=Paper=
{{Paper
|id=Vol-2832/paper02
|storemode=property
|title=Coverage of the coronavirus pandemic through entropy measures
|pdfUrl=https://ceur-ws.org/Vol-2832/paper02.pdf
|volume=Vol-2832
|authors=Vladimir N. Soloviev,Andrii O. Bielinskyi,Natalia A. Kharadzjan
}}
==Coverage of the coronavirus pandemic through entropy measures==
https://ceur-ws.org/Vol-2832/paper02.pdf
Coverage of the coronavirus pandemic through
entropy measures
Vladimir N. Solovieva,b , Andrii O. Bielinskyia and Natalia A. Kharadzjana
a
Kryvyi Rih State Pedagogical University, 54 Gagarin Ave., Kryvyi Rih, 50086, Ukraine
b
Bohdan Khmelnitsky National University of Cherkasy, 81 Shevchenko Blvd., Cherkasy, 18031, Ukraine
Abstract
The rapidly evolving coronavirus pandemic brings a devastating effect on the entire world and its econ-
omy as a whole. Further instability related to COVID-19 will negatively affect not only on companies and
financial markets, but also on traders and investors that have been interested in saving their investment,
minimizing risks, and making decisions such as how to manage their resources, how much to consume
and save, when to buy or sell stocks, etc., and these decisions depend on the expectation of when to
expect next critical change. Trying to help people in their subsequent decisions, we demonstrate the
possibility of constructing indicators of critical and crash phenomena on the example of Bitcoin mar-
ket crashes for further demonstration of their efficiency on the crash that is related to the coronavirus
pandemic. For this purpose, the methods of the theory of complex systems have been used. Since the
theory of complex systems has quite an extensive toolkit for exploring the nonlinear complex system,
we take a look at the application of the concept of entropy in finance and use this concept to construct
6 effective entropy measures: Shannon entropy, Approximate entropy, Permutation entropy, and 3 Re-
currence based entropies. We provide computational results that prove that these indicators could have
been used to identify the beginning of the crash and predict the future course of events associated with
the current pandemic.
Keywords
coronavirus, Bitcoin, cryptocurrency, crash, critical event, measures of complexity, entropy, indicator-precursor
1. Introduction
The novel coronavirus outbreak (COVID-19) quickly became a catastrophic challenge for the
whole world, and it would be even more of a “black swan” than the global financial crisis and
Great recession of 2008-2009 [1]. While the governments are focused on saving lives and pre-
venting the virus from spreading further, at the moment, it continuous to led to a substantial
disruption of the world financial system. Furthermore, economic worries are also increasing
dramatically. At first, people have worried about potentially become one of the infected. Then,
CS&SE@SW 2020: 3rd Workshop for Young Scientists in Computer Science & Software Engineering, November 27,
2020, Kryvyi Rih, Ukraine
" vnsoloviev2016@gmail.com (V.N. Soloviev); krivogame@gmail.com (A.O. Bielinskyi);
n.a.kharadzjan@gmail.com (N.A. Kharadzjan)
~ https://kdpu.edu.ua/personal/vmsoloviov.html (V.N. Soloviev);
https://kdpu.edu.ua/personal/nakharadzhian.html (N.A. Kharadzjan)
0000-0002-4945-202X (V.N. Soloviev); 0000-0002-2821-2895 (A.O. Bielinskyi); 0000-0001-9193-755X (N.A.
Kharadzjan)
© 2020 Copyright for this paper by its authors.
Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR
Workshop
Proceedings
http://ceur-ws.org
ISSN 1613-0073 CEUR Workshop Proceedings (CEUR-WS.org)
when the situation escalated, people started to worry about the impact on their economic sit-
uation [2, 3, 4]. To prevent exponential growth of the disease incidence bans on traveling the
world, visiting public places, and businesses were established that had particularly a destructive
impact on financial markets, and the cryptocurrency market, showing that for most systems it
was unexpectable. Subsequent days on the market promise to be more volatile and riskier than
before. And it seems that Bitcoin is going to fall into a new Great recession along with the most
influential stock indexes. As it was noted by Pier Francesco Procacci, Carolyn E. Phelan and
Tomaso Aste [5], the crisis-state representative and eventually becomes extremely dominant
in March, but a lot of factors have to be included to make an adequate analysis of this problem.
