About non-parametric identification of T-processes

This paper is devoted to the construction of a new class of models under incomplete information. We are talking about multidimensional inertia-free objects for the case when the components of the output vector are stochastically dependent, and the character of this dependence is unknown a priori. The study of a multidimensional object inevitably leads to a system of implicit dependencies of the output variables of the object from the input variables, but in this case this dependence extends to some components of the output vector. The key issue in this situation is the definition of the nature of this dependence for which the presence of a priori information is necessary to some extent. Taking into account that the main purpose of the model of such objects is the prediction of output variables with known input, it is necessary to solve a system of nonlinear implicit equations whose form is unknown at the initial stage of the identifica- tion problem, but only that one or another output component depends on other variables which determine the state of the object. Thus, a rather nontrivial situation arises for the solution of a system of implicit nonlinear equations under condi- tions when there are no usual equations. Consequently, the model of the object (and this is a main identification task) cannot be constructed in the same way as is accepted in the existing theory of identification as a result of a lack of a priori information. If it was possible to parametrize the system of nonlinear equations, then at a known input it would be necessary to solve this system, since in this case it is known, once the parameterization step is overcome. The main content of this article is the solution of the identification problem, in the presence of T-processes, and while the pa- rametrization stage can not be overcome without additional a priori information about the process under investigation. In this connection, the scheme for solving a system of non-linear equations (which are unknown) can be represented in the form of some successive algorithmic chain. First, a vector of discrepancies is formed on the basis of the available training sample including observations of all components of the input and output variables. And after that, the evalua- tion of the output of the object with known values of the input variables is based on the Nadaraya-Watson estimates. Thus, for given values of the input variables of the T-process, we can carry out a procedure of estimating the forecast of the output variables. Numerous computational experiments on the study of the proposed T-models have shown their rather high effi- ciency. The article presents the results of computational experiments illustrating the effectiveness of the proposed tech- nology of forecasting the values of output variables on the known input.