Abstract
An approach is proposed to assess the quality of stationary Markov models without absorbing states on the basis of a measure of statistical stability: the description is formulated and its properties are determined. It is shown that the estimates of statistical stability of models were raised by different authors, either as a methodological aspect of the model quality, or within the framework of other model properties. When solving practical problems of simulation, for example, based on Markov models, there is a pronounced problem of ensuring the dimension of the required samples. On the basis of the introduced formulations, a constructive approach to solving the problems of sample size optimization and statistical volatility analysis of the Markov model to the emerging anomalies with restrictions on the accuracy of the results is proposed, which ensures the required reliability and the exclusion of non-functional redundancy.
 To analyze the type of transitions in the transition matrix, a measure of its divergence (normalized and centered) is introduced. This measure does not have the completeness of the description and is used as an illustrative characteristic of the models of a certain property. The estimation of the divergence of transition matrices can be useful in the study of models with high sensitivity of detection of the studied properties of objects. The key stages of the approach associated with the study of quasi-homogeneous models are formulated.
 Quantitative estimates of statistical stability and statistical volatility of the model are proposed on the example of modeling a real technical object with failures, recovery and prevention. The effectiveness of the proposed approaches in solving the problem of statistical stability analysis in the problems of qualimetric analysis of quasi-homogeneous models of complex systems is shown. On the basis of the offered constructive approach the operational tool of decision-making on parametric and functional adjustment of difficult technical objects on long-term and short-term prospects is received.
Highlights
СТАЦИОНАРНЫХ МАРКОВСКИХ МОДЕЛЕЙПредложен подход для оценки качества стационарных Марковских моделей без поглощающих состояний на основе меры статистической устойчивости: формулируется описание меры и определяются ее свойства
На примере моделирования реального технического объекта с отказами, восстановлениями и профилактикой предложены количественные оценки статистической устойчивости и статистической волатильности модели
An approach is proposed to assess the quality of stationary Markov models without absorbing states on the basis of a measure of statistical stability: the description is formulated and its properties are determined
Summary
Предложен подход для оценки качества стационарных Марковских моделей без поглощающих состояний на основе меры статистической устойчивости: формулируется описание меры и определяются ее свойства. Вопросы оценки статистической устойчивости моделей поднимались разными авторами либо как методологический аспект качества модели, либо в рамках других модельных свойств. K — число запусков модели, за которое система из первоначального состояния перейдет в терминальное состояние, описываемое матрицей Pijk ; PijMm — «восстановленная» матрица переходных вероятностей, содержащая статистические оценки Pij , (именно эта матрица будет использована для анализа статистической устойчивости модели Mx ); — матрица результатов моделирования, (например: финальные вероятности переходов; некоторая мера отклонения числа совпавших элементов Pij и PijMm и т.п.). Свойство 2 может быть распространено на случай, когда сравниваются модели для одинаковых доверительных вероятностей p1M1 , но при k1M1 k2M1.
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