=Paper= {{Paper |id=Vol-2667/paper64 |storemode=property |title=Descriptive model of temporal features of multivariate time series based on granulation |pdfUrl=https://ceur-ws.org/Vol-2667/paper64.pdf |volume=Vol-2667 |authors=Tatiana Afanasieva,Irina Moshkina }} ==Descriptive model of temporal features of multivariate time series based on granulation == https://ceur-ws.org/Vol-2667/paper64.pdf

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

Abstract—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,
Keywords—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 © 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 = (𝑥𝑡𝑗 ), (𝑗 = 1,2, … , 𝑚; 𝑡 = 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, 𝑚 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 «behavior» 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 «behavior» for one-
mining were discussed by I. Batyrshin and L. Sheremetov in
dimensional TS 𝑥𝑡𝑗 ∈ 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, … , 𝑟, 𝑟 < 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 𝑥𝑡𝑗 . 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 𝑥𝑡𝑗 is a linguistic label 𝑦 ∈ 𝑌,
method of granulation of quantifiers in propositions. An 𝑌 = {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 ′Growth', ′Fall′ and
intermediate quantifiers (a constituent of fuzzy natural ′Chaotic Fluctuation′ correspond to non-stationary behavior
logic) and generalized Aristotle's syllogisms was showed. of TS, while ‘Systematic Fluctuation’ and ‘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 𝑥𝑡𝑗 ∈ 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 ∈ 𝑌, 𝑌 = {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 = {𝑦𝑘 |𝑘 = 1,2, … , 𝑘𝑛},
of a process or of object represented by one-dimensional TS where 𝑘𝑛 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 𝑥𝑡𝑗 ∈ 𝑋,
associated with domain semantics. Therefore, before characterizing its behavior on the interval 𝑡 = 1,2, … , 𝑛 by
conducting a diagnostic or predictive analysis for them, it is GT, is kn-dimensional vector 𝐻𝑗 = (ℎ𝑗𝑘 ), (𝑘 = 1,2, … , 𝑘𝑛,

VI International Conference on "Information Technology and Nanotechnology" (ITNT-2020) 288
Data Science

where ℎ𝑗𝑘 = 𝜇𝑦𝑘 (𝑥𝑡𝑗 )), where 𝜇𝑦𝑘 (𝑥𝑡𝑗 ) denotes a degree This knowledge is expressed in a concise linguistic form,
understandable to the expert and useful for methods of
of belonging TS 𝑥𝑡𝑗 to 𝑦𝑘 ∈ 𝑌. 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 𝑥𝑡𝑗 to 𝑦𝑘 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:
ℎ𝑗𝑘 = 𝜇𝑦𝑘 (𝑥𝑡𝑗 ).
{𝑥𝑡 |𝑥𝑡 ∈ 𝑊, 𝑊 ⊆ ℝ, 𝑡 = 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 𝑟 linguistic terms:
𝐿𝑜𝑐(𝑋) = {𝐿𝑗 , 𝜇𝑗 |𝑗 = 1,2, … , 𝑚}, (1) 𝑍̃ = {𝑧̃𝑖 |𝑖 = 1,2, … , 𝑟, 𝑟 < 𝑛}. (8)
𝐿𝑗 = 𝑦𝑐 , 𝜇𝑗 = ℎ𝑗𝑐 , 𝑐 = 𝑎𝑟𝑔𝑚𝑎𝑥𝑘=1,2,…,𝑘𝑛 (ℎ𝑗𝑘 ). Note the number of generated fuzzy terms 𝑟 of linguistic
(2) variable 𝑍̃ for each TS could be set by an expert or determine
automatically.