The doctrine of the unity of the scientific method states that for the study of events in socio-
economic systems, the same methods and criteria as those used in the study of natural phe-
nomena are applicable. The increasing mathematical knowledge of complex structure provides
us with such methods and tools that are interconnected and can be used to study so relevant
and “almost unpredictable” situations as the coronavirus pandemic. By almost unpredictable,
we mean that although past experience and data are not so relevant in this case, we can use
some mathematical methods (models) that are useful in current situations, including current
statistics, relevant parameters, and even the inner complexity of the system. As an example,
Alexis Akira Toda [6] estimated the Susceptible-Infected- Recovered (SIR) epidemic model for
COVID-19 because of its desire to help individuals in managing their investments. Govern-
ments also need help in making an informed decision on imposing travel restrictions, social
distancing, closure of schools and businesses, etc., and, mainly, for how long [7]. In the pa-
per of Bohdan M. Pavlyshenko [8] was studied different regression approaches for modeling
COVID-19 spread and its impact on the stock market. The logistic curve model was used
under Bayesian regression for predictive analytics of the coronavirus spread. The obtained
results showed that different crises with different reasons have a different impact on the same
stocks. Bayesian inference makes it possible to analyze the uncertainty of crisis. Michele Cos-
tola, Matteo Iacopini and Carlo R. M. A. Santagiustina [9] regarding the public concern of 6
countries during the outbreak of COVID-19 find that the Italian index in the current situation
is relevant in explaining index returns for other studied countries. Assuming that COVID-19
has a deterministic, exogenous impact on the market, Karina Arias-Calluari, Fernando Alonso-
Marroquin, Morteza Nattagh-Najafi and Michael Harré [10] forecast it with a model of the
stochastic and systematic risk. Here, such a model is assumed to be q-Gaussian diffusion pro-
cess which is accompanied by three spatio-temporal regimes. In this case, the results were
achieved with 85% accuracy. Presented results are promising not only for markets but also
for risk control in other areas such as seismology, communication networks, etc. In the paper
[11] the bandwidth and the discount factor are proposed to minimize a criterion consistent
with the traditional requirements of the validation of a probability density forecast. They use
Kolmogorov-Smirnov statistic and a discrepancy statistic to build a quantitative criterion of the
accuracy of pdf which they try to maximize when selecting the bandwidth and the discount
factor of their time-varying pdf. Such an approach allows exposing an accurate chronology of
the current pandemic. Ayoub Ammy-Driss and Matthieu Garcin [12] explore novel pandemic
using two efficiency indicators: the Hurst exponent and the memory parameter of a fractional
Lévy-stable motion. Presented results highlight the occurrence of inefficiency at the almost be-
ginning of the crisis for US indices. Asian and Australian indices are seemed to be less affected
25
during this period, i.e., their inefficiency is even questionable. A. Fronzetti Colladon, S. Grassi,
F. Ravazzolo and F. Violante [13] present a new textual data index, which assesses the related
most usable keywords and semantic network position, for predicting stock market data. They
apply it to the Italian press and use it for predicting recent periods of Italian stock, including the
COVID-19 crisis. According to the results, it can be observed that the index is characterized by
strong predictability. Unfortunately, by the luck of observation and potentially diseased people
that are surrounding us at the moment, it is still difficult to predict further dynamic during the
pandemic.
As it can be observed, markets have seen significant numbers of investors selling off and
rebalancing their portfolios with less risky assets. That has been leading to large losses and
high volatilities, typical of crisis periods. The economy key for preventing such activity may lie
in cryptocurrency and constructing effective indicators of possible critical states that will help
investors and traders fill in safety. Bitcoin, which is associated with the whole crypto market,
has such properties as detachment and independence from the standard financial market and
the proclaimed properties that should make it serve as the ‘digital gold’ [14]. As was shown
by Ladislav Kristoufek [15], Bitcoin promises to be a safe-haven asset with its low correlation
with gold, S&P 500, Dow Jones Industrial Average, and other authoritative stock indices even
in the extreme events. But authors please not overestimate the cryptocurrency since according
to their calculations and, obviously, the current structure of the system, gold remains more
significant. But for ten years, this token has been discussed by many people, it has experienced
a lot in such a short period, many people believe in it, and it has managed to form a fairly
complex and self-organized system. The integrated actions from real-world merge in such
dynamics and relevant information that is encoded in Bitcoin’s time series can be extracted
[16, 17, 18]. In the context of volatile financial markets, it is important to select such measures
of complexity that will be able to notify us of upcoming abnormal events in the form of crises
at an early stage.
In this article, we:
• present such measures;
• study critical and crash phenomena that have taken place in the cryptocurrency market;
• try to understand whether a crash caused by the coronavirus pandemic could have been
identified and predicted by such informative indicators or not.
According to our goals and actions, the paper is structured as follows. In Section 2, we pre-
sented a brief overview of the studies in this field of science. In Section 3, relying on these
researches and the experience of other scientists, we present our classification of Bitcoin’s
crises for the period from 1 January 2013 to 9 April 2020. In Section 4, we describe the ap-
plied methods and present some empirical results with their subsequent description. Section 5
concludes.
