Mathematical methods of geoinformatics. III. Fuzzy comparisons and recognition of anomalies in time series
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CYBERNETICS MATHEMATICAL METHODS OF GEOINFORMATICS. III. FUZZY COMPARISONS AND RECOGNITION OF ANOMALIES IN TIME SERIES1 A. D. Gvishiani,a S. M. Agayan,a Sh. R. Bogoutdinov, a J. Zlotnicki,b and J. Bonninc
UDC 519.68
The search for anomalies in time series by methods of fuzzy logic is further explored. The algorithms DRAS and FLARS underlying these methods are further developed in the form of the algorithm FCARS that is completely based on fuzzy comparisons. Keywords: record, fuzzy logic, straightening, activity, signal. 1. DISCRETE MATHEMATICAL ANALYSIS In investigating data represented by time series, two successively solved problems are distinguished: the signal detection against the background of noise and recognition of abnormal fragments of the signal detected. This article is devoted to the latter problem that is important in investigations connected with retrospective analysis and estimation of possible occurrence of various natural phenomena (earthquakes, volcanic eruptions, magnetic storms, etc.). In this case, the initial information consists of observable time series of geophysical data [19–22]. This work continues the series of articles [1–10] in which a new approach is described that is developed by the authors and is oriented toward the investigation of anomalies (zones of increased activity). In this approach, an attempt is made to model discrete analogues of fundamental concepts of mathematical analysis, for example, limit, continuity, smoothness, connectivity and monotony, extremum, inflection, convexity, etc. This give ground to use the term “discrete mathematical analysis” (DMA) in the substantiations of new algorithms constructed below. The starting point of this modeling is a sufficiently “soft” (in the L. A. Zadeh opinion [24]) character of perception of discreteness properties by man. In fact, an experienced researcher, as a rule, efficiently clusterizes and detects thickenings and traces and finds anomalies in two- or three-dimensional arrays and in small-size time series. The objective of DMA is to extend the execution of these operations to higher dimensions and larger sizes of processed data. The technical basis of DMA consists of fuzzy mathematics and fuzzy logic that possess expressive capabilities for translation of human ideas and speculations into a formal computer language. Figure 1 presents the scheme of our approach to the construction of DMA. The three upper blocks of the scheme are specific to the formal foundations of DMA and conclude with the definition of a finite limit. Note that, of course, we consider that the introduced definition of a finite limit is only one of possible definitions. The density construction based on it allows one to introduce a new interpretation of the concept of a thickening, a cluster, and a trace in multidimensional discrete spaces. It is the first series (the left branch presented in the scheme) of DMA algorithms that is adjusted to the search for them, namely, “Crystal,” “Monolith,” “Rodin,” and “Tracing” [10, 1–3]. The other branches of the DMA
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