Descriptive model of temporal features of multivariate time series based on granulation Tatiana Afanasieva Irina Moshkina Ulyanovsk State Technical University Ulyanovsk State Technical University Ulyanovsk, Russia Ulyanovsk, Russia tv.afanasjeva@gmail.com timina_i@mail.ru Abstractβ€”Modern systems are characterized by high rates obtained by granulating MTS at the micro and macro levels and volumes of receipt of numerical data. The number of determined by the consideration aspect. indicators of economical, biological, and technical systems, including Autonomous ones, is increasing, generating large Usually, the representation of features is considered as a amounts of numerical data of observation in real time. These set (or vector) of numerical attributes, each of which data have a multidimensional structure and binding to time numerically summarizes a separate feature of a one- points, which allows us to consider them in the form of dimensional time series (TS). This representation does not numerical multivariate time series. As part of the descriptive take into account the features of the two-dimensional analysis of these data, the article presents new model of structure of the MTS, which allows one to extract more representation of local features, considered at different levels complex structures in the form of micro and macro granules of granulation, in respect to temporal features of a multivariate and on this basis describe its local and global features of time series in terms of general tendencies. For this purpose, the temporal patterns, local and global tendencies, fuzzy and provisions of the theory of fuzzy sets and fuzzy time series were associative rules. In this study, granulation refers to the applied in descriptive model, which provided a linguistic automatic processing of MTS to extract features aimed at description of tendencies, understandable to the expert. understanding its behavior, according to the approach of R. Carried out results in modelling of local feature in terms of Yager and J. Kacprzyk [1]. The granular presentation of tendency in descriptive analysis of COVID-19 spread showed MTS will allow describing its features within the part of one effectiveness and operability of proposed approach. methodological basis, will reduce the dimension of MTS, Keywordsβ€”multivariate time series, fuzzy time series, develop new methods for their classifying, predicting, granulation, general tendency clustering, and on this basis, deepen scientific knowledge in a subject-oriented field. I. INTRODUCTION Considering MTS as the object of descriptive analysis, it Data sets of numerical data in the form of numerical should be noted that linguistic interpretation of the extracted multidimensional time series (MTS), describing the behavior granules representing the temporal features of MTS is most of complex objects, are a source of hidden knowledge required for domain experts. Such a linguistic interpretation necessary when analyzing the feature of processes in many can be obtained by combining domain-specific knowledge in applied systems, including telecommunications, industry, the field of MTS analysis and fuzzy models integrating healthcare, meteorology, biology, sociology, public numerical and linguistic values. The use of fuzzy models is administration, medicine, computer networks and financial caused, on the one hand, by the need to represent temporal applications. By feature we mean a characteristic, distinctive features MTS that contain inaccuracies and distortions, and, property of object that distinguishes it from other objects or on the other hand, by the ability to obtain interpreted determines its similarity with other objects. The specified information granules. This is in demand by domain experts, semantics of feature of objects allows us to distinguish two analysts, and intellectual assistants to select and apply ways in the analysis of features of objects: analysis of the adequate models in the subsequent stages of the analysis of features of an individual object and analysis of the features complex objects presented by MTS. of a set of objects. In each of these areas, one can formulate a typical set of stages of the analysis of features, such as The goal of the paper is to develop a descriptive model descriptive, diagnostic, predictive, prescriptive, and cognitive for mining and representing temporal features of MTS based analysis. In this case, descriptive (descriptor) analysis