Formal Concept Analysis Applied to a Longitudinal Study of COVID-19

Paulo Lana, Cristiane Nobre, Luis Zarate and Mark Song

Instituto de Ci

encias Exatas e Inform

atica, Pontif

ıcia Universidade Cat

olica de Minas Gerais, Brazil

Keywords:

Formal Concept Analysis, Triadic Concept Analysis, Association Rules, Implications, COVID-19.

Abstract:

The COVID-19 pandemic, and consequently the difﬁculty of obtaining feedback on the effectiveness of con-

tamination prevention methods, has caused an increased need to produce a relevant and consistent analysis

from collected data. Through Formal Concept Analysis, applying the triadic approach, called Triadic Con-

cept Analysis (TCA), it is possible to evaluate the correlation between prevention measures and the number

of contaminated people by performing concept extraction and implication rules. The advantage of using this

method is the possibility of correlating the waves, which allows us to explain and understand the evolution of

the data over the collection waves, helping us draw a more assertive conclusion from the data analyzed. This

paper uses the data collected from the 2020 National Population Survey of Nigeria to depict how Nigerian

society’s essential and everyday behaviors impacted the evolution of the COVID-19 pandemic in that country.

The results obtained from this research can assist governments, and public entities in developing better public

policies to combat highly infectious diseases. Furthermore, it provides practical evidence of how TCA can be

applied, bringing beneﬁts to different areas and ﬁelds of science.

1 INTRODUCTION

Formal Concept Analysis (FCA) (Wille, 1982) was

introduced in 1982 by Rudolf Wille as a derivation

of concept hierarchy from a set of objects and their

properties. The main objectives are to extract and rep-

resent knowledge in an efﬁcient way that aids and cor-

roborates decision-making from database analysis.

On the fundamental theorem of triadic concepts

and contexts, Wille (1982), providing considerable

contributions to the intelligent and adequate database

modeling for the extraction of triadic concepts, devel-

oped the Triadic Concept Analysis (TCA).

The developed work has shown that FCA’s qual-

itative concept data analysis capability can be com-

bined with a more quantitative approach, investigat-

ing data correlations that are not a priori visible or

easy to link.

The main beneﬁt of choosing to use TCA in this

research is in the correlation between different peri-

ods of data collection, represented by the longitudinal

base analyzed here.

As we will further explain in the theoretical foun-

dation, triadic rules consist of a context formed from

a quadruple T = (G, M, B,Y ) where G, M, B are, re-

spectively, sets of objects, attributes, and conditions

belonging to the ternary relation Y ⊆ (G × M × B).

This relation is interpreted as: object g has attribute

m under condition b.

Thus, in any longitudinal databases, the advantage

of using an analysis such as TCA is to make it possi-

ble to explain the evolution of variables between col-

lection waves, clarifying possible causes of change,

such as temporal behaviors, impacts of external fac-

tors, assisting in a more accurate conclusion of the

data analyzed.

In this work, Triadic Formal Analysis is applied to

behavioral data collected from a society widely im-

pacted by the COVID-19 pandemic, speciﬁcally the

society of Nigeria. Through this analysis it is possible

to generate knowledge, portraying within the popula-

tion context of this country, the relationship between

basic hygiene and public health actions that can in-

terfere in the contamination and dissemination of the

disease.

The analysis of behavioral relations such as wash-

ing hands with soap, the use of masks in public, the

need for medical treatment, and the frequency of chil-

dren/adolescents with school activities that can affect

the spread of COVID-19 is made, helping in indirect

actions that aim at the containment and prevention of

contamination.

The data collected and analyzed bring about com-

plex relationships due to the variety of question cat-

148

Lana, P., Nobre, C., Zarate, L. and Song, M.

Formal Concept Analysis Applied to a Longitudinal Study of COVID-19.

DOI: 10.5220/0011036000003179

In Proceedings of the 24th International Conference on Enterprise Information Systems (ICEIS 2022) - Volume 1, pages 148-154

ISBN: 978-989-758-569-2; ISSN: 2184-4992

egories and possible answers, the number of inter-

viewees, and the several collection waves carried

out. However, after a reﬁnement, they can be ex-

plored, through the Formal Analysis of Triadic Con-

cepts (TCA), to obtain results derived from several

combinations that optimize actions and health poli-

cies adopted to ﬁght the pandemic.

