## Abstract

**This article discusses the problem** of extraction of such set of pattern features that is informative and stable with respect of stochastic noise. This is done through the stochastic information measure.

## Introduction

**In describing of non deterministic system we** must take into account the nature of its uncertainty. In pattern recognition uncertainty could have both probabilistic nature, that defined by some additive measure, and more "imprecise" nature, which can be described, for example, by using monotonous nonadditive measures. The feature extraction is the important task of pattern recognition in general and image analyses in particular. This task consists to extraction of some minimal set of features or combinations of them from a set of features with given informativeness, which would have sufficient integral informativeness in solution of given problem of recognition. The are traditional approaches to solving this problem, such as correlation analysis (principal components analysis, etc.), discriminant analysis [2].

**In this article we concretized the general problem** of informative features extraction as follows. Suppose that pattern X is defined by ordered set X = We call every subseta representation of pattern X. The problem is to find such representation A that it will be minimal on the one hand and it will be near to X on the other hand. Elements in the X may have different priorities, in other words, they may have different informativeness. Therefore, we will establish correspondence between the elementand the nonnegative number – a feature, that characterize the importance of the element x for the representation of pattern X. We will determine the degree of representativeness of the setfor pattern X using the set functionand require from the set function ^ to satisfy all the axioms of monotonous measure: 1)

In addition the measure ^ meets the certain additional conditions related to the specific task of pattern recognition. We will call such measures monotonous information measures. In [1] information measures were used to set and solve the task of finding the most minimal informative polygonal representation of the curve.

**In certain tasks of pattern recognition**, in particular, in image processing, image analysis and image recognition random nature of image features can be caused by some noisy effects. For example, if the pattern is a discrete plane curve that extracted on the image and features are some characteristics of curve points (eg, feature is a estimation of curvature in given point of discrete curve [3]), then a random character of features (eg, curvature) is caused by the noise of image. In this case the expectation E [M(A)] characterize the level of infor-mativeness of represenationand the variancecharacterize the level of stability of representation to noise pattern. Then there is the problem of finding the most stable and informative representationof the pattern X. The complexity of solutions of this problem will be determined by the degree of dependence of random features of each other. Stochastic measure of informativeness M will be additive measure if the features are independent random variables. Otherwise, the measure would be nonadditive measure. In this article mentioned task will be discussed and resolved in the case of probabilistic independence of random features. The case of random dependency of features is investigated in [4].

## The Average Set Function of Information of Pattern

**In general case the feature may be depend from all or some** elements from the representation A on which it calculated i.e.. For example, let be a discrete plane close curve:and , whereis a point that follows from the point g in ordered representation A oris a some estimation of curvature of plane discrete curve that is calculated inneighbourhood of point g [3].

The degree of completeness of representationin describing the pattern X with respect to featuremay be assigned with the follow set function

which we call averaged set function of information of pattern X (with respect to feature w). We putif the valueis not defined for set A (in particular

Example 1. Let X = r be a discrete plane close curve:

Also we introduce the set functions , where L(B) and S(B) are perimeter and square correspondingly of figure, that is limited by polygon with vertex in points of ordered set B. Thenandare averaged set functions of information of curvewherefor set functionand

for set functionis a radius vector of point g G A with respect to arbitrary point O. Note that iiL is a monotonous measure, and is is a monotonous measure too if the domain bounded by polygons with vertexes in the points of discrete curve r is convex.

There is a question when the averaged set functions of information will be monotonous measure? It can be easily shown that follow proposition is true.

**Proposition 1.** Averaged set functions of information ^ on 2X of form (1) are a monotonous measure iff ^ obeys the following condition

and for any set

and

**Corollary 1**. Iffor alland .then set function

of form (1) is an additive measure on 2X.

Letbe a some representation of pattern X. We denote

Corollary 2. Letand then set functionof form (1) is a monotonous measure iff

for allandIn particular, letfor all

andthen set functionof form (1) is a monotonous measure iffor all

andwhereis a point that follows from the point x in ordered representation A.