Here 𝐿𝑗 ∈ 𝑌 denotes linguistic label having maximal
membership degree 𝜇𝑗 among other labels 𝑦𝑘 ∈ 𝑌, 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 𝑥𝑡 into a fuzzy TS 𝑋̃𝑡 , we use
it possible to use it in a diagnostic, predictive and the NFLX-transforming TS («conversion from numeric to
prescriptive analysis of the underlying process or object. At fuzzy linguistic» values) according to expression (4) as was
the descriptive stage, the frequency analysis of linguistic described in [22]:
labels in 𝐿𝑜𝑐(𝑋) may provide the knowledge about global 𝑁𝐿𝐹𝑋: { 𝑥𝑡 |𝑡 = 1,2, … , 𝑛} ⟼ {𝑋̃𝑡 | 𝑡 = 1,2, … , 𝑛},
feature of MTS temporal behavior in general. (9)
Using this approach, the MTS global features could be The fuzzy TS 𝑋̃𝑡 is formed as follows:
extracted in respect to stationary or non-stationary MTS
temporal behavior. Also, such summing propositions could 𝜇𝑥̃𝑡 (𝑥𝑡 ) = 𝑚𝑎𝑥𝑖=1,2,…,𝑟 (𝜇𝑧̃𝑖 (𝑥𝑡 ) , 𝑠 ∈ {1, 2, … , 𝑟}, (10)
be formed as “In MTS all tendencies referred to Fall”, “In
MTS less than half tendencies referred to Chaotic 𝑥̃𝑡 = 𝑧̃𝑠 , 𝑠 = 𝑎𝑟𝑔𝑚𝑎𝑥𝑖=1,2,…,𝑟 (𝜇𝑧̃𝑖 (𝑥𝑡 )),
Fluctuation” and others to describe temporal changes in MTS (11)
using general tendency. The techniques of such 𝑋̃𝑡 = {𝑥̃𝑡 , 𝜇𝑥̃𝑡 (𝑥𝑡 )| 𝑡 = 1,2, … , 𝑛}, (12)
summarization were considered in [1,18, 21].
where 𝑥̃𝑡 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 𝜇𝑥̃𝑡 (𝑥𝑡 ) is the
MTS temporal behavior in terms of GT by following
sequence of expressions: degree of belonging 𝑥𝑡 to this linguistic term at time t.

𝑥𝑡𝑗 = 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
𝑋̃𝑗𝑡 = 𝑓2(𝑥𝑡𝑗 , 𝑍̃), (4) ordered by increasing their indices 𝑖 (according to
assumptions about the linguistic variable).
𝐿𝑜𝑐(𝑋) = 𝑓3(𝑋̃𝑡𝑗 , 𝑌), (5)
Then the values of two neighboring fuzzy values 𝑥̃𝑡 and
𝐿 = 𝑓4(𝐿𝑗 ). (6) 𝑥̃𝑡−1 in fuzzy TS may be represented by linguistic labels as
follows for: 𝑡 = 2,3, … , 𝑛:
In this descriptive model (3-6), transformations 𝑓1 and
𝑓2 refer to micro granulation of numeric MTS, and the result 𝑥̃𝑡−1 = 𝑧̃𝑠(𝑡−1) , (13)
of the transformation (4) is a fuzzy time series 𝑋̃𝑡𝑗 , obtained
𝑥̃𝑡 = 𝑧̃𝑣(𝑡) , (14)
for a one-dimensional 𝑗-th TS 𝑥𝑡𝑗 ∈ 𝑋. Micro granulation is
considered as the process of creating small granules by where 𝑠(𝑡 − 1) and 𝑣(𝑡) 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 “fragmentation” between the MTS and its (𝑡 − 1) and 𝑡 respectively.