The result intended with this study is the applica-

tion of the TCA in a questionnaire of basic public hy-

giene and health behaviors. Thus, this work brings as

a result rules (implications) regarding the behavior of

society by means of health policies and the evolution

of the pandemic.

Such results can direct, corroborate and open pos-

sibilities for taking actions that add value, either

by optimizing the ﬂow of data in a given network

or by increasing conﬁdence in decision making for

strengthening or weakening of certain policies to ﬁght

pandemics and diseases.

This work is structured by presenting a brief intro-

duction to Triadic Formal Concept Analysis (TCA).

After the TCA introduction, the database processing

steps are described, alongside the results of the meth-

ods applied. At the end of this work, over the con-

clusion of the analysis, the contributions of the study

are revelead as to how critical gaps could be ﬁlled

on information that can be used by the Nigerian gov-

ernment. This is relevant to assist the development

of public policies to mitigate the negative impacts of

COVID-19 on its population.

2 BACKGROUND

Formal Concept Analysis (FCA) is a mathematical

formalism that deals with data represented in two di-

mensions, relating objects and their attributes. Triadic

Formal Concept Analysis (TCA) extends FCA, trans-

forming the binary relation into a ternary considera-

tion by introducing a dimension called condition or

mode, establishing a link between objects, attributes,

and conditions.

This approach allows the identiﬁcation of factors

related to object changes over time. Moreover, the tri-

adic approach contributes to databases from longitu-

dinal studies by considering the time factor as a con-

dition for analyzing the variation of the relationship

between objects and attributes. In this way, the triadic

implication rules also explore the temporal evolution

of the relations studied.

Thus, with the results from the TCA, it is possi-

ble to indicate points of improvement and failure re-

garding the impacts from health policies and the dis-

covery of previously unnoticed relationships. Never-

theless, the gains for the increase in the quality of life

and health of the population, in general, are countless,

even if we consider only the impacts coming from the

discoveries of new relations and correlations.

Formal Concept Analysis involves different com-

ponents such as: Formal Context, formal concept, and

rules.

A formal Context K is a triple K = (G, M, I),

where G is a set of objects, M is a set of attributes and

I is the incidence relation (I ⊆ G × M) that assigns an

object certain characteristics.

K = (G, M, I) (1)

I ⊆ G × M (2)

If an object (g ∈ G) and an attribute (m ∈ M ) are

in relation I, this is represented by ((g,m) ∈ I) or gIm

and is interpreted as the object g has the attribute m.

From a set of objects A ⊆ G, coming from a for-

mal context K, it is interesting to recognize which at-

tributes B ⊆ M are shared by all objects in A. Anal-

ogously, determine for a set B ⊆ M, which objects

A ⊆ G have in common the attributes deﬁned in B.

The search for these answers leads to derivation oper-

ators, which are formally deﬁned as:

:= {m ∈ M | gIm ∀ g ∈ A} (3)

:= {g ∈ G | gImIm; ∀ m ∈ B} (4)

From K, it is possible to extract association rules

dependent on elements in a set of M. An association

between attributes of M is a pair (X, Y), X, Y ∈ M,

receiving the notation X → Y. Association rules, in

turn, reveal frequent patterns in data.

From FCA, we extend the deﬁnition to encom-

pass a third dimension resulting in a triadic context

T , which is now a quadruple T = (G, M, B, Y ) where

G, M, B are, respectively, sets of objects, attributes

and conditions pertaining to the ternary relation Y ⊆

(G × M × B). This relation is interpreted as: object g

has attribute m under condition b.

Table 1 presents a triadic context similar to the one

analyzed in this paper.

T = (G, M, B, Y ) (5)

Y ⊆ (G × M × B) (6)

Table 1: Example of triadic context.

Cicle 1 Cicle 2 Cicle 3

Obj p1 p2 p3 p1 p2 p3 p1 p2 p3

1 x x x x x x x

2 x x x x x

3 x x

4 x x x x x x

5 x x x x x

6 x x x x x

One can describe two types of triadic associa-

tion rules that can be extracted from the context

Formal Concept Analysis Applied to a Longitudinal Study of COVID-19

149

K := (G, M, B, Y ) (Biedermann, 1997): Biedermann

Conditional Attribute Association Rule (BCAAR) and

Biedermann Attributional Condition Association Rule

(BACAR) (Zhuk et al., 2014).