is the on fuzzy time series, granulation and tendency. first stage that determines the effectiveness of subsequent II. RELATED WORKS stages of the analysis of objects. Features of MTS are usually represented as numerical The main task of descriptive analysis of MTS can be characteristics by mapping to a low-dimensional feature considered as the task of extracting, describing and object- space using various transforms, such as locality oriented interpretation of its features observed in a given preserving projections (LPP)[2], which preserves the nearest time interval, and is to answer the question "What neighbor relation, singular value decomposition (SVD)[3], happened?". The MTS data structure is complex. Therefore, and multidimensional wavelet transforms [4]. However, the when extracting and analyzing the features of such structures, it is advisable to consider MTS in various aspects attributes thus obtained may not have a semantic both as a separate complex object with global (integrative) interpretation and may not express the inherent features of features, and as a set of one-dimensional time series (TS) MTS behavior. Chris Aldrich shows the challenges and forming it. At the same time, the one-dimensional TS can makes a review of approaches to extracting the MTS also be described on the basis of its global, local and features in the problem of defect detection in real dynamic temporal features. This allows us to consider the features of systems based on principal component analysis (PCA) [5]. MTS from the point of view of global and local granules The author considers classical approaches for extracting features with respect to MTS, considered as a sequence of Copyright Β© 2020 for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0) Data Science images. However, the characteristics obtained in these necessary to find out and to describe their temporal approaches are global, which usually lose local data properties, which we will call features. In our study, we characteristics. focus on one point of view for all TS of MTS, which consists In order to extract local features from the MTS data, in its temporal features, presented by a linguistic description some scientists are expanding the methods of representing of the TS tendency extracted using fuzzy TS [19]. This the features of time series by a combination of shapelets of approach corresponds to provisions of the work [14]. We various variables [6] to generate associative rules and in the apply fuzzy representation of TS to deal with uncertainty in tasks of early classifying of MTS. Note that this approach data produced by noise and to create meaningful granules in linguistic form of MTS behavior. That description for each does not take into account the interpretation of the TS represents its global feature and, at the same time, the behavioral characteristics in terms of tendencies of MTS. In local MTS feature. In a sense, such a representation of the addition, studies in the work [7] have shown that there are temporal properties of MTS corresponds to the result of deviations between the extracted shapelets and the essential visual analysis of MTS by an expert. We consider the features of MTS, therefore, the shapelets cannot fully summarization of the set of these local features as a global express the essential characteristics of multidimensional feature of MTS. time-series data. The application of fuzzy transformations and rules for Let 𝑋 = (π‘₯𝑑𝑗 ), (𝑗 = 1,2, … , π‘š; 𝑑 = 1,2, … , 𝑛) be extracting static features of TS was considered in [8–10]. It numeric MTS. Here 𝑗 is index of one-dimensional TS, π‘š is uses the numerical characteristics of TS, such as average, number of one-dimensional numeric TS in MTS and 𝑛 is variation, minimum and range, which are too common for number of observations. TS. Granulation methods, which are based on the theory of To represent the local feature of MTS, we use the fuzzy sets [11], are used in TS analysis and decision making Β«behaviorΒ» characteristic with respect to the general [12-15]. A scaling and granulation of linear trend patterns tendency of the 𝑗 -th TS, for which this feature will be using fuzzy models for producing interpretable TS segments global. in different aspects of perception based time series data To describe the global feature of Β«behaviorΒ» for one- mining were discussed by I. Batyrshin and L. Sheremetov in dimensional TS π‘₯𝑑𝑗 ∈ 𝑋 we use the concept of general the work [14]. In the problem of representing features of TS by granules in terms of fuzzy values and tendencies was tendency [16] introduced for fuzzy TS [19], where fuzzy TS is understood as TS, the levels (values) of which are studied and applied in software engineering domains [12, presented by fuzzy sets forming some linguistic variable 13]. 