micro granules, is established. Macro granulation establishes
the “generalization” 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 𝛼𝑔𝑟𝑜𝑤𝑡ℎ > 2 ∗ 𝛼𝑓𝑎𝑙𝑙 , then
intensity of change of two neighboring fuzzy values in fuzzy 𝑦2 = '𝐺𝑟𝑜𝑤𝑡ℎ′, 𝜇𝑦1 (𝑥𝑡 ) = 0 ,
TS for observation 𝑡 is presented as: 𝛼𝑔𝑟𝑜𝑤𝑡ℎ 𝛼𝑓𝑎𝑙𝑙
𝜇𝑦2 (𝑥𝑡 ) = , 𝜇𝑦3 (𝑥𝑡 ) =
(𝑟−1)∗(𝑛−1)
,
(𝑟−1)∗(𝑛−1)
𝛼𝑡 = 𝑣(𝑡) − 𝑠(𝑡 − 1), 𝑡 = 2,3, … , 𝑛. (15)
Step 2.3. If 𝛼𝑓𝑎𝑙𝑙 > 2 ∗ 𝛼𝑔𝑟𝑜𝑤𝑡ℎ , then 𝑦3 = '𝐹𝑎𝑙𝑙′ ,
Thus, at the stage of micro granulation of MTS, for each 𝛼𝑔𝑟𝑜𝑤𝑡ℎ 𝛼𝑓𝑎𝑙𝑙
value of TS, we obtain the degree of its belonging 𝜇𝑥̃𝑡 (𝑥𝑡 ), 𝜇𝑦1 𝑡 ) = 0, 𝜇𝑦2 (𝑥𝑡 ) =
(𝑥 , 𝜇𝑦3 (𝑥𝑡 ) =
(𝑟−1)∗(𝑛−1) (𝑟−1)∗(𝑛−1)
,
the corresponding linguistic label 𝑥̃𝑡 , and the intensity of
changes in neighboring values 𝛼𝑡 . Step 2.4. If (0,85 ∗ 𝛼𝑓𝑎𝑙𝑙 < 𝛼𝑔𝑟𝑜𝑤𝑡ℎ < 1,15 ∗ 𝛼𝑓𝑎𝑙𝑙 )
𝑜𝑟 (0,85 ∗ 𝛼𝑔𝑟𝑜𝑤𝑡ℎ < 𝛼𝑓𝑎𝑙𝑙 < 1,15 ∗ 𝛼𝑔𝑟𝑜𝑤𝑡ℎ ),
B. Macro granulation in MTS
then 𝑦4 = '𝑆𝑦𝑠𝑡𝑒𝑚𝑎𝑡𝑖𝑐 𝐹𝑙𝑢𝑐𝑡𝑢𝑎𝑡𝑖𝑜𝑛′,
Macro granulation is considered as process of combining 𝛼𝑔𝑟𝑜𝑤𝑡ℎ 𝛼𝑓𝑎𝑙𝑙
𝜇𝑦1 (𝑥𝑡 ) = 0, 𝜇𝑦2 (𝑥𝑡 ) = (𝑟−1)∗(𝑛−1) , 𝜇𝑦3 (𝑥𝑡 ) = (𝑟−1)∗(𝑛−1) ,
micro granules into larger ones, obtained by expressions (1)
an (2). Using the proposed MTS descriptive model, macro 𝜇𝑦4 (𝑥𝑡 ) = 1, else 𝑦5 = '𝐶ℎ𝑎𝑜𝑡ic Fluctuation', 𝜇𝑦1 (𝑥𝑡 ) =
𝛼𝑔𝑟𝑜𝑤𝑡ℎ 𝛼𝑓𝑎𝑙𝑙
granulation is considered according to expression (3) to 0, 𝜇𝑦2 (𝑥𝑡 ) = , 𝜇𝑦3 (𝑥𝑡 ) =
(𝑟−1)∗(𝑛−1)
,(𝑟−1)∗(𝑛−1)
produce local features of MTS temporal behavior in terms 𝛼𝑔𝑟𝑜𝑤𝑡ℎ 𝛼𝑓𝑎𝑙𝑙
of GT. Since determine the membership functions 𝜇𝑦𝑘 for 𝜇𝑦2 (𝑥𝑡 ) = (𝑟−1)∗(𝑛−1) , 𝜇𝑦3 (𝑥𝑡 ) = (𝑟−1)∗(𝑛−1) , 𝜇𝑦4 (𝑥𝑡 ) = 0,
linguistic terms of variable 𝑌 is challenge we propose the 𝜇𝑦5 (𝑥𝑡 ) = 1.
approach to calculate the degree of belonging TS 𝑥𝑡𝑗 ∈ 𝑋 to
Step 3. Determining global feature of TS in terms of GT.
each 𝑦𝑘 ∈ 𝑌.
