Decomposition of conditional probability for high-order symbolic Markov chains

TL;DR

This paper introduces a decomposition method for the conditional probability function of high-order symbolic Markov chains, enabling better sequence modeling by capturing complex correlations.

Contribution

It develops a novel decomposition procedure representing the conditional probability as a sum of multi-linear memory functions, bridging likelihood estimation and additive Markov chains.

Findings

Memory functions are expressed in terms of high-order correlations at weak dependencies.

The method allows sequential sequence generation considering increasing correlation complexity.

Potential application in neural network training approximation.

Abstract

The main goal of the paper is to develop an estimate for the conditional probability function of random stationary ergodic symbolic sequences with elements belonging to a finite alphabet. We elaborate a decomposition procedure for the conditional probability function of sequences considered as the high-order Markov chains. We represent the conditional probability function as the sum of multi-linear memory function monomials of different orders (from zero up to the chain order). This allows us to construct artificial sequences by method of successive iterations taking into account at each step of iterations increasingly more high correlations among random elements. At weak correlations, the memory functions are uniquely expressed in terms of the high-order symbolic correlation functions. The proposed method fills up the gap between two approaches: the likelihood estimation and the additive Markov chains. The obtained results might be used for sequential approximation of artificial neural networks training.

Equations

$$\mathbb{S}=a_0,a_1,a_2,...$$

$$\mathcal{A}=\{{\alpha }^1,{\alpha }^2,...,{\alpha }^m\},a_i\in \mathcal{A},i\in {\mathbb{N}}_{+}=\{0,1,\mathrm{2...}\}.$$

$$P(a_i={\alpha }^k|\dots ,a_{i-2},a_{i-1})$$

$$=P(a_i={\alpha }^k|a_{i-N},\dots ,a_{i-2},a_{i-1}).$$

$$C(r_1,r_2,\dots ,r_{k-1})$$

$$=\overline{[a_0-\overline{a})][a_{r_1}-\overline{a})]\dots [a_{r_1+\dots +r_{k-1}}-\overline{a}]},$$

$$C_{{\beta }_1,...{\beta }_k}(r_1,r_2,\dots ,r_{k-1})$$

$$=$$

$$P(a_i=\alpha |a_{i-N},\dots ,a_{i-1})=\frac{P(a_{i-N},\dots ,a_{i-1},\alpha )}{P(a_{i-N},\dots ,a_{i-1})},$$

$$N<L_{max}\sim \frac{\ln M}{\ln m}.$$

$$P_{add}=P^{(1)}(a_i=\alpha |a_{i-N},\dots ,a_{i-2},a_{i-1})$$

$$=p_{\alpha }+\sum _{r=1}^N\sum _{\beta \in \mathcal{A}}{F}_{\alpha \beta }(r)[\delta (a_{i-r},\beta )-p_{\beta }].$$

$$p_{\beta }=\overline{\delta (a_i,\beta )}.$$

$$C_{\alpha \beta }(r)=\sum _{r^{\prime }=1}^N\sum _{\gamma \in \mathcal{A}}{C}_{\alpha \gamma }(r-r^{\prime })F_{\beta \gamma }(r^{\prime }).$$

$$C_{\alpha\beta}(r)\!=\!\overline{\big{[}\delta(a_{i},\alpha)-p_{\alpha}\big{]}\!\big{[}\delta(a_{i+r},\beta)-p_{\beta}\big{]}}.$$

$$Dist=\overline{[\delta (a_i,\alpha )-P(a_i=\alpha |a_{i-N}^{i-1}){]}^2}.$$

$$P(.|.)=P(a_i=\alpha |a_{i-N}^{i-1})$$

$$=\sum _{{\beta }_1...{\beta }_N\in \mathcal{A}}\sum _{r_1...r_N}{F}_{\alpha ;{\beta }_1...{\beta }_N}(r_1,...,r_N)\prod _{s=0}^N\delta (a_{i-r_s},{\beta }_s).$$

$$F_{\alpha ;{\beta }_1...{\beta }_N}(r_1,...,r_N)=\sum _{k=0}^N{F}_{\alpha ;{\beta }_{i_1}...{\beta }_{i_k}}(r_{i_1},...,r_{i_k}),$$

$$r_1⩽r_{i_1}<r_{i_2}<...<r_{i_k}⩽r_N,$$

$$P(a_i=\alpha |a_{i-N}^{i-1})=\sum _{k=0}^N{Q}^{(k)}(.|.)$$

$$Q^{(0)}=p_{\alpha },$$

$$Q^{(1)}=\sum _{\beta \in \mathcal{A}}\sum _{r=1}^N{F}_{\alpha \beta }(r)[\delta (a_{i-r},\beta )-p_{\beta }].$$

$$Q^{(k)}(.|.)=\sum _{{\beta }_1...{\beta }_k\in \mathcal{A}}\sum _{1⩽r_1<...<r_k⩽N}{F}_{\alpha ;{\beta }_1...{\beta }_k}(r_1,...,r_k)$$

