Modeling Seasonality of Emotional Tension in Social Media

Abstract

Social media has become an almost unlimited resource for studying social processes. Seasonality is a phenomenon that significantly affects many physical and mental states. Modeling collective emotional seasonal changes is a challenging task for the technical, social, and humanities sciences. This is due to the laboriousness and complexity of obtaining a sufficient amount of data, processing and evaluating them, and presenting the results. At the same time, understanding the annual dynamics of collective sentiment provides us with important insights into collective behavior, especially in various crises or disasters. In our study, we propose a scheme for identifying and evaluating signs of the seasonal rise and fall of emotional tension based on social media texts. The analysis is based on Russian-language comments in VKontakte social network communities devoted to city news and the events of a small town in the Nizhny Novgorod region, Russia. Workflow steps include a statistical method for categorizing data, exploratory analysis to identify common patterns, data aggregation for modeling seasonal changes, the identification of typical data properties through clustering, and the formulation and validation of seasonality criteria. As a result of seasonality modeling, it is shown that the calendar seasonal model corresponds to the data, and the dynamics of emotional tension correlate with the seasons. The proposed methodology is useful for a wide range of social practice issues, such as monitoring public opinion or assessing irregular shifts in mass emotions.

Full text

Citation: Nosov, A.; Kuznetsova, Y.;
Stankevich, M.; Smirnov, I.; Grigoriev,
O. Modeling Seasonality of Emotional
Tension in Social Media. Computers
2024,13, 3. https://doi.org/10.3390/
computers13010003
Academic Editor: Paolo Bellavista
Received: 9 November 2023
Revised: 13 December 2023
Accepted: 20 December 2023
Published: 22 December 2023
Copyright: © 2023 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license (https://
creativecommons.org/licenses/by/
4.0/).
computers
Article
Modeling Seasonality of Emotional Tension in Social Media
Alexey Nosov , Yulia Kuznetsova , Maksim Stankevich , Ivan Smirnov * and Oleg Grigoriev
Federal Research Center “Computer Science and Control” of Russian Academy of Sciences,
119333 Moscow, Russia; nosov@isa.ru (A.N.); kuzjum@yandex.ru (Y.K.); stankevich@isa.ru (M.S.);
oleggpolikvart@yandex.ru (O.G.)
*Correspondence: ivs@isa.ru
Abstract: Social media has become an almost unlimited resource for studying social processes.
Seasonality is a phenomenon that significantly affects many physical and mental states. Modeling
collective emotional seasonal changes is a challenging task for the technical, social, and humanities
sciences. This is due to the laboriousness and complexity of obtaining a sufficient amount of data,
processing and evaluating them, and presenting the results. At the same time, understanding the
annual dynamics of collective sentiment provides us with important insights into collective behavior,
especially in various crises or disasters. In our study, we propose a scheme for identifying and
evaluating signs of the seasonal rise and fall of emotional tension based on social media texts. The
analysis is based on Russian-language comments in VKontakte social network communities devoted
to city news and the events of a small town in the Nizhny Novgorod region, Russia. Workflow steps
include a statistical method for categorizing data, exploratory analysis to identify common patterns,
data aggregation for modeling seasonal changes, the identification of typical data properties through
clustering, and the formulation and validation of seasonality criteria. As a result of seasonality
modeling, it is shown that the calendar seasonal model corresponds to the data, and the dynamics
of emotional tension correlate with the seasons. The proposed methodology is useful for a wide
range of social practice issues, such as monitoring public opinion or assessing irregular shifts in
mass emotions.
Keywords: social media texts; emotional tension; seasonality modeling; statistical method; aggregation;
clustering; data mining
1. Introduction
Emotions attract close attention because they make a decisive contribution to attitudes
toward events, objects, and people. Motivation and behavior are largely determined
emotionally at both the individual and collective levels, so understanding emotional
phenomena provides a more complete picture of social processes and massactions. Tracking
the changes in collective emotions allows us to predict people’s reactions to events [
1
] and
prevent negative scenarios [
2
]. Monitoring based on data from social networks favorably
differs from classical sociological and statistical methods in its unobtrusiveness and minimal
time lag between an emotional shift and its identification. This expands the possibilities of
a prompt response to the growth of negative moods in society [
3
]. In addition, several areas
of public interest, including health, finance, entertainment, advertising, and culture, could
potentially benefit from measuring human emotions on social media [
4
]. The importance
of understanding collective sentiment increases dramatically in a crisis situation, as shown
by natural and man-made disasters, as well as the COVID-19 pandemic [5,6].
In our work, mood is considered as a complex long-lived emotional phenomenon,
the main component of which is emotional tension. Due to its duration, unlike the emo-
tion itself, mood has long-term dynamics. There is some evidence that collective senti-
ment is subject to seasonal fluctuations, with fluctuations occurring from year to year.
Computers 2024,13, 3. https://doi.org/10.3390/computers13010003 https://www.mdpi.com/journal/computers

Computers 2024,13, 3 2 of 24
If this is true, then the seasonality of public sentiment should be taken into account in
socio-political practice.
The focus of the study is on emotional tension as the simplest component of collective
mood, and we do not consider emotional investment in mood in this paper. The purpose
of our study is to answer the following questions:
•
What methods are applicable to detect patterns of variation in multiple assessments of
a population’s psychological states when observed over time?
•
Do collective emotional tensions in reality have seasonal variations that can be tracked
through social media content analysis?
To study the seasonality of emotional tension, we used text comments from users of
the social network VKontakte. In particular, we collected data from the largest communities
dedicated to local news and events in a small town in the Nizhny Novgorod region
of Russia.
Modern monitoring tools allow for the collection and storage of data in the form
of time series. Time series contain the necessary information about the dynamics of the
processes generating them. General methods of time series analysis, as a rule, make it
possible to answer many of the questions related to the nature of the occurrence and
seasonality of the processes being studied.
The distinctive features of the time series obtained as a result of monitoring social
networks are discreteness, non-stationarity, and high sensitivity to data changes, which
complicates their modeling and the use of traditional methods of analysis [7–11].
We used a combined approach, which is more typical of data science. We combined
different data analysis techniques into a single workflow. In this approach, we do not
model the time series but take its values as the domain to define the functions that are used
for analysis, thus eliminating the problem of data sensitivity.
The workflow includes steps such as a statistical method for categorizing data, ex-
ploratory data analysis (EDA), feature selection and identification of common patterns
according to a new target variable, aggregation of data to model seasonal changes, identifica-
tion of typical data properties through clustering, analysis of cluster properties, formulation,
and the validation of seasonality criteria. As part of the workflow, an approach to mod-
eling seasonality in a class of time-aggregated models is proposed, and the conditions
for their compliance with the data and the criterion of seasonality on a specific dataset
are formulated.
2. Related Works
Certain researchers have noted that conceptual inconsistencies have hampered progress
in the field of mood research. Some streamlining of terminology and differentiation of
emotional phenomena—which, in many NLP studies, are arbitrarily referred to as inten-
tions, beliefs, feelings, emotions, mood, and sentiment—could play a significant role in
improving the efficiency of analysis [
12
]. In particular, ref. [
13
] emphasized the difference
between background sentiment (or mood) and rapid shifts in sentiment (or emotion), as
well as the impossibility of accurately identifying sentiment shifts because the patterns of
background sentiment evolution are largely ignored by existing methods.
The following is a summary of three issues related to mood research: the characteristics
and structure of mood; the mood markers used in NLP; and mood duration and dynamics
(including methods for recording mood swings).
Currently, the emotionality of online content is mainly studied in the form of transient
emotions, while the background mood remains out of sight. Mood and emotion differ in
many dimensions such as clarity, duration, intensity, stability, causality, and control [
14
].
For example, emotion is more intense, but mood is longer in time; emotion is triggered by
a specific event or incident, but mood does not necessarily need a contextual stimulus; and
mood is strongly influenced by several factors, such as environment, physiology, or mental
state [
12
]. In addition, emotion is primarily associated with positive valence, while mood is
primarily associated with negative valence [14].