𝑍̃ = {𝑧̃𝑖 |𝑖 = 1,2, … , π‘Ÿ, π‘Ÿ < 𝑛} [20, 22]. This linguistic The book [15] notes that TS granulation is the most variable should be built on the set of admissible values of W adequate method for extracting TS features in the temporal of each numerical one-dimensional TS π‘₯𝑑𝑗 . It is assumed that and spatial aspects. Another interesting approach is related the indices 𝑖 of fuzzy labels 𝑧̃𝑖 correspond to partially to the clustering of granules represented in symbolic form. ordered intervals on W, that are carriers of fuzzy labels 𝑧̃𝑖 . The granular representation of TS was studied in the prediction problem in the work [16-17]. The application of Definition 1. The general tendency (GT) of a one- linguistic summary to granular data is given in [1] as a dimensional TS π‘₯𝑑𝑗 is a linguistic label 𝑦 ∈ π‘Œ, method of granulation of quantifiers in propositions. An π‘Œ = {Stability, Growth, Fall, Systematic Fluctuation𝑠, algorithm for finding intervals of monotonous behavior of Chaotic Fluctuations} expressing its temporal behavior in TS was suggested in [18] and then approach to automatic total. summarization of information on time series based on We assume that general tendencies β€²Growth', β€²Fallβ€² and intermediate quantifiers (a constituent of fuzzy natural β€²Chaotic Fluctuationβ€² correspond to non-stationary behavior logic) and generalized Aristotle's syllogisms was showed. of TS, while β€˜Systematic Fluctuation’ and β€˜Stability’ The analysis of the current state of the MTS features characterize in some sense its stationary property. representation area in the MTS granulation problem for Representation of TS behavior in the form of GT terms is extracting local features allows us to draw the following common to all one-dimensional TS and provides additional conclusion. Models for representing features of MTS are knowledge about temporal changes, useful both for experts and for automation further analysis. Therefore, in this study under development, while the fuzzy models are a promising the local features of MTS are considered as the set of the approach due their opportunity to give linguistic above linguistic terms related to each numerical one- interpretation of features in different levels of TS dimensional TS π‘₯𝑑𝑗 ∈ 𝑋. granulation. In real application the set of labels for linguistic III. DESCRIPTIVE MODEL OF FEATURES OF MTS TEMPORAL describing TS general tendency could be expanded by new BEHAVIOR ones or reduced as well. We consider MTS as an abstract object, representing Definition 2. The general linear tendency of TS is a some observation of a set of changing characteristics of linguistic label 𝑦 ∈ π‘Œ, π‘Œ = {Stability, Growth, Fall} process or of object, of which we do not know anything and expressing its temporal behavior in total. Below we suggest assume some noise in their values. Changing characteristics the following designation of GT Y = {π‘¦π‘˜ |π‘˜ = 1,2, … , π‘˜π‘›}, of a process or of object represented by one-dimensional TS where π‘˜π‘› is equal to quantity of GT labels. and their properties could be considered from different points of view, consequently they may have different interpretation Definition 3. The global feature of the 𝑗-th TS π‘₯𝑑𝑗 ∈ 𝑋, associated with domain semantics. Therefore, before characterizing its behavior on the interval 𝑑 = 1,2, … , 𝑛 by conducting a diagnostic or predictive analysis for them, it is GT, is kn-dimensional vector 𝐻𝑗 = (β„Žπ‘—π‘˜ ), (π‘˜ = 1,2, … , π‘˜π‘›, VI International Conference on "Information Technology and Nanotechnology" (ITNT-2020) 288 Data Science where β„Žπ‘—π‘˜ = πœ‡π‘¦π‘˜ (π‘₯𝑑𝑗 )), where πœ‡π‘¦π‘˜ (π‘₯𝑑𝑗 ) denotes a degree This knowledge is expressed in a concise linguistic form, understandable to the expert and useful for methods of of belonging TS π‘₯𝑑𝑗 to π‘¦π‘˜ ∈ π‘Œ. diagnostic, predictive, prescriptive and cognitive analysis. Then, to determine TS global feature in terms of GT the A. Micro granulation in MTS membership degree of TS π‘₯𝑑𝑗 to π‘¦π‘˜ should be calculated. Let us consider micro granulation of numeric MTS as the Since there is challenge to create membership