=Paper=
{{Paper
|id=Vol-2713/paper01
|storemode=property
|title=Recurrence plot-based analysis of financial-economic crashes
|pdfUrl=https://ceur-ws.org/Vol-2713/paper01.pdf
|volume=Vol-2713
|authors=Vladimir Soloviev,Oleksandr Serdiuk,Serhiy Semerikov,Arnold Kiv
|dblpUrl=https://dblp.org/rec/conf/m3e2/SolovievSSK20
}}
==Recurrence plot-based analysis of financial-economic crashes==
https://ceur-ws.org/Vol-2713/paper01.pdf
21
Recurrence plot-based analysis of financial-economic
crashes
Vladimir Soloviev1,2[0000-0002-4945-202X], Oleksandr Serdiuk2[0000-0002-1230-0305],
Serhiy Semerikov1,3,4[0000-0003-0789-0272] and Arnold Kiv5[0000-0002-0991-2343]
1 Kryvyi Rih State Pedagogical University, 54 Gagarin Ave., Kryvyi Rih, 50086, Ukraine
vnsoloviev2016@gmail.com, semerikov@gmail.com
2 The Bohdan Khmelnytsky National University of Cherkasy,
81 Shevchenka Blvd., 18031, Cherkasy, Ukraine
serdyuk@ukr.net
3 Kryvyi Rih National University, 11 Vitalii Matusevych Str., Kryvyi Rih, 50027, Ukraine
4 Institute of Information Technologies and Learning Tools of NAES of Ukraine,
9 M. Berlynskoho Str., Kyiv, 04060, Ukraine
5 Ben-Gurion University of the Negev, P.O.B. 653, Beer Sheva, 8410501, Israel
kiv.arnold20@gmail.com
Abstract. The article considers the possibility of analyzing the dynamics of
changes in the characteristics of time series obtained on the basis of recurrence
plots. The possibility of using the studied indicators to determine the presence of
critical phenomena in economic systems is considered. Based on the analysis of
economic time series of different nature, the suitability of the studied
characteristics for the identification of critical phenomena is assessed. The
description of recurrence diagrams and characteristics of time series that can be
obtained on their basis is given. An analysis of seven characteristics of time
series, including the coefficient of self-similarity, the coefficient of predictability,
entropy, laminarity, is carried out. For the entropy characteristic, several options
for its calculation are considered, each of which allows the one to get its own
information about the state of the economic system. The possibility of using the
studied characteristics as precursors of critical phenomena in economic systems
is analyzed. We have demonstrated that the entropy analysis of financial time
series in phase space reveals the characteristic recurrent properties of complex
systems. The recurrence entropy methodology has several advantages compared
to the traditional recurrence entropy defined in the literature, namely, the correct
evaluation of the chaoticity level of the signal, the weak dependence on
parameters. The characteristics were studied on the basis of daily values of the
Dow Jones index for the period from 1990 to 2019 and daily values of oil prices
for the period from 1987 to 2019. The behavior of recurrence entropy during
critical phenomena in the stock markets of the USA, Germany and France was
studied separately. As a result of the study, it was determined that delay time
measure, determinism and laminarity can be used as indicators of critical
phenomena. It turned out that recurrence entropy, unlike other entropy indicators
of complexity, is an indicator and an early precursor of crisis phenomena. The
ways of further research are outlined.
___________________
Copyright © 2020 for this paper by its authors. Use permitted under Creative Commons License
Attribution 4.0 International (CC BY 4.0).
22
Keywords: complex systems, recurrence entropy, indicator-predictor of
crashes.
1 Introduction
During last few decades the behavior of global financial system attracted considerable
attention. Strong sharp fluctuations in stock prices lead to sudden trend switches in a
number of stocks and continue to have a huge impact on the world economy causing
the instability in it with regard to normal and natural disturbances [18]. The reason of
this problem is the crisis of methodology modeling, forecasting and interpretation of
socio-economic realities. 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 phenomena are applicable. Similar idea has attracted
considerable attention by the community from different branches of science in recent
years [6].
