The object of research
Technology for constructing mathematical models of functioning systems in the form of regression equations.
The problem to be solved
The known scheme for constructing a regression equation based on the least squares method works well if the number of experiments significantly exceeds the number of estimated components of the regression equation. Deterioration of the ratio between the number of estimated coefficients and the number of experiments leads to negative consequences for the following reasons: the variances of the values of the coefficient estimate vector components lying on the main diagonal of the covariance error matrix increase; the number of degrees of freedom decreases, resulting in an expansion of the confidence intervals covering the true values of regression equation coefficients.
The proposed solution and expected effects
To solve the problem, a procedure for artificial orthogonalization of the results of a passive experiment is proposed, which is an alternative to the existing ones, consisting either in increasing the number of experiments or in reducing the number of estimated model parameters. The task of orthogonalization is to transform the measurement results of factors so that the matrix combining them is orthogonal.
The proposed procedure includes the following steps:
– normalization of real measurements of factors to the interval [–1;1], forming a hypercube in the m-dimensional factor space with the center at the origin and the edge length equal to two;
– dividing the set of passive experiment results into subsets according to a specially introduced rule forming hypercubes in the factor space;
– construction of a piecewise linear description of the response function for each obtained hypercube, from which the values of the response function at the points corresponding to the hypercube vertices are calculated. The combination of these values forms the design of an active orthogonalized full factorial experiment (OFFE).
A procedure for constructing a replica-like orthogonal design symmetric to the center of the experiment is proposed, based on solving a multi-index mathematical programming problem. This procedure eliminates the difficulties in constructing a piecewise linear description of the response function in the factor space domains with few or no experimental points.
The proposed technology for processing the results of a passive experiment allows constructing a regression equation in a situation where the total number of experiments is not enough to build an adequate model. Transformation of a passive experiment into an active one, followed by forming a replica-like orthogonal representative sub-design, allows for an independent elimination of insignificant factors and interactions. This simplifies the structure of the estimated regression equation and increases its accuracy.
Scope and ways to apply the results
Regression equations, as a product of the proposed procedure, can be used to diagnose objects of various nature, predict the properties and characteristics of functioning technical, economic, social systems, and solve technological problems.