## The Stochastic Additive Average Information Measure

Letbe a averaged information measure onof form (1) and for allandIn other words the feature value at point x does not depend from considered representation. The estimation of curvatureneighbourhood of point g for discrete plane curve is an example of this feature. Then (see corollary 1) the measure

, is an additive measure. We suppose that feature characteristics will be independence random variables In this case the value of information measure for fixed setwill be random variable too. The set function M(A) will be

an additive measure for any random event. We considered the example of this situation. Suppose that discrete plane curvewas subjected to additive stochastic noncorrelated noise. In result we get a random curvewhere

are random noncorrelated variables and

a2[€k] = a2y k. Suppose that there is such basic set B C r for which random variables ^(g), g G B, are independent. In this case the feature characteristics w(G) = S7 (g) will be random variables just as the value of measure M(A), A G 2B, is.

Let E [f?(x)] = mx, a2 [ft(x)] = aX. We will investigate numerical characteristics of random additive measure M(A), A G 2X.

### The Numerical Characteristics of Stochastic Additive Average Information Measure

Let us find the mathematical expectation of random variable M(A) with a fixed A G 2X. Random variable M(A) is equal to a quotient two random variables € = £xeA n(x) and n = Zxex n(x).

**Lemma 1.** Let € and n be random variables that taking values in the intervals l^, ln respectively on positive semiaxis and ln C ((1 — J)E [n], (1 + £)E [n]), l£ C (E [€] — [n] , E [€]+ ^E [n]). Then it is valid the following formulas for mean and variance of distribution of ^ respectively

where K [£, n] is a covariation of random variables € and n, could meet the certain additional conditions re i.e. K [€, n] = E [(€ — E [€]) (n — E [n])]/ ri, rx are the residuals that depend on numerical characteristics of £ and rj and \r\\ < -p^r •

**Notice that,** a2 [M(X\A)] = a2 [1 — M(A)] = a2 [M(A)] for A G 2X .We will use formulas (4) and (5) without their residuals. Respective values E [M(A)] = E [M(A)] — ri, a2 [M(A)] = a2 [M(A)] — r2 we will call estimations of numerical characteristics. The class of random variables {M(A) : A G 2X} satisfies all the requirements for a finitely additive stochastic measure [5]: 1) E[M2(A)] < to for all A G 2X; 2) M(A) is an almost probably finitely additive measure.

Note that mathematical expectation E [M(A)] define the set function on 2X andSince S(A) and D (A) are additive set function then measure E [M(A)] will be additive too.

### Finding of Optimal Stable Pattern Representation

We set a problem of finding such representation B of pattern X, which cardinality is less or equal to the given numberfor which the summarized variancewill be minimal but the sum of squares of mathematical expectations of all representationswill be maximal. The value of summarized variance characterizes the stability of representation and all its subset to noise level of pattern and it depends also from number of elements in representation. The more elements in the presentation we have, then the summarized value of variance is greater. We will use mathematical expectations of nonnormalized information measures, instead of , for simplification of computations. We introduce follow criteria:

Then it is necessary to find such set, for which

We simplify function f (B). Let

Then it is necessary to find such set, for which

We simplify function f (B). Let

**Proposition 2**. If feature characteristics of pattern X are independent random variables then following equality is true for every

**Corollary 3.** If

As

then following corollary is true.

**Corollary 4. If**

We will use the "inclusion-exclusion" procedure for finding of optimal representation which minimize the function f (B). We estimate the variation of function f (B) to the exclusion of element x from B and inclusion of element y G X\B.

Theorem 1. If feature characteristics of pattern X are independent random variables then following asymptotic equality

where is true for every

where

is true for every

Corollary 5.

Formulas from theorem 1 and corollary 5 may be use for construction of algorithmic procedures for finding of representation B, which minimize the function f. LetThe algorithm for finding of representation B of cardinality k which minimize the function f (if random variables are independent), consist of following steps:

1) we select the set B0 consisting of k elements of pattern X with maximal values of informativenessas a initial representation;

2) we compute the valuefrom Theorem 1 and we find argmin, if this pair is exist. Then

is a new representation for whichaccurate

within to small values of second order. This step will be repeating so long as will be pairs

Ifthen following conclusion follows from the corollary 5, if we disregard small values: the optimal representation B with cardinality is less or equal k, which minimize the function f will be consist of elements with the greatest values(on the assumption of random variablesare independent).