GMDH-BASED OPTIMAL SET FEATURES DETERMINATION IN DISCRIMINANT ANALYSIS Anatation. The task of searching optimum on complexity discriminant function is considered. Criteria of quality of the discriminant functions developed in the Group Method of Data Handling

Anatation. The task of searching optimum on complexity discriminant function is considered. Criteria of quality of the discriminant functions developed in the Group Method of Data Handling are described: the criterion based on a partition of observations on training and testing samples, and criterion of sliding examination. The tasks of this class belong to pattern recognition problems under the condition of structural uncertainty, which were considered by academician A.G. Ivakhnenko as long ago as 60–70-th of the last century as actual problems of an engineering cybernetics.

In this way, observations, which are serially excluded from training subsamples, are used as testing observations. In the literature, these ways are traditionally treated as heuristic methods though the fact of existence in them of optimum set of features repeatedly proved by a method of statistical tests. In the Group Method of Data Handling (GMDH), analytical research of these two ways is carried out [1][2][3][4][5][6]. For the decision of a task of the discriminant analysis in conditions of structural uncertainty except for a way of comparison discriminant functions it is required to specify algorithm of generation of various combinations of the features included in discriminant functions. Algorithms, which are based on principles GMDH, are developed [7][8]. It is supposed, that  © Sarychev A.P., Sarycheva L.V., 2019 as such method is chosen the complete sorting-out of all possible combinations of features.
According to principles of modeling in the GMDH for prove of adequacy of criterion it is necessary: 1) to calculate mathematical expectation of researched criterion for given structure of model; 2) to research behavior of mathematical expectation of this criterion depending on structure of models; 3) to prove existence of model of optimum complexity; 4) to receive a condition of a reduction (simplification) of model of optimum complexity.   (   T  II  II   T  I  I   1  II  I where kA v are matrices of deviations of observations kA V from estimates kÃ v ] ,..., , In formula (5) (2), we obtain for ) The validity of theorem follows from the validity of the following: 1) the estimates obtained on subsamples A and B are independent; 2) the estimate (3) and estimate (4) are independent; 3) matrix A S is random ) ( s s matrix which has the Wishart distribution with r degrees of freedom. Definition 1. The optimal set components (set features) is defined as Definition 2. Optimal discriminant function with respect to the number and composition of the components is defined as the Fisher discriminant function constructed on the set of components OPT V .
We proved that optimal set of components exist and formulated the conditions under which the optimal discriminant function is simplified in number of the features included in it. For this purpose, it was investigated It is possible to divide set of components X into the following nonin- (where  is the empty set) is the set of components whose mathematical expectation satisfy Our analytical investigations confirm these empirically determined regularities about the existence of the discriminant function optimal by the number and composition of components. Let's formulate the conditions of reduction (simplification) optimal discriminant function for a special case of an independent feature. Let the set of V is those, that is carried out (9) According to the above-mentioned lemmas for Mahalanobis distances of sets V and o X the ratio where is the observation numbered h in the first group, centered about the estimate be the Mahalanobis distance for the set of components V , n n n   II

Theorem 2. For the random variable
The validity of theorem follows from the validity of the following:  We proved that optimal set of components exist in the way that con- Our analytical investigations confirm these empirically determined regularities about the existence of the discriminant function optimal by the number and composition of components. Let's formulate the conditions of reduction (simplification) optimal discriminant function for a special case of an independent feature. Let the set of V is those, that is carried out where o o X x  (one feature is missed). Taking into account (25), we receive    Conclusion. The two methods for comparison of the discriminant functions are proved. The first method based on dividing of the initial data sample on training and testing subsamples and second method based on sliding examination. In spite of successful use of these ways in practice and repeated confirmation of its efficiency by the method of statistical test, it was considered traditionally as heuristic method.
It is shown that under condition of structural uncertainty and the absence of a priori estimates of parameters of general sets these methods make it possible to solve the problem of search of the discriminant function of optimal complexity. Conditions of reduction (simplification) of discriminant function, which is optimal by structure of features, are revealed. It is shown, as these conditions depend on volumes samples and parameters of general sets, i.e. on mathematical expectations and covariance matrices of features.