$$\times \{\prod _{s=1}^k[\delta (a_{i-r_s},{\beta }_s)-p_{{\beta }_s}]-C_{{\beta }_k...{\beta }_1}(r_k-r_{k-1},...,r_k-r_1)\}.$$

$$C_{\alpha \beta \gamma \delta }(2,2,-3)=$$

$$\displaystyle\!\!\!\!\!\overline{\big{[}\delta(a_{0},\alpha)\!-\!p_{\alpha}\big{]}\!\big{[}\delta(a_{2},\beta)\!-\!p_{\beta}\big{]}\big{[}\delta(a_{4},\gamma)\!-\!p_{\gamma}\big{]}\!\big{[}\delta(a_{1},\delta)\!-\!p_{\delta}\big{]}}$$

$$\displaystyle\!\!\!\!\!=\!\!\overline{\big{[}\delta(a_{0},\alpha)\!-\!p_{\alpha}\big{]}\!\!\big{[}\delta(a_{1},\delta)\!-\!p_{\delta}\big{]}\!\!\big{[}\delta(a_{2},\beta)\!-\!p_{\beta}\big{]}\!\!\big{[}\delta(a_{4},\gamma)\!-\!p_{\gamma}\big{]}\!}$$

$$=C_{\alpha \delta \beta \gamma }(1,1,2).$$

$$\sum _{{\alpha }_m\in \mathcal{A}}{C}_{{\alpha }_1,...{\alpha }_k}(r_1,\dots ,r_{k-1})=0,1⩽m⩽k.$$

$$Q^{(2)}(.|.)=Q^{(2)}(a_i=\alpha |a_{i-N}^{i-1})$$

$$=\sum _{\beta ,\gamma \in \mathcal{A}}\sum _{1⩽r_1<r_2⩽N}{F}_{\alpha ;\beta \gamma }(r_1,r_2)\times$$

$$\{[\delta (a_{i-r_1},\beta )-p_{\beta }][\delta (a_{i-r_2},\gamma )-p_{\gamma }]-C_{\gamma \beta }(r_2-r_1)\}.$$

$$P_{bilin}(.|.)=P^{(2)}(.|.)=P^{(1)}(.|.)+Q^{(2)}(.|.).$$

$$\overline{[\delta (a_0,{\beta }_1)-p_{{\beta }_1}]\dots [\delta (a_{r_1+\dots +r_{k-1}},{\beta }_k)-p_{{\beta }_k}]}$$

$$=\overline{[\delta (a_0,{\beta }_1)-p_{{\beta }_1}]\dots [P(a_{r_1+\dots +r_{k-1}}={\beta }_k|\cdot )-p_{{\beta }_k}]},$$

Full text

I Introduction

Systems with long-range interactions (and/or sequences with long-range memory) and natural sequences with nontrivial information content have been the focus of a large number of studies in different fields of science for the past several decades.

Such random sequences are the subject of study of the algorithmic (Kolmogorov-Solomonoff-Chaitin) complexity, information theory, compressibility of digital data, statistical inference problem, computability, data compression Salomon , natural language processing Mann , artificial intelligence Wiss and have many application aspects.

Random sequences with a finite number of state space exist as natural sequences (DNA or natural language texts) or arise as a result of coarse-grained mapping of the evolution of the chaotic dynamical system into a string of symbols Eren ; Lind . The items of sequence can also be phonemes, syllables, words or DNA’s base pairs according to the application.

A standard method of understanding and describing statistical properties of a given random symbolic sequence of data requires the estimation of the joint probability function of $L$-words (subsequences of the length $L$) occurring. Reliable estimations for the probability can be achieved only for small $L$ because the number $m^{L}$ (where $m$ is the finite-alphabet length) of different words of the length $L$ has to be much less than the total number of words $m^{L}\ll M-L$ in the whole sequence of the length $M$. This is the crucial point because usually the correlation lengths of natural sequences of interest are of the same order as the length of sequence, whereas the last inequality can be fulfilled for the words maximal lengths $L_{max}\lesssim 10$ only.

The main purpose and results of the paper can be formulated as following. We elaborate a decomposition procedure for the conditional probability function of sequences considered as the high-order Markov chains. We represent the conditional probability function as the sum of multi-linear monomials of different orders (from zero up to the chain order). At weak correlations, the memory functions are uniquely expressed in terms of the high-order symbolic correlation functions.

At last, we reveal close connection between our analytical Markov chain approach and artificial neuron network models.

II Symbolic Markov chains

Consider a semi-infinite random stationary ergodic sequence $\mathbb{S}$ of symbols (letters, characters) $a_{i}$,

[TABLE]

taken from the finite alphabet

[TABLE]

We use the notation $a_{i}$ to indicate a position $i$ of the symbol $a$ in the chain and the unified notation $\alpha^{k}$ to stress the value of the symbol $a\in\mathcal{A}$. We also use the personified notation for the symbols $a$ of the same alphabet, $\mathcal{A}=\{\alpha,\beta,...,\omega\}.$

We suppose that the symbolic sequence $\mathbb{S}$ is the high-order Markov chain. The sequence $\mathbb{S}$ is the Markov chain if it possesses the following property: the probability of symbol $a_{i}$ to have a certain value $\alpha^{k}\in\mathcal{A}$ under the condition that all previous symbols are fixed depends only on $N$ previous symbols,

[TABLE]

There are many other terms for such sequences. They are also referred to as: categorical Hoss , higher-order Raftery ; Seifert , the multi- or the $N$-step RewUAMM ; UYa Markov’s chains. One of the most important and interesting application of the symbolic sequences is the probabilistic language model specializing in predicting the next item in such a sequence by means of $N$ previous known symbols. Here the Markov chain is known as the $N$-gram model.