functions for process of transforming its set of one-dimensional numerical linguistic terms in Y in next Section we propose the TS into fuzzy TS according to expression (4). We denote technique of micro and macro granulating to obtain TS some one-dimensional numeric TS included in the MTS as global feature and calculate the degrees of belonging follows: β„Žπ‘—π‘˜ = πœ‡π‘¦π‘˜ (π‘₯𝑑𝑗 ). {π‘₯𝑑 |π‘₯𝑑 ∈ π‘Š, π‘Š βŠ† ℝ, 𝑑 = 1,2, … , 𝑛}. (7) Definition 4. The descriptive model of local feature of MTS X presented in linguistic terms of GT is the set of TS Suppose that a linguistic variable 𝑍̃ [20, 22] is created on the global features, presented by following expression: set W (domain of TS values) with π‘Ÿ linguistic terms: πΏπ‘œπ‘(𝑋) = {𝐿𝑗 , πœ‡π‘— |𝑗 = 1,2, … , π‘š}, (1) 𝑍̃ = {𝑧̃𝑖 |𝑖 = 1,2, … , π‘Ÿ, π‘Ÿ < 𝑛}. (8) 𝐿𝑗 = 𝑦𝑐 , πœ‡π‘— = β„Žπ‘—π‘ , 𝑐 = π‘Žπ‘Ÿπ‘”π‘šπ‘Žπ‘₯π‘˜=1,2,…,π‘˜π‘› (β„Žπ‘—π‘˜ ). Note the number of generated fuzzy terms π‘Ÿ of linguistic (2) variable 𝑍̃ for each TS could be set by an expert or determine automatically. Here 𝐿𝑗 ∈ π‘Œ denotes linguistic label having maximal membership degree πœ‡π‘— among other labels π‘¦π‘˜ ∈ π‘Œ, and 𝑐 is We assume the set W is covered by partially ordered the number of this label. intervals and each linguistic term 𝑧̃𝑖 ∈ 𝑍̃ is constrained by its corresponding interval. The proposed descriptive model of local feature of GT represents its behavior generically and concisely and makes To convert a numerical TS π‘₯𝑑 into a fuzzy TS 𝑋̃𝑑 , we use it possible to use it in a diagnostic, predictive and the NFLX-transforming TS (Β«conversion from numeric to prescriptive analysis of the underlying process or object. At fuzzy linguisticΒ» values) according to expression (4) as was the descriptive stage, the frequency analysis of linguistic described in [22]: labels in πΏπ‘œπ‘(𝑋) may provide the knowledge about global 𝑁𝐿𝐹𝑋: { π‘₯𝑑 |𝑑 = 1,2, … , 𝑛} ⟼ {𝑋̃𝑑 | 𝑑 = 1,2, … , 𝑛}, feature of MTS temporal behavior in general. (9) Using this approach, the MTS global features could be The fuzzy TS 𝑋̃𝑑 is formed as follows: extracted in respect to stationary or non-stationary MTS temporal behavior. Also, such summing propositions could πœ‡π‘₯̃𝑑 (π‘₯𝑑 ) = π‘šπ‘Žπ‘₯𝑖=1,2,…,π‘Ÿ (πœ‡π‘§Μƒπ‘– (π‘₯𝑑 ) , 𝑠 ∈ {1, 2, … , π‘Ÿ}, (10) be formed as β€œIn MTS all tendencies referred to Fall”, β€œIn MTS less than half tendencies referred to Chaotic π‘₯̃𝑑 = 𝑧̃𝑠 , 𝑠 = π‘Žπ‘Ÿπ‘”π‘šπ‘Žπ‘₯𝑖=1,2,…,π‘Ÿ (πœ‡π‘§Μƒπ‘– (π‘₯𝑑 )), Fluctuation” and others to describe temporal changes in MTS (11) using general tendency. The techniques of such 𝑋̃𝑑 = {π‘₯̃𝑑 , πœ‡π‘₯̃𝑑 (π‘₯𝑑 )| 𝑑 = 1,2, … , 𝑛}, (12) summarization were considered in [1,18, 21]. where π‘₯̃𝑑 is a linguistic term equal to a linguistic term 𝑧̃𝑠 Based on the introduced concepts of local and global with a maximum degree of membership for TS at time t, 𝑠 features, we define a process of descriptive modeling of is the number of this linguistic term, and πœ‡π‘₯̃𝑑 (π‘₯𝑑 ) is the MTS temporal behavior in terms of GT by following sequence of expressions: degree of belonging π‘₯𝑑 to this linguistic term at time t. π‘₯𝑑𝑗 = 𝑓1(𝑋), (3) In that way the fuzzy values of numeric TS are formed using a linguistic variable 𝑍̃, the fuzzy terms of the latter are 𝑋̃𝑗𝑑 = 𝑓2(π‘₯𝑑𝑗 , 𝑍̃), (4) ordered by increasing their indices 𝑖 (according to assumptions about the linguistic variable). πΏπ‘œπ‘(𝑋) = 𝑓3(𝑋̃𝑑𝑗 , π‘Œ), (5) Then the values of two neighboring fuzzy values π‘₯̃𝑑 and 𝐿 = 𝑓4(𝐿𝑗 ). (6) π‘₯Μƒπ‘‘βˆ’1 in fuzzy TS may be represented by linguistic labels as follows for: 𝑑 = 2,3, … , 𝑛: In this descriptive model (3-6), transformations 𝑓1 and 𝑓2 refer to micro granulation of numeric MTS, and the result π‘₯Μƒπ‘‘βˆ’1 = 𝑧̃𝑠(π‘‘βˆ’1) , (13) of the transformation (4) is a fuzzy time series 𝑋̃𝑑𝑗 , obtained π‘₯̃𝑑 = 𝑧̃𝑣(𝑑) , (14) for a one-dimensional 𝑗-th TS π‘₯𝑑𝑗 ∈ 𝑋. Micro granulation is considered as the process of creating small granules by where 𝑠(𝑑 βˆ’ 1) and 𝑣(𝑑) denote the indices of fuzzy labels decomposing MTS into components. In this case, the of the linguistic variable 𝑍̃ , associated with time instants relationship of β€œfragmentation” between the MTS and its (𝑑 βˆ’ 1) and 𝑑 respectively. micro granules, is established. Macro granulation establishes the β€œgeneralization” relation and is represented by Since fuzzy terms in the linguistic variable 𝑍̃ are ordered transformations 𝑓3 and 𝑓4, which form larger granules by indices (according to the assumptions about the