Computers 2024,13, 3 3 of 24
Under the names of popular forms of sentiment [
15
], aggregate mood [
16
], collective
mood [
17
], collective sentiment [
18
], background sentiment [
13
], etc., two types of mass
mood studies can be distinguished: (1) where mood is analyzed in terms of positive
and negative polarity, and the mood dynamics are related to the movement between
the poles [
18
–
21
];
(2) where mood
is studied as a set of feelings involving more than
one emotion [
22
], and the mood dynamics are seen as changes in the corresponding
emotions [
23
]. In such approaches, mood does not differ from emotion in duration and is
labeled as a rapidly changing attribute [
16
] with a minute [
24
], hourly [
25
], daily, or—at
best—weekly cycle of change
[26–28]
. But, psychologically, it is more correct to say that
mood is stable for several weeks, which provides a basis for calling it a “chronic” emotional
state [29].
In terms of duration, mood can be recognized as a psychological phenomenon with an
annual cycle of change. Indeed, it is believed that mood and related behavior are strongly
dependent on the time of year [
30
]. A strong argument for the seasonality of emotional
fluctuations is Seasonal Affective Disorder (SAD)—a recurrent type of major depression.
Typically, SAD begins in the fall and continues through the winter months. Less commonly,
SAD causes depression in spring or early summer. Symptoms consist of a sad mood and
low energy [31].
In 2012, the prevalence of diagnosed SAD ranged from 1% to 10% of the global
population, and, in temperate zones, from 3% to 10%. Subsyndromal SAD with blurred
symptoms was found in 6% to 20% of the temperate population [
32
]. The minimum
percentage of people with syndromic and sub-syndromic forms of SAD in Russia as a
whole was at least 9%. We do not know how this ratio has changed over the past decade,
but it is unlikely that its decrease, if any, has been significant. In addition, in the fall and
spring periods, exacerbations of other mental illnesses occur, which gives an additional
surge of depressive, asthenic, neurotic, and hypochondriacal symptomatology in the patient
population [
30
,
33
]. Thus, it is difficult to accurately estimate how many social media users
suffer from various endogenous seasonal mood swings and how much they contribute to
the total amount of content generated.
Many of the studies reviewed in [
34
] empirically proved that seasonal mood swings are
common in the general population as well. For example, nearly 50% of non-depressed peo-
ple reported experiencing some depressive symptoms in winter, and it seemed that almost
everyone had the most happiness in spring; however, it was also found that worthlessness,
suicidality, and aggression have a significant connection with the seasons [34–37].
In terms of user-generated content, an analysis of 509 million tweets written by
2.4 million people in 84 countries showed that a shorter day length is associated with less
positive sentiment in tweets [
38
], and another analysis of 800 million tweets in the UK
revealed peak sadness in winter [
39
]. A study of Russian users’ search queries, using the
Google Trends application, for “depression”, “anxiety”, “panic attack”, etc., showed that
seasonal variations in web searches repeat the spring–autumn peaks and summer–winter
valleys of depressive disorders and anxiety–depressive disorders [
40
]. In contrast, the study
of [
18
] did not statistically support seasonal sentiment changes in structurally stable Twitter
communities. The authors suggested that, when sentiment in a community temporarily
deviates strongly from its normal level, it can usually be associated with a significant
identifiable event that has affected the community, sometimes an external news event—in
other words, the detected spikes are emotion-dependent rather than mood-dependent.
Thus, it is still unclear whether seasonality is more or less intrinsic to the general population,
as some researchers claim, or whether it can only be detected in fairly specific groups, as
others believe.
For NLP, sentiment remains a very noisy signal due to the subtlety of human lan-
guage [
18
]. Indeed, many effective tools have been proposed to analyze sentiment in social
media based on machine learning or lexicon [
41
]. However, although mood word lists, idioms,
emoticons, negation words, linguistic rules, and mood polarity classification algorithms [
23
]

Computers 2024,13, 3 4 of 24
have been used to extract emotions from user content, negation, irony, metaphorical, and
contextual ways of expressing attitudes interfere with the analysis results.
In addition, the textual expression of mood is colored by a person’s peculiar vocabulary
and style, as well as by the social context, including social norms, history, and common
understanding [
15
]. As for machine learning, recently, there have been doubts about its
direct suitability for solving many of the problems of the socio-humanities in general and
text analysis in particular [42,43].
In contrast to approaches to mood as a generalized bipolar emotion or as a combination
of emotions, our study treats mood structurally, as a variable complex of emotions and
emotional tension [
44
]. Emotional tension is a less significant and well-defined component
of mood than emotion, and it is experienced as a state ranging from apathy to agitation [
45
].
In a social context, increases in public emotional tension in the form of mass forms of
hostility, social anxiety, panic, hysteria, and aggression are associated with irrational
collective behavior, such as social protest [
46
]. For our purposes, we rely on the tradition
of assessing emotional tension as a component of mood, and this is embedded in the
widely used Profile of Mood States (POMS) questionnaire [
47
]. Studies using the POMS
questionnaire have shown that emotional tension is an attribute of both individual and
group mood [48–50].
The advantage of assessing emotional tension, rather than mood per se, is that it can
be extracted by simpler and more reliable means than tone dictionaries and other lexical
tools. Namely, a tense emotional state is revealed by the correlation of parts of speech—
verbs, nouns, adjectives—and their forms in user-generated content [
51
]. Such markers are
less dependent on the topic and form of communication and are much less consciously
controlled, which increases the reliability of the results. We investigated emotional tensions
in social media using the Trager coefficient, or the ratio of verbs to adjectives in text.
The Trager coefficient was proposed to measure the level of a person’s emotional
stability [
52
]. Its norm is close to 1 (more precisely, 1.34 ± 0.05), and values above the
norm indicate emotional arousal and other sthenic states. Low values indicate insecurity,
dependence, and anxiety [
53
]. Trager’s coefficient correlates with mental stress [
54
], sui-
cidality [
55
], schizophrenia and clinical depression [
56
,
57
], expressed civil identity [
58
],
insincerity in communication [
59
], etc., and it can also be used directly to assess emotional
tension [
52
]. We believe that the fluctuations in the Trager coefficient in user-generated
content reflect the dynamics of emotional tension in a user’s mood structure.
The study of mood as a long-lasting emotional state requires special methods that
allow for capturing and reflecting on the temporal patterns of ongoing processes. A
useful tool for this purpose can be variation, the significance of which is now well known
from the classical works of W. Shewhart, which laid the foundation for the widespread
use of the statistical method for the continuous monitoring and diagnosis of ongoing
processes [
60
,
61
] and statistical process control (SPC) [
62
]. It is the variation in the Trager
coefficient described above and mentioned in [
52
] that is an indicator of the emotional state,
so the use of Shewhart control charts are appropriate in this case.
Through using control charts based on variation values, one can divide the entire
observation period into days differing in emotional intensity, and one can then attempt to
identify seasonal patterns by examining the resulting categorical time series. This partition
plays a key role in our scheme for seasonality searching since, after transforming the
data into a categorical series, we can obtain a randomized sequence with more stable
patterns than in the original data. The transition to a categorical time series makes it
possible to confidently use both well-known methods for working with such objects, such
as statistical methods [
63
–
65
] and newly developed ones. A general theory of such series is
currently being actively developed [
66
,
67
]; researchers propose various practical techniques
depending on the software used [
68
], some of which may be suitable for finding seasonality.
Different types of aggregation are often used to detect seasonality, i.e., by calendar
period or by selected observations, within a sliding window [
69
]. In the second case, the

