Інтерполяція випадкового поля для області спостережень у вигляді системи вкладених прямокутників
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Date
2018
Authors
Флоренко, Анастасія
Щестюк, Наталія
Заєць, Настасія
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Abstract
Дослiджено задачу оцiнювання лiнiйних функцiоналiв вiд невiдомих значень однорiдного випадкового
поля ξ(x, y) для областi K за спостереженнями поля ξ(x, y) в точках (x, y) ∈ Z2 \ K. Знайдено формули
для обчислення середньоквадратичної похибки та спектральної характеристики оптимальної лiнiйної
оцiнки функцiонала у випадку областi спостережень у виглядi системи вкладених прямокутникiв.
Forecasting of static processes and estimation of random fields of a different nature is becoming more widespread among scientists of different specialties, and a new branch of science appears with its specific methodology. That problems of estimation of the unknown values of random fields are generalization of problems of extrapolation, interpolation and filtering of stochastic processes. The study of the dependence of the obtained formulas on the geometry and the number of embeds are the topical problems in the field of the forecasting theory, in geology, geodesy, and some other directions. The methods of solution of linear estimation problems for stochastic processes and random fields were developed by A. M. Kolmogorov [2], A. M. Yaglom [3]. Traditional methods of solution of these problems are employed under the condition that spectral densities are known exactly. The case of estimating the unknown values of a random field for an area that represents a system of embedded rectangles is of interest in the study of random fields with peculiarities. The problem is the estimation of linear functionals which depend on the unknown values of a homogeneous random field ξ(x, y) in the region K from observations of ξ(x, y) at points (x, y) ∈ Z2 \ K, where K is a region that represents the union of the edges of the rectangles mx × my, with the number of rectangles - sx, lx and ly spaced between the attachments on the X and Y axes, respectively. That is, we find a value Akξ from the class of linear functionals, which minimizes the value of the mean square error ∆ = M|Akξ − Akξ|2. To solve this problem, we used a classical method of projections in the Hilbert space. Formulas for calculating the mean square errors and the spectral characteristics of the optimal linear estimate of functionals are derived in the case when the areas of observations represent a system of embedded rectangles.
Forecasting of static processes and estimation of random fields of a different nature is becoming more widespread among scientists of different specialties, and a new branch of science appears with its specific methodology. That problems of estimation of the unknown values of random fields are generalization of problems of extrapolation, interpolation and filtering of stochastic processes. The study of the dependence of the obtained formulas on the geometry and the number of embeds are the topical problems in the field of the forecasting theory, in geology, geodesy, and some other directions. The methods of solution of linear estimation problems for stochastic processes and random fields were developed by A. M. Kolmogorov [2], A. M. Yaglom [3]. Traditional methods of solution of these problems are employed under the condition that spectral densities are known exactly. The case of estimating the unknown values of a random field for an area that represents a system of embedded rectangles is of interest in the study of random fields with peculiarities. The problem is the estimation of linear functionals which depend on the unknown values of a homogeneous random field ξ(x, y) in the region K from observations of ξ(x, y) at points (x, y) ∈ Z2 \ K, where K is a region that represents the union of the edges of the rectangles mx × my, with the number of rectangles - sx, lx and ly spaced between the attachments on the X and Y axes, respectively. That is, we find a value Akξ from the class of linear functionals, which minimizes the value of the mean square error ∆ = M|Akξ − Akξ|2. To solve this problem, we used a classical method of projections in the Hilbert space. Formulas for calculating the mean square errors and the spectral characteristics of the optimal linear estimate of functionals are derived in the case when the areas of observations represent a system of embedded rectangles.
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Keywords
iнтерполяцiя, спектральна щiльнiсть, випадкове поле, оцiнка функцiонала, стаття, interpolation, spectral density, random field, estimation of a functional
Citation
Флоренко А. С. Інтерполяція випадкового поля для області спостережень у вигляді системи вкладених прямокутників / Флоренко А. С., Щестюк Н. Ю., Заєць Н. В. // Могилянський математичний журнал : науковий журнал. - 2018. - Т. 1. - С. 49-53.