The BCAAR rule is represented by: (R → S) C,

where R, S ⊆ M and C ⊆ B. That is, each object pos-

sessing all attributes in R also possesses all attributes

in S under a condition C, with one support (sup.) and

one conﬁdence (conf.).

The BACAR rule is represented by: P → Q N

(sup.,conf.), where P, Q ⊆ B and N ⊆ M. That is, ev-

ery object under the conditions in P will also be under

the conditions in Q on attribute N, with one support

(sup.) and one conﬁdence (conf.).

The support corresponds to the proportion of ob-

jects in the subset g ∈ G that satisfy the implication

P → Q, relative to the total number of objects |G| of

the formal context K (Equation 7) where (

) corre-

sponds to the derivation operator.

Support(P → Q) =

|(P ∪ {Q})

|G|

(7)

Conﬁdence corresponds to the ratio of objects g ∈

G that contain P, which also contain Q, to the total

number of objects |G| (Equation 8).

Con f (P → Q) =

|(P ∪ {Q})

Support(P → Q)

Support(P)

(8)

3 RELATED WORKS

This paper utilizes the Formal Analysis of Triadic

Concepts, and the approach and rationale for this un-

derstanding are applied through a practical and the-

oretical bias. Related work on this research topic is

presented below.

Wei et al. (2018) analyzed the Triadic approach of

Formal Concept Analysis (TCA), from four aspects:

(i) basics of triadic concept analysis, (ii) triadic impli-

cations and triadic association rules, (iii) triadic factor

analysis, and, (iv) triadic fuzzy concept analysis.

The addition offered by Konecny and Osicka

(2010) on the theoretical research and the seek sub-

sidies for applications of TCA, aims to better under-

stand the general approach on TCA and its fundamen-

tals. (Konecny and Osicka, 2010) discusses Triadic

Concept Analysis (TCA), presenting it as an exten-

sion of Formal Concept Analysis (FCA), dyadic case,

through the optic of conditions, besides just weight-

ing on objects and attributes relations.

The work of Ganter and Obiedkov (2004) head to-

wards a more practical bias of the research on Tri-

adic Concept Analysis, considering the various im-

plications of its use in multiple scenarios. The work

addresses the various possibilities of deﬁning impli-

cations of a triadic formal context. Due to the vast

different interests one can have for a given triadic con-

text, the authors aim to present compact descriptions

and incorporate them into an algorithm that generates

implications from triadic contexts. Examples of this

variety of interests in a triadic context is the work pre-

sented in Silva et al. (2017), where social networks

interactions are in focus, and in Ferreira et al. (2017),

in which molecular biology challenges are faced.

Kis et al. (2016) reveals the development of a tool

that provides a visualization for dyadic and triadic

concepts, focusing on a navigation paradigm for tri-

adic contexts. The authors take into consideration

the complex contexts of application of Triadic Con-

cept Analysis, given the importance of the existence

of methods and procedures for the development and

improvement of analyses.

Biedermann (1997), also presents contexts and

ways of applying Triadic Concept Analysis. The

work is dedicated to explain different types of infor-

mation and knowledge that can be read in triadic di-

agrams. These labeled line diagrams graphically rep-

resent the conceptual structure of triadic contexts that

can be represented as three-dimensional. The author

discusses throughout the paper how one should inter-

pret such diagrams and how such dyadic conceptual

structures can be determined within the triadic dia-

grams.

Bringing fundamentals and subsidies for the de-

termination of Triadic Concept Analysis guidelines in

various scenarios and circumstances, similiar to Ana-

nias et al. (2019), Dau and Wille (2000) developed a

study on modal applications for the understanding of

triadic contexts, having as perspective the triadic con-

text as sets of formal objects, formal attributes and

formal conditions, along with the formalization of the

ternary relation, indicating when an object has an at-

tribute under a certain condition.

4 METHODOLOGY

4.1 Database

The database used in this paper was obtained from the

COVID-19 National Longitudinal Phone Survey 2020

(Nigeria COVID-19 NLPS), a survey applied on indi-

viduals residing in Nigeria, conducted by the National

Bureau of Statistics (NBS), an afﬁliate of the Federal

Government of Nigeria and produced by the World

Bank.