Computers 2024,13, 3 5 of 24
choice of periods is associated with an integer optimization problem (a review of methods
for solving such problems is given in [70]).
Seasonality modeling in the absence of comparative time series models is possible
using clustering. Practical methods of clustering, in particular clustering based on model
fitting, are given in [69] and are more fully presented in [70].
In our study, within the framework of the workflow, we identified the main problems
that arise when determining the presence of seasonal changes in collective emotional
tension. Using the example of a specific dataset, we present a possible way of solving
them in this particular case. Moreover, because of the key transition to a categorical time
series, it is possible to apply the abovementioned methods that are aimed at solving similar
problems in the general case. Thus, we have answered the first question posed in this study.
3. Materials and Methods
3.1. Data and Text Processing
Data for the study were collected from the 5 largest and most active communities of
the VKontakte social network, which are dedicated to local city news and events in a town
with a population of less than 100 thousand in the Nizhny Novgorod region of Russia. The
number of subscribers in these social media communities varies from 12 to 43 thousand.
The data were collected using the official API of VKontakte. All posts and comments
from 21 December 2019 to 5 March 2023 were downloaded. We did not download informa-
tion about specific users’ data. Only the message type (comment or post), message text,
and date were retained. A total of 83,125 posts and 662,881 comments were collected. In the
VKontakte social network, posts are messages written by community administrators—in
our case, they are mostly news related to local events. Comments, on the other hand, are
messages from the subscribers of these communities who comment on the posts. Only
comments were considered for further analysis as they contain the text of users, while posts
are published on behalf of the community.
During the data cleaning process, we removed all comments that did not contain
letters of the Russian alphabet, removed all characters that were not letters of the Russian
alphabet or punctuation marks, and removed outliers that exceeded the 99th percentile for
the number of characters and the number of sentences. The parameters of the collected
data after cleaning are presented in Table 1.
Table 1. Parameters of the collected data.
Parameters Values
Number of days 1171
Number of comments 606,638
Min and max comments per date 19 (min)–2976 (max)
Avg. comments per date 518
Avg. character count per comment 68.95
Avg. word count per comment 13.63
Avg. sentence count per comment 1.64
In the next step, each comment was analyzed using TITANIS [
71
], which computes
multi-level linguistic markers of texts. Those text markers that required a morphological
annotation of words were based on the results of a MyStem [
72
] analysis. The Trager
coefficient was calculated by dividing the number of verbs by the number of adjectives in
each comment. These values were averaged for each day, thus forming the final dataset on
which further experiments were performed. The dataset with the average Trager coefficient
by date is available in [73].

Computers 2024,13, 3 6 of 24
3.2. Dataset Specification: Main Properties and Features
Let us consider the main characteristics of the data and their properties, and let us also
use exploratory analysis to identify existing patterns in the data and attributes that may
indicate the presence of seasonality.
3.2.1. Descriptive Statistics of the Dataset
The dataset comprises daily calculations of the Trager coefficient values recorded
from 21 December 2019 to 5 March 2023. We have 1171 observations, arranged by time.
The descriptive statistics show the proximity of the dataset to a normal distribution (see
Table 2).
Table 2. Descriptive statistics of the dataset.
Statistics Values
Min. 0.5209
Max. 1.6797
Mean 1.2089
Median 1.2153
Mode 0.5209
Std. dev. 0.1237
Range 1.1588
Skewness −0.3716
Excess kurtosis 0.9129
Descriptive statistics provide a more detailed understanding of the characteristics of
the data under study.
3.2.2. Main Properties
Figure 1shows the fit of the dataset to a normal distribution.
0.6 0.8 1 1.2 1.4 1.6
Data
0
0.5
1
1.5
2
2.5
3
3.5
Density
Original data
Normal Distribution Fit
Figure 1. This histogram visualizes the proximity of the dataset to a normal distribution. The
MATLAB Distribution Fitter was used to fit the normal distributions to the data with Parameter
Estimate: mu = 1.20889 (Std. Err. 0.00361506) and sigma = 0.123707 (Std. Err. 0.00255787).
By direct calculation, we verified that 68.66% of the data fit within one standard
deviation of the mean, 95.64% were within two standard deviations of the mean, and