The increasing mathematical knowledge of the complex structures of nonlinear
systems has provided successful tools to the understanding of irregular space and
temporal behaviors displayed by collected data in all applied sciences. Time series
analysis has turned to be a key issue providing the most direct link between nonlinear
dynamics and the real world [9]. Among the many methods of analysis of complex
nonlinear, non-stationary emergent signals, which are the signals of complex systems,
those that adequately reflect the spatial and temporal manifestations of complexity are
especially popular [17]. In this case, the search for such quantitative measures of
complexity that adequately reflect the dynamics of processes taking place in a complex
system is relevant. Financial systems being complex dynamic objects exhibit
unexpected critical phenomena, which are most clearly manifested in the form of
crashes. Over the past 20 years, these are the global currency crisis of 1998, the collapse
of the dotcoms 2001, the global financial crisis of 2008, the European debt crisis of
2012, the Chinese crisis of 2015-2016 and the crisis of the US stock market in early
2019 [16]. For this reason, it is extremely important to highlight such measures of
complexity that are sensitive to critical phenomena and can serve as their predictors [2;
21].
In this paper, we will consider the possibilities of new entropy indicators of the
systems complexity, calculated in the phase space, and examine their capabilities with
respect to the prevention of crisis phenomena.
2 Research methods
2.1 The recurrence plots
In recent years, new quantifiers of nonlinear time series analysis have appeared based
on properties of phasespace recurrences [13]. According to stochastic extensions to
Taken’s embedding theorems the embedding of a time series in phase space can be
carried out by forming time-delayed vectors ⃗ = [ , , ,..., ( ) ] for
23
each value xn in the time series, where m is the embedding dimension, and τ is the
embedding delay. These parameters are obtained by systematic search for the optimal
set. Figure 1 shows a phase portrait of the normalized logarithmic returns of the time
series of Bitcoin (BTC) prices for the period July 17, 2010 to August 30, 2019.
Fig. 1. A phase portrait of the normalized logarithmic returns of the daily values BTC/$ for the
period July 17, 2010 to August 30, 2019.
A modern visualization method known as recurrence plot (RP), and is constructed from
the recurrence matrix ⃗ defined as ⃗ ( ) = − − , ∈ ,
, = 1,2, . . . , , where and represent the dynamical state at time i and j, is the
Heaviside function, M is length of the analyzed time series and is the threshold or
vicinity parameter, consisting of a maximum distance between two points in a trajectory
such that both points can be considered recurrent to each other.
The recurrence plot for the phase portrait of figure 1 is presented in figure 2.
The graphical representation of the RP allows to derive qualitative characterizations
of the dynamical systems. 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)
developed by Webber and Zbilut [20] and later by Marwan et al. [13]. Based on the
statistical properties of the recurrence plot, a large number of quantifiers have been
developed to analyze details of a RP. Many of them deal with statistical properties such
as mean size, maximum size, frequency of occurrence of diagonal, vertical or horizontal
recurrence lines.
The appearance of the recurrence diagram allows us to judge the nature of the
processes occurring in the system, the presence and influence of noise, states of
repetition and fading (laminarity), the implementation during the evolution of the
system of abrupt changes (extreme events). Thus, a visual assessment of the diagrams
can give an idea of the evolution of the studied trajectory. There are two main classes
of image structure: typology, represented by large-scale structures, and texture
24
(texture), formed by small-scale structures. Topology gives a general idea of the nature
of the process. A detailed examination of recurrence diagrams allows you to identify
small-scale structures – a texture that consists of simple points, diagonal, horizontal and
vertical lines. Combinations of vertical and horizontal lines form rectangular clusters
of points.
Fig. 2. Recurrence plot of daily values of BTC/$ price fluctuations.
Webber and Zbilut developed a tool for calculating a number of measures based on the
calculation of the density of recurrence points and the construction of the frequency
distribution of diagonal line lengths: recurrence rate (RR), determinism (DET),
divergence (DIV), entropy (ENTR), trend (TREND), laminarity (LAM), trapping time
(TT). The calculation of these measures in the submatrices of the recurrence chart along
the identity line shows the behavior of these measures over time. Some studies of these
measures have shown that their application can help identify bifurcation points, chaos-
order transitions.
Let Rij=1 if (i, j) are recurrent, otherwise Rij=0; and Dij=1 if (i, j) and (i+1, j+1) or
(i–1, j–1) are recurrent, otherwise Dij=0.
Now the coefficients of self-similarity and predictability will be, respectively, equal
∑,
= ∑, and =∑ .
,
To demonstrate the values of the described indicators, we used 3 time series with
1/f-noise with =0.5, =0.75, =1.0, a time series of sine values and the same series
with mixed values. The time series with 1/f-noise are shown in figure 3.
