z

← Older revision		Revision as of 13:35, 21 April 2009
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	\quad \mbox{with} \ c_i \not= c_j, i \not= j.		\quad \mbox{with} \ c_i \not= c_j, i \not= j.
	\quad (\mbox{We note that for} k_s = 0 : \sum_{i=1}^{k_5}		\quad (\mbox{We note that for}\quad k_s = 0 : \sum_{i=1}^{k_5}
	\frac{a_{ji}}{(x-b_{ji})^{c_s}}		\frac{a_{ji}}{(x-b_{ji})^{c_s}})

imported>Anonymous at 13:34, 21 April 2009

2009-04-21T13:34:50Z

← Older revision		Revision as of 13:34, 21 April 2009
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	dx		dx

	\quad \mbox{with} \ c_i \not= c_j		\quad \mbox{with} \ c_i \not= c_j, i \not= j.
			\quad (\mbox{We note that for} k_s = 0 : \sum_{i=1}^{k_5}
			\frac{a_{ji}}{(x-b_{ji})^{c_s}}