Computers 2024,13, 3 7 of 24
99.57% were within three standard deviations of the mean, which agrees quite well with the
three-sigma rule (68–95–99.7). Table 3shows the distribution of data by the area delineated
by standard deviations: the number of records with values below one std (1 Std), one to two
std (2 Std), two to three std (3 Std), and more than three std (out of 3 Std); the percentage of
records by area; and the cumulative percentage by the areas of the data that fall within the
intervals of up to one std, up to two std, and up to three std.
Table 3. Data distribution through the standard deviation areas.
Areas Number of Records Percentage by Area Cumulative Percentage
1 Std 804 68.6593 68.6593
2 Std 316 26.9854 95.6447
3 Std 46 3.9283 99.5730
Out of 3 Std 5 0.4270 —
Total 1171 100.00
According to the three-sigma rule, the five values in the last row of Table 2in the
area “Out of 3 Std dev” are potential outliers or anomalies. These values do not affect the
overall statistical properties, which are determined by more than 99% of the data. Here, we
consider them to be simply values outside the standard deviation of the mean.
The properties of the studied dataset, recorded in Table 3, satisfy the basic control
chart [
74
] and allow us to use the ideas of statistical method [
61
] for its dichotomy into
categories. The simplest SPC method is a control chart, which presents the values grouped
around a mean and the control limits. This is also known as the Shewhart [
75
] control
chart. Using the control limits of the basic control chart, we transformed the data into a
categorical time series, thereby assigning the category “white” to the values in the first area
“1 Std dev” and the category “black” to the values in the remaining areas of Table 3.
Remark 1. Note that through the key role of the dichotomy in the entire workflow, after transforming
the data into a binary time series (categorical or, alternatively, a count series of zeros and ones), we
obtain randomized sequences with more consistent patterns than those in the original data. Here, it
is possible to use both purely statistical methods [
63
,
64
]and machine learning methods. Also, of
undoubted interest, are the recent works by [
68
]on practical techniques with ordinal series and the
works of [76,77]with count time series.
3.2.3. Dataset Features
Let us denote
D
as the time-ordered set of all records of the Trager coefficient values.
Let us divide this set into two subsets: subset
W
(which contains records with values within
one standard deviation of the mean) and
B=D\W
. Now, we have set
W
with moderate
variance and set
B
with high variance. Therefore,
D={W
,
B}
. According to Table 2, we
see that, in set
D
, 68.66% of the elements are elements of set
W
and 31.34% of the elements
are elements of set
B
. Therefore, we can determine the Base Level (
BL
) of set
D
as 68.66%
W
(“Whites”) vs. 31.34%
B
(“Blacks”). For any subset
D′={W′
,
B′}
of
D
similarly, we can
determine the white level vs. black level (
BW
) as a percentage of the number of elements
D′
for
W′
and
B′
; thus, we can talk about the white-to-black ratio for
D′
. Now, we can
compare the white-to-black ratios with the Base Level as BW −BL for different samples.
Three-month cumulative samples for the entire observation period were considered
according to seasonal, quarterly, and off-season samples: from February to April, from May
to July, from August to October, and from November to January. Such samples reflected all
the possible seasonal changes in the annual cycle.
The following charts show how the features of the
D
dataset manifested themselves in
deviations in the white-to-black ratio from the Base Level for different calendar periods.
The diagram in Figure 2is consistent with our idea of seasonal changes in emotional
tension, i.e., that the persistence of emotional tension in winter and summer is statistically

Computers 2024,13, 3 8 of 24
higher than in spring and fall. This suggests that the data properties may show signs of
seasonality in the form of alternating dominant colors in a white-to-black ratio. If we take
this as a sign of seasonality, then, in Figure 3, we can see signs of seasonality in the quarterly
samples. Additionally, in Figure 4, we can see that the alternation in color dominance was
already broken.
2.8222
-2.3549
4.8915
-3.0915
-2.8222
2.3549
-4.8915
3.0915
Winters Springs Summers Falls
Seasonal samples for 2020-2023
-6
-4
-2
0
2
4
6
Deviations from Base Level (%)
Figure 2. The samples were based on records from 1 March 2020 to 28 February 2023. In the chart, the
numbers at the ends of the bars indicate the deviation in the white
−
to
−
black ratios from the Base
Level (
BW −BL
). The seasonal samples from 2020 to 2023 show an alternation in white
−
to
−
black
ratios, with black dominance in the spring and fall and white dominance in the winter and summer.
The presence of a characteristic in a cumulative sample does not mean that the sea-
sonality property will be present in annual samples; meanwhile, the opposite is true,
seasonality in the annual samples manifested itself in the aggregate ones, and the character-
istic must be present in them. The disruption of the alternation in the off-season samples
meant that these periods would not be considered further and that we can limit ourselves
to annual seasonal and quarterly samples.
The following diagrams detail the distributions of subsets
W′
and
B′
for these calendar
periods relative to the BW variable.
In Figures 5and 6, we see the manifestation of patterns in the alternation of changes
in the white-to-black ratio over the selected periods. Such patterns can serve as signs of
seasonality in the samples under consideration, but they do not provide an unambiguous
answer about the nature of seasonality since they manifests themselves in both seasonal
and quarterly intra-annual periods.
As a result of the primary analysis, patterns were identified in the data, thus indicating
the presence of a certain common property (let us call it “seasonality”). This property of
the data manifested itself in the form of alternating changes in white-to-black ratios in intra-
year periods, both in the seasonal and quarterly samples, which did not allow them to be
unambiguously localized in time and only on the basis of the considered statistical samples.
Features of the dataset allowed us to distinguish two types of patterns: one type based
on calendar seasonal samples (Figure 5) and the other based on calendar quarterly samples
(Figure 6).

Computers 2024,13, 3 9 of 24
2.1894
-6.0219
6.3407
-0.54332
-2.1894
6.0219
-6.3407
0.54332
1st quarter 2nd quarter 3rd quarter 4th quarter
Quarterly samples for 2020-2022
-6
-4
-2
0
2
4
6
Deviations from Base Level (%)
Figure 3. The samples are based on records from 1 January 2020 to 31 December 2022. In the graph,
the numbers at the ends of the bars indicate the deviation of the white
−
to
−
black ratios from the Base
Level (
BW −BL
). The quarterly samples from 2020 to 2022 show an alternation in white
−
to
−
black
ratios, with black dominance in the second and fourth quarters and white dominance in the first and
third quarters. From the first to the third quarter, the amplitude of deviations increases and falls
sharply in the fourth quarter.
1.1169
-1.6303
0.54363
1.6306
-1.1169
1.6303
-0.54363
-1.6306
FebApr MayJul AugOct NovJan
Off-season samples for 2020-2023
-6
-4
-2
0
2
4
6
Deviations from Base Level (%)
Figure 4. The samples were based on records from 1 February 2020 to 31 January 2023. The numbers
at the ends of the bars indicate the deviation in white
−
to
−
black ratios from the Base Level (
BW −BL
).
In the off-season samples, the alternation in the white
−
to
−
black ratio was broken. The predominance
of white over black began in August and continued until January.
Next, we modeled seasonality in the data, obtained a description of this property,
and determined the criterion by which this property could be uniquely identified in the
detected patterns. Let us check the feasibility of the criterion for each type of pattern, and,

Computers 2024,13, 3 10 of 24
if the criterion satisfies any type, we will consider a seasonality model based on this type of
pattern, check it for compliance with the data and for its adequacy in our understanding
of seasonality.
Springs 2020
Summer 2020
Fall 2020
Winter 2021
Springs 2021
Summer 2021
Fall 2021
Winter 2022
Springs 2022
Summer 2022
Fall 2022
Winter 2023
Seasonal samples for 2020-2023
0
10
20
30
40
50
60
70
80
White and Black levels (%)
Figure 5. The samples were based on records from 1 March 2020 to 28 February 2023. The seasonal
alternation in white
−
to
−
black ratios (
BW
) was typical from spring 2020 to spring 2021. The intervals
from summer 2021 to autumn 2021 were approximately equal. Winter 2022 was an exception, after
which the alternation was restored.
1st quarter 2020
2nd quarter 2020
3rd quarter 2020
4th quarter 2020
1st quarter 2021
2nd quarter 2021
3rd quarter 2021
4th quarter 2021
1st quarter 2022
2nd quarter 2022
3rd quarter 2022
4th quarter 2022
Quarterly samples for 2020-2022
0
10
20
30
40
50
60
70
80
White and Black levels (%)
Figure 6. The samples were based on records from 1 January 2020 to 31 December 2022. The quarterly
alternation in white
−
to
−
black ratios (
BW
) was typical from the first quarter of 2020 to the fourth
quarter of 2021. The alternation sequence was broken from the fourth quarter of 2021 to the second
quarter of 2022. The alternation was restored from the second quarter of 2022 to the fourth quarter of
2022, but with a noticeably smaller amplitude.