The figure 4 shows the graphs of the RR values for the demonstration time series.
As can be seen from the figure for more ordered rows (another words the rows with
less noise), the value of RR is higher.
25
Fig. 3. The time series with 1/f-noise with =0.5, =0.75, =1.0, and the time series of sine
values and the mixed sin values.
Fig. 4. Self-similarity (RR) of 1/f-noise and sin time series.
The figure 5 shows the graphs of the DET values for the demonstration time series. As
can be seen from the figure for more ordered rows (another words the rows with less
noise), the value of DET is higher.
If Ni is the number of diagonal lines, and li is the length of the i-th diagonal line, then
the length of the longest diagonal line is determined by the expression
= ( ; = 1, . . . , ).
26
Fig. 5. Determinism (DET) of 1/f-noise and sin time series.
A diagonal line of length l means that the segment of the trajectory is close for l steps
of time to another segment of the trajectory at another time; therefore, these lines are
associated with the divergence of the trajectory segments.
∑
∑ ()
The average length of the diagonal line = ∑ is the average time during
∑ ()
which the two segments of the trajectory are close to each other, and can be interpreted
as the average time of the forecast.
∑
∑ ( )
The average length of vertical structures is given by expression = ∑
∑ ( )
and is called the delay time or capture time. Its calculation also requires consideration
of the minimum length , as in the case of LAM. The TT estimates the average time
that the system will be in a certain state, or how long this state will be captured.
The figures 6 and 7 present graphs of L and TT calculation for the demonstration
time series.
As the measures can refer to the diagonal and/or horizontal lines on the recurrence
map, at the same time, there are vertical lines with appropriate measures.
The total number of vertical length lines in RP is given by the histogram
( )=∑, (1 − , )(1 − , )∏ , .
Similar to the definition of determinism, the ratio of recurrent points that form
vertical structures to a complete set of recurrent points can be calculated as
∑
∑ ( )
= ∑
. This measure is called laminarity (LAM). Laminarity calculations
( )
are performed for those that exceed the minimum length . For recurrence maps
27
often take =2. The value of LAM decreases if RP consists of more single recurrent
points than vertical structures.
Fig. 6. The diagonal lines measure (L) of 1/f-noise and sin time series.
Fig. 7. The delay time measure (TT) of 1/f-noise and sin time series.
An example of LAM calculation for the demonstration time series, is given in figure 8.
In contrast to quantitative measures based on diagonal lines, the measures just
introduced can be applied to chaos-chaos transitions. The last two parameters
characterize two different typical time intervals during which the trajectories are close
28
to . Their window dynamics allows the one to track the time component of recurrence
maps.
Fig. 8. The laminarity (LAM) of daily values of BTC/$ fluctuations and returns of fluctuations.
2.2 The recurrence-based entropy
An important class of recurrence quantifiers are those that try to capture the level of
complexity of a signal. As an example, we mention the already known entropy based
on diagonal lines statistics. This quantity has been correlated with others dynamical
quantifiers as, for example, the largest Lyapunov exponent, since both capture
properties of the complexity level of the dynamics. The vertical (horizontal) lines in
are associated to laminar states, common in intermittent dynamics [13]. It was reported
the use of the distribution of diagonal lines ( ) for a different quantifier of recurrences,
based on the Shannon entropy [13]. If we choose a distribution of diagonals
( ) = ( )/ ∑ ( ) for the maximum length of the diagonal lines, then we get
one of the known quantitative indicators of recurrence analysis:
∑
= −∑ () ( ). However, as follows from the analysis of entropy
indicators, the results are not always possible to coordinate with the proposed models.
To the pleasure of the researchers, it turned out that depending on the technology of
using the properties of the recurrence of the phase space, different types of recurrence
entropies are distinguished [7].
2.3 Recurrence probability (period) density entropy
Recurrence probability (or period) density entropy (RPDEn) is useful for characterizing
the extent to which a time series repeats the same sequence [1; 11; 15] and is, like the
ENTR a quantitative characteristic of recurrence analysis. Around each point xn in the
29
phase space, an -neighbourhood (an m-dimensional ball with this radius) is formed,
and every time the time series returns to this ball, after having left it, the time difference
T between successive returns is recorded in a histogram. This histogram is normalized
to sum to unity, to form an estimate of the recurrence period density function P(T). The
∑
normalized entropy of this density = −∑ ( ) ( ) / ln is the
RPDEn value, where Tmax is the largest recurrence value.
