The discrete mathematical analysis (DMA) is a series of algorithms aimed at the solution of basic problems of data analysis: clustering and tracing in multidimensional arrays, morphological analysis of reliefs, search for anomalies and trends in records etc. All the DMA algorithms are of universal nature, joined by the same formal foundation, based, in its turn, on fuzzy logic (FL) and fuzzy mathematics (FM). The current study finalizes the search for the anomalies in one-dimensional time series within the scope of DMA: here the initial concept of an interpreter’s logic gets its additional development. First, the formal expert’s opinions are more fully expressed, and this is realized with the more complex measures of activity (the concept of straightenings (Gvishiani et al. 2003; Gvishiani et al. 2004; Zlotnicki et al. 2005) is replaced by the measures of activity which come to the fore): second, for the junction of anomalies, a recently created DPS (Discrete Perfect Sets) algorithm is used DPS (Discrete Perfect Sets) (Agayan et al. 2011; Agayan et al. 2014)

The method of analyzing data known as Discrete Mathematical Analysis (DMA) incorporates fuzzy mathematics and logic. This paper focuses on applying DMA to study the morphology of time series by utilizing the language of fuzzy mathematics. The morphological characteristics of the time series, such as background, slopes, and vertices, are considered fuzzy sets within the domain of its definition. This allows for the use of fuzzy logic in examining the morphology of time series, ultimately leading to the detection of anomalies.

The article addresses the issue of clustering of multidimensional data arrays with a noise using the methods of discrete mathematical analysis (DMA clustering). The theory of DMA clustering through the logical densities calculus is detailed, and the new algorithm Linear Discrete Perfect Sets (LDPS) is described. The main objective of the LDPS algorithm is to identify linearly stretched anomalies in a multidimensional array of geo-spatial data (geophysical fields, geochemistry, satellite images, local topography, maps of recent crustal movements, seismic monitoring data, etc.). These types of anomalies are associated with tectonic structures in the upper part of the Earth’s crust and pose the biggest threat for integrity of the isolation properties of the geological environment, including in regions of high-level radioactive waste disposal. The main advantage of the LDPS algorithm as compared to other cluster analysis algorithms that may be used in arrays with a noise is that it is more focused on searching for clusters that are linear. The LDPS algorithm can apply not only in the analysis of spatial natural objects and fields but also to elongated lineament structures.

The results presented in this paper are obtained as part of the continued development and research of clustering algorithms based on the discrete mathematical analysis. The article briefly describes the theory of Discrete Perfect Sets (DPS-sets) that is the basis for the construction of DPS-clustering algorithms. The main task of the previously constructed DPS-algorithms is to search for clusters in multidimensional arrays with noise. DPS-algorithms have two stages: the first stage is the recognition of the maximum perfect set of a given density level from the initial array, the second stage is the partitioning of the result of the first stage into connected components, which are considered to be clusters. Study of qualities of DPS-algorithms showed that, in a number of situations in the first stage, the result does not include all clusters which have practical sense. In the second stage, partitioning into connected components can lead to unnecessarily small clusters. Simple variation of parameters in DPS-algorithms does not allow for eliminating these drawbacks. The present paper is devoted to the construction on the basis of DPS-algorithms of their new versions, more free from these drawbacks.

The article is focused on the approach based on the discrete mathematical analysis conception and continues a series of studies related to the application of the previously developed methodology to geophysical data analysis. The main idea of the study is the modification of earlier conceptions regarding the interpreter’s logic that allows introducing a multiscale approach and performing the time series analysis using the activity measure plots, implying the vertical scale. This approach was used to study the morphology of several intense geomagnetic storms at the final stages of the 23rd and 24th solar activity cycles. Geomagnetic observatory data and interplanetary magnetic field parameters as well as the solar wind flux speed and proton density were analyzed for each of the studied storms using the activity measures. The developed methods, applied to geomagnetic storm morphological analysis, displayed good results in revealing the decreases and increases in various durations and intensities during storms, detecting low-amplitude disturbances, and storm sudden commencement recognition. The results provide an opportunity to analyze any physical data using a unified scale and, in particular, to implement this approach to geomagnetic activity studies.