Computers 2024,13, 3 11 of 24
3.3. Seasonality Modeling
Model Aggregated by Time-Duration
Recall that the dataset was divided into two subsets of white and black days
(
D={W
,
B}
), and its Base Level (
BL
) was 68.6593% white versus 31.3407% black, which is
defined as
BL =100 ·(|w|,|b|)
|d|,
where |w|,|b|, and |d|are the cardinalities of the sets W,B, and D, respectively.
The time-aggregated model (TAM) for dataset Dis
T=
k
∑
i=1
Ti, (1)
where
Ti
is the selected time intervals consisting of the number of full days, so the
Ti
for
each i=1, ..., kis an integer.
The number of elements in
Ti
is represented by a pair
(Wi
,
Bi)
of white and black days;
therefore, for the cardinalities of the sets
|Ti|=|wi|+|bi|
and for each
i=
1,
. . .
,
k
, the
white and black levels (BW)for Tiwere defined as a vector as follows:
BW(i) = 100 ·(|wi|,|bi|)
|Ti|, (2)
where
(|wi|
,
|bi|)
is a 2-vector composed of the cardinalities of the sets
Wi
and
Bi
, and the
white and black level for TAM is
BW =1
k·
k
∑
i=1
BW(i).
The condition for the model to match the data is
∥BL −BW ∥ → min
T, (3)
where BL is the Base Level of dataset D, and ∥ · ∥ is the Euclidean norm of the vector.
3.4. Uniformly Aggregated TAMs
Let k∈N,|d|=k·∆+r,∆∈N,r∈N0,r<∆.
TAM, with the cardinalities of the sets
|Ti|=|wi|+|bi|=∆
for any
i=
1,
. . .
,
k
, is
uniformly aggregated if r=0.
Statement 1. Any uniformly aggregated TAM on some dataset D matches to the dataset.
Proof of Statement 1. If r=0, then
BW =100 ·"1
k
k
∑
i=1
(|wi|,|bi|)
∆#=100 ·"k
∑
i=1
(|wi|,|bi|)
k·∆#=
=100 ·




k
∑
i=1(|wi|,|bi|)
|d|




=100 ·(|w|,|b|)
|d|=BL,
and so ∥BL −BW ∥=0.
Remark 2. The time series is a trivial TAM case with ∆=1.

Computers 2024,13, 3 12 of 24
Consider the subsets of the dataset
D
of 1170 days (for example, the records from
21 December 2019 to 4 March 2023, or the records from 22 December 2019 to 5 March 2023).
For two subsets, the cardinality d=k·∆and their TAMs were as follows:
Ti=∆,i=1, . . ., k, (4)
which corresponded with k=13 and ∆=90.
Uniformly aggregated TAMs are useful for learning datasets because they can be
extended to an arbitrary dataset. Let us designate the first model for the dataset with
records from 21 December 2019 to 4 March 2023 as TAM1, and let us set the second model
for the dataset with the records from 22 December 2019 to 5 March 2023 as TAM2. Let us
see how these models behave across the entire dataset D.
3.5. Extension of Uniformly Aggregated TAMs on a Dataset
For dataset
D
with 1171 records from 21 December 2019 to 5 March 2023, we have
d=k·∆+r
,
k=
13,
∆=
90, and
r=
1. There are two ways to extend the uniformly
aggregated TAM that is represented by (4) into the
D
dataset. We can interpret the TAM as
a uniform
k
-lattice on top of the data with cells
Ti
and add the missing data to the beginning
or end of the lattice (by expanding cell
T1
or cell
Tk
), or by moving the lattice to the left or
right and resizing the first and last cells.
3.5.1. Extending the Model by Adding Data
Uniformly aggregated TAM1 specifies the distribution of the white and black days in
the form
{Wi
,
Bi}
on a dataset of 1170 days. Consider the extended TAM1 with additional
data on the right as follows:
|Ti|
k−1
i=1=∆,|Tk|=∆+r. (5)
Here, the
r
elements need to be added to the last cell
Tk
, and, for the extended model,
the new number of elements in
Tk
will be
(|wk|+|x|
,
|bk|+|x|)
, where
x
is a logical
r
-vector
of 0 and 1, which indicates the presence of white and black elements in an additional
interval of length
r
. The extended model does not have to fit the data exactly, but, for TAM,
the error in condition (3) can be expressed in terms of the model parameters.
Now, we can express the Base Level of Dfor this decomposition as
BL ="1
k·
k−1
∑
i=1
(|wi|,|bi|)
(∆+r/k)+(|wk|+|x|,|bk|+|x|)
k·∆+r#·100, (6)
and the white and black levels for extended TAM1 as
BW =1
k·
k
∑
i=1
BW(i) = "1
k·
k−1
∑
i=1
(|wi|,|bi|)
∆+(|wk|+|x|,|bk|+|x|)
k·(∆+r)#·100. (7)
From (6) and (7), we can obtain
BL −BW
100 =1
k
k−1
∑
i=1
(|wi|,|bi|)
(∆+r/k)+(|wk|+|x|,|bk|+|x|)
k·∆+r−
−1
k
k−1
∑
i=1
(|wi|,|bi|)
∆−(|wk|+|x|,|bk|+|x|)
k·(∆+r),

Computers 2024,13, 3 13 of 24
and, after bringing similar ones forward, we can find the error in matching the model to
the data in the form
err =




−rk−1
∑
i=1(|wi|,|bi|)
k2·∆2+rk +(|wk|+|x|,|bk|+|x|)·r·(k−1)
(k·∆+r)·k·(∆+r)




·100. (8)
Similarly, for the extended TAM2 with additional data on the left, we can obtain
|Ti|
k
i=2=∆,|T1|=∆+r, (9)
and we can find the error in matching the model to the data in the form
err =




−rk
∑
i=2(|wi|,|bi|)
k2·∆2+rk +(|w1|+|x|,|b1|+|x|)·r·(k−1)
(k·∆+r)·k·(∆+r)




