Discrete Mathematical Analysis Research Articles

Regression Derivatives and Their Application in the Study of Magnetic Storms

Discrete Mathematical Analysis (DMA) is a data analysis method that uses fuzzy mathematics and fuzzy logic. DMA involves the active participation of the researcher in the study of records, offering technologies and algorithms for analyzing records through the properties of interest to the researcher. In the present work, such properties are related to regression derivatives, and the results obtained are applied to magnetic records. The possibilities of the method in the morphological analysis of geomagnetic storms are demonstrated on the example of three strongest storms that have occurred since the beginning of the current 25th solar cycle.

Time Series Analysis by Fuzzy Logic Methods

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.

Течение пандемии COVID-19 2020-2022 гг.: описание методами математической статистики и дискретного математического анализа

The paper describes the course of the COVID-19 pandemic using a combination of mathematical statistics and discrete mathematical analysis (DMA) methods. The method of regression derivatives and FCARS algorithm as components of DMA will be for the first time tested outside of geophysics problems. The algorithm is applied to time series of the number of new cases of COVID-19 infections per day for some regions of Russia and the Republic of Austria. This allowed to assess the nature and anomalies of pandemic spread as well as restrictive measures and decisions taken in terms of the administration of countries and territories. It was shown that these methods can be used to identify time intervals of change in the nature of the incidence rate and areas with the most severe course of the epidemic. This made it possible to identify the most significant restrictive measures that allowed to reduce the growth of the disease.

DPS Clustering: New Results

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.

Поиск зон экстремальности на основе алгоритмов дискретного математического анализа для выявления рисков при бурении по геофизическим данным

Despite the impressive list of examples of the integration of pattern recognition theory into various activities in the development of oil and gas fields, the authors propose a fundamentally new approach to the use of artificial intelligence. The paper considers in detail the algorithm for searching for extremity zones, based on discrete mathematical analysis (DMA), as applied to the problem of identifying geological hazards. The application of the method is shown on models of the physical properties of rocks reconstructed from seismic data. Potentially, it can also be applied directly to the wave seismic field to identify objects.

LINEAR RECURRENCE RELATONS AND ORDINARY GENERATING FUNCTIONS APPLIED ON MODELING PROCESSES IN CONTROL THEORY

In this paper we apply multistep recurrence relations, as one of very simple and useful mathematical models. It is an efficient tool for solving many problems in mathematics, science, and technics. We also use generating functions, as a connection between real number sequences and real functions, and as a very smooth and efficient connection between the discrete mathematics and (continual) mathematical analysis. We present an application of multistep homogenous linear recurrence relations for modelling some processes in the control theory. Further on, we use the ordinary generating function aiming to find appropriate formulae for calculating members of an appropriate recurrence sequence. Finally, we show the application of this novel mathematical approach on one real example in the control theory.

Dynamic Activity Index for Feature Engineering of Geodynamic Data for Safe Underground Isolation of High-Level Radioactive Waste

In this study, we developed a new approach for feature engineering in geosciences. The main focus of this study was feature engineering based on the implementation of the dynamic activity index (MDAI) as a function of the anomaly of the spatial distribution of data, using systems and discrete mathematical analysis. The methodology for calculating MDAI by groups, geomorphological variability, the density of tectonic faults, stress-strain state, and magnetic field anomalies, is presented herein for a specific area. A detailed analysis of the correlation matrix of MDAI revealed weak correlations between the development features. This showed that the considered properties of the geological environment are independent sets and can be used in the analysis of its geodynamic stability. As a result, it was found that most of the territory where high-level radioactive waste (HLRW) disposal is currently planned is in a relatively stable zone.

Assessment of the Influence of Astronomical Cyclicity on Sedimentation Processes in the Eastern Paratethys Based on Paleomagnetic Measurements Using Discrete Mathematical Analysis

The introduction of modern methods for the mathematical processing of geological data is one of the promising areas of study and development in the field of geosciences. For example, today mathematical geology makes it possible to reliably identify astronomical cycles by measuring the scalar magnetic parameters of rocks (magnetic susceptibility). The main aim of this study is to develop a mathematical tool for identifying stable oscillation cycles (periods) in the dataset of the magnetic susceptibility of rocks in a geological section. The author’s method (algorithm) is based on the concept of discrete mathematical analysis—an innovative mathematical approach to the analysis of discrete geological and geophysical data. Its reliability is also demonstrated, by comparison with the results obtained by classical methods: Fourier analysis, Lomb periodogram, and REDFIT. The proposed algorithm was applied by the authors to analyze the material of field geological studies of the Zhelezny Rog section (Taman Peninsula). As a result, stable cycles were determined for the Pontian and Lower Maeotian sedimentary strata of the Black Sea Basin (Paratethys).

