ESTIMATING WITH A GIVEN ACCURACY OF THE COEFFICIENTS AT NONLINEAR TERMS OF UNIVARIATE POLYNOMIAL REGRESSION USING A SMALL NUMBER OF TESTS IN AN ARBITRARY LIMITED ACTIVE EXPERIMENT
Abstract
We substantiate the structure of the efficient numerical axis segment an active experiment on which allows finding estimates of the coefficients for nonlinear terms of univariate polynomial regression with high accuracy using normalized orthogonal Forsyth polynomials with a sufficiently small number of experiments. For the case when an active experiment can be executed on a numerical axis segment that does not satisfy these conditions, we substantiate the possibility of conducting a virtual active experiment on an efficient interval of the numerical axis. According to the results of the experiment, we find estimates for nonlinear terms of the univariate polynomial regression under research as a solution of a linear equalities system with an upper non-degenerate triangular matrix of constraints. Thus, to solve the problem of estimating the coefficients for nonlinear terms of univariate polynomial regression, it is necessary to choose an efficient interval of the numerical axis, set the minimum required number of values of the scalar variable which belong to this segment and guarantee a given value of the variance of estimates for nonlinear terms of univariate polynomial regression using normalized orthogonal polynomials of Forsythe. Next, it is necessary to find with sufficient accuracy all the coefficients of the normalized orthogonal polynomials of Forsythe for the given values of the scalar variable. The resulting set of normalized orthogonal polynomials of Forsythe allows us to estimate with a given accuracy the coefficients of nonlinear terms of univariate polynomial regression in an arbitrary limited active experiment: the range of the scalar variable values can be an arbitrary segment of the numerical axis. We propose to find an estimate of the constant and of the coefficient at the linear term of univariate polynomial regression by solving the linear univariate regression problem using ordinary least squares method in active experiment conditions. Author and his students shown in previous publications that the estimation of the coefficients for nonlinear terms of multivariate polynomial regression is reduced to the sequential construction of univariate regressions and the solution of the corresponding systems of linear equalities. Thus, the results of the paper qualitatively increase the efficiency of finding estimates of the coefficients for nonlinear terms of multivariate polynomial regression given by a redundant representation.
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