26
2. Review of the previous studies
For its short history of existence, bitcoin has experienced many events, periodic rises and sud-
den dips in specific periods, regulatory actions, and discussions about whether it can become
a universally mature commodity around the world or not, and therefore its largely unexplored
dynamics is still presenting new opportunities and challenges for traders, economists, and re-
searchers from different fields of science to highlight chaos that is hidden in it. Although
officially bitcoin is considered to be a commodity rather than a currency, the comprehensive
analysis of the applicability of the instruments which have been used for mature financial mar-
kets for a long time can be made for cryptocurrencies and, namely, for the Bitcoin market.
A vast amount of different methods, as an example, from the theory of complexity, the pur-
pose of which is to quantify the degree of complexity of systems obtained from various sources
of nature, can be applied in our study. Such applications have been studied intensively for an
economic behavior system. As an example, Miguel Henry and George Judge [19] used an infor-
mation theoretic-symbolic logic approach which is based on Shannon’s information entropy
and called Permutation entropy (PEn). The entropy is applied to the Dow Jones Industrial Av-
erage to extract information from this complex economic system. The result demonstrates
the ability of the PEn method to detect the degree of disorder and uncertainty for the specific
time that is explored. In such paper, [20] presented by Higor Sigaki, Matjaž Perc, and Haroldo
Valentin Ribeiro, the PEn and statistical complexity over sliding time-window of daily clos-
ing price log-returns are used to quantify the dynamic efficiency of more than four hundred
cryptocurrencies. Authors address to the efficient market hypothesis when the values of two
statistical measures within a time-window cannot be distinguished from those obtained by
chance. They find that 37% of the cryptocurrencies in their study stay efficient over 80% of the
time, whereas 20% are informationally inefficient in less than 20% of the time. Moreover, the
market capitalization is not correlated with their efficiency. Performed analysis of information
efficiency over time reveals that different currencies with similar temporal patterns form four
clusters, and it is seen that more young currencies tend to follow the trend of the most leading
currencies.
Steve M. Pincus and Rudolf E. Kalman [21], considering both empirical data and models, in-
cluding composite indices, individual stock prices, the random-walk hypothesis, Black- Sholes,
and fractional Brownian motion models to demonstrate the benefits of Approximate entropy
(ApEn), a quantitative measure of sequential irregularity. On the example of the applied mod-
els and empirical data, the authors presented that ApEn can be readily applied to the classical
econometric modeling apparatus. This research the usefulness of ApEn on the example of
three major events of the stock market crash in the US, Japan, and India. During the major
crashes, there is significant evidence of a decline of ApEn during and pre-crash periods. Based
on the presented results, their research concludes that ApEn can serve as a base for a good
trading system. This article [22] gives evidence of the usefulness of Approximate Entropy. The
researchers quantify the existence of patterns in evolving data series. In general, scientists
observe that the degree of predictability increases in times of crisis. However, the presented
results do not demonstrate that the regularity techniques studied in this paper can serve to
predict an imminent crash.
Also, there is a paper [23] that is dedicated to Ethereum. Here, the concept of entropy is
27
applied for characterizing the nonlinear properties of the cryptocurrencies. For their goal,
Shannon, Tsallis, Rényi, and Approximate entropies are estimated. From their empirical re-
sults, it is obtained that all entropies are positive. Of great interest is the results of ApEn
which demonstrates larger value for Ethereum than for Bitcoin. In this case, it concludes that
Ethereum has higher volatility. The same result for other measures. Daniel Traian Pele and
Miruna Mazurencu [24] investigate the ability of several econometrical models to forecast value
at risk for a sample of daily time series of cryptocurrency returns. Using high-frequency data
for Bitcoin, they estimate the entropy of the intraday distribution of log-returns through the
symbolic time series analysis (STSA), producing low-resolution data from high-resolution data.
Their results show that entropy has strong explanatory power for the quantiles of the distri-
bution of the daily returns. They confirm the hypothesis that there is a strong correlation
between the daily logarithmic price of Bitcoin and the entropy of intraday returns Based on
Christoffersen’s tests for Value at Risk (VaR) backtesting, they conclude that the VaR forecast
built upon the entropy of intraday returns is the best, compared to the forecasts provided by
the classical GARCH models.
During our investigation, we found that many theses and papers that have been studied
the dynamics of financial markets rather than cryptocurrencies using the measures from the
theory of complexity. Thus, we are interested in contributing to the study of the dynamics
of a potentially strong currency that affects the world economy and faces the same problems
as even the most quoted financial markets. Thus, the construction of predictive models and
measures of unexpectable and critical events in the cryptocurrency market remains relevant.