2.4 Recurrence entropy
Recent works [3; 12] presents a slightly different technique for calculating recurrence
entropy using a novel way to extract information from the recurrence matrix. The
authors have generalize these concepts recurrence defining recurrence microstates ( )
as all possible cross-recurrence states among two randomly selected short sequences of
N consecutive points in a K ( ≥ ) length time series, namely ( ) are × small
binary matrices. The total number of microstates for a given N is = 2 . The
microstates are populated by random samples obtained from the recurrence matrix
such that = ∑ , where is the number of times that a microstate i is observed.
For = / , the probability related to the microstate i, we define an entropy of the
RP associated with the probabilities of occurrence of a microstate as
( )=∑ .
3 The recurrence plot-based measures for crash time
series
The behavior of the measures described in section 2.1 was conducted on the basis of
daily data of the Dow Jones Industrial Average (DJIA), taken for the period from 1990
to 2019, and daily values of the price on the spot oil market, taken for the period from
1987 to 2019, in order to assess the dynamics of changes in the values of indicators in
certain critical periods of economic systems, tables of critical and crisis phenomena in
the relevant markets were compiled.
Table 1 lists the critical and crisis phenomena in the DJIA for the period under study.
Calculations of investigating measures of complexity were carried out within the
framework of a moving (sliding) window algorithm. For this purpose, the part of the
time series (window), for which there were measures of complexity, was selected, then
the window was displaced along 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 dynamics of the behavior of complexity with
changes in the time series. If this or that measures of complexity behaves in a definite
way for all periods of crisis, for example, decreases or increases during the pre-crisis
period, then it can serve as an indicator or precursor of such a crisis phenomenon.
30
Table 1. Critical and crisis phenomena in the DJIA for the period from 01.01.1990 to
01.06.2019.
N Time period Duration, days Falling rate, %
1 17.07.1990-23.08.1990 28 17.21
2 01.10.1997-21.10.1997 15 12.43
3 17.08.1998-31.08.1998 11 18.44
4 14.08.2002-01.10.2002 34 19.52
5 16.10.2008-15.12.2008 42 30.21
6 09.08.2011-22.09.2011 32 11.94
7 18.08.2015-25.08.2015 6 10.53
8 29.12.2015-20.01.2016 16 11.02
9 03.12.2018-24.12.2018 15 15.62
In figure 9 presents the results of calculations the measures RR, DET, L, TT, and LAM
for the DJIA values database. The calculations were carried out for a moving window
size of 500 days and a step of 1 day.
Fig. 9. Window recurrence plot-based measures of complexity for the crashes presented in table
1. The start point of the crash is marked.
The figure shows that the value of the measure of self-similarity (RR) in 3 cases out of
9 decreases during the critical phenomenon (events 1, 2, 4, 5), and in 2 cases out of 9
is in the local minimum (events 3, 6). Note that these critical phenomena are quite long
with a length of 11 to 42 days, and also have a large percentage of falls: from 12 to
30%. Phenomena 7, 8, 9, which the indicator did not feel, have a shorter duration and
a lower percentage of fall.
The determinism (DET) measure shows a tendency to fall during all the studied
critical phenomena, which, given the stable result, can serve as an indicator of a critical
31
phenomenon. Moreover, in 3 cases out of 9 during the critical phenomenon, the
indicator is at the local maximum, after which it begins to decrease. Thus, the DET
measure can be an indicator of the beginning of a critical phenomenon.
We have noted the same behavior of the determinism and the laminarity measures,
the form of graphs of the dynamics of which is very similar. Thus, it is enough to choose
only one of the two measures to build economic instruments.
Finally, the form of graphs for the average length of lines L and delay time TT is the
same, which allows using only one of these measures in the research. The values of the
measures decrease during 7 of the 9 studied critical phenomena, and during the
remaining 2 phenomena the values are at the point of local maximum, after which they
begin to fall.
Therefore, based on the analysis of the values of the DJIA index, the following
intermediate conclusions were obtained:
1. Only 3 measures out of the 5 studied can be used in research, because there are
measures with the same form of graphs, due to which only one can be chosen.