·100. (10)
Substituting the parameter values
k=
13,
∆=
90, and
r=
1 into
Expressions (8) and (10)
,
we obtain for the extended TAM1 (denote as ExTAM1) the error in matching the model to the
data in (8), which is equal 0.0033. For the extended TAM2 (denoted as ExTAM2), the error
in matching the model to the data in (10) is equal 0.0060. Thus, ExTAM1 fits the data on the
set Dslightly better than ExTAM2.
3.5.2. Expanding the Model by Shifting a Uniform Lattice on the Dataset
The principle of moving windows can be implemented in TAM by shifting a uniform
lattice on the dataset. Let us consider a uniformly aggregated model with finite cells that
are shifted by a distance l, where lis an integer of the form
|Ti|
k−1
i=2=∆,|T1|=∆−l,|Tk|=∆+r+l. (11)
This is similar to moving a conveyor belt, i.e., when a lattice is moved to the left by a
distance
l
, the first cell
T1
goes beyond the boundary of the set
D
and its size decreases by
l
,
but, at the same time, the last cell
Tk
at the right end of the lattice increases by the same
amount l.
For each
l
, we obtain a new distribution of white and black days across cells
Ti
in the form
{Wi
,
Bi}
on a dataset
D
; thus, we can express the BaseLevel of
D
for each
decomposition as
BL ="(|w1|,|b1|)
k·∆+r+1
k·
k−1
∑
i=2
(|wi|,|bi|)
(∆+r/k)+(|wk|,|bk|)
k·∆+r#·100,
and the white and black levels for (11) as
BW =1
k·
k
∑
i=1
BW(i) = "(|w1|,|b1|)
k·(∆−l)+1
k·
k−1
∑
i=2
(|wi|,|bi|)
∆+(|wk|,|bk|)
k·(∆+l+r)#·100.

Computers 2024,13, 3 14 of 24
The condition for the model to match the data in (3) takes the form
BL −BW
100 =(|w1|,|b1|)
k·∆+r+1
k·
k−1
∑
i=2
(|wi|,|bi|)
(∆+r/k)+(|wk|,|bk|)
k·∆+r−
−(|w1|,|b1|)
k·(∆−l)−1
k·
k−1
∑
i=2
(|wi|,|bi|)
∆−(|wk|,|bk|)
k·(∆+l+r),
and, after bringing similar ones forward, we can find an expression that connects the error
in matching the model with the data with the model parameters as follows:
err =100 ·




−rk−1
∑
i=2(|wi|,|bi|)
k2·∆2+rk 




+
+100 ·"−(|w1|,|b1|)·(k·l+r)
(k·∆+r)·k·(∆−l)+(|wk|,|bk|)·k·(l+r)−r
(k·∆+r)·k·(∆+l+r)#. (12)
Remark 3. The shift method can produce models equivalent to the extended models. Assuming
l=
0in (11), we have
T1=∆
and
|Tk|=∆+r
, so the model turns into
|Ti|
k−1
i=1=∆
and
|Tk|=∆+r
. This is what exactly gives (5) and vice versa. When shifting to the right by
(−l)
with
l=r, we obtain |Tk|=∆, in which |Ti|
k
i=2=∆, T1=∆+l, which is what exactly matches (9).
Note that the shift method identifies models that better fit the data. For example, a
model of type (11) with parameter values
k=
13,
∆=
90,
r=
1, and
l=
3 (denoted as
ShTAM3) from the expression (12) has a model-to-data matching error of 0.0015. A model
of type (11) with parameter values
k=
13,
∆=
90,
r=
1, and
l=
10 (denoted as ShTAM10)
from Expression (12) has a best model-to-data fitting error of ≈0.0002.
3.5.3. Evaluating and Comparing Models on the Real Dataset
Let us estimate the fitting errors of the seasonality models on the entire dataset
D
,
which was described above. Let us consider models with parameter values
k=
13,
∆=
90,
and
r=
1, as well as vary the parameter
l
from
−
41 to 41 so that the minimum sizes of the
end cells are not less than 50 (the starting point
l=
0 here corresponds to 90 days from the
moment of the first record). The best-fitting models with matching errors of less than 0.01
are presented in Table 4.
Note the three isolated local minima with
l=
10,
l=−
12, and
l=−
19, as well as a
robust local minimum with l=3.
Let
L
denote the set consisting of the parameter values of the best-fitting models from
Table 4. As such, L={−19, −12, [−6, 6], 9, 10}.
Let
Yl= (yl
1
,
. . .
,
yl
k)
be an output of the shift model (11) with parameters
l
and
k
, then
yl
i=BW(i)
,
i=
1,
. . .
,
k
, where the
BW(i)
for each
i
is defined in (2) for the corresponding
model.
Let us form a cluster on the set of the best-fitting models using the idea of the centroid
method. In the first iteration, we build a centroid
ˆ
Yc=1
|L| ∑
l∈L
Yl(13)
by averaging the white and black levels for all the models from set
L
. The R-squared (
R2
)
and MAPE metrics were used to estimate the distance
ρ(Yl
,
ˆ
Yc)
for each model from the
set Lto the centroid.

Computers 2024,13, 3 15 of 24
Table 4. Ranking models according to the data fit.
Shift Parameter Value First Cell Size (Days) Fitting Error
l=10 80 0.0002
l=3 87 0.0015
l=−19 109 0.0018
l=5 85 0.0019
l=4 86 0.0022
l=−5 95 0.0022
l=−4 94 0.0027
l=1 89 0.0028
l=−12 102 0.0028
l=2 88 0.0029
l=0 90 0.0033
l=9 81 0.0043
l=−3 93 0.0056
l=−1 91 0.0060
l=−2 92 0.0067
l=6 84 0.0090
l=−6 96 0.0090
After the first iteration, the models with parameter values
l=−
19 (
R2=
0.8717) and
l=−12 (R2=0.68751) were excluded as the two worst on the R2-metric.
In the second iteration, two of the models with fit errors of 0.0181 and 0.180 were added
because their parameter values of
l=
7 and
l=
8 were inside the new cluster. Models with
parameter values of
l=
5 (
R2=
0.9714),
l=
9 (
R2=
0.9766),
l=
10 (
R2=
0.9654), and
l=−6 (R2=0.9790) were then excluded based on the R2-metric being less than 0.98.
At the third iteration, a final cluster
L={[−
5, 8
]} \ {
5
}
of thirteen models (
R2>
0.98)
was formed (Table 5) around a centroid whose matching error to the data was 0.0016.
Table 5. Comparison of models by their closeness to the centroid.
Shift Parameter R2Metric MAPE * White Level MAPE Black Level
l=8 0.9818 1.1586 2.6558
l=7 0.9939 0.6037 1.3471
l=6 0.9864 0.9170 2.2199
l=4 0.9866 0.8843 2.2868
l=3 0.9877 0.9261 2.1152
l=2 0.9924 0.6691 1.5474
l=1 0.9887 0.7373 1.7731
l=0 0.9860 0.9214 2.1533
l=−1 0.9876 0.9762 2.2509
l=−2 0.9931 0.6314 1.4705
l=−3 0.9941 0.6325 1.3964
l=−4 0.9934 0.6737 1.4017
l=−5 0.9896 0.7878 1.7520
* Mean absolute percentage error. The MAPE metric estimates the magnitude of the error as a percentage of the
size of the variable being estimated, so the metric values for the white and black levels are different. Here, we
present the metric values separately for the white and black levels for a more convenient intuitive perception of
the nature of the error.
A visualization of the final centroid using a bar chart gives a characteristic visual
picture of the dataset in Figure 7, which most closely matches the distribution of black and
white days for the models in the cluster.