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150

The survey, which collects data monthly, aims to

monitor the socio-economic effects of the pandemic

COVID-19 with real-time evolution. The study is part

of the World Bank’s Living Standards Measurement

Survey program to provide support to countries to

help mitigate the spread and impact of the new coro-

navirus disease.

The data collection is done through several tele-

phone calls to households located throughout Nigeria.

Each month, Nigerian households across the country

are asked about topics that are likely to be affected by

COVID-19 restrictions. Thus, the Nigeria COVID-19

NLPS intends to monitor the socio-economic effects

of the COVID-19 pandemic that will contribute to the

development of policies that will mitigate the negative

impacts on the population.

This paper performs the triadic analysis applied to

this base, using data collected in 3 waves of monthly

interviews, which had as areas of interest (i) access to

basic services, (ii) behavior, (iii) education, and (iv)

access to medical treatment.

The survey was conducted with 1,800 individuals

who are components and residents of the population

of Nigeria, aged eighteen (18) years and above.

In order to analyze the areas of interest mentioned,

four questions were selected related to areas of behav-

ior and status in society, such as: washing hands with

soap, use of mask in public, need for medical treat-

ment, and frequency of children and adolescents with

school activities.

The resulting analysis will allow us to associate

such behaviors with the spread of COVID-19 and the

socioeconomic impacts on the country.

The ﬁrst wave of Nigeria COVID-19 NLPS tele-

phone interviews considered for the present work was

conducted between June 2 and 16, 2020. The second

took place between July 2-16, 2020, and the third be-

tween the dates of August 9-24, 2020.

To bring the intended result in the present paper,

relating the behavioral results of the Nigeria COVID-

19 NLPS survey to the spread of COVID-19 during

June, July, and August 2020 in Nigeria, the base Nige-

ria: Coronavirus Pandemic Country Proﬁle was used,

which determines the estimate of the effective repro-

duction rate (R) of COVID-19. The reproduction

rate represents the average number of new infections

caused by a single infected individual.

The aforementioned statistics and survey Nige-

ria: Coronavirus Pandemic Country Proﬁle, was con-

ducted by Our World in Data. The database provided,

in addition to the many linked graphs, uses the com-

plete set of data on conﬁrmed cases and deaths from

Johns Hopkins University (JHU) and the European

Centre for Disease Prevention and Control (ECDC).

This Johns Hopkins University dataset is main-

tained by the Center for Systems Science and Engi-

neering (CSSE). Since January 22, 2020, it has pub-

lished updates on conﬁrmed cases and deaths for all

countries and updates its data several times a day. The

data used from this database comes from the govern-

ment, national and subnational agencies in Nigeria.

4.2 Methods

To achieve the goal of the paper, it was necessary to

perform data transformation in a triadic context, us-

ing the Nigeria COVID-19 NLPS database as input

to the concept extraction algorithm. To do this, the

following process was carried out:

1) A preprocessing of the base was performed.

Relevant and consistent questions were ﬁltered and

selected between the collection waves. After selecting

the questions, the respective answers were discretized

according to each context, so that there was no in-

terference in the conﬁguration and purpose of each

question.

2) Grouping the preprocessed data into their

waves. After preparing the questions and discretizing

the answers, the data from each wave were grouped in

the same table, keeping the reference of each wave for

later interpretation of the implication rules generated.

3) For the rule generation the Lattice Miner 2.0

tool was used, a data mining prototype developed un-

der the supervision of Professor Rokia Missaoui by

the LARIM research laboratory of the Universit

e du

ebec (Missaoui and Emamirad, 2017). It is a public

domain Java platform whose main functions include

all the low-level operations and structures to represent

and manipulate input data, structures, and association

rules. The platform allows the generation of clusters,

called formal concepts, and association rules, includ-

ing logical implications, given a binary relationship

between a collection of objects and a set of attributes

or properties.

Among the functions made available by Lattice

Miner 2.0, this work focuses on the use of the

rules proposed by Biedermann (1997), these are:

Biedermann Conditional Attribute Association Rule

(BCAAR) and Biedermann Attributional Condition

Association Rule (BACAR).