3. Data preparation and classification
The ecosystem of cryptocurrencies has been growing at an increasing pace. More and more
of them become tradable and begin to inspire confidence for many people. Such events as
the coronavirus threat that are presented to be unpredictable, break this confidence and the
subsequent anxiety of traders shape the extent of economic worries and collapse.
Previously, we conducted research on the dynamics of both stock and cryptocurrency mar-
kets where were provided results confirming the effectiveness of using quantitative methods
of the theory of complexity which can serve as a base for estimation of indicators-precursors
of the critical states. This paper presents a comparative analysis of the measures based on the
concept of entropy: Shannon, Approximate, Sample, Permutation, and Recurrence based en-
tropies. Here, each entropy is applied to the entire market and separate bubble, crashes, and
critical events that have taken place in this market. Thus, we advanced into action and set the
tasks:
• Classification of such bubbles, critical events and crashes.
• Construction of such indicators that will predict crashes, critical events in order to give
investors and ordinary users the opportunity to trade in this market.
At the moment, there are various research papers on what crises and crashes are and how
to classify such interruptions in the market of cryptocurrencies. Taking into account the ex-
perience of previous researchers and our own [25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36],
28
we present our classification of such leaps and falls from the previous articles [28, 29, 30, 33]
on the cryptocurrency topic, relying on Bitcoin time series during the period (01.01.2013 –
09.04.2020) of verifiable fixed daily values of the Bitcoin price (BTC) (https://finance.yahoo.
com/cryptocurrencies).
For our classification, crashes are short, time-localized drops, with the strong losing of price
per each day, which are formed as a result of the bubble. Critical events are those falls that could
go on for a long period, and at the same time, they were not caused by a bubble. The bubble
is increasing in the price of the cryptocurrency that could be caused by certain speculative
moments. Therefore, according to our classification of the event with the number (1-3, 6-8,
12-15, 17) are the crashes, all the rest - critical events. More detailed information about crises,
crashes, and their classification following these definitions are given in table 1.
Table 1
BTC HISTORICAL CORRECTIONS. LIST OF BITCOIN MAJOR CORRECTIONS ⩾ 20% SINCE APRIL
2013
Days Bitcoin Bitcoin
Decline, Decline,
№ Name in High Low
% $
correction Price, $ Price, $
1 08.04.2013-15.04.2013 8 230.00 68.36 70 161.64
2 04.12.2013-18.12.2013 15 1237.66 540.97 56 696.69
3 05.02.2014-25.02.2014 21 904.52 135.77 85 768.75
4 12.11.2014-14.01.2015 64 432.02 164.91 62 267.11
5 26.01.2015-31.01.2015 5 269.18 218.51 20 50.67
6 09.11.2015-11.11.2015 3 380.22 304.70 20 75.52
7 18.06.2016-21.06.2016 4 761.03 590.55 22 170.48
8 04.01.2017-11.01.2017 8 1135.41 785.42 30 349.99
9 03.03.2017-24.03.2017 22 1283.30 939.70 27 343.60
10 10.06.2017-15.07.2017 36 2973.44 1914.08 36 1059.36
11 31.08.2017-13.09.2017 13 4921.85 3243.08 34 1678.77
12 16.12.2017-22.12.2017 7 19345.49 13664.9 29 5680.53
13 13.11.2018-26.11.2018 13 6339.17 3784.59 40 2554.58
14 09.07.2019-16.07.2019 7 12567.02 9423.44 25 3143.58
15 22.09.2019-29.09.2019 7 10036.98 8065.26 20 1971.7
16 27.10.2019-24.11.2019 28 9551.71 7047.92 26 2503.79
17 06.03.2020-16.03.2020 11 9122.55 5014.48 45 4108.07
Accordingly, during this period in the Bitcoin market, many crashes and critical events shook
it. Thus, considering them, we emphasize 17 periods on Bitcoin time series, whose falling we
predict by our indicators, relying on normalized returns, where usual returns are calculated as:
𝐺 (𝑡) = ln 𝑥 (𝑡 + 𝛿𝑡) − ln 𝑥 (𝑡) ≅ [𝑥 (𝑡 + 𝛿𝑡) − 𝑥 (𝑡)] / 𝑥 (𝑡)
and normalized returns as:
𝑔 (𝑡) ≅ [𝐺 (𝑡) − ⟨𝐺⟩] / 𝜎 ,
where 𝜎 is a standard deviation of G , 𝛿𝑡 is a time lag (here 𝛿𝑡 = 1), and ⟨…⟩ denotes the
average over the period under study.