Computers 2024,13, 3 16 of 24
3.5.4. Cluster Properties
The models united in a cluster have common properties. As an example, Figure 8
shows bar charts with a similar pattern for the two models from the cluster with parameters
l=8 and l=−5, which were located at opposite ends of the cluster.
Figures 7and 8clearly show stable alternations in white-to-black ratios on the intervals
T1
–
T4
and
T9
–
T13
, which persist when the data distributions over the intervals change with
the model. It was also clearly visible that the main differences in the ratio of white and
black appeared at the intervals
T5
—
T8
, which is when small fluctuations were observed in
the level of white and black leading to visual changes in this part of the picture. This was
true for all models in the cluster.
1234567891011 12 13
Average pattern for cluster models
0
10
20
30
40
50
60
70
80
White and Black levels (%)
Figure 7. The centroid was an array
(ˆ
yc
1
,
. . .
,
ˆ
yc
k)
, the values of the components of which were
determined by the formula (13) by averaging the white and black levels for all the models from the
set
L
with
k=
13. Therefore, the
x−
axis shows the indices of the array elements, and the
y−
axis
shows the values of the array elements. For each index
i=
1, ...,
k
, the value of the array element
ˆ
yc
i
was a pair consisting of white and black values corresponding to this index.
Let us look, in detail, at the behavior of the cluster models in the range
T5
–
T8
. Figure 9
graphically shows fragments
(yl
5
,
. . .
,
yl
8)
of the output values
Yl
for all the thirteen cluster
models, i.e., for all l∈ L.
Figure 9highlights two “trends”—the blue line of the centroid, which determines the
direction for most models, and the yellow line of the “boundary” model with parameter
l=4, for which all values (y4
5, . . ., y4
8)were equal.
Let us denote this extreme value as
ζ
and focus on an important property of the cluster,
which we describe for white levels in the output as follows:
(wl
i≥ζ,i=5, 7
wl
i≤ζ,i=6, 8 ∀l∈ L. (14)
The conditions (14) indicated that the alternations in white-to-black ratios were not
violated in the range of T5—T8.

Computers 2024,13, 3 17 of 24
T1T2T3T4T5T6T7T8T9T10 T11 T12 T13
90-days T-intervals with modeling parameter l = 8
0
10
20
30
40
50
60
70
80
White and Black levels (%)
(a)
T1T2T3T4T5T6T7T8T9T10 T11 T12 T13
90-days T-intervals with modeling parameter l = -5
0
10
20
30
40
50
60
70
80
White and Black levels (%)
(b)
Figure 8. Black and white ratios on the
T−
intervals. The
x−
axis shows the intervals
Ti
corresponding
to the model, and the
y−
axis shows the white and black levels for
Ti
, which was determined by
Formula (2). (a) Model with parameter value l=8; (b) model with parameter value l=−5.
T5T6T7T8
T5 - T 8 intervals sample
72
72.5
73
73.5
74
74.5
75
75.5
76
76.5
77
White Level (%)
centroid
"l = 4"
Figure 9. Comparative behavior of the intervals from
T5
to
T8
. Values of the white levels in the output
components
(yl
5
,
. . .
,
yl
8)
in the points
T5
,
. . .
,
T8
for all
l∈ L
. Some of the lines overlapped each other,
so there appeared to be fewer than thirteen lines.
Taking into account Equation (14), the alternation conditions on the entire dataset, i.e.,
in the range T1—T13, for the model with parameter lwere written in the form
wl
i−wl
i−1≤0, ∀i=2n, or (15)
wl
j−wl
j+1≥0, ∀j=2n−1, n=1, . . ., [k/2], (16)
where [·]denotes the integer part of the real number.
It was directly verified that Conditions (15) and (16) were true for all l∈ L.
Let us formalize this result in the form of a criterion that allows us to unambiguously
determine whether a given model of type (1) corresponds to the seasonality property of the
data. As such, the following theorem is true.

Computers 2024,13, 3 18 of 24
Theorem 1 (Data seasonality compliance criteria).Let us say that the time-aggregated model (1)
with parameter
k=
13 and 50
≤ |Ti| ≤
130 for all
i=
1,
. . .
,
k
, corresponds to the data seasonality
on dataset
D
. Then, if the white-to-black ratio
BW
in Equation (2) for
T
satisfy one of Conditions (17)
or (18), we have
(−1)α−1·[BWα(i)−BWα(i−1)]≤0, ∀i=2n(17)
(−1)α−1·[BWα(j)−BWα(j+1)]≥0, ∀j=2n−1, (18)
where n =1, . . ., [k/2]and α= (1, 2), and [·]denotes the integer part of the real number.
Theorem 1establishes the necessary conditions for the pattern to correspond to the
data seasonality property.
4. Results
At the beginning of this paper, the objectives of our study were formulated to answer
two questions:
1.
What methods are applicable to detect patterns of variation in multiple assessments
of a population’s psychological states when observed over time?
2.
Do collective emotional tensions in reality have seasonal variations that can be tracked
through social media content analysis?
To answer the first question, let us list the workflow steps used in this paper to identify
the patterns of change in the psychological state of the population: statistical method and
exploratory data analysis based on descriptive statistics; definition of new functions to
identify patterns in the data; formulation of the modeling problem; data aggregation; use
of the clustering method to identify typical properties of the data; formulation of the soft
sign criterion of “seasonality” (based on the analysis of data typical properties); and the
demonstration of the manifestation of “data-seasonality” in the calendar seasonal model.
The affirmative answer to the second question is based on the following results:
•
As a result of the analysis of the statistical data, features of the data array were
identified that make it possible to display mass emotional tension in the ratio of whites
and blacks in selected calendar periods;
•
The proposal of an approach to model seasonality in a class of time-aggregate models;
•
Within the framework of the above proposed approach, it was shown that the data
were characterized by the property of “data seasonality”, and the description of
this property was obtained in the form of a stable pattern for all models from the
found cluster;
•
Based on the identified features of data seasonality, a criterion for matching this
property for other time aggregated models was formulated.
In a direct test of the seasonality of the compliance criteria (Theorem 1), it was shown
that the pattern on calendar seasonal samples (Figure 5) satisfies condition (17), while
the seasonality data fit criterion was not met for the pattern on calendar quarterly sam-
ples (Figure 6).
The diagram in Figure 10 shows a calendar seasonal model pattern that extended
to the entirety of dataset
D
, which corresponded with the data seasonality property (see
Figure 7for visual confirmation).
This means that the calendar seasonal pattern was consistent with the data, and
the alternating white-to-black ratios fit well with our understanding of how the overall
dynamics of emotional tension correlate with the seasons.

