On the possibility of mathematical modelling of the evolution of the polysemy of natural language signs with using of non-stationary birth-death processes

We consider the possibility of mathematical modeling of the evolution of polysemy of ensemble of signs of natural language by means of non-stationary processes of birth and death. We showed that an adequate mathematical model of polysemy of ensemble of signs might be built on the base of hidden non-stationary model of the birth and death processes of the meanings of linguistic signs. We assume exponential decay of the intensities of the processes of birth and death: ) = X0 ep- t -10 , = Ц0 p (t - t0 V2 , where t is the current time; t₀ is the time moment when the sign appears in the ensemble; X₀, ц₀ are the initial values of intensities of the processes of birth and death; т = G / X₀, т₂ = G / ц₀ are time decay constants of intensities, and G is the average number of meanings, which the sign may birth and lose during his life: G = jX(t)dt = X₀tj , G = |ц(()dt = ц₀т₂ . We received the conditional (with fixed parameters t₀, X₀, Ц₀, G) probability distribution of states n of this process: · ^ f и(Л \ c (t) Г (a (t) +1) f b (t) f X ^ k=0 k !(n - k )!Г(п₀а (t)-k +1)) 1 -b{t ц(<) Pn (t|S) = exp -(1 - b (t)) (1 - b (t)) where b(() = expf- jц(()dt^ , c(()=(t)-;(). In the hidden model of the statistical ensemble of processes of birth and death the parameters t₀, X₀, ц₀, G of each individual process (of each linguistic sign) randomly vary in relation of each to other, subject to certain distribution laws. Under the assumption of a Pois-son distribution of the flow of signs, the distribution density of the parameter t0 can be considered as uniform on a large enough time interval, while the distributions of parameters X₀, ц₀, G are unknown. Unconditional probability distribution P_n(t) of the state n of an ensemble of the processes of birth-death (of the polysemy of an ensemble of signs) at moment t is the mathematical expectation of the conditional distribution P_n(t|0) over the distribution of parameters t₀, X₀, ц₀, G. We have solved the task of estimation of the parameter distributions (for identifying of hidden model) according to the empirical polysemy distribution P_ne obtained from a representative dictionary, with the subsequent calculation of the optimal theoretical distribution Pn(t). As an identification criterion (criterion of proximity of distribution), we select a logarithmic RMS criterion of type: =5) J log Pn (t) - log P_m ( ) log Pns (t) j=± z n₀ n=1 ^ min convenient for large (several orders of magnitude) changes in distributions for different n. The criterion was implemented on example of using of the dictionary of Pushkin's language. We obtain a good agreement of distributions P_n(t) and P_ne that confirms the possibility of using of hidden mathematical model of non-stationary process of birth-death for the simulation of polysemy evolution of the ensemble of signs of natural language.

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