As a rule, the statistical properties of random sequences are determined by correlation functions, however, in distinction from the ordinary, numeric correlation functions,

[TABLE]

the symbolic correlation functions of the $k$th order is given by the following expression,

[TABLE]

The $\overline{overlines}$ mean a statistical average over an ensemble of sequences. It can be replaced by the average along the chain or by the arithmetic, Cesàro’s average for numerical purposes. Note, that in some sense symbolic correlation functions-matrices are more general construction than numeric correlation functions. They can describe in more detail even numeric sequences.

II.1 Likelihood estimation

If the sequence, the statistical properties of which we would like to analyze, is given, then the conditional probability function (CPF) of the $N$th order can be found by a standard method known under the name of the likelihood estimation,

[TABLE]

where $P(a_{i-N},\ldots,a_{i-1},\alpha)$ and $P(a_{i-N},\ldots,a_{i-1})$ are the probabilities of the $(N+1)$-subsequence $a_{i-N},\ldots,a_{i-1},\alpha$ and $N$-subsequence $a_{i-N},\ldots,a_{i-1}$ occurring, respectively. Hereafter, we often drop the superscript $k$ from $\alpha^{k}$ to simplify the notations.

III Formulation of the problem

The conditional probability function completely determines all statistical properties of the random chain and the method of its generation. Equation (6) says that the CPF is determined if we know the probability of $(N+1)$-words occurring, the words containing $(N+1)$ symbols without omissions among their numbers. This instrument perfectly works if the random chain under study satisfies the inequality

[TABLE]

For $N>L_{max}$ this method represents adequately statistical properties of the chain at $L<L_{max}$ only. The main idea of the present paper is to elaborate a method, which would allow to use the information contained in the “perforated”, holed words such as, e.g., $(a_{i}a_{i+3}),(a_{i}a_{i+5}),(a_{i}a_{i+N}),(a_{i}a_{i+5}a_{i+8})$ and so on.

III.1 Additive high-order Markov chains

The unsatisfactory conclusion expressed by inequality Eq. (7), often referred to as the problem of the “curse of dimensionality”, induced us to propose MUYaG ; muya an alternative method based on the additive high-order Markov chain, where the conditional probability function takes on the specific, simplified, “linear form” with respect to the random variables $a_{i}$

[TABLE]

Here $p_{\beta}$ is the relative number of symbols $\beta$ in the chain, or their probabilities of occurring,

[TABLE]

The Kronecker delta-symbol $\delta(.,.)$ plays the role of the characteristic function of the random variable $a_{i}$ and converts symbols to numbers. The additivity of the chain means that the “previous” symbols $a_{i-N},\ldots,a_{i-2},a_{i-1}\equiv a_{i-N}^{i-1}$ exert an independent effect on the probability of the “final“, generated symbol $a_{i}=\alpha$ occurring. The first term in the right-hand side of Eq. (III.1) is responsible for the correct reproduction of statistical properties of uncorrelated sequences, the second one takes into account and correctly reproduces correlation properties of the chain up to the second order. The higher order correlation functions are not independent anymore. We cannot control them and reproduce correctly by means of the memory function $F(r)$.

There were suggested two methods for finding the memory functions $F_{\alpha\beta}(r)$ of a sequence with a known pair correlation functions. The first one is a completely probabilistic straightforward calculation analogous to that presented in MUYaG . Its modification is used below while obtaining Eqs. (25) and (V). For any values of $\alpha,\beta\in\mathcal{A}$ and $r\geqslant 1$ the relationship between the correlation and memory functions was obtained,

[TABLE]

Here the two-point (binary, pair) symbolic correlation function is the particular case of definition (5),

[TABLE]

The second method muya for deriving Eq. (10) is based on the minimization of the “distance” between the conditional probability function, containing the sought-for memory function, and the given sequence $\mathbb{S}$ of symbols with a known correlation function,

[TABLE]

Let us note, that in the considered case the two-point quantities, the memory and correlation functions, are not obliged to satisfy strong inequality (7). In other words the additive Markov chain, (III.1), can describe and predict two-point statistical properties of random sequences at distances longer than that based on Eq. (4). At the same time, the model of random sequences based on the likelihood estimation works more satisfactory at short distances; see, e.g., the result of the DNA’s entropy estimation MU ; DNA16Mus , where the discrepancy between the two theories is evident. The next step to improve the prediction quality of symbols in random chain, which we present here, is based on the $k$-linear conditional probability function, which should close the mentioned gap and, probably, explain a very astonishing fact why binary correlations are so important and so long.