A Multiscale Approach to Geomagnetic Storm Morphology Analysis Based on DMA Activity Measures

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.

Theoretical Framework for Determination of Linear Structures in Multidimensional Geodynamic Data Arrays

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.

Regression differentiation and regression integration of finite series

Data analysis is a complex and multi-dimensional concept. It is explained by both the objective complexity of the data itself and the subjective nature of the expert analyzing them. Therefore, adequate formalization of this requires a completely new apparatus, on the one hand, capable of overcoming the objective complexity of data (irregularity and inaccuracy), and, on the other hand, the vague nature of expert judgment. The development of Discrete Mathematical Analysis (DMA) is an important step in this direction. DMA is highly expert oriented and occupies an intermediate position in data analysis between strict mathematical methods (statistical analysis, TFA, etc.) and soft combinatorial methods (simulation modeling, neural networks, etc.). This paper proposes new mathematical constructions of regression derivatives and regression integrals for discrete time series defined in general on an irregular grid. An important role in their study is played by the recently created by the authors projection method for solving systems of linear algebraic equations described at the end of this paper. The obtained constructions of regression derivatives and regression integrals have a hierarchical character in the spirit of wavelet and fractal analysis. The results of this work define another direction in further studies, namely, the propagation of regression differentiation and regression integration in finite mathematics under scenarios of the classical one.

Problem of Recognition of Strong-Earthquake-Prone Areas: a State-of-the-Art Review

Abstract—The paper addresses almost half a century’s development of pattern recognition algorithms’ application for solving the problem of determination of strong earthquake-prone areas. This approach was named Earthquake-Prone Areas (EPA). The pattern recognition algorithms applied for this purpose, the studied regions, and the methods for assessing the reliability of the obtained results including the theory of dynamic and limit recognition problems are considered. A recently developed alternative method for solving the problem by identifying the clusters of earthquake epicenters is also presented. This method is based on the approaches of the Discrete Mathematical Analysis (DMA) and implemented in the form of the algorithmic system named Formalized Clustering and Zoning (FCAZ). The comparison of the results obtained by the EPA approach and FCAZ system shows their good consistency which provides an additional argument in favor of their reliability. The possibilities for further development and joint application of the EPA approach and FCAZ system and creating, on this basis, an integrated method for systems analysis with the inclusion of artificial intelligence are outlined. In case of success, this method is expected to be used in seismic hazard assessment and planning of earthquake-resistant construction.

Распознавание образов в задачах оценки сейсмической опасности

Статья посвящена некоторым аспектам применения алгоритмов распознавания образов при решении задач определения мест возможного возникновения сильных землетрясений, что может быть использовано для оценки сейсмической опасности. Приведены основные принципы имеющего многолетнюю историю подхода к распознаванию мест сильных землетрясений (РМСЗ) рассматриваемого региона на базе схемы его морфоструктурного районирования с применением алгоритмов «Кора-3» и «Хемминг». Дан обзор полученных в этом направлении результатов и работ по разработке новых алгоритмов, основанных, в частности, на дискретном математическом анализе. Отмечено использование подходов распознавания образов для создания алгоритмов среднесрочного прогноза землетрясений, с помощью которых может быть получена оперативная оценка сейсмической опасности. Рассмотрено применение Общего закона подобия для землетрясений для оценки сейсмической опасности и рисков с учетом результатов РМСЗ. Приведен обзор результатов РМСЗ и оценки сейсмической опасности и рисков для региона Кавказа.

