| Popis: |
The solution of many problems of assessing and predicting the state of complex systems of different nature, operating in a multifactorial environment, is reduced to finding a relationship that connects the values of the monitored parameters of the environment with the value of some resulting parameter – the response characterizing the state of the system. As a mathematical description of such systems, the Kolmogorov-Gabor regression polynomial is often used, the estimates of the coefficients of which are determined by the least squares methods. However, in the case of a small sample of initial data, practical difficulties arise in the implementation of this approach. They are associated with an unsatisfactory ratio between the number of estimated parameters N = 2m and the number of experiments performed N. The problems arising in this case are solved either by increasing the number of experiments, which is not always possible, or by a reasonable decrease in the number of estimated parameters. One of the promising ways to reduce the dimension of the vector of estimates of the coefficients of the Kolmogorov-Gabor polynomial is the artificial orthogonalization of the results of a passive experiment. But the direct use of this technology in real conditions is difficult due to the uncertainty regarding the numerical values of the experimental results. A variant of solving this problem, described in this study, consists in applying the method proposed in it for estimating the coefficients of the regression equation in conditions of fuzzy initial data using orthogonalization of a passive experiment. The essence of the method is that for the regions adjacent to the vertices of the hypercube of the scaled factor space, a local description of the response function behavior is constructed. This is done using a nominal linear in parameters and factors in the form of regression, the parameters of which are determined by the least squares method, and instead of the output variable, the membership function of fuzzy measurement results in Gaussian form is used. For cases where the number of experimental points in some subdomain adjacent to its vertex turns out to be small, as well as with a different number of them in each of the subdomains, a procedure for forming a truncated replica-like orthogonal subdesign is proposed. A feature of the solutions formed in this case is that they are obtained by removing a rather rigid assumption about the equality of the measurement errors of the response function at all observation points. Due to this, the proposed method makes it possible to solve the problem associated with the estimation of the coefficients of the Kolmogorov-Gabor regression polynomial in the case of a fuzzy description of the output variable based on a small sample of experimental data. |