The ﬁrst, BCAAR, takes the form (A1 →

A2)C(sup, con f ), where A1, A2 ⊆ M and C ⊆ B. This

rule indicates that every time A1 occurs in the condi-

tions C, then A2 also occurs, with support sup and

conﬁdence conf. The second rule used is BACAR,

which takes the form (C1 → C2)A(sup, con f ), where

C1, C2 ⊆ B and A ⊆ M, that is, every time C1 occurs

for all attributes in A, then for the condition C2, there

Formal Concept Analysis Applied to a Longitudinal Study of COVID-19

151

is an occurrence of the same attributes, with support

sup and conﬁdence conf.

5 RESULTS

From the triadic analysis performed on the database,

it was possible to relate how the various basic,

everyday behaviors of the Nigerian population can

affect the spread of COVID-19.

Considering:

w1: Collection wave 1 conducted in June 2020;

w2: collection wave 2 conducted in July 2020, and

w3: Collection wave 3 carried out in August 2020.

We could obtain the follwing rules:

BCAAR Implication Rules

1 : (B → D) w1 [sup = 53, 6% con f = 68, 5%]

2 : (B → A) w3 [sup = 70, 7% con f = 92, 4%]

3 : (B → D) w3 [sup = 50, 5% con f = 66, 0%]

4 : (A → D) w2 [sup = 51, 0% con f = 58, 8%]

5 : (A → B) w2 [sup = 74, 6% con f = 86, 0%]

6 : (D → B) w1 [sup = 53, 6% con f = 84, 1%]

7 : (A → B) w3 [sup = 70, 7% con f = 85, 7%]

8 : (A → D) w3 [sup = 52, 8% con f = 64, 1%]

9 : (D → A) w2 [sup = 51, 0% con f = 89, 1%]

10 : (B → A) w2 [sup = 74, 6% con f = 94, 0%]

11 : (D → B) w3 [sup = 50, 5% con f = 83, 2%]

12 : (D → A) w3 [sup = 52, 8% con f = 87, 1%]

For this research, the questions asked to the

nigerian population were:

A: Washing hands with soap after going in public;

B: Wearing a mask in public;

C: Need for medical treatment in the last 7 days;

D: Children and adolescents with school activi-

ties;

The value of 50% was used as lower limits of

support and conﬁdence in the generation of the

BCAAR implication rules. With the results in hand,

for interpretation and exposition in this paper, we

kept the 50% for support and considered only the

rules above 75% conﬁdence.

From rule 5 it can be seen that during wave 2 (w2,

conducted in June), 74.6% of the individuals inter-

viewed who took care to wash their hands with soap

in public (A) also took precautions by wearing masks

(B). Note that of those who washed their hands in

public, 86.06% certainly wore masks.

From rule 9, it is observed that during collection

wave 2, in June 2020, 51% of the individuals living

with children and adolescents with school activities

(D) were careful to wash their hands after going in

public (A). Of these respondents, 89.1% were sure to

wash their hands with soap after going out in public.

From the generated rule 10, 74.6% of individuals

who wore a mask in public (B), 94.5% were associ-

ated with washing their hands with soap after going

out in public (A), in wave 2.

It is observed by rule 11, through the data col-

lected during wave 3 (August 2020), that 50.5% of the

interviewed individuals live with children and adoles-

cents with school activities (D). Of this percentage,

it is perceived that 83.2% were certain to have worn

masks in public (B).

The aforementionted rules implies a direct re-

lationship between individuals who washed their

hands after going out in public, who also used masks

in public, and in which families kept children and

adolescents with school activities, during July and

August.

BACAR Implication Rules

1 : (w1 → w3) B [sup = 64, 5% con f = 82, 5%]

2 : (w1 → w2) B [sup = 63, 5% con f = 81, 1%]

3 : (w3 → w1) B [sup = 64, 5% con f = 84, 3%]

4 : (w3 → w2) B [sup = 61, 1% con f = 79, 9%]

5 : (w2 → w3) A [sup = 71, 5% con f = 82, 5%]

6 : (w3 → w2) A [sup = 71, 5% con f = 86, 8%]

7 : (w2 → w1) B [sup = 63, 5% con f = 80, 0%]

8 : (w2 → w3) B [sup = 61, 1% con f = 77, 0%]

9 : (w1w3 → w2) B [sup = 52, 4% con f = 81, 2%]

10 : (w1w2 → w3) B [sup = 52, 4% con f = 82, 5%]

11 : (w2w3 → w1) B [sup = 52, 4% con f = 85, 7%]

Similarly applied to the BCAAR rules, no rules

were generated with support and conﬁdence lower

than 50%. Also, as with the previous interpretation,

only rules with percentages above 50 support and 75

conﬁdence were taken into account.