IV Decomposition of the conditional probability function

Equation (III.1) can be considered as an approximate model expression simplifying the general form of the conditional probability function. As a matter of fact the conditional probability (6) of the symbolic sequence of random variables $a_{i}\in\mathcal{A}$ can be represented exactly as a finite polynomial series containing $N$ Kroneker delta-symbols: a specific decomposed form of the CPF, which expresses some “independence” of the random variables $a$ and spatial coordinates $i$,

[TABLE]

Here the arguments $r_{1},...,r_{N}$ of the function $F_{\alpha;\beta_{1}...\beta_{N}}(r_{1},...,r_{N})$, supposed to be ordered $r_{1}<r_{2}<...<r_{N-1}<r_{N}$, indicate the distances between the final “generated” symbol $a_{i}=\alpha$ and symbols $a_{i-1},...,a_{i-N}$. It is clear that there is one-to-one correspondence between $P(a_{i}=\alpha|a_{i-N}^{i-1})$ and the function $F_{\alpha;\beta_{1}...\beta_{N}}(r_{1},...,r_{N})$, which is referred to as the generalized memory function. The characteristic functions $\delta(a_{i-r_{s}},\beta_{s})$ play the pole of basis and the generalized memory functions are coordinates of the CPF. We hope the reader paid attention to the difference between $\alpha^{k},k\in(1,...,m)$ and $\alpha_{s}$, where subscript $s,s\in(1,...,N)$ enumerate different sets of letters.

It is helpful to decouple the memory function $F_{\alpha;\beta_{1}...\beta_{N}}(r_{1},...,r_{N})$ and present it in the form of the sum of memory functions of $k$th order, $F^{(k)}=F_{\alpha;\beta_{i_{1}}...\beta_{i_{k}}}(r_{i_{1}},...,r_{i_{k}})$,

[TABLE]

where all symbols $r_{i}$ at the right hand side of Eq. (14) are different, ordered

[TABLE]

and contain all different sets $\{r_{i_{1}},\!...,r_{i_{k}}\}$ picked out from the set $\{r_{1},...,r_{N}\}$. Inserting (14) in (13) we get

[TABLE]

with the following definitions of $Q^{(0)}$ and $Q^{(1)}$, see Eqs. (III.1) and  (13),

[TABLE]

and

[TABLE]

The general term of $k$th order, referred in multi-linear algebra to as a $k$-linear form, is

[TABLE]

In Eq. (19) we have added the term $C_{\beta_{1}...\beta_{k}}$ to provide the equality $\overline{Q^{(k)}(.|.)}=0$ for all $k=1,2,\ldots,N$.

Definition Eq. (5) is correct for $r_{i}>0$, $i=1,\ldots,k-1$. If some arguments of the correlation function in Eq. (5) are negative one should interpret it by the following way, which is referred highord to as the ”collating”; we should order the arguments of the function according to definition (5). For example,

[TABLE]

We used this method to represent the correlation function in Eq. (19) in the collating form.

The function $C_{\alpha_{1},...\alpha_{k}}$ depends on $k$ arguments, but the Markov chain under consideration is supposed to be homogenous. Then function $C_{\alpha_{1},...\alpha_{k}}$ depends on $k-1$ arguments, the differences between the indexes, $r_{1},r_{2},\ldots,r_{k-1}$, of neighbor symbols. A trivial property of the function $C_{\alpha_{1},...\alpha_{k}}(r_{1},\ldots,r_{k-1})$ is

[TABLE]

The last term $Q^{(N)}$ in Eq. (16) contains all different arguments $r_{1},...,r_{k}$. For each fixed set of symbols $\alpha;\beta_{1}...\beta_{N}$ there is just one function-constant $F_{\alpha;\beta_{1}...\beta_{N}}(1,...,N)$.

The k-linear conditional probability function $P^{(k)}(.|.)$ in the form of Eqs. (16),  (19) can reproduce correctly the correlations of the Markov chain up to the $k$th order. For the value of $k=N$ the function $P^{(N)}(.|.)$ represents exactly the function $P(a_{i}=\alpha|a_{i-N}^{i-1})$, Eq. (6).

Thus, the conditional probability function $P(a_{i}=\alpha^{k}|a_{i-N}^{i-1})$ is presented as decomposition of multi-liner form into $k$-linear subspaces. Hosseinia, Leb and Zideka Hoss proved rigorously that the conditional probability can be written as a linear combination of the monomials of past process responses for the Markov chain. Earlier this idea was presented in Besag’s paper Besag .

The utility of the decomposition procedure can be explained in the following way. First, the partial terms $Q^{(k)}$ of the CPF are mainly responsible for reproducing the correlation properties of $k$th order, see below Eq. (33) and second, can be considered as asymptotic successive approximation for the CPF.

V Bilinear CPF

The model of the additive high-order Markov chain is well studied RewUAMM . In this Section we examine high-order Markov chains with a bilinear conditional probability function.

