Problems of Further Development of the Group Method of Data Handling Algorithms . Part I

The GMDH algorithms for solving interpolation problems of artificial intelligence differ from each other in the form of the reference function and the iteration rules of the multilayer model structure. In some multilayer algorithms, the number of terms in the iteration rule is constant, which leads to the skipping of some models. In the algorithm called combinatorial, the iteration rule increases by one term when passing to each next row, which ensures an exhaustive search through all of the equations. For exact and complete data, the minimum of the external criterion is nonsharp, and to determine an optimal method, extrapolation of the locus of points of the minimum of the external criterion should be performed. A comparison of linear, polynomial, and ratio-polynomial (with respect to the coefficients) functions may give a method for improving the accuracy of problem solutions. To reduce computational time, a threshold GMDH algorithm is developed which preliminarily estimates the effectiveness of the input variables at the information level and searches for model-candidates based on the most effective input variables (arguments or features). Received December 27, 1999 1. PREPROCESSING OF THE INITIAL DATA SAMPLE It is expedient to accept the following procedure for preprocessing large samples of initial data. (1) The “wild points” (the values of variables are obviously impossible) are removed and replaced by the tripled mean deviation. (2) The mean value of the variables is calculated after the wild points are removed. (3) The missing values in the sample are replaced by the mean value. (4) The quantitative variables are normalized to fit into the range between 0 and 1. (5) Each qualitative variable is assigned the value 0 or 1, depending on its class. If it is required to recognize several patterns, the sample is divided into several subsamples, and the patterns or classes are recognized pairwise. 2. ACHIEVEMENT OF THE MODEL UNIQUENESS AND ESTIMATION OF ITS REPRESENTATIVENESS In recursive search modeling methods, the initial data are represented in the sample of experimental data as a table with N rows, which are called observations, or realizations, or images (in pattern recognition). The model sought for is an equation that expresses the value of the output variable through the current and past (i.e., retarded) values of the input variables. In the GMDH algorithms, the model is sought in the form of a linear polynomial. The complexity of the model structure is determined by the number of terms in the polynomial. The optimal model (most accurate under the preset noise level) corresponds to a minimum of the external accuracy criterion. We use the term specific complexity for the ratio of the number of terms in the polynomial S describing the model to the number M + 1 of the input variables (primary and secondary features) increased by one (to take into account the presence of the free term). The graphically represented dependence of the external criterion on the specific complexity of the model answers all questions about the number of minima in the model and the sufficiency or representativeness of the sample. The number of models that can be obtained based on a given sample equals the number of sufficiently sharp minima of the dependence. A sample is considered sufficient or representative if it gives only one sharp minimum. The uniqueness of the minimum can be achieved by dividing the sample into parts according to the clusters of the optimal physical clustering. The resulting sample must contain only the rows included in the first cluster (in decreasing order of the number of points). For the second model, the points of the second cluster are used, etc. The model is considered most representative if it gives the absolute minimum of the external criterion. MATHEMATICAL THEORY OF PATTERN RECOGNITION