29
Further calculations were carried out within the framework of the algorithm of a moving
window. For this purpose, the part of the time series (window), for which there were calcu-
lated measures of complexity, was selected, then the window of a length 100 was displaced
along with the time series in a one-day increment and the procedure repeated until all the
studied series had exhausted. Further, comparing the dynamics of the actual time series and
the corresponding measures of complexity, we can judge the characteristic changes in the dy-
namics of the behavior of complexity with changes in the cryptocurrency. The key idea here
is the hypothesis that the complexity of the system before the crashes and the actual periods
of crashes must change. This should signal the corresponding degree of complexity if they are
able to quantify certain patterns of a complex system. A significant advantage of the introduced
measures is their dynamism, that is, the ability to monitor the change in time of the chosen
measure and compare it with the corresponding dynamics of the output time series. This al-
lowed us to compare the critical changes in the dynamics of the system, which is described by
the time series, with the characteristic changes of concrete measures of complexity. It turned
out that quantitative measures of complexity respond to critical changes in the dynamics of
a complex system, which allows them to be used in the diagnostic process and prediction of
future changes.
Moreover, using the measures already mentioned, we are trying to understand whether cer-
tain patterns or information were hidden in the market that could have enabled us to predict
the current crisis that was caused by the coronavirus pandemic, or not. Therefore, in the next
section, we give a brief description of the applied methods and present the empirical results.
For further analysis, the computations in a rolling window algorithm with a length of 100
and a step of 1 for the entire time series and local crashes we made. From the classification
table, we selected 4 crashes with numbers 1, 3, and 7. This choice was due to the fact that the
current crisis is essentially a crash according to our classification, and therefore for convenient
presentation of complexity measures in further, we should select a few of any crashes, the
nature of which in some sense would be similar to the main crash of the presented article. For
the entire Bitcoin time series, each crash and the critical event will be marked by the arrow in
accordance with the table. For local crashes, the beginning of each crash has to be indicated
by our measures in the time of 100.
4. Related methods
Nowadays, the most important quantity that allows us to parameterize complexity in deter-
ministic or random processes is entropy. Originally, it was introduced by Rudolf Clausius [37],
in the context of classical thermodynamics, where according to his definition, entropy tends
to increase within an isolated system, forming the generalized second law of thermodynamics.
Then, the definition of entropy was extended by Boltzmann and Gibbs (BG) [38, 39], linking
it to molecular disorder and chaos, to make it suitable for statistical mechanics, where they
combined the notion of entropy and probability [40].
After the fundamental paper of Claude Elwood Shannon [41] in the context of information
theory, its notion was significantly redefined. After this, it has been evolved along with dif-
ferent ways and successful enough used for the research of economic systems [42, 43, 44]. In
30
what follows, the theoretical background and empirical results for the most popular entropy
measures are presented.
4.1. Shannon entropy
Shannon entropy (ShEn) was proposed as a measure of uncertainty by its author – Claude El-
wood Shannon in his famous paper “A Mathematical Theory of Communication” [41], where,
at first, its purpose was to quantify the degree of ‘lost information’ in phone-line signals. His
contribution has proved that this approach can be generalized for any series where probabil-
ities exist. Comparatively to the entropy of Clausius and Boltzmann that were valid only for
thermodynamic systems, it was significant progress. Formally, his approach can be briefly de-
fined as the average amount of ‘information’ and ‘uncertainty’ encoded in patterns recorded
from a signal, or message. Other interpretations refer to entropy as a measure of ‘chaos’ or dis-
order in a system. During decades the generalization of Shannon’s entropy has been applied
to various domains, particularly to the financial sector.
The general approach can be described as follows. Formally, we represent the underlying
dynamic state of the system in probability distribution form P and then ShEn S with an arbitrary
base (i.e. 2, 𝑒, 10) is defined as:
𝑁
𝑆 [P] = − ∑ 𝑝𝑖 log 𝑝𝑖 , (1)
𝑖=1
where 𝑝𝑖 represents the probability that price 𝑖 occurs in the sample’s distribution of the Bitcoin
time series, and 𝑁 is the total amount of data in our system. When dealing with continuous
probability distributions with a density function 𝑓 (𝑥), we can define the entropy as:
+∞
𝐻 (𝑓 ) = − ∫ 𝑓 (𝑥) log 𝑓 (𝑥). (2)
−∞
According to the approach, the negative log increases with rarer events due to the information
that is encoded in them (i.e. they surprise when they occur). Thus, when all 𝑝𝑖 ’s have the
same value, i.e. where all values are equally probable, and 𝑆 [P] reaches its minimum for more
structured time series (events that are more certain). Equation 2 is obeyed to the same rules as
discrete version of this method. In the figure 1 are the empirical results for ShEn, for the entire
time series (1a), and local crashes (1b).
31
(a) (b)
Figure 1: ShEn dynamics along with the entire time series of Bitcoin (a). The dynamics of ShEn for
the local crashes (b) in accordance with the table 1
4.2. Approximate entropy
To gain more detail analysis of the complex financial systems, it is known other entropy meth-
ods have become known, particularly, ApEn developed by Steve M. Pincus [45] for measuring
regularity in a time series.