linguistic characterizing of MTS temporal behavior in the form of its variable), we use these indices to determine the intensity of local and global features in terms of GT. Based on this change for two neighboring fuzzy values of fuzzy TS in descriptive model, knowledge about the local and global direction of their increasing and decreasing. We suppose that features of MTS, characterizing its behavior, is extracted. between two neighboring fuzzy values there can also be no VI International Conference on "Information Technology and Nanotechnology" (ITNT-2020) 289 Data Science changes. Taking in account the expressions (13) and (14), the Step 2.2. If π›Όπ‘”π‘Ÿπ‘œπ‘€π‘‘β„Ž > 2 βˆ— π›Όπ‘“π‘Žπ‘™π‘™ , then intensity of change of two neighboring fuzzy values in fuzzy 𝑦2 = 'πΊπ‘Ÿπ‘œπ‘€π‘‘β„Žβ€², πœ‡π‘¦1 (π‘₯𝑑 ) = 0 , TS for observation 𝑑 is presented as: π›Όπ‘”π‘Ÿπ‘œπ‘€π‘‘β„Ž π›Όπ‘“π‘Žπ‘™π‘™ πœ‡π‘¦2 (π‘₯𝑑 ) = , πœ‡π‘¦3 (π‘₯𝑑 ) = (π‘Ÿβˆ’1)βˆ—(π‘›βˆ’1) , (π‘Ÿβˆ’1)βˆ—(π‘›βˆ’1) 𝛼𝑑 = 𝑣(𝑑) βˆ’ 𝑠(𝑑 βˆ’ 1), 𝑑 = 2,3, … , 𝑛. (15) Step 2.3. If π›Όπ‘“π‘Žπ‘™π‘™ > 2 βˆ— π›Όπ‘”π‘Ÿπ‘œπ‘€π‘‘β„Ž , then 𝑦3 = 'πΉπ‘Žπ‘™π‘™β€² , Thus, at the stage of micro granulation of MTS, for each π›Όπ‘”π‘Ÿπ‘œπ‘€π‘‘β„Ž π›Όπ‘“π‘Žπ‘™π‘™ value of TS, we obtain the degree of its belonging πœ‡π‘₯̃𝑑 (π‘₯𝑑 ), πœ‡π‘¦1 𝑑 ) = 0, πœ‡π‘¦2 (π‘₯𝑑 ) = (π‘₯ , πœ‡π‘¦3 (π‘₯𝑑 ) = (π‘Ÿβˆ’1)βˆ—(π‘›βˆ’1) (π‘Ÿβˆ’1)βˆ—(π‘›βˆ’1) , the corresponding linguistic label π‘₯̃𝑑 , and the intensity of changes in neighboring values 𝛼𝑑 . Step 2.4. If (0,85 βˆ— π›Όπ‘“π‘Žπ‘™π‘™ < π›Όπ‘”π‘Ÿπ‘œπ‘€π‘‘β„Ž < 1,15 βˆ— π›Όπ‘“π‘Žπ‘™π‘™ ) π‘œπ‘Ÿ (0,85 βˆ— π›Όπ‘”π‘Ÿπ‘œπ‘€π‘‘β„Ž < π›Όπ‘“π‘Žπ‘™π‘™ < 1,15 βˆ— π›Όπ‘”π‘Ÿπ‘œπ‘€π‘‘β„Ž ), B. Macro granulation in MTS then 𝑦4 = 'π‘†π‘¦π‘ π‘‘π‘’π‘šπ‘Žπ‘‘π‘–π‘ πΉπ‘™π‘’π‘π‘‘π‘’π‘Žπ‘‘π‘–π‘œπ‘›β€², Macro granulation is considered as process of combining π›Όπ‘”π‘Ÿπ‘œπ‘€π‘‘β„Ž π›Όπ‘“π‘Žπ‘™π‘™ πœ‡π‘¦1 (π‘₯𝑑 ) = 0, πœ‡π‘¦2 (π‘₯𝑑 ) = (π‘Ÿβˆ’1)βˆ—(π‘›βˆ’1) , πœ‡π‘¦3 (π‘₯𝑑 ) = (π‘Ÿβˆ’1)βˆ—(π‘›βˆ’1) , micro granules into larger ones, obtained by expressions (1) an (2). Using the proposed MTS descriptive model, macro πœ‡π‘¦4 (π‘₯𝑑 ) = 1, else 𝑦5 = 'πΆβ„Žπ‘Žπ‘œπ‘‘ic Fluctuation', πœ‡π‘¦1 (π‘₯𝑑 ) = π›Όπ‘”π‘Ÿπ‘œπ‘€π‘‘β„Ž π›Όπ‘“π‘Žπ‘™π‘™ granulation is considered according to expression (3) to 0, πœ‡π‘¦2 (π‘₯𝑑 ) = , πœ‡π‘¦3 (π‘₯𝑑 ) = (π‘Ÿβˆ’1)βˆ—(π‘›βˆ’1) ,(π‘Ÿβˆ’1)βˆ—(π‘›βˆ’1) produce local features of MTS temporal behavior in terms π›Όπ‘”π‘Ÿπ‘œπ‘€π‘‘β„Ž π›Όπ‘“π‘Žπ‘™π‘™ of GT. Since determine the membership functions πœ‡π‘¦π‘˜ for πœ‡π‘¦2 (π‘₯𝑑 ) = (π‘Ÿβˆ’1)βˆ—(π‘›βˆ’1) , πœ‡π‘¦3 (π‘₯𝑑 ) = (π‘Ÿβˆ’1)βˆ—(π‘›βˆ’1) , πœ‡π‘¦4 (π‘₯𝑑 ) = 0, linguistic terms of variable π‘Œ is challenge we propose the πœ‡π‘¦5 (π‘₯𝑑 ) = 1. approach to calculate the degree of belonging TS π‘₯𝑑𝑗 ∈ 𝑋 to Step 3. Determining global feature of TS in terms of GT. each π‘¦π‘˜ ∈ π‘Œ. Step 3.1. Calculating the index of linguistic label for TS In this Section the technique for assessing GT 𝐿𝑗 as with maxim membership degree: global characteristic of each numeric TS π‘₯𝑑𝑗 ∈ 𝑋 is 𝑐 = π‘Žπ‘Ÿπ‘”π‘šπ‘Žπ‘₯ {πœ‡π‘¦π‘˜ (π‘₯𝑑 )} , πœ‡π‘— = πœ‡π‘¦π‘ (π‘₯𝑑 ). presented using fuzzy TS (see expression (12)), the indices π‘˜=1,2,…,5 of its two neighboring fuzzy values and the set of linguistic Step 3.2. Determining the linguistic term of GT of j-th labels TS π‘₯𝑑 : 𝐿𝑗 = 𝑦𝑐 . π‘Œ = {Stability, Growth, Fall, Systematic Fluctuation, Chaotic Fluctuation}. Step 4. Repeat Steps 1-3 for m TS of MTS and The task is to determine GT 𝐿𝑗 ∈ π‘Œ (see definition 3) for determine its local feature: TS which is presented by fuzzy TS using linguistic variable πΏπ‘œπ‘(𝑋) = {𝐿𝑗 , πœ‡π‘— |𝑗 = 1,2, … , π‘š}. 𝑍̃ and to describe the local feature of MTS in respect to definition 4. Consequently, the membership degrees IV. DESCRIPTIVE MODELING OF COVID-19 USING GRANULATION AND GENERAL TENDENCIES πœ‡π‘¦π‘˜ (π‘₯𝑑𝑗 ) of TS π‘₯𝑑𝑗 to π‘¦π‘˜ ∈ π‘Œ, π‘˜ = 1,2, . . ,5 should be calculated. To illustrate the practical application of the proposed For this purpose, we suggest rule-based technique of model of local feature of MTS in terms of GT, let us consider assessing local features of MTS in terms of GT which an example of descriptive analysis of MTS formed by includes following steps: COVID-19 [23] indicators observed in the local territorial region to understand how a pandemic spreads there. Given Step 1. Pre-processing. that the nature and behavior of COVID-19 is poorly Step 1.1. Micro granulation of MTS according to understood, and many