Reconstruction of multidimensional functional relations from samples of limited volume

Abstract

We shall examine a typical multidimensional regression reconstruction problem [1‐7]. We assume that the object of study can be described by some ensemble of numerical parameters. The investigator is faced with the problem of predicting the values of certain parameters from the values of others and determining the error of the prediction. In the general case, the construction of the multiple nonlinear regression equations with the use of the analytic methods of mathematical statistics is usually not possible. Therefore the empirical methods that yield adequate results are often used. These methods can be reduced to three variants: the all regressions method; the exclusion of variables method; the inclusion of variables method. We note that many problems of the reconstruction of relations can be reduced to the same mathematical scheme—minimizing the average risk with respect to the empirical values [1]. We shall use for the solution of this problem the all‐ regressions method, which is based on a multifactor model of the form: y = a 0 f 1 ( x 1 ) f 2 ( x 2 ) × … × f n ( x n ). Thus, the approximating function is the product of the smooth one-dimensional basis functions f i ( x i ), i , …, n . In the classical approach, the all regressions method is used to determine the unknown functions f i ( x i ) in this expression, the parameters of the approximating functions are determined with the aid of the least squares method, and the criterion for selecting the unknown relation is the coefficient of the mean square approximation error. We shall examine a modified variant of the multifactor multiple nonlinear regression reconstruction model, the basis of which is the algorithm for constructing the polynomial approximation of a onedimensional functional relation with the aid of the method of structural minimization of the empirical risk [1].