When calculating it, given 𝑁 data points {𝑥 (𝑖 ) | 𝑖 = 1, … , 𝑁 } are transformed into subvectors
⃗ (𝑖) ∈ ℜ𝑑𝐸 , where each of those subvectors has [𝑥(𝑖 ), 𝑥(𝑖 + 1), … , 𝑥(𝑖 + 𝑑𝐸 − 1)] for each 𝑖, 1 ≤
X
𝑖 ≤ 𝑁 −𝑚+1. Correspondingly, for further estnimations, we would like to calculate a probability
of finding such patterns whose Chebyshev distance 𝑑[X ⃗ (𝑗)] does not exceed a positive real
⃗ (𝑖), X
number 𝑟:
𝑁 −𝑑𝐸 +1
𝐶𝑖𝑑𝐸 (𝑟) = (𝑁 − 𝑑𝐸 + 1)−1 ∑ (𝑟 − 𝑑[X
⃗ (𝑖), X
⃗ (𝑗)])
𝑗=1
where (⋅) is the Heviside function which count the number of instances 𝑑[X
⃗ (𝑖), X
⃗ (𝑗)] ≤ 𝑟.
Next, we estimate
𝑁 −𝑑𝐸 +1
𝐹 𝑑𝐸 (𝑟) = (𝑁 − 𝑑𝐸 + 1)−1 ∑ ln(𝐶𝑖𝑑𝐸 (𝑟)),
𝑖=1
and ApEn of a corresponding time series (for fixed 𝑑𝐸 and 𝑟) measures the logarithmic likeli-
hood that patterns that are close for 𝑑𝐸 adjacent observations remain close on the next com-
parison:
𝐴𝑝𝐸𝑛(𝑑𝐸 , 𝑟, 𝑁 ) = 𝐹 𝑑𝐸 (𝑟) − 𝐹 𝑑𝐸 +1 (𝑟). (3)
If equation (3) is small, then we should expect high reqularity in our data (sequences remain
close to each other), and extreme value of ApEn indicates independent sequential processes.
The calculation results for the full time series (2a) and the local ApEn values (2b) are pre-
sented in figure 2.
32
(a) (b)
Figure 2: ApEn dynamics along with the entire time series of Bitcoin (a). The dynamics of ApEn for
the local crashes (b) in accordance with the table 1
4.3. Permutation entropy
PEn, according to the previous approach, is a complexity measure that is related to the funda-
mental Information theory and entropy proposed by Shannon. Such a tool was proposed by
C. Bandt and B. Pompe [46], which is characterized by its simplicity, computational speed that
does not require some prior knowledge about the system, strongly describes nonlinear chaotic
regimes. Also, it is characterized by its robustness to noise [47, 48] and invariance to nonlinear
monotonous transformations [49]. The combination of entropy and symbolic dynamics turned
out to be fruitful for analyzing the disorder for the time series of any nature without losing
their temporal information. According to this method, we need to consider “ordinal patterns”
that consider the order among time series and relative amplitude of values instead of individual
values. For evaluating PEn, at first, we need to consider a time series {𝑥(𝑖) | 𝑖 = 1, … , 𝑁 } which
relevant details can be “revealed” in 𝑑𝐸 -dimensional vector
⃗ (𝑖) = [𝑥(𝑖), 𝑥(𝑖 + 𝜏 ), … , 𝑥(𝑖 + (𝑑𝐸 − 1)𝜏 )] ,
X
where 𝑖 = 1, 2, … , 𝑁 − (𝑑𝐸 − 1)𝜏 , and 𝜏 is an embedding delay of our time delayed vector.
After it, we map By the ordinal pattern, related to time 𝑡, we consider the permutation pattern
𝜋𝑙 (𝑡) = (𝑘0 , 𝑘1 , … , 𝑘𝑑𝐸−1 ) where 1 ≤ 𝑙 ≤ 𝑚! if the following condition is satisfied:
𝑥(𝑗 + 𝑘0 𝜏 ) ≤ 𝑥(𝑗 + 𝑘1 𝜏 ) ≤ … ≤ 𝑥(𝑗 + 𝑘𝑑𝐸 −1 𝜏 ). (4)
Then, regarding the probability distribution 𝑃 of each ordinal pattern, we finally define the
normalized permutation entropy as:
− ∑𝑑𝑙=1
𝐸!
𝑝𝑙 ln 𝑝𝑙
𝐸𝑠 [𝑃] = , (5)
ln 𝑑𝐸 !
where 𝑝𝑙 is the relative frequency of each ordinal pattern.
