Порівняння потужності критеріїв наявності тренду в часових рядах

Abstract

The subject of research is methods, mathematical models, and methods of continuous analysis of a multidimensional set of output variables and state variables of power and energy installations built on the basis of gas turbine engines, which in general constitute time series. The purpose of this work is to establish the power of trend and randomness criteria by statistical modeling of time series with a linear trend and the use of known trend and randomness statistics to establish their empirical distributions and operational characteristics for comparing trend criteria by their power. The tasks faced by the developers were to determine the empirical distributions of known parametric and non-parametric trend statistics when applying the linear trend model in superposition with a random component, and set the level of errors of the first kind (false solution), with a given level of errors of the second kind (false anxiety). The methods that were used to achieve the established goal of the research: general methods of trend analysis, methods of applied statistics, and methods of conducting computer experiments. The results of the research provide a rationale for the approach to establishing the power of known criteria of trend and randomness. The limitation of the known methods of applied statistics is that it is theoretically only possible to refute the hypothesis regarding the randomness of the initial data at a certain level of significance, which determines the level of errors of the second kind (false alarms). Establishing the level of errors of the first kind (wrong decision) poses significant difficulties, because in the presence of a trend, the time series can no longer be stationary. But it is the statistical level of such errors that actually determines the strength of the criteria for the presence of a trend in the time series. The resolution of this contradiction is proposed by means of statistical modeling of time series with a linear trend and the use of well-known trend and randomness statistics to establish their empirical distributions and operational characteristics and to compare trend criteria according to their power. Statistical modeling was performed for a number of trend and randomness statistics, namely: the most common parametric statistics: correlation criterion and its modifications, Fisher's criterion, and Student's criterion; and non-parametric Wald-Wolfowitz criteria; Bartles; as well as the inversion criterion. According to the results of statistical modeling, it was established that the Student's criterion is the most powerful of the parametric criteria, and the inversion criterion is the most powerful of the non-parametric criteria. It is understood that such conclusions are valid when the assumptions regarding the initial statistical model of data generation in the form of a superposition of a linear trend and a random component as a sample from the general population of independent and normally distributed random variables and the corresponding algorithm for processing time series counts for the formation of decisive statistics are fulfilled. The scientific novelty of the obtained results lies in the fact that for the first time, the issue of comparing the power of parametric and non-parametric criteria of trend and randomness with respect to the applied model of data generation in the form of a linear trend in superposition with a random component was considered. The practical significance of the obtained results lies in the fact that the research results make it possible to choose an appropriate criterion based on its power for solving applied tasks of monitoring the technical condition of power and energy installations built on the basis of gas turbine engines.