Identification of locally-adaptive regression models with triangular indicator functions

Abstract

The basic idea of constructing locally-adaptive regression models (LAR models) consists in the use of regressors defined on the local subregions of factor values. The belonging of factor values to a particular local subdomain is set by indicator functions. Indicator functions by their nature are close to the well-known concepts of membership functions from the theory of fuzzy systems (Fuzzy Systems). As a rule, to provide the required smoothness of the required dependence of the response on the acting factors such local subdomains are defined with overlapping - in the form of the so-called fuzzy partitions. Type or type of indicator functions may be very different: triangular, trapezoidal, and non-linear. Specifying one or another type of indicator function determines the scheme of weighing local models. Each indicator function must be defined for the entire range of the corresponding factor. Triangular-type functions are used as indicator functions in this work. Linear factor models are considered as local models. It is noted that in their original form the proposed LAR models are not identifiable. The issue of identification of such models in the case of joint estimation of all parameters is considered. The procedure of model reduction is introduced. The resulting model is written out in the space of functions that allow estimation. In the case of dividing the domain of factor determination into two, three or four fuzzy partitions we propose the basis of functions allowing evaluation. The results of computational experiment on regression dependence reconstruction by ordinary polynomials of different degrees and by LAR models are given. The efficiency of LAR models in comparison with polynomials of degree 3 and 4 is noted.