TWO-DIMENSIONAL SINGULAR DECOMPOSITION OF TIME SERIES COMPONENTS
DOI:
https://doi.org/10.32782/KNTU2618-0340/2021.4.2.1.6Keywords:
time series; trend; trend setting methods; correlation matrix; statistical modeling; diagnosticsAbstract
Trend analysis has now emerged as an independent section of applied statistics due to the specifics of the research object and the importance of the tasks being solved. Trend analysis methods are widely used in econometrics, diagnostics, climatology, medicine, and other industries. In technical applications, trend analysis is an integral part of a set of methods for diagnosing the state of complex equipment complexes. The criteria of trend and randomness make it possible to establish, at a given level of significance, the fact of the onset and development of unfavorable trends during the operation of complex technical objects in their life cycle. Isolation and study of the trend makes it possible to forecast its development for the upcoming period of operation in order to implement the strategy of operation based on the technical condition. One of the most advanced methods of trend analysis is the method of decomposing a time series into orthogonal components. The algorithmic basis of this approach is factor analysis and the method of principal components. The advantage of orthogonal decomposition, in comparison with other methods of trend analysis, is the ability to predict the development of a trend. The methods SSA, catarpiller and others proposed on this basis are scalar and do not take into account the multidimensionality of the variables of the technical state of complex objects. Therefore, studies aimed at expanding the methods of trend analysis to multidimensional time series are relevant and in demand in practice. The aim of the work is to improve the approach to the analysis of multidimensional time series, which are formed by the parameters of registration of the technical state of complex objects to be diagnosed. The main idea of the proposed approach is to combine time series into twodimensional ones, form a rectangular complex-valued trajectory matrix, and study the distributions of eigenvalues and eigenvectors of the unitary correlation matrix. It has been established that if the first eigenvalue of the unitary correlation matrix is many times greater than its other eigenvalues, and, at the same time, the statistical hypothesis of the equal correlation of its rows is confirmed, then the first centered component of the complex time series by the first principal component is a two-dimensional trend. Moreover, this trend component and the moving average of this series do not have statistically significant differences. It was found that when pairwise combining a multidimensional set of time series into complex two-dimensional ones, and their sequential orthogonal decomposition, it becomes possible to divide the trends of a group of parameters of a multidimensional object into statistically related and having a common cause of occurrence.
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