# Dynamics of ternary statistical experiments with equilibrium state

aa r X i v : . [ s t a t . O T ] J un Dynamics of ternary statistical experimentswith equilibrium state

M.L.Bertotti ♭ , S.O. Dovgyi ♯ , D. Koroliouk ♯♭ Free University of Bozen-Bolzano, Faculty of Science and Technology, PiazzaUniversita’ 5, 39100 Bozen-Bolzano, Italy. ♯ Institute of telecommunications and global information space Ukr.Acad.Sci.,Chokolovskiy Boulevard 13, 03110 Kiev, Ukraine.

Abstract.

We study the scenarios of the dynamics of ternary statistical experiments,modeled by means of diﬀerence equations. The important features are a balance conditionand the existence of a steady-state (equilibrium). We give a classiﬁcation of scenarios ofthe model’s evolution which are signiﬁcantly diﬀerent between them, depending on thedomain of the values of the model basic parameters V and ρ (see Proposition 1). Key words: ternary statistical experiment, persistent regression, equilibrium state,limit behaviour, classiﬁcation of scenarios.

Equilibrium processes are common in complex systems and play important role in manymechanisms of interaction and self-regulation. Studying these processes, one should de-velop an adequate method of description, analysis and prediction of the behavior of suchsystems, taking into account the action of a wide variety of external factors. In the taskof monitoring the dynamics of equilibrium system , one can study the frequencies (ofconcentrations) of a ﬁxed set of alternative features in increasing discrete time instants(also called stages).Such a frequency model has numerous practical interpretations such as the concen-tration of a substance in chemical reactions, the establishment or breaking of chemicalbonds between molecules, the interaction between the elements that make up a complexsystem, the learning process of a team, etc.In particular, if there are three alternatives, the model is called a ternary statisti-cal experiment and determines the frequency of occurrence of one of the three possibleattributes. So the sum of the three frequencies is always 1.The existence of three alternative features means that the analysis can actually becarried out for two independent frequencies. However, for the sake of analysis symmetry,it is advisable, in the case of three frequencies, to analyze vector statistical experiments,which dimension coincides with the total number of features. For this reason, ternarystatistical experiments with three possible attributes are deﬁned by three-dimensionalvectors.In this setting, the mathematical model can be built as statistical experiments in whichthe frequency of the presence of attributes depends on the set of their frequency in thesystem at the previous stage. 1he equilibrium of a system is determined by the form of this dependence, whichis called the regression function with directing parameters that provide a change thefrequency of a given attribute in proportion to its frequency simultaneously with a changein the frequencies of other attributes (alternatives).In the works [1], [2], [3], [4], [5] a model of binary statistical experiments with persistentregression and equilibrium is considered for several aspects of investigation. The presentwork considers the ternary model in a three-dimensional scheme of discrete stationaryMarkov diﬀusion, deﬁned by a vector diﬀerence stochastic equation. The classiﬁcationof ternary statistical experiments limit dynamics is given in base of speciﬁc law of largenumbers by passing to a double limit by the sample volume and by the discrete timeparameter.

We consider statistical experiments (SE) with persistent linear regression [1] with addi-tional alternatives.The basic idea of the model construction is to choose a main factor that determinesthe essential state of SE, supplemented by additional alternatives in the way that theaggregation of the principal factor and its complementary alternatives completely describethe dynamics of SE on time.The basic characteristic of the main factor and of the additional alternatives are theirprobabilities (frequencies): P of the main factor and P , P of the additional alternatives,for which the balance condition takes place: P + P + P = 1 . (1)The dynamics of SE characteristics is determined by a linear regression function [2]which speciﬁes the values of SE characteristics in the next stage of observation, for givenvalue probability at the present stage.Consider a sequence of SE characteristics values which depends on the stage of obser-vation, or, equivalently, on a discrete time parameter k ≥ P ( k ) := ( P ( k ) , P ( k ) , P ( k )) , k ≥ , and their increments at k -th time instant:∆ P ( k + 1) := P ( k + 1) − P ( k ) , k ≥ . The linear regression function of increments is deﬁned by a matrix which is generated bydirecting action parameters: ∆ P ( k + 1) = − b V P ( k ) , k ≥ , (2)where b V := [ b V mn ; 0 ≤ m, n ≤ , b V mm = 2 V m , b V mn = − V n , ≤ n ≤ , n = m. (3)2he directing action parameters V , V , V satisfy the following inequality [1]: | V m | ≤ , ≤ m ≤ . (4)An important feature of SE is the presence of a steady state ρ (equilibrium), which isdetermined by zero of the regression function of increments : b V ρ = 0 , (5)or in scalar form: b V m ρ := b V m ρ + b V m ρ + b V m ρ = 0 , ≤ m ≤ . (6)Of course, the following balance condition takes place: ρ + ρ + ρ = 1 . (7)Next, we consider the ﬂuctuations probabilities relative to equilibrium value b P m ( k ) := P m ( k ) − ρ m , ≤ m ≤ . (8) The basic assumption.

