Studying the solutions to the parametric identification of the models of distributed dynamic processes
Abstract
Identifying the parameters of dynamic distributed systems is an important task which has technical, economic, and social applications. When modelling the objects of such applications, multidimensional autoregression equations are normally used with regressors located at adjacent nodes of spatial coordinates. Obviously, in this case there is usually a significant correlation between the regressors. Here, we observe a quasimulticollinearity effect which results in an overestimated value of the standard error of the estimation of the autoregression parameters and biased estimates of the model’s parameters. Methods of improving the quality of statistical estimates include a large number of those that reduce the standard error and increase the bias, and vice versa, such as the method of instrumental variables, ridge regression, etc. Thus, the presence of two components of error results in the need to find a compromise between bias and variance, a dilemma well-known in machine learning. In our study we focused on the multiple autoregression equations obtained by means of approximation of homogeneous partial differential equations with constant parameters to the difference equations with the property of conservativeness. A difference scheme is considered conservative if it preserves the same conservation laws on the grid as in the original differential problem. The article presents the results of a comparative analysis of the following solutions to the problem of parametric identification: the ordinary least squares (OLS) method, ridge regression, and two methods of reducing the dimension suggested by the authors. The comparative analysis of the application of the studied identification methods to the estimation of the parameters showed a significant dependence of the quality of the assessment on the intensity of the observation interference. With low interference, all the considered methods are effective. With an increase in the intensity of interference, only the author’s method of reducing the dimension taking into account the conservativeness of the difference scheme provides satisfactory results.
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2. Bezruchko B. P., Smirnov D. A. (2006) Modern problems of modeling from time series. Proceedings of the Sa-Raevskogo University, series “Physics”. Vol. 6, No. 1-2. P. 3–27.
3. Guo L. Z., Billings S. A., Coca D. (2010) Identification of partial differential equation models for a class of mul-tiscale spatio-temporal dynamical systems. International Journal of Control, No 83:1. P. 40–48. DOI: 10.1080/00207170903085597
4. Matveev M. G., Mikhailov V. V., Sirota E. A. (2016) Combined prognostic model of a nonstationary multidimensional time series for constructing a spatial profile of atmospheric temperature. Vol. 22. No. 2. P. 89–94.
5. Xiaolei Xun, Jiguo Cao, Bani Mallick, Raymond J. Car-roll, Arnab Maity (2013) Parameter Estimation of Partial Differential Equation Models. Journal of the American Statistical Association. No 108:503. P. 1009–1020, DOI: 10.1080/01621459.2013.794730
6. Nosko V. P. (2002 ) Econometrica. Introduction to the regression analysis of time series. Moscow : NFPK. 273 p.
7. Gareth J., Witten D., Hastie Tr., Tibshirani R. (2013) An Introduction to Statisti-cal Learning. Springer. P. 40–48.
8. Matveev M. G., Kopytin A. V., Sirota E. A., Kopytina E. A. (2017) Modeling of Nonsta-tionary Distributed Processes on the Basis of Multidimensional Time Series. Procedia Engineering. – No.201. P. 1–862. DOI: 10.1016/j. proeng. 2017. 09. 643
9. Samarsky A. A. Theory of difference schemes. Moscow : Nauka, 1978. 376 p.
10. Matveev M. G., Sirota E. A. (2020) Analysis of the properties of the OLS-estimators in case of elimination of multi-collinearity in the problems of parametric identification of distributed dynamic processes. Bulletin of Voronezh state University, series “System analysis and information technologies”. No 2. P. 15–22.
11. Matveev M. G., Sirota E. A. (2021) Analysis and investigation of the conservativeness condition in the problem of parametric identification of dynamic distributed processes. Journal of Physics: Conference Series, No 1902. 012079. DOI:10.1088/1742-6596/1902/1/012079
12. Kopytin A. V., Kopytina E. A., Matveev M. G. (2018) Application of the expanded Kalman filter for identifying the parameters of a distributed dynamical system. Vestnik VSU, series “System Analysis and Information Technologies”. No. 3. P. 44–50.
13. Barseghyan A. A., Kupriyanov M. S., Kholod I. I., Tess M. D., Elizarov S. I. (2009 ) Analysis of data and processes: textbook. Manual. 3rd ed., reprint. and additional. St. Petersburg : BHV-Petersburg. 512 p.
14. Tibshirani R. J. (1996) Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society. Series B (Methodological). Vol. 58, No. 1. P. 267–288.
15. Ayvazyan S. A., Bukhstaber V. M., Enyukov I. S., Meshalkin L. D. (1989) Applied statistics. Classification and reduction of dimension. Moscow : Finance and Statistics. 607 p.
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