33
The same idea has the permutation entropy. With the much lower entropy value, we get
a more predictable and regular sequence of the data. Therefore, PEn gives a measure of the
departure of the time series from a complete noise and stochastic time series.
There must be predefined appropriate parameters on which PEn relying, namely, the em-
bedding dimension 𝑑𝐸 is a paramount of importance because it determines 𝑑𝐸 possible states
for the appropriate probability distribution. With small values such as 1 or 2, parameter 𝑑𝐸
will not work because there are only few distinct states. Furthermore, for obtaining reliable
statistics and better detecting the dynamic structure of data, 𝑑𝐸 should be relevant to the length
of the time series or less [50]. For our experiments, 𝑑𝐸 ∈ {3, 4} and 𝜏 ∈ {2, 3} indicate the best
results. Hence, in figure 3 we can observe the empirical results for PEn where it serves as
indicator-precursor of the possible unusual states.
(a) (b)
Figure 3: PEn dynamics along with the entire time series of Bitcoin (a). The dynamics of PEn along
with the local crashes (b) in accordance with the table 1
4.4. Recurrence based entropies
The corresponding measure of entropy is related to the recurrence properties that may be
peculiar for the nonlinear complex system. A method that allows us to visualize and understand
the recurrence behavior of the system is the recurrence plot (RP). A recurrence plot reflects
the binary similarities of pairs of vectors in phase space. In fact, we visually represent the
reconstructed 𝑑𝐸 -dimensional phase space of the system according to Takens’ theorem [51] by
a square similarity matrix that can be defined as:
⃗ (𝑖) − X
𝑅𝑖,𝑗 = (𝜖 − ‖X ⃗ (𝑗) ‖),
where 𝑖, 𝑗 = 1, … , 𝑁 ; 𝜖 is a threshold that determines the similarity of two probably neigh-
borhood trajectories, and previously mentioned is a Heaviside function that represents an
answer in binary form; ‖ ‖ is the Chebyshev distance, and 𝑁 is the size of the analyzed data. If
we keep the fixed radius condition and use 𝐿∞ -norm, as a result, the binary matrix captures a
total 𝑁 2 similarity values. Further recurrence plot is the representation of similar vector pairs
that are represented by black dots, whereas cells referring to dissimilar pairs of vectors are
presented by white dots.
34
Such representation of the systems can be, certainly, useful. But it has a limitation. As an
example, with the increasing size of a plot, it is hard to display it graphically as a whole. It
has to be resized, which in turn will lead to new artifacts and distortion of the patterns. These
types of plots may cause incorrect interpretation.
However, to enable an objective assessment, the graphical representation of RP allows us to
derive qualitative characterizations of the dynamical systems within a recurrence plot. For the
quantitative description of the dynamics, the small-scale patterns in the RP can be used, such
as diagonal and vertical lines. The histograms of the lengths of these lines are the base of the
recurrence quantification analysis (RQA) [52, 53, 54].
A large number of different approaches have been developed to obtain as much as possible
information about the nature of the studied phase space. An important class of recurrence
quantifiers is those that try to capture the level of complexity of a signal. In accordance with
this study, the entropy diagonal line histogram (DLEn) is of the greatest interest which uses the
Shannon entropy of the distribution of diagonal lines 𝑃(𝑙) to determine the complexity of the
diagonal structures within the recurrence plot. One of the most know quantitative indicators
of the recurrence analysis can be defined as:
𝑙=𝑙𝑚𝑎𝑥
𝐷𝐿𝐸𝑛 = − ∑ 𝑝(𝑙) log 𝑝(𝑙)
𝑙=𝑙𝑚𝑖𝑛
and
𝑃(𝑙)
𝑝(𝑙) = ,
∑𝑁𝑙=𝑙𝑚𝑖𝑛 𝑃(𝑙)
where 𝑝(𝑙) captures the probability that the diagonal line has the exactly length 𝑙, and 𝐷𝐿𝐸𝑛
reflects the complexity of deterministic structure in the system. Further calculations were pro-
vided and presented in figure 4 for both Bitcoin time series 4a and its several local crashes 4b.
(a) (b)
Figure 4: DLEn dynamics along with the entire time series of Bitcoin (a). The dynamics of DLEn for
the local crashes (b) in accordance with the table 1
However, as follows from the analysis of the entropy indicators, the results may differ for
different data preparation. Thus, further in the paper, we take into account two types of en-
35
tropy based on the general Shannon’s approach: recurrence period density entropy (RPDEn) and
recurrence entropy (RecEn).