The SE dynamics is determined by a diﬀerence equation for themain factor probabilities b P ( k ), and by the probabilities of additional alternatives b P ( k )and b P ( k ) ∆ b P ( k + 1) = − b V b P ( k ) , k ≥ , (9)or in scalar form:∆ b P m ( k + 1) = b V m b P ( k ) + b V m b P ( k ) + b V m b P ( k ) , ≤ m ≤ , k ≥ . (10)Also the initial values have to be ﬁxed: b P (0) = ( b P (0) , b P (0) , b P (0)) . Remark . Considering equations (5) - (6) and the balance condition (7), we have explicitformulas for equilibrium: ρ m = V − m /V , ≤ m ≤ ,V := V − + V − + V − , or in other form: ρ = V V /V , ρ = V V /V , ρ = V V /V,V := V V + V V + V V . (11)The validity of the formulas (11) can be easily conﬁrmed by their substitution inequations (6) - (7). Obviously this holds true under the additional condition: V = 0. Remark . The dynamics determination by the linear regression function (9) - (10) inregression model of statistic experiments, does not envolves the balance condition (1),and the equilibrium (5) with additional restrictions:0 ≤ P m ( k ) ≤ , ≤ m ≤ , k ≥ ≤ ρ m ≤ , ≤ m ≤ The model interpretation

The model of SE is constructed in several stages. First, the main factor should be chosen,characterized by probability (or frequency, concentration etc.). So there exist supplemen-tary alternatives, whose probabilities are complement to the main factor probability. Inparticular, having only one alternative, the classiﬁcation of binary models are consideredin [6] (see also [7], [8]). The presence of two or more alternatives brings more diﬃcultiesin the analysis of SE.With a full set of SE characteristics, the probabilities of the main factor and of addi-tional alternatives satisfy the balance condition (1) or, equivalently, the balance condition(7) and the dynamics of the main factor probability P , as well as of supplementary factors P , P is given by the following diﬀerence equations for the probabilities of ﬂuctuationsfor all k ≥

0: ∆ b P ( k + 1) = V b P ( k ) + V b P ( k ) − V b P ( k ) , ∆ b P ( k + 1) = V b P ( k ) + V b P ( k ) − V b P ( k ) , ∆ b P ( k + 1) = V b P ( k ) + V b P ( k ) − V b P ( k ) . (12)The increment of probabilities ﬂuctuations of the main and supplementary factors∆ b P m ( k + 1) := b P m ( k + 1) − b P m ( k ) , ≤ m ≤ , k ≥ , is determined by the values of directing action parameters V , V , V . Remark . The ﬂuctuations of probabilities in (8) satisfy the balance condition: b P ( k ) + b P ( k ) + b P ( k ) = 0 , k ≥ , (13)and by formula (8) one has:∆ b P m ( k ) = ∆ P m ( k ) , ≤ m ≤ , k ≥ . (14)The equation (12) characterizes two basic principles of alternatives interaction: stim-ulation (positive term) and containment (negative term). The existence of an equilibrium point for the ﬂuctuations increments regression function(5) provides the possibility to analyze the dynamics of SE (by k → ∞ ) in view of thepossible directing parameter values which satisfy the constraint (4).The dynamics of the main factor probability is described by several scenarios . Proposition . The main factor probability P ( k ), k ≥