Estimation of the parameters of the regression equation under the conditions of a small sample
DOI:
https://doi.org/10.15587/1729-4061.2009.216542Keywords:
regression equation, passive experiment, artificial orthogonalization, replica-like symmetric orthogonal design, multi-index mathematical programming problemAbstract
The well-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 components of the vector of estimates of the coefficients lying on the main diagonal of the covariance matrix of errors increase; the number of degrees of freedom decreases, which leads to an expansion of the confidence intervals covering the true values of the coefficients of the regression equation.
To improve the situation in this work, a procedure for artificial orthogonalization of the results of a passive experiment is proposed, which is an alternative to the existing ones, which consists either in increasing the number of experiments or in reducing the number of estimated parameters of the model. The task of orthogonalization is to transform the results of measurements of factors in such a way that the matrix combining them would be orthogonal.
The proposed procedure includes the following steps:
- normalization of real measurements of factors to the interval [-1; 1], which forms a hypercube in -dimensional factor space with center at the origin and length of edges equal to two,
- splitting the set of results of a passive experiment into subsets according to the rule, in which the following designations are accepted: the j-th point in the normalized space of factors; ; – -th vertex of the hypercube, ,
- for each obtained hypercube, a piecewise-linear description of the response function is constructed, on the basis of which the values of the response function are calculated at the points corresponding to the vertices of the hypercube. The combination of these values forms the design of an active orthogonalized complete factorial experiment (OCFE).
A procedure for constructing a replica-like orthogonal design symmetric with respect to the center of the experiment is proposed, based on solving a multi-index problem of mathematical programming. This procedure eliminates the difficulties arising in the process of constructing a piecewise-linear description of the response function in those regions of the factor space in which the number of experimental points is small or absent.
The proposed technology for processing the results of a passive experiment makes it possible to construct a regression equation in a situation where the total number of experiments is insufficient to build an adequate model. Transformation of a passive experiment into an active one, followed by the formation of a replica-like orthogonal representative subdesign, allows independent screening of insignificant factors and interactions. This simplifies the structure of the estimated regression equation and improves its accuracyDownloads
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