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 𝑥𝑡𝑗 ∈ 𝑋 is 𝑐 = 𝑎𝑟𝑔𝑚𝑎𝑥 {𝜇𝑦𝑘 (𝑥𝑡 )} , 𝜇𝑗 = 𝜇𝑦𝑐 (𝑥𝑡 ).
presented using fuzzy TS (see expression (12)), the indices 𝑘=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 𝑥𝑡 :
𝐿𝑗 = 𝑦𝑐 .
𝑌 = {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 ∈ 𝑌 (see definition 3) for determine its local feature:
TS which is presented by fuzzy TS using linguistic variable 𝐿𝑜𝑐(𝑋) = {𝐿𝑗 , 𝜇𝑗 |𝑗 = 1,2, … , 𝑚}.
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
𝜇𝑦𝑘 (𝑥𝑡𝑗 ) of TS 𝑥𝑡𝑗 to 𝑦𝑘 ∈ 𝑌, 𝑘 = 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 𝑥𝑡 ∈ 𝑋. regarding the intensity and management of quarantine
activities, many researchers and ordinary people are
Step 1.2. Based on the values 𝛼𝑡 , 𝑡 = 2,3, … , 𝑛, interested in the question of when and by what signs it can be
calculated according to expression (15), for a TS 𝑥𝑡 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
𝐹𝑜𝑟 𝑡 = 2,3, … , 𝑛 in TS of the indicators of this pandemic over some temporal
interval, it is possible to make decision and informed
𝐼𝑓 𝛼𝑡 > 0, 𝑡ℎ𝑒𝑛 𝛼𝑔𝑟𝑜𝑤𝑡ℎ = 𝛼𝑔𝑟𝑜𝑤𝑡ℎ + 𝑎𝑏𝑠(𝛼𝑡 ), recommendations on the weakening of quarantine measures.
𝐼𝑓 𝛼𝑡 < 0, 𝑡ℎ𝑒𝑛 𝛼𝑓𝑎𝑙𝑙 = 𝛼𝑓𝑎𝑙𝑙 + 𝑎𝑏𝑠(𝛼𝑡 ). 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 𝑌: number of patients recovered (𝑆𝑟) and the total number of
𝐹𝑜𝑟 𝑘 = 1,2, … ,5 𝜇𝑦𝑘 (𝑥𝑡 ) = 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 (𝛼𝑔𝑟𝑜𝑤𝑡ℎ = 0 and 𝛼𝑓𝑎𝑙𝑙 = 0) , then Using micro and macro granulation of MTS, we extract
𝑦1 = '𝑆𝑡𝑎𝑏𝑖𝑙𝑖𝑡𝑦′, 𝜇𝑦1 (𝑥𝑡 ) = 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 = { 𝑟(𝑡), 𝐾𝑟(𝑡), 𝑛(𝑡), 𝑎(𝑡), 𝐾𝑣(𝑡), 𝐾𝑠(𝑡), 𝐾𝑛(𝑡), 𝐾𝑎(𝑡)}.
