On correctness of the problems of nonlinear regression in case of monitoring natural and man-made objects

Authors

  • V. S. Mostovoy Institute of Geophysics of the National Academy of Sciences of Ukraine, Ukraine
  • S. V. Mostovoy Institute of Geophysics of the National Academy of Sciences of Ukraine, Ukraine

DOI:

https://doi.org/10.24028/gzh.0203-3100.v34i2.2012.116626

Abstract

Under consideration there is a compliance with observed data and nonlinear models of monitoring. These models are based on superposition of oscillators with free parameters. Optimal estimation of free parameters of model which enter into model both linearly and nonlinearly, we shall consider as a problem of nonlinear regression. The optimality is understood in sense of a global minimum of an objective functional. The point in space of possible values of free parameters of model in which criterion has a global minimum is accepted as the optimal solution of a problem. For the chosen nonlinear mathematical models it is necessary to find out the questions connected with existence of the solution, its uniqueness, and stability of the solution depending on initial data. The last circumstance is especially important, as the algorithms constructed on the basis of these models, are concentrated on direct processing of field supervision. It means dependence on characteristics of the measuring equipment, errors of measurement and to accompanying by background noises. Separation of linear and nonlinear parameters with the purpose of calculation process optimization is offered for construction of optimal estimations model parameters. By search quasi-optimal solutions such division allows to use for the Monte-Carlo technique simulation only nonlinear parameters. Linearly entering parameters are defined by the solution of system of the linear equations. Thus, dimension of a search problem of optimal estimations is decreased on a size of a linear parameters vector dimension.

References

Виноградов И. М. Математическая энциклопедия. Т. 4. - Москва: Сов. энциклопедия, 1977. - 742 с.

Мостовой В. С. Математическая модель накопления сейсмических сигналов при активном мониторинге // Докл. НАН Украины. - 2008а. - № 4. - С. 132-136.

Мостовой В. С. Оптимальное обнаружение сигналов на фоне микросейсмического шума // Доп. НАН Украины. - 2008б. - № 1. - С. 106-110.

Мостовой В. С., Мостовой С. В. Вариационный подход к решению обратной задачи при накоплении сейсмических сигналов в активном мониторинге // Докл. НАН Украины. - 2008. - № 8. - С. 113-116.

Мостовой В. С., Мостовой С. В. Математическое моделирование оценки старения природных и техногенных объектов в системах мониторинга // Докл. НАН Украины. - 2011а. - № 7. - С. 114-118.

Мостовой В. С., Мостовой С. В. Оптимальные оценки нелинейных параметров в моделях сейсмоакустического мониторинга // Докл. НАН Украины. - 2011б. - № 8. - С. 103-107.

Мостовой В. С, Мостовой С. В., Кондра С. М., Страшко Ж. С. Оценка информативных параметров состояния строительных конструкций в режиме мониторинга // Промышленное строительство и инженерные сооружения. - 2011. - № 1. - С. 7-13.

Bethea R. M., Duran B. S., Boullion T. L. Statistical Methods for Engineers and Scientists. - New York: Marcel Dekker, 1985. - 447 p.

Evans L. C. Partial Differential Equations // Amer. Math.Soc. - 1998. - 19. - 662 p.

Levenberg K. A. Method for the Solution of Certain Non-Linear Problems in Least Squares // Quart. Appl. Math. - 1944. - № 2. - Р. 164-168.

Marquardt D. An Algorithm for Least-Squares Estimation of Nonlinear Parameters // SIAM J. Appl. Math. - 1963. - 11. - Р. 431-441.

Published

2012-04-01

How to Cite

Mostovoy, V. S., & Mostovoy, S. V. (2012). On correctness of the problems of nonlinear regression in case of monitoring natural and man-made objects. Geofizicheskiy Zhurnal, 34(2), 140–143. https://doi.org/10.24028/gzh.0203-3100.v34i2.2012.116626

Issue

Section

Scientific Reports