The right-hand side of Eq. (III.1) contains two first terms of asymptotic expression of the exact form, Eq. (13). The next term $Q^{(2)}$ is

[TABLE]

The conditional probability function, which contains linear term $P_{add}=P^{(1)}(.|.)$ and bilinear function $Q_{2}$ (see Eqs. (III.1) and (V)) defines the Markov chain with the bilinear memory function,

[TABLE]

It is possible to find the recurrence relations for the correlation functions of the $N$-step bilinear Markov chain. For this purpose first of all we should calculate explicitly the average value of symbol $a_{r_{1}+\ldots+r_{k-1}}$ in Eq. (5). Taking into account the equation $P(a=\alpha|\cdot)+P(a\neq\alpha|\cdot)=1$ we can rewrite Eq. (5) for arbitrary $k\geqslant 2$ in the form,

[TABLE]

[TABLE]

where CPF $P(a_{r_{1}+\ldots+r_{k-1}}=\alpha_{k}|\cdot)$ is given by Eq. (13). In that way, we can obtain the fundamental recurrence relation connecting the correlation functions of different orders $k$. Here we restrict ourselves by presenting these equations for the correlation functions $C_{\alpha\beta}(r)$ and $C_{\alpha\beta\gamma}(r_{1},r_{2})$ and bilinear CPF, Eq. (23),

[TABLE]

Equation for $C_{\alpha\beta\gamma}(r_{1},r_{2})$ reads

[TABLE]

The other way to get Eqs. (25) and (V) is based on the minimization of the “distance”, Eq. (12).

The system of equations (V) allows us to find the unknown memory functions $F_{\alpha\beta}(r)$ and $F_{\alpha\beta\gamma}(r_{1},r_{2})$ for consecutive constructing a representative random sequence with given correlation functions of the second and third orders. The memory functions should be expressed by means of the probability $p_{\alpha}$ and the correlation functions $C_{\alpha\beta}(r)$ and $C_{\alpha\beta\gamma}(r_{1},r_{2})$, which can be found numerically by means of analysis of a given random chain.

Equations (25) and (V) enable us to understand, that higher-order correlators and all correlation properties of higher orders are not independent anymore. We cannot control them and reproduce correctly by means of the memory function $F_{\alpha\beta}(r)$ because the latter is completely determined by the pair correlation function. Really, when we generate an additive Markov chain, the second order memory function $F_{\gamma;\eta\varepsilon}(r^{\prime}_{1},r^{\prime}_{2})$ equals zero. Correlation function of the third order $C_{\alpha\beta\eta}(r_{1},r_{2})$, determined by Eq. (V), depends on $F_{\beta\gamma}(r_{1})$.

We can make a similar conclusion about $k$th order memory functions. They generate sequences for which we control the $k$th order correlation function. The $N$ step Markov chain with $k$-linear memory function allows us to reproduce correctly the chains up to the correlation function of $k$th order.

VI Approximate solution of equations

Equations (25) and (V) can be analytically solved only in some particular cases: for one- or two-step chains, the Markov chain with a step-wise memory function and so on. Here we give their approximate solution supposing that correlations in the sequence are not too strong (in amplitude, but not in length) and the alphabet $\mathcal{A}$ contains many letters. In order to formulate these conditions we introduce the normalized symbolic correlation function defined by

[TABLE]

If correlations in the random chain are not strong, it is plausible to suppose that the all components of the normalized correlation function with $r\neq 0$ are small with respect to $K_{\alpha\beta}(0)=1$.

Neglecting the second term in the right-hand side of Eq. (25) (correctness of this approximation is explained below, after Eq. (34)) we get Eq. (10). The solution of this equation can be written in the form

[TABLE]

if in definition (27) of $C_{\alpha\beta}(0)$ we can neglect the term $p_{\alpha}p_{\beta}$ with respect to $p_{\alpha}$. This is possible if the dimension $|\mathcal{A}|$ of alphabet $\mathcal{A}$ satisfies the condition

[TABLE]

so that all probabilities $p_{\alpha}$ are small.

It is easy to see that after substituting Eq. (28) into  (III.1), we can rewrite the additive conditional probability in the intuitively clear form,

[TABLE]

which explains probabilistic meaning of Eq. (28) – in this approximation each symbol $a_{i-r},\,1\leqslant r\leqslant N$ effects independently on the probability to generate $a_{i}$.

Our analysis of Eq. (V) shows that we can neglect the first term in RHS of Eq. (V) with respect to the term $C_{\alpha\beta\gamma}(r_{1},r_{2})$ in LHS because $F_{\gamma\eta}(r^{\prime}_{1})$ contains only nondiagonal small component of $C_{\eta\gamma}(r^{\prime}_{1})$). By the same reason the term $C_{\alpha\beta}(r_{1})C_{\varepsilon\eta}(r^{\prime}_{2}-r^{\prime}_{1})$ is small with respect to $C_{\alpha\beta\varepsilon\eta}(r_{1},r_{2}-r^{\prime}_{2},r^{\prime}_{2}-r^{\prime}_{1})$. The last statement follows from estimation of the correlator $C_{\alpha\beta\varepsilon\eta}(r_{1},r_{2}-r^{\prime}_{2},r^{\prime}_{2}-r^{\prime}_{1})$. Its largest unique component satisfying the conditions $r_{2}^{\prime}\geqslant r_{1}^{\prime}+1,r^{\prime}_{1}\geqslant 1$ is $C_{\alpha\beta\alpha\beta}(r_{1},-r_{1},r_{1})$ at $r^{\prime}_{1}=r_{2},r^{\prime}_{2}=r_{1}+r_{2}$. Thus, Eq. (V) reduces to

