Construction of a Multivariate Polynomial Given by a Redundant Description in Stochastic and Deterministic Formulations Using an Active Experiment

Abstract

We present the methods for constructing a multivariate polynomial given by a redundant representation based on the results of a limited active experiment. We solve the problem in two formulations. The first is the problem of constructing a multivariate polynomial regression given by a redundant representation based on the results of a limited active experiment. The solution method is based on the previous results of Professor A. A. Pavlov and his students showing the fundamental possibility of reducing this problem to the sequential construction of univariate polynomial regressions and solving the corresponding nondegenerate systems of linear equations. There are two modifications of this method. The second modification is based on proving for an arbitrary limited active experiment the possibility of using only one set of normalized orthogonal polynomials of Forsythe. The second formulation refers to the solution of this problem for a particular but sufficient from the practical point of view case when an unknown implementation of a random variable is not added to the initial measurement results during an active experiment. This method is a modification of the solution method for the multivariate polynomial regression problem. Also, we used the main results of the general theory (which reduces the multivariate polynomial regression problem solving to the sequential construction of univariate polynomial regressions and solution of corresponding nondegenerate systems of linear equations) to consider and strictly substantiate fairly wide from the practical point of view particular cases leading to estimating the coefficients at nonlinear terms of the multivariate polynomial regression as a solution of linear equations with a single variable.

Efficiency Substantiation for a Synthetical Method of Constructing a Multivariate Polynomial Regression Given by a Redundant Representation

“…Aggregated modified algorithmic scheme of the first sub-algorithm [13]. The l -th step ( 1 L l  , where 1 L is the total number of nonlinear components in (12), each of which contains at least one scalar variable raised to a power greater than or equal to two) is implemented for the next nonlinear term of the MPR (12), whose coefficient was not estimated at the previous steps of the first sub-algorithm and that contains a scalar variable to the maximum power. Let's denote it as x takes the value according to (7), (8), and the other scalar variables in all tests take fixed values.…”

“…The right-hand parts of the obtained nondegenerate systems of linear equations are the estimates for nonlinear terms of the UPRs. Their solutions are the estimates of the corresponding coefficients for nonlinear terms of the MPR (12).…”

“…Aggregated modified scheme of the second sub-algorithm [13]. The l -th step ( 2 L l  , where 2 L is the number of nonlinear components in (12) of the form …”

“…Estimation of coefficients for nonlinear members of a UPR using a single set of NOPFs. It was shown in [12] that the estimates of the coefficients for nonlinear terms of a UPR (1) at arbitrary values of d c  can be found using only a single set of NOPFs based on the results of the following virtual experiment. We introduce a virtual deterministic scalar variable z and for its values…”

“…, corresponds to the most common practical case. All the results obtained below are trivially extended to arbitrary real values of the numbers [13] implements the methodology of reducing the estimation of nonlinear members of an MPR (12) to the sequential construction of UPRs and the solution of the corresponding systems of linear equations, the variables of which are the estimates for the nonlinear members of the MPR (12).…”