Computers 2024,13, 3 19 of 24
Winter 2020
Springs 2020
Summer 2020
Fall 2020
Winter 2021
Springs 2021
Summer 2021
Fall 2021
Winter 2022
Springs 2022
Summer 2022
Fall 2022
Winter 2023
Seasonal pattern for 2019-2023
0
10
20
30
40
50
60
70
80
White and Black levels (%)
Figure 10. Seasonal pattern according to the calendar seasonal model across the dataset based on
records from 21 December 2019 to 5 March 2023.
5. Discussion
From a psychological point of view, two of the results were the most significant.
First, the obtained data confirmed the general seasonal dynamics of emotional tension
to be traceable in the analysis of network communications. Indicators of emotional tension
stability were statistically significantly higher in winter and summer than in spring and
fall (see Figures 2and 10). We found that the indicator of collective emotional tension
varied strongly from day to day more often in spring and fall than in winter and summer.
The revealed dynamics corresponded well with the above-described trends of changes in
emotional state, which were revealed in psychiatric and psychological practice or in the
course of sociological surveys.
Second, we found the absence of spring and fall peaks in the dynamics of emotional
tension in 2021. In contrast to 2020 and 2022 (where there were pronounced differences
between the more stable winter and summer on the one hand, and the more volatile spring
and fall on the other), the level of differences in emotion tension in 2021 remained relatively
unchanged across all four seasons. The available data did not allow us to infer the nature
of this equalization. The cause could be either constant fatigue and apathy or, conversely,
constant excitement and overexcitement during the second year of the pandemic. Thus
far, we can only point to an atypical pattern of seasonal dynamics of emotional tension in
2021 if we take the winter–spring and summer–fall differences as typical. Based on the idea
of the endogenous nature of seasonal fluctuations of mood (and emotional tension as its
component), we can assume that in the first year the dynamics are still intact, and in the
third year it is somewhat restored.
A pandemic is a prolonged stressor that disrupts the normal life of the population and
undoubtedly affects mass psychiatric conditions. At the outset of the pandemic, an exces-
sive impact on mental functioning was identified and a further increase in psychopathologic
symptoms was predicted [
78
]. However, defense mechanisms (such as threat underestima-
tion [79] or humor [80]) kept psychological states inert for some time. A possible basis for
the recovery of mass mental functions is adaptation to prolonged stressors [
21
]. Thus, the
detected pattern of collective emotional tension can be explained as a result of the action of
defense mechanisms in 2020, the disorder of adaptation of the population to a long-term
stressor in 2021, and gradual adaptation to extreme conditions in 2022.

Computers 2024,13, 3 20 of 24
The presented scheme for determining seasonality may be of interest for various social
practices. For example, by accumulating data on the severity of fluctuations in emotional
tension in different regions, it will be possible to identify regions with an increased risk of
chaotic mass behavior during periods of seasonal exacerbations. It is also useful to predict
the possible deterioration of the collective emotional state in order to optimize the work
of various social services that may face an increased flow of requests during unfavorable
periods. It is possible to link the seasonality of emotional tension with the manifestation
of mass somatic or mental disorders affecting the economic and social functioning of
regions, etc.
Limitations and Future Work
This study raised many questions. We studied collective sentiment averaged over a
large number of social media users. We do not know whether only users with pronounced
emotional seasonal shifts affected the overall emotional tension in the network while others
did not affect the tension at all, or whether all users contributed to some degree to the
overall emotional tension online. To clarify this question, a special longitudinal study
involving the identification of people with different emotional statuses is needed. For a
meaningful characterization of “black days”, it is necessary to distinguish between days
in which an instability of emotional tension is caused by a significant upward trend of
the Trager coefficient (spikes of overexcitement) or a significant downward trend (spikes
of apathy). Verification of the identified seasonal trends is possible both with the help of
other methods for assessing emotional tension in online communication and by building
up more texts for analysis. A promising direction for further research is to determine the
emotional component of mood in addition to the assessment of emotional tension.
It should be noted that this study was conducted using data from one local social
media community, which could potentially introduce bias. Thus, observations in other
communities could show a different picture. In addition, this paper only considers the
Trager coefficient to assess emotional tension, whereas our method could potentially be
applied to other normally distributed psycholinguistic parameters. These limitations
should be addressed in future work.
6. Conclusions
In this paper, we proposed a combined approach for detecting the seasonality of
emotional tension in social media based on the statistical method for data categorization,
exploratory data analysis to identify general patterns, modeling seasonality in the class of
time-aggregated models (TAM), and identifying typical properties of TAM data using the
clustering method. It was shown that the dynamics of emotional tension correlate with
the seasons of the year. To the best of our knowledge, this is the first study to investigate
the task of detecting seasonality in emotional tension using social media data. To assess
emotional tension, we used the Trager coefficient, which has not previously been used in
this task. We look forward to using the proposed method to account for the seasonality of
emotional tension when analyzing relationships between non-seasonal shifts in emotional
tension, specific events, and the content of social media texts.
Our results suggest that the emotional tension manifested in social media communica-
tion tends to fluctuate with the seasons, but strong and long-lasting stressors can distort
seasonality. The proposed methodology for identifying, evaluating, and displaying the
dynamics of emotional tension based on network communications demonstrates sensitivity
to both stable annual fluctuations and the external influences that disrupt them. However,
further development of the method will require more data, longer observation periods, and
data from other social media communities.
Author Contributions: Conceptualization, Y.K. and I.S.; methodology, A.N. and Y.K.; software, M.S.;
validation, A.N.; formal analysis, A.N.; investigation, A.N.; data curation, M.S.; writing—original
draft preparation, A.N., Y.K. and M.S.; writing—review and editing, I.S. and O.G.; visualization, A.N.;

Computers 2024,13, 3 21 of 24
supervision, I.S.; project administration, I.S. and O.G.; funding acquisition, O.G. All authors have
read and agreed to the published version of the manuscript.
Funding: This work was supported by the Ministry of Science and Higher Education of the Russian
Federation (project No. 075-15-2020-799).
Institutional Review Board Statement: Ethical approval is not required for this paper, as the data
used in this paper were collected and processed in accordance with the Federal Law of the Russian
Federation No. 149-FZ of 27 July 2006 “On Information, Information Technologies and Protection of
Information”. The data presented in this study contain only numerical values and cannot be used to
reveal any real identities.
Data Availability Statement: The data presented in this study are openly available in the Hugging
Face repository at https://doi.org/10.57967/hf/1475 (Accessed on 21 December 2023).
Conflicts of Interest: The authors declare no conflicts of interest.
Abbreviations
The following abbreviations are used in this manuscript:
EDA Exploratory Data Analysis
NLP Natural Language Processing
SPC Statistical Process Control
TAM Time-Aggregated Model
SAD Seasonal Affective Disorder
API Application Programming Interface
MAPE Mean Absolute Percentage Error
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