[TABLE]

Taking into account Eq. (29) and neglecting correlations while calculating $C_{\alpha\beta\alpha\beta}(r_{1},-r_{1},r_{1})$ we get

[TABLE]

Equation (13) for the conditional probability function of the symbolic high-order Markov chain with bilinear memory function (in the first approximation with respect to the small parameters $|C_{\alpha\beta}(r)|\ll 1,\,r\neq 0$ and a multi-letter alphabet, $p_{\alpha}\ll 1$ ) takes the form

[TABLE]

VII K-linear form of the CPF

Equations (28) and (32) show that we can hope to obtain the similar expressions for generalized memory functions of $k$th order expressed by means of correlation function. Really, the result of such calculations can be presented in the form

[TABLE]

To obtain this result let us summarize the main steps of this procedure:

(i) calculate the correlation function of $(k+1)$th order, $C_{\alpha_{1}...\alpha_{k+1}}(r_{1},...,r_{k})$, $1\leqslant k\leqslant N$;

(ii) while calculating use $P(.|.)=P^{(N)}(.|.)=\sum_{k=0}^{N}Q^{(k)}(.|.)$.

(iii) as a result the correlation function is presented as a sum of the memory functions of different order from 1 to $N$ with coefficients $C_{\alpha^{\prime}_{1}\alpha^{\prime}_{2}...}(r^{\prime}_{1},r^{\prime}_{k},...)$.

(iv) in the main term of the sum containing $\sum_{r^{\prime},\beta}F_{\alpha_{k+1};\beta_{1}...\beta_{k}}(r^{\prime}_{1},...,r^{\prime}_{k})\times C_{\alpha_{1}...\alpha_{k}\beta_{k}...\beta_{1}}(r_{1},...,r_{k-1},r_{k}-r^{\prime}_{k},r^{\prime}_{k-1}-r^{\prime}_{k-2},r^{\prime}_{2}-r^{\prime}_{1})$ find the maximal term $C_{\alpha_{1}...\beta_{1}}(r_{1},...,r^{\prime}_{2}\!-\!r^{\prime}_{1})$. It is maximal if in two increasing sequences $0,r_{1},\!...\!,r_{k-1}$ and $r_{k}\!-\!r^{\prime}_{k},r^{\prime}_{k-1}\!-\!r^{\prime}_{k-2},r^{\prime}_{2}\!-\!r^{\prime}_{1}$ (rewritten accordingly to the collating procedure as $0,r_{1},r_{1}+r_{2},...,r_{1}+r_{2}+...+r_{k-1}$ and $r_{1}+...+r_{k}-r^{\prime}_{k},r_{1}+...+r_{k}-r^{\prime}_{k-1},...,r_{1}+...+r_{k}-r^{\prime}_{1}$) there is one-to-one correspondence among their terms: $r^{\prime}_{1}=r_{k},r^{\prime}_{2}=r_{k}+r_{k-1}$ and $r^{\prime}_{k}=r_{k}+...+r_{1}$ .

(iv) neglact correlations while obtaing $C_{\alpha_{1}...\beta_{1}}(r_{1},...,r^{\prime}_{2}\!-\!r^{\prime}_{1})=\prod_{s=1}^{k}p_{\alpha_{s}}\delta(\alpha_{s},\beta_{s})$;

(v) all others terms containing $F_{\alpha_{s+1};\beta_{1}...\beta_{s}}(r^{\prime}_{1},...,r^{\prime}_{s})$ with $s\neq k$ are small with respect to that taken into account; they contain additional small factors $p^{r},r\geqslant 1$;

Thus, the conditional probability function Eq. (13) for the symbolic high-order Markov chain in the first approximation with respect to the small parameters $|C_{\alpha\beta}(r)|\ll 1,\,r\neq 0$ and a multi-letter alphabet, $p_{\alpha}\ll 1$  is expressed by means of “experimentally” measured quantities, i.e., the correlation functions. Taking into account property (21), $\sum_{\alpha_{m}}C_{\alpha_{1},...\alpha_{k}}(r_{1},\ldots,r_{k-1})=0$, it is convenient to present the final main result of the paper in the form of series Eq. (16), where we should substitute Eq. (19) and replace the memory function by Eq. (34),

[TABLE]

Equation (VII) provides a tool for constructing weak correlated sequences with given, prescribed correlation functions. Note that the $i$-independence of the function $P(a_{i}=\alpha|a_{i-N}^{i-1})$ provides homogeneity and stationarity of the sequence under consideration. According to the Markov theorem (see, e.g., Ref. shir ), the finiteness of $N$ together with the strict inequalities

[TABLE]

provides ergodicity of the random sequence.