From rule 1, it can be seen that among the 64.5%

of individuals interviewed who had taken precautions

by wearing masks in public (B) in wave 2, in June

2020, 82.5% continued to wear their masks in wave

3, in August of the same year.

It is observed from rule 2 that among the 63.5%

individuals who were wearing masks in public (B) in

wave 2, in June 2020, 81.1% maintained the same

protective behavior in wave 2, the following month,

in July.

Also, employing rule 8, it is possible to iden-

tify that of the 61.1% of respondents who adhered to

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152

Figure 1: Covid Rate.

wearing masks in July 2020 (wave 2), 77.0% contin-

ued wearing the masks in public also in the subse-

quent month, during wave 3.

Finally, from rule 10, it is noted that 52.4% of

the interviewed individuals who were wearing masks

in public continuously between the months of June

and July 2020 (w1 and w2), 82.5% continued with the

protection method during the month of August of the

same year (w3).

It is possible to observe that the behavior of indi-

viduals who used a mask in June, in the wave w1, was

maintained during July and August, waves w2 and w3

respectively. Rule 5 also enforces basic service access

behavior:

5 : (w2 → w3) A [sup = 71.5%; con f = 82.5%]

Using rule 5, it can be seen that of the 71.5% in-

dividuals who followed the recommendation to wash

their hands with soap after going in public in July

2021 (w2), 82.5% maintained this good practice in

August (w3).

When comparing such implication rules with the

results found in Nigeria (Figure 1): Coronavirus Pan-

demic Country Proﬁle survey on the estimation of the

effective reproduction rate (R) of COVID-19, we can

observe that during June, July, and August, there was

a decrease in R rate:

It is possible to infer, when compiling such

databases, that the adherence of individuals in public

policies to basic behavioral services, such as wearing

masks and washing hands after going public, had an

impact in preventing the spread of new cases of infec-

tion by an individual contaminated by COVID-19.

Yet another interpretation we can draw from

BCAAR rules 5, 9, 10, 11, we ﬁnd that keeping chil-

dren and adolescents in school activities may have

assisted in the adherence of families to the public

health recommendations put forth by the government

of Nigeria.

Thus, by analyzing these implications, it is possi-

ble to generate practical results, increasingly reliable,

to support the construction of mitigatory and pre-

ventive actions and policies for the contamination of

COVID-19, helping the government to obtain greater

adherence of the population in restrictive actions with

low socioeconomic impact and that have positive re-

sults in combating COVID-19.

Formal Concept Analysis Applied to a Longitudinal Study of COVID-19

153

6 CONCLUSION AND FUTURE

WORK

The intention of the work, after careful selection of

the results, is to understand the importance of the ex-

tracted knowledge on mortality rates, infection rates,

and the number of cases of COVID-19 during the col-

lection period.

In addition, the research seeks to highlight the

contrasts of the results obtained with information

from the Nigeria: Coronavirus Pandemic Country

Proﬁle” database, maintained and updated daily by

the organization ”Our World In Data”.

This database provides statistical information

about the progression of COVID-19, with data such

as the number of deaths, number of conﬁrmed cases,

mortality rate, and number of tests performed.

Thus, in face of the proposed analyses, the present

work aimed to expand the applicability of the Triadic

Concepts Analysis.

It turned out to be an efﬁcient and useful approach

to ﬁnd aspects not easily identiﬁed at ﬁrst, related to

several contexts of knowledge areas, as health and so-

cial information related to pandemics, diseases, and

social behaviors.

Such results, by bringing correlations between

population behaviors and dissemination of diseases,

can be very useful in the basis and rationale for

making government decisions of great impact on the

health of the world population.

The study, therefore, by performing the complex

analysis of data not directly linked, ﬁnding their cor-

relation, can help governments and public entities to

develop better public policies to combat highly infec-

tious diseases and promote sanitation.

Thus, it is expected that several areas of study and

scenarios will beneﬁt from this methodology in their

investigations and analysis, relying on the data inter-

pretation enabled through TCA.

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