countries have different policies expressions (7-15) and consideration j-th TS π‘₯𝑑 ∈ 𝑋. regarding the intensity and management of quarantine activities, many researchers and ordinary people are Step 1.2. Based on the values 𝛼𝑑 , 𝑑 = 2,3, … , 𝑛, interested in the question of when and by what signs it can be calculated according to expression (15), for a TS π‘₯𝑑 judged that the activity of COVID-19 is reduced. determining its total intensities of changes for growth and Most researchers suggest evaluating tendencies in for fall: COVID-19 prevalence rates [24]. Considering the tendencies πΉπ‘œπ‘Ÿ 𝑑 = 2,3, … , 𝑛 in TS of the indicators of this pandemic over some temporal interval, it is possible to make decision and informed 𝐼𝑓 𝛼𝑑 > 0, π‘‘β„Žπ‘’π‘› π›Όπ‘”π‘Ÿπ‘œπ‘€π‘‘β„Ž = π›Όπ‘”π‘Ÿπ‘œπ‘€π‘‘β„Ž + π‘Žπ‘π‘ (𝛼𝑑 ), recommendations on the weakening of quarantine measures. 𝐼𝑓 𝛼𝑑 < 0, π‘‘β„Žπ‘’π‘› π›Όπ‘“π‘Žπ‘™π‘™ = π›Όπ‘“π‘Žπ‘™π‘™ + π‘Žπ‘π‘ (𝛼𝑑 ). In our study, the MTS characterizing COVID-19 spreading is defined by a set of TS that represent daily changes in the Step 1.3. Initialization of membership degrees for all total number of detected cases of infection (𝑆𝑣), the total linguistic labels in π‘Œ: number of patients recovered (π‘†π‘Ÿ) and the total number of πΉπ‘œπ‘Ÿ π‘˜ = 1,2, … ,5 πœ‡π‘¦π‘˜ (π‘₯𝑑 ) = 0. patients who died ( 𝑆𝑑 ). As an example of descriptive Step 2. Assessing membership degrees and linguistic analysis, we focus on analyzing, extracting and interpretation the tendencies of such MTS, which describe the prevalence labels of global feature of TS temporal behavior in GT of COVID-19 in the city of Moscow of Russian Federation terms. from March 26, 2020 to May 3, 2020 [25]. Step 2.1. If (π›Όπ‘”π‘Ÿπ‘œπ‘€π‘‘β„Ž = 0 and π›Όπ‘“π‘Žπ‘™π‘™ = 0) , then Using micro and macro granulation of MTS, we extract 𝑦1 = 'π‘†π‘‘π‘Žπ‘π‘–π‘™π‘–π‘‘π‘¦β€², πœ‡π‘¦1 (π‘₯𝑑 ) = 1, the global features of its indicators and describe the local feature of COVID-19 activity in terms of the GT with meaningful interpretation. These features, expressed VI International Conference on "Information Technology and Nanotechnology" (ITNT-2020) 290 Data Science linguistically, will be focused on summarizing the dynamics intervals of values, which are necessary for the automatic of COVID-19 spread and the dynamics characterizing to determination of tendencies. Based on the introduced some extent the formation of collective immunity. For this, indicators, for descriptive analysis of the dynamics of we use, based on the main indicators presented at [25], the COVID-19 activity in Moscow, the following MTS was new ones grouped into two types: (1) characteristics of the formed: spread of COVID-19 and (2) characteristics of patient recovery. 𝑋 = { π‘Ÿ(𝑑), πΎπ‘Ÿ(𝑑), 𝑛(𝑑), π‘Ž(𝑑), 𝐾𝑣(𝑑), 𝐾𝑠(𝑑), 𝐾𝑛(𝑑), πΎπ‘Ž(𝑑)}. To extract its micro granules in the form of fuzzy TS In the experimental study of descriptive analysis of values, the NFLX-transform was used. For this purpose, a COVID-19 activity in Moscow the following variables and preliminary linguistic variable 𝑍̃ with ten fuzzy sets was indicators were used: determined for each of the eight time series included in the 𝑑 – this is number of daily observation, 𝑑 = 1,2, … ,39. MTS. When modeling fuzzy terms, triangular membership 𝑆𝑣(𝑑), π‘†π‘Ÿ(𝑑), π‘†π‘Ÿ(𝑑) describe the total number of cases functions were used, which were built on partially ordered per day of infection, recovery cases and death, respectively. intervals of the same length. The universal set of each π‘†π‘Ž(𝑑) is TS of the total number of active cases, π‘†π‘Ž(𝑑) = linguistic variable was determined on the basis of an 𝑆𝑣(𝑑) βˆ’ π‘†π‘Ÿ(𝑑) βˆ’ 𝑆𝑑(𝑑). extended range between the maximum and minimum values 𝑆𝑛(𝑑) designates the daily total number of new of each derived indicator, as described in the work [26]. At infections: 𝑆𝑛(𝑑) = 𝑆𝑣(𝑑) βˆ’ 𝑆𝑣(𝑑 βˆ’ 1). the stage of macro granulation, to each component of the 𝑛(𝑑), π‘Ÿ(𝑑), π‘Ž(𝑑) present the number per day fixed of new MTS, the linguistic characteristic of its GT was determined, cases of infection, recovery and active, respectively. The which made it possible to determine the local feature of the increase in TS of daily infections, deaths, and active analyzed MTS, presented in Table 1. infections indicates a negative trend, while the fluctuation The data from table 1 show that 75% of the trends in the trend can be interpreted as a sign of a transition to a positive dynamics of COVID-19 