Математические методы анализа и прогноза афтершоков землетрясений: необходимость смены парадигмы

Анализ и прогноз афтершоков сильных землетрясений в мировой практике в настоящее время основан исключительно на стохастических моделях развития афтершокового процесса. Это дает возможность использования статистических методов анализа, а также применять в прогнозе ”сценарный” подход путем многократного генерирования случайных последовательностей афтершоков и подсчета частоты повторения интересующих событий. Исследования по проекту РНФ ”Создание информационной системы автоматической оценки сейсмической опасности после сильных землетрясений по данным геофизического мониторинга” в 2016-2018 гг. показали однако, что эффективность таких подходов имеет существенные ограничения. В статье дается критический обзор статистических методов анализа и прогноза афтершоков, интерпретируются пределы эффективности прогнозов при использовании стандартных подходов, приводится обоснование необходимости смены парадигмы. В качестве одного из направлений поиска предлагается применение методов Дискретного математического анализа (ДМА), разрабатываемых академиком А.Д. Гвишиани и его научной школой. Очевидное преимущество такого подхода продемонстрировано на примере простого алгоритма идентификации афтершоков с использованием аппарата нечетких сравнений.

Accuracy Estimation of the Modern Core Magnetic Field Models Using DMA-Methods for Recognition of the Decreased Geomagnetic Activity in Magnetic Observatory Data

A new approach to identify the signals of the Earth’s main magnetic field (core field) based on the magnetic observatory data processing is suggested. The algorithms implemented in the approach are based on the Discrete mathematical analysis (DMA). The developed method is used to analyze the data from 49 midand low-latitude observatories of the INTERMAGNET network collected in 2011–2015. The results are compared with the classical method for determining the periods of low magnetic activity of external origin which is adopted by the International Association for Geomagnetism and Aeronomy (IAGA). The advantages of the suggested new approach are demonstrated. Based on the data records for the selected time intervals, the time series of the core field components and their secular variations are obtained for each observatory. These data are compared to the values predicted by the most accurate core field models: SIFM, CHAOS-6, and EMM-2015. The accuracy of the models is estimated using a set of statistical parameters: Pearson’s coefficient of linear correlation, Spearman’s and Kendall’s coefficients of rank correlation, the mean and median values over the data sets, and the mean difference between the data obtained by the suggested method from observatory measurements and the model predictions.

On the monitoring of seismic activity using the algorithms of discrete mathematical analysis

On the monitoring of seismic activity using the algorithms of discrete mathematical analysis

A new approach to monitoring seismic activity: California case study

Preliminary work on the creation of a method for monitoring of seismic activity, on the basis of discrete mathematical analysis for further seismic hazard assessment of a territory, is presented. The method developed is tested for the territory of California and adjacent areas of the United States. The results obtained are analyzed for the period of 1980–2015. The nonrandomness of the results is shown by means of an error chart.

The Study of Time Series Using the DMA Methods and Geophysical Applications

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).

Weighted gravitational time series smoothing

This article continues the series of papers by the authors on the new universal DMA-smoothing of time series, originally intended for the analysis of geophysical time series obtained in the framework of discrete mathematical analysis (DMA), developed by GC RAS. We formulated the general concept of weighted DMA-smoothing, constructed and analyzed one of its variants. This is the e-book version of the article, published in the Russian Journal of Earth Sciences (doi:10.2205/2014ES000543). It is generated from the original source file using LaTeX’s epub.cls class.

Survey of Geomagnetic Observations Made in the Northern Sector of Russia and New Methods for Analysing Them

An overview of the geomagnetic observations made in the northern part of Russia is presented from a historical perspective. Several stations were deployed on the territory of the former Soviet Union during the International Geophysical Year, 1957–1958, with the active participation and guidance of the Interagency Geophysical Committee which is inherited by the Geophysical Center of the Russian Academy of Sciences (GC RAS). In the 1990s, the majority of these stations, especially those in the remoter regions, were closed. Nowadays, the geomagnetic network, including the observatories of the INTERMAGNET program, has been restored. Examples of high-latitude geomagnetic variations in the Russian longitudinal sector are shown, and maps and trends of the secular variation over the territory of Russia presented. Particular attention is paid to the automated processing of data and to the analysis methods used. To process the growing amount of high-resolution geomagnetic data, sophisticated mathematical methods based on the fuzzy logic approach and new discrete mathematical analysis algorithms have been developed. The formal methods and algorithms for recognizing both artificial and natural disturbances in the magnetograms are described.