We see that if the correlations are week, all terms of the CPF are independent of each other. If, e.g., we generate a sequence using the terms of zero order and bilinear one we find that all correlators are equal to zero except the third order correlator. In the general case it is not correct. When, e.g., we generate a sequence with an additive memory function there appear correlations of all order, not only pair ones.

VIII What to do if $p_{\alpha}$ are not small

If the real sequence under study does not satisfy the condition $p_{\alpha}\ll 1$, as, for example, it is for nucleotide DNA sequences, where all four probabilities $p$ of the different nucleotide occurring is of the order $1/4$, we cannot apply Eq. (VII) for obtaining the CPF by means of correlation functions. To make this possible we should decrease the probabilities $p$. For this purpose we could use the idea proposed by Jiménez-Montaño, Ebeling, and others Jim ; Jim2 ; Rapp who suggested coding schemes of the non-sequential recursive pair substitution. Each successive substitution is accompanied by decrease of probability $p_{\widetilde{\alpha}}$, where $\widetilde{\alpha}$ belong to a new extended alphabet.

IX Artificial neural network

Above we presented the analytical method of finding the memory functions $F_{\alpha;\beta_{1}...\beta_{N}}(r_{1},...,r_{N})$ and expressed them in terms of correlation functions. In this section, we expose briefly numerical method of network training – estimation of unknown parameters in a network. The result of this procedure should be the values of the matrix-functions $F(.)$.

According to definition Haykin , artificial neural networks (ANN) are a family of connectionist models used to estimate or approximate functions (in our case it is $P(a_{i}=\alpha|a_{i-N}^{i-1})$) that can depend on a large number of generally unknown parameters. Artificial neural networks are generally presented as systems of interconnected nodes or “neurons”. The connections have numeric weights that can be tuned based on experience, making neural nets adaptive to inputs and capable of learning. These definitions correspond to our goal to numerically estimating the unknown functions-matrices $F_{\alpha;\beta_{1}...\beta_{N}}(r_{1},...,r_{N})$ depending on a great number of parameters, say $N\sim 10^{5}$ or more.

We mentioned that the equations for the memory functions $F_{\alpha;\beta_{1}...\beta_{N}}(r_{1},...,r_{N})$ can be obtained analytically by means of minimization of the distance (12) (known in the ANN theory under the names of cost function or average system error) between desired and actual neuron output values, the elements of real referent sequence and the CPF. The same distance can be used for purposes of numerical finding unknown quantity – generalized memory matrices $F(.)$ – under the network training.

The considered problem with (potentially) given random sequence falls within the paradigm of supervised learning, which can be thought of as learning with a “teacher”. In supervised learning, each example is a pair consisting of an input vector object, $a_{i-N}^{i}$, and a desired output value $a_{i}=\alpha$ or, more precisely, their conditional probabilities $P(a_{i}=\alpha|a_{i-N}^{i-1}$). A supervised learning algorithm analyzes the training data and produces an inferred function, which can be used for mapping new examples. An optimal scenario will allow for the algorithm to correctly determine the class of memory functions.

A commonly used mean-squared error tries to minimize the average squared error between the network’s output and the target value over all the example pairs. When one tries to minimize this cost using gradient descent for the class of neural networks called multilayered perceptrons, one obtains the common and well-known backpropagation algorithm for training neural networks.

A number of supervised learning methods have been introduced in the last two decade. In the Caruana and Niculescu-Mizil paper Caruana a reader can find a large-scale empirical comparison between ten supervised learning methods: SVMs, neural nets, and so on.

X Numerical simulations

In this section, to verify our analytical results, we give examples of numerical generation of random sequences with the state space of dimension $|\mathcal{A}|=2$ (the symbols-numbers of the sequence can only take on the two values: [math] or $1$). Let us note, that for a binary sequence there is no distinction between symbolic and numeric approaches: all symbolic correlation functions can be expressed by means of numeric ones and vice versa. More detailed explanation is outlined in Appendix.

It is supposed that the modeled statistical properties of the random chain are determined by the probability of the symbols occurring, $p_{\alpha}=1/2$, the additive part of the memory function:

[TABLE]

and the exponential (with respect to the both arguments) bilinear part of the memory function:

[TABLE]

with the truncated parameter $N=20$, playing the role of the memory length.

These memory functions allow us to calculate conditional probability function (13), consisting in this case of three partes: the zero-order function, $p_{\alpha}=1/2$, the first-order additive function $Q^{(1)}$, and the second order bilinear function $Q^{(2)}$, where the last two terms are given by expressions  (18) and (V). Using $p_{\alpha}$ and Eqs. (37) and (38), it is possible to build up four different CPFs. Taking, e.g., the zero-order function only, we obtain uncorrelated sequence.

We construct numerically three different random sequences of length $10^{8}$. The first one is generated by the additive probability function $Q^{(1)}$, Eq.(37), the second sequence are obtained with the bilinear part, Eq.(38) (but without additive part) and the third chain, the most general in our case, is get by the CPF contained both terms (37) and (38). All these CPFs contain, evidently, the zero-order term, $p_{\alpha}$.