activity in Moscow are positive trend. The downward trend in 𝑛(𝑑) and π‘Ž(𝑑) will show a according to the descriptive model of local features of MTS. positive trend. It is understood that the increase in the It can be noted that according to the indicators number of recovered patients is a good trend. characterizing the recovery of patients in this study, all 𝐾𝑣(𝑑) = π‘†π‘Ž(𝑑)/𝑆𝑣(𝑑) determines TS of proportion of trends are positive. total active cases in relation to all cases of infection. A decrease in this fraction indicates that the distribution TABLE I. RESULTS OF A DESCRIPTIVE ANALYSIS OF MTS BY GT, activity of COVID-19 is reduced. This indicates a positive CHARACTERIZING THE DYNAMIC OF COVID-19 ACTIVITY IN MOSCOW trend. MTS Linguistic label Interpretation of πΎπ‘Ÿ(𝑑) = π‘†π‘Ÿ(𝑑)/π‘†π‘Ž(𝑑) – this is TS of proportion of total of GT GT cases of recovery in relation to active cases. The growth of π‘Ÿ(𝑑) Growth Positive this share shows that the number of ill and received πΎπ‘Ÿ(𝑑) Growth Positive immunity increases, which is positive in terms of the 𝑛(𝑑) Growth Negative formation of collective immunity. 𝐾𝑠(𝑑) = π‘†π‘Ž(𝑑 + 𝑝)/π‘†π‘Ž(𝑑) is the coefficient of the π‘Ž(𝑑) Growth Negative delayed effect of total active cases π‘†π‘Ž(𝑑) per day with 𝐾𝑣(𝑑) Fall Positive number t on the total active cases that occur by the end of the incubation period π‘†π‘Ž(𝑑 + 𝑝) (according to WHO [23], 𝐾𝑠(𝑑) Fall Positive the duration of incubation period 𝑝 can be up to 14 days). A 𝐾𝑛(𝑑) Fall Positive decrease in this coefficient indicates that the activity of infection from active cases is reduced. This indicates a πΎπ‘Ž(𝑑) Fall Positive positive trend. 𝐾𝑛(𝑑) = 𝑆𝑛(𝑑 + 𝑝)/π‘†π‘Ž(𝑑) is the coefficient of the According to the last column of Table 1, we can conclude delayed effect of the total active cases of π‘†π‘Ž(𝑑) per day that in Moscow by May 3, 2020, only 67% of the distribution with number t on the total new cases of 𝑆𝑛(𝑑 + 𝑝) that occur indicators of COVID-19 had a positive trend. Negative at the end of the incubation period. A decrease values in this dynamics trends were observed in the rates of new and active coefficient indicates that the activity of infection from active cases of COVID-19 infection recorded daily. It can be cases is reduced. This indicates a positive trend. assumed that this is due to several reasons, among which πΎπ‘Ž(𝑑) = π‘Ž(𝑑 + 𝑝)/π‘Ž(𝑑) determines the coefficient of the should be noted an increase in the number of tests conducted in Moscow. To clarify this, it is necessary to conduct an delayed effect of daily recorded active cases π‘Ž(𝑑) per day additional analysis, which may be the subject of a new study. with number 𝑑 on the occurrence of daily recorded active cases π‘Ž(𝑑 + 𝑝) that occur at the end of the incubation V. CONCLUSION period. A decrease in this coefficient indicates that the The authors propose an approach to descriptive activity of infection from active cases is reduced. This modelling of local feature of MTS that characterizes its indicates a positive trend. behavior in terms of general tendency. The positions and the To our mind introduced above indicators are necessary descriptor model of the MTS, as well as expressions, in order to be able, on the one hand, to extract additional allowing to generate a linguistic description of its local information about the positive or negative dynamics of feature, are considered. The proposed approach is COVID-19 activity, and on the other hand, in order to be characterized by the use of granulation tools MTS, fuzzy TS able to construct linguistic variables and fuzzy sets on and concept of general tendency, which allows you to extract VI International Conference on "Information Technology and Nanotechnology" (ITNT-2020) 291 Data Science interpretable knowledge that is useful for further analysis of [10] W. Pedrycz and S.M. Chen, β€œModeling and Applications,” A behavior of processes and objects. Application of the Computational Intelligence Perspective. Intelligent Systems Reference Library, vol. 47, pp. 404, 2013. proposed model of local feature in terms of GT in descriptive [11] L.A. Zadeh, β€œToward a theory of fuzzy information granulation and analysis of MTS of COVID-19 spread in Moscow showed its its centrality in human reasoning and fuzzy logic,” Fuzzy Sets and effectiveness and operability while automatically monitoring Systems, vol. 90, pp. 111-127, 1997. the situation. Moreover, the obtained knowledge is useful in [12] T. Afanasieva, N. Yarushkina and I. Sibirev, β€œTime Series making decision corresponding to decline quarantine Clustering using Numerical and Fuzzy Representations,” Proc. of activity. Joint 17th World Congress of international Fuzzy Systems Association and 9th International Conference on Soft Computing and ACKNOWLEDGMENT Intelligent Systems (IFSA-SCIS), 2017. [13] T. Afanasieva and V. Moiseev, β€œFramework for Assessing The authors acknowledge the reported study was funded Professional Growth of Software Developers,” Advancies of by RFBR, as a part of project β„– 19-07-00999, and was Intelligent Systems and Computing, vol. 874, pp. 46-50, 2019. funded by RFBR and Ulyanovsk Region, as a part of project [14] I. Batyrshin and L. Sheremetov, β€œPerception Based Time Series Data β„– 19-47-730001. Mining for Decision Making,” Advances in Soft Computing, vol. 42, pp. 209-219, 2007. REFERENCES [15] W. Pedrycz, A. Skowron and V. Kreinovich, β€œHandbook of Granular [1] J. Kacprzyk and R. Yager, β€œLinguistic summaries of data using Computing,” Willey, 2008. fuzzy logic,” International Journal of General System, vol. 2, pp. 133- [16] N. Yarushkina, T. Afanasieva, D. Zavarzin and G. Guskov, β€œFuzzy 154, 2001. trends data mining in knowledge discovery process,” Creativity in [2] X. He and P. Niyogi, β€œLocality Preserving Projections (LPP),” Proc. Intelligent, Technologies and Data Science, vol. 535, pp. 115-123, of the 16th International Conference on Neural Information 2015. Processing Systems, pp. 153-160, 2003. [17] T. Afanasieva, N. Yarushkina and G. Guskov, β€œThe Study OF Basic [3] J. BrzeziΕ„ska, β€œSingular Value Decomposition Approaches in A Fuzzy Time series Forecasting models,” Uncertainty modelling in Correspondence Analysis with The Use of R,” Folia Oeconomica Knowledge Engineering and Decision Making, vol. 10, pp. 295-300, Stetinensia, vol. 18, no. 2, pp.178-189, 2018. 2016. [4] S.T. Ali, J.P. Antoine and J.P. Gazeau, β€œMultidimensional Wavelets,” [18] V. Novak, β€œLinguistic characterization of time series,” Fuzzy Sets and Coherent States, Wavelets and Their Generalizations. Graduate Texts Systems, vol. 285, pp. 52-72, 2016. in Contemporary Physics, pp. 307-330, 2000. [19] Q. Song and B. Chissom, β€œFuzzy time series and its models,” Fuzzy [5] C. Aldrich, β€œProcess Fault Diagnosis for Continuous Dynamic Sets and Systems, vol. 54, pp. 269-277, 1993. Systems Over Multivariate Time Series, in Time Series Analysis - [20] L.A. Zadeh, β€œThe concept of a linguistic variable and its application Data, Methods, and Applications,” 2019. DOI: to approximate reasoningβ€”I,” Information Sciences, vol. 8, no. 3, pp. 10.5772/intechopen.85456. 199-249, 1975. [6] Y. Lin, H. Chen, V. S. Tseng and J. Pei, β€œReliable Early [21] T. Afanasieva, A. Shutov and E. Efremova, β€œThe Methodology of Classification on Multivariate Time Series with Numerical and Descriptive Analysis of Multidimensional Data based on Combining Categorical Attributes,” PAKDD, vol. 1, LNAI 9077, pp. 199-211, of Intelligent Technologies,” 2020, in press. 2015. DOI: 10.1007/978-3-319-18038-0_16. [22] T. Afanasieva, Y. Egorov and N. Savinov, β€œAbout Transformations of [7] G. He, W, Zhao, X, Xia, R. Peng and X. Wu, β€œEnsemble of Shapelet a Numerical Time Series using a Linguistic Variable,” Advancies of based Classifiers on Inter- class and Intra-classImbalanced Intelligent Systems and Computing, vol. 679, pp. 226-233, 2018. Multivariate Time Series at the Early Stage,” Soft Computing, vol. 23, pp. 6097-6114, 2019. DOI: 10.1007/s00500-018-3261-3. [23] World Health Organization [Online]. URL: https://www.who.int/ emergencies/diseases/novel-coronavirus-2019. [8] H.B. Sandya, P. Hemanth Kumar, H. Bhudiraja and S.K. Rao, β€œFuzzy Rule Based Feature Extraction and Classification of Time Series [24] Worldometer [Online]. URL: https://www.worldometers.info/ Signal,” International Journal of Soft Computing and Engineering coronavirus/#countries. (IJSCE), vol. 3, no. 2, pp. 2231-2307, 2013. [25] Coronavirus in Moscow [Online]. URL: https://coronavirusstat.ru/ [9] I. Perfileva, V. Novak and A. Dvorak, β€œFuzzy transform in the country/moskva/. analysis of data,” International Journal of Appr. Reasoning, vol. 48, [26] Q. Song, β€œForecasting enrollments with fuzzy time series-Part I,” no. 1, pp. 36-46, 2008. Fuzzy Sets and Systems, vol. 54, no. 1, pp. 1-9, 1993. VI International Conference on "Information Technology and Nanotechnology" (ITNT-2020) 292