Non-parametric algorithms of reconstruction of mutually ambiguous functions from observations

We consider the task of reconstruction of the regression function from observations with errors. Under parametric uncertainty conditions this problem is solved in the following sequence: first the type of regression function with accu- racy to parameters is set, then the next stage is the estimation of these parameters based on training sample elements. The main problem that arises is choosing a parametric structure, i. e. the choice of parameters with an accuracy to the vector of parameters. At the same time more or less inaccuracy can be allowed, descriptions of many variables func- tions with accuracy to parameters cause particular difficulties. Another known way of solving such problems, which is the nonparametric estimation of regression function from observations, in this case the stage of choosing a parametric equation of the regression function is missing. A number of publications is devoted to this area including monographs where the results are in most cases related to the asymptotic properties of the regression function. The article considers the task of reconstruction of mutually ambiguous functions of many arguments from observa- tions with random errors in the conditions of nonparametric uncertainty. This problem has been insufficiently studied, although it has a significant importance in the identification and control of objects of a Wiener and Hammerstein class. The control theory widely uses already known mutually ambiguous specifications that describe the work of items with a loop of hysteresis, backlashes and others. Some modifications of nonparametric estimates of mutually ambiguous fea- tures including multidimensional are given. A series of computing experiments have been conducted where for simplic- ity reasons the simpliest mutually ambiguous curves were taken, parametric structure of these curves for the algorithms was unknown, only observation was known. Numerical studies covered two cases: different sample sizes and various disturbances affecting the studied processes. The reconstruction of mutually ambiguous dependency plays an important role in the development of robots and various robotic systems moving on in an undefined or unknown terrain. As sepa- rate blocks the considered algorithms can be useful in devices that are used in the aerospace industry.