The RPDEn is the quantitative measure of the recurrence analysis that is useful for char-
acterizing the periodicity or absolutely random processes in the time series. It is useful for
quantifying the degree of repetitiveness [55, 56]. Considering embedded data points X ⃗ (𝑖) and
suitable threshold 𝜖 in 𝑑𝐸 -dimensional space, we are following forward in time until it has
left this ball of radius 𝜖. Subsequently, the time 𝑗 at which the trajectory first returns to this
ball is recorded, and the time difference 𝑇 of these two states is recorded. The procedure is
repeated for all states of the embedded vector, forming a histogram of recurrence times 𝑅(𝑇 ).
The histogram is then normalized to give the recurrence time probability density:
𝑅(𝑇𝑖 )
𝑃(𝑇𝑖 ) = 𝑇𝑚𝑎𝑥
,
∑𝑖=1 𝑅(𝑇𝑖 )
where 𝑇𝑚𝑎𝑥 = max {𝑇𝑖 }. The normalized entropy of the obtained density can be defined as:
− ∑𝑇𝑖=1
𝑚𝑎𝑥
𝑃(𝑇𝑖 ) ln 𝑃(𝑇𝑖 )
𝑅𝑃𝐷𝐸𝑛 = . (6)
ln 𝑇𝑚𝑎𝑥
In fact, based on the length of the sequences of neighboring points in the phase space: the more
points are neighborhoods, the lower the value of the entropy according to equation (6). The
comparing of RPDEn and the Bitcoin’s critical states can be seen in figure 5.
(a) (b)
Figure 5: RPDEn dynamics along with the entire time series of Bitcoin (a). The dynamics of RPDEn
for the local crashes (b) in accordance with the table 1
However, recent articles [57, 58] present a slightly different technique for calculating recur-
rent entropy using a novel way to extract information from the recurrence matrix. To properly
define it, we need to define the microstates 𝐹 (𝜖) for the RP that are associated with features
of the dynamics of the time series. Selecting the appropriate metric and using the Heaviside
function, we evaluate the matrices of dimension 𝑁 × 𝑁 that are sampled from the RP. The total
number of microstates for a given 𝑁 is 𝑁𝑚𝑠 = 2𝑁 . The microstates are populated by 𝑁 random
2
36
samples obtained from the recurrence matrix such that 𝑁 = ∑𝑁𝑖=1𝑚𝑠 𝑛𝑖 , where 𝑛𝑖 is the number
of times that a microstate 𝑖 is observed.
The probability of occurrence of the related microstate 𝑖 can be obtained as 𝑃𝑖 = 𝑛𝑖 ⋅ (𝑁 ). The
RecEn of the RP associated with the probability distribution of the corresponding microstates
is given by the following equation:
𝑁𝑚𝑠
𝑅𝑒𝑐𝐸𝑛 = ∑ 𝑃𝑖 ln 𝑃𝑖 . (7)
𝑖=1
In figure 6 we can see the performance of RecEn accordingly to the described above method.
(a) (b)
Figure 6: RecEn dynamics along with the entire time series of Bitcoin (a). The dynamics of RecEn for
the local crashes (b) in accordance with the table 1
5. Conclusions
Definitely, the situation with coronavirus is of paramount importance and is of significant dan-
ger. From the literature overview, we have understood that the peak and ultimate duration of
the outbreak is going to be undetermined for a long time. Yesterday‘s, today‘s, and tomorrow‘s
events associated with this pandemic will not disappear without a trace, but will also affect the
fate of both individuals and the States in which we live in the long term.
In order to give reliable, powerful, and simple indicators- precursors that are able to minimize
further losses as a result of critical changes, we addressed the theory of complexity and the
methods of nonlinear dynamics that can identify special trajectories in the complex dynamics
and classify them.
The obtained quantitative methods were applied to classified crashes of the Bitcoin market,
where it was seen that these indicators can be used to protect yourself from the upcoming
critical change. Due to the nature of some crashes and critical events, at any moment there
does not exist such a predictive model that could foreshadow them in advance. Thus, the
most understandable crashes were selected along with the last critical fall that was caused
by COVID-19 due to the desire to understand whether this crash was predictable or not. We
37
tested 6 measures of complexity on the familiar and past crises, along with the latest ongoing
crash, to demonstrate their effectiveness. According to our result, it can be said that general
Shannon entropy might be a robust indicator of crashes and critical events not only in the stock
market but also in the cryptocurrency market, whereas 5 other measures can serve as efficient
harbingers of these events.
Apparently, the impact of the pandemic was reflected in the cryptocurrency market, and
therefore, the beginning of the subsequent crisis could be predicted using the appropriate in-
dicators of the theory of complexity. In our further studies, we are going to continue exploring
and analyzing other methods from the theory of complexity, and particularly, make research on
the fields of artificial intelligence, machine learning, and deep learning [59, 60, 61, 62, 63, 64, 65].
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October 2020 Through 22 October 2020.
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