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
𝑡 – this is number of daily observation, 𝑡 = 1,2, … ,39. MTS. When modeling fuzzy terms, triangular membership
𝑆𝑣(𝑡), 𝑆𝑟(𝑡), 𝑆𝑟(𝑡) 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
𝑆𝑎(𝑡) is TS of the total number of active cases, 𝑆𝑎(𝑡) = linguistic variable was determined on the basis of an
𝑆𝑣(𝑡) − 𝑆𝑟(𝑡) − 𝑆𝑑(𝑡). extended range between the maximum and minimum values
𝑆𝑛(𝑡) designates the daily total number of new of each derived indicator, as described in the work [26]. At
infections: 𝑆𝑛(𝑡) = 𝑆𝑣(𝑡) − 𝑆𝑣(𝑡 − 1). the stage of macro granulation, to each component of the
𝑛(𝑡), 𝑟(𝑡), 𝑎(𝑡) 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(𝑡) and 𝑎(𝑡) 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(𝑡) = 𝑆𝑎(𝑡)/𝑆𝑣(𝑡) 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
𝐾𝑟(𝑡) = 𝑆𝑟(𝑡)/𝑆𝑎(𝑡) – this is TS of proportion of total of GT GT
cases of recovery in relation to active cases. The growth of 𝑟(𝑡) Growth Positive
this share shows that the number of ill and received 𝐾𝑟(𝑡) Growth Positive
immunity increases, which is positive in terms of the
𝑛(𝑡) Growth Negative
formation of collective immunity.
𝐾𝑠(𝑡) = 𝑆𝑎(𝑡 + 𝑝)/𝑆𝑎(𝑡) is the coefficient of the 𝑎(𝑡) Growth Negative
delayed effect of total active cases 𝑆𝑎(𝑡) per day with
𝐾𝑣(𝑡) Fall Positive
number t on the total active cases that occur by the end of
the incubation period 𝑆𝑎(𝑡 + 𝑝) (according to WHO [23], 𝐾𝑠(𝑡) Fall Positive
the duration of incubation period 𝑝 can be up to 14 days). A 𝐾𝑛(𝑡) Fall Positive
decrease in this coefficient indicates that the activity of
infection from active cases is reduced. This indicates a 𝐾𝑎(𝑡) Fall Positive
positive trend.
𝐾𝑛(𝑡) = 𝑆𝑛(𝑡 + 𝑝)/𝑆𝑎(𝑡) is the coefficient of the According to the last column of Table 1, we can conclude
delayed effect of the total active cases of 𝑆𝑎(𝑡) per day that in Moscow by May 3, 2020, only 67% of the distribution
with number t on the total new cases of 𝑆𝑛(𝑡 + 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
𝐾𝑎(𝑡) = 𝑎(𝑡 + 𝑝)/𝑎(𝑡) 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 𝑎(𝑡) per day
additional analysis, which may be the subject of a new study.
with number 𝑡 on the occurrence of daily recorded active
cases 𝑎(𝑡 + 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, “Modeling 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, “Toward 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, “Time 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, “Framework for Assessing
The authors acknowledge the reported study was funded Professional Growth of Software Developers,” Advancies of
by RFBR, as a part of project № 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, “Perception Based Time Series Data
№ 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, “Handbook of Granular
[1] J. Kacprzyk and R. Yager, “Linguistic 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, “Fuzzy
154, 2001. trends data mining in knowledge discovery process,” Creativity in
[2] X. He and P. Niyogi, “Locality 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, “The Study OF Basic
[3] J. Brzezińska, “Singular 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, “Multidimensional Wavelets,” [18] V. Novak, “Linguistic 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, “Fuzzy time series and its models,” Fuzzy
[5] C. Aldrich, “Process 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, “The concept of a linguistic variable and its application
Data, Methods, and Applications,” 2019. DOI: to approximate reasoning—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, “Reliable Early [21] T. Afanasieva, A. Shutov and E. Efremova, “The 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, “About Transformations of
[7] G. He, W, Zhao, X, Xia, R. Peng and X. Wu, “Ensemble 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, “Fuzzy
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, “Fuzzy transform in the country/moskva/.
analysis of data,” International Journal of Appr. Reasoning, vol. 48, [26] Q. Song, “Forecasting 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