Since the sequences are prepared, we can calculate their correlation functions. In Fig. 1, the obtained correlators are presented by the dots. At the same time, using memory functions Eqs. (37) and (38) and solving iteratively the system of equations Eqs. (25, V) with respect to $C_{\alpha\beta}(r)$ and $C_{\alpha\beta\gamma}(r_{1},r_{2})$, we obtain the correlation functions presented in Fig. 1 ($C_{\alpha\beta}(r)=C_{11}(r)=C_{2}(r)$) by the curves.

The middle curve and dots correspond to the sequence generated with the additive memory function. The bottom curve and dots describe a sequence based on the bilinear part of the memory function. The upper curve and dots present $C_{2}$ of the sequence obtained with the additive and bilinear memory functions simultaneously.

From Eqs. (25, V) and numerical simulations, it follows that the correlation functions are entangled and intricate. The additive part of the memory function is responsible for not only second-order correlation function, the bilinear part of the memory function effect on not only ternary correlation function — they are mixed up in the system (of equations): each memory function affects each correlator. The additive and bilinear parts of the memory function is mainly responsible for for second-order and third-order correlation functions, correspondingly, for a limiting case of small correlations predominantly. Note, the “small” values of the correlation functions $C_{2}$ of order of $10^{-3}$ are really not small. We chosen the normalization coefficients of the memory functions (37) and (38) in such way to provide extremely strong persistent correlations with the CPF $P^{(2)}(a_{i}=\alpha|a_{i-N}^{i-1})$, Eq. (23).

In the case of purely additive MF, the form of the correlator is close to the shape of the memory function (middle curve-dots in Fig. 1), but has a significant “tail” at distance $10\lesssim r\lesssim 15$. This the phenomenon of “lag” of a correlator with respect to a memory function, we have already seen in previous studies  muya . In the absence of the additive MF (bottom curve in Fig. 1), the pair additive correlator is not equal to zero and has a shape defined by solution of Eqs. (25) and (V). In the case of the simultaneous generation with additive and bilinear MFs, the pair correlator becomes significantly different from the previous cases.

The two dimensional surface for the third-order correlator $C(r_{1},r_{2})$, obtained by solution of Eqs. (25) and (V) with bilinear part of the memory function Eq. (38) only is shown in Fig. 2. The comparison of this function (line) and numerically found the third-order correlation function $C(r_{1},r_{2})$ (dots) of the binary sequence for the fixed coordinate $r_{1}=1$ is presented in Fig. 3.

A good agreement between the analytical and numerical calculations of the correlation functions in all three cases gives a reason to believe that the system of equations and the generation procedure based on the bilinear form of the CPF are well grounded.

XI Conclusion

The decomposition procedure for the CPF of symbolic random sequences considered as the high-order Markov chains with long-range memory is obtained. We represented the conditional probability function as the sum of multi-linear memory function monomials of different orders. It allows us to build artificial sequences by method of successive iterations taking into account at each step of iterations more and more high correlations among random elements. At weak correlations, the memory functions are uniquely expressed in terms of high-order symbolic correlation functions. So we have filled the theoretical gap between the methods of additive Markov chain and likelihood estimation. The obtained results might be used for sequential approximation of artificial neural networks training.

XII Appendix

Binary random sequences always can be considered as numeric and by this reason equations (25) and  (V) can be considerably simplified to a numeric form. Let us consider symbolic correlator (5) taken at $\alpha_{1}=\alpha_{2}=\ldots=\alpha_{k}=1$. In this case it is possible to put $\delta(a_{r},\alpha)=a_{r}$, whereupon the symbolic correlation function takes on the form identical to the numeric one,

[TABLE]

To calculate a symbolic correlator with arbitrary indexes it is sufficient to note that the term $\delta(a_{r},\alpha)-p_{\alpha}$ changes its sign when one replaces $1$ to [math] or [math] to $1$, $\delta(a_{r},\alpha)-p_{\alpha}=(1-a_{r})-p_{0}=-(a_{r}-p_{1})$. Each such replacement changes a sign of the correlator. Consequently, its value depends on the number $q$ of symbols “0” among $\alpha_{1},...\alpha_{k}$,

[TABLE]

Using the same reasoning, we find a simplified expression for $\delta(a_{r},\alpha)-p_{\alpha}$ and the CPF,

[TABLE]

where

[TABLE]

Let us rewrite Eq. (25) for the element $C_{11}(r)$. We put $\alpha=\beta=1$ and take a sum over $\gamma=\{0,1\}$. Then, at the LHS we obtain the numeric correlator $C(r)$; at the RHS, we express all symbolic correlators by means of numeric ones, Eq. (40). After that, all symbolic memory functions turn out to be grouped in combinations, which give numeric memory functions, e.g.,

[TABLE]

In the same way we manipulate with the second term in Eq. (25) and introduce the second order numeric memory function

[TABLE]

As a result, equations (25) and  (V) take on the following simplified formes convenient for describing binary sequences:

[TABLE]

[TABLE]