2. The most sensitive to critical phenomena are the measures of DET, L, TT, and LAM.
During calculations, we have used a window with a width of 500 points. In order to
determine the degree of influence of the window width on the dynamics of the analyzed
characteristics, we repeated the calculations using a window width of 250 points. In
figure 10 shows graphs of the dynamics of three measures, RR, DET and LAM, obtained
for windows with a width of 500 and 250 points. A comparative analysis of the graphs
led to the conclusion that although they differ to some extent, however, both graphs
retain properties that are essential for our analysis, namely: the starting points of critical
phenomena fall into local extremes or areas of decreasing values. This allows us to
hypothesize the informativeness of the results, the possibility of using different window
widths in the calculation procedure (but not arbitrary width!), and further search for the
optimal value of the parameter to minimize calculations while maintaining the
informativeness of the result.
Often, time series studies also use not the initial data, but the returns, which for the
time series xi, i=1,..,M, are calculated using the expression = . In figure 11
presents the results of calculations the measures RR, DET, L, TT, and LAM for the DJIA
returns database. The calculations were carried out for a moving window size of 500
days and a step of 1 day.
The figure 11 clearly shows approximately the same behavior for all studied
indicators. Note the chaotic behavior, that is, the moments of the beginning of critical
phenomena exist both on the areas of growth of indicators and on the areas of their
decline. Therefore, based on the obtained partial result, we can hypothesize the
impossibility of using returns as indicators of critical phenomena for measures based
on recurrent plots.
In studying the behavior of the measures described above on the spot oil market
prices, critical phenomena were used, the list of which is given in table 2.
32
Fig. 10. Window recurrence plot-based measures of complexity for the crashes presented in
table 1. The window width is 250 in the window moving procedure. The start point of the crash
is marked.
Fig. 11. Window recurrence plot-based measures of complexity for the crashes presented in
table 1 using the returns of DJIA. The start point of the crash is marked.
In figure 12 presents the results of calculations the measures RR, DET, L, TT, and LAM
for the spot oil price values database. The calculations were carried out for a moving
window size of 500 days and a step of 1 day.
33
Table 2. Critical and crisis phenomena on the spot oil market for the period from 01.01.1987 to
01.06.2019.
N Time period Duration, days Falling rate, %
1 09.12.1987-21.12.1987 9 18
2 11.10.1990-23.08.1990 8 31
3 17.11.1993-17.12.1993 22 18
4 11.04.1996-05.06.1996 38 22
5 30.09.1998-25.11.1998 40 33
6 07.03.2000-10.04.2000 24 29
7 27.11.2000-20.12.2000 17 29
8 14.09.2001-24.09.2011 6 27
9 12.03.2003-21.03.2003 7 28
10 26.10.2004-10.12.2004 31 28
11 07.08.2006-17.11.2006 73 27
12 03.07.2008-23.12.2008 120 80
13 03.05.2010-25.052010 16 25
14 29.04.2011-17.05.2011 12 15
15 24.02.2012-28.06.2012 87 27
16 06.09.2013-27.11.2013 58 17
17 20.06.2014-29.01.2015 152 59
18 03.11.2015-20.01.2016 52 44
19 03.10.2018-27.12.2018 56 41
Fig. 12. Window recurrence plot-based measures of complexity for the crashes presented in
table 2. The time series of the spot oil price is used. The start point of the crash is marked.
The figure 12 shows that the value of the measure of self-similarity (RR) in 9 cases out
of 19 decreases during the critical phenomenon, in 5 cases out of 19 is in the local
minimum, and in 2 cases out of 19 is in the local maximum. In other cases, the critical
phenomenon is on the area of growth of the indicator. Interestingly, the falling rate of
critical phenomena, the onset of which goes to the local extremum, is about 30%.
34
The determinism measure and the laminarity measure, as in the case for DJIA time
series, have very similar dynamics. The studied critical phenomena are mainly in the
areas of falling indicators: 14 out of 19. Practically the absence of existence indicators
in local extrema indicates the impossibility of their use as precursors, but only as
indicators of possible critical phenomena.
Measures of average line length (L) and time delay (TT) also coincide in their
dynamics. In 10 cases out of 19 at the time of the critical phenomenon, the indicators
of measures are decreasing, and in 5 cases out of 19 are in the local maximum. Given
that only in one case the indicators of measures are in the local minimum, it is possible
to hypothesize the possibility of interpreting the indicators not only as indicators but
also as precursors with a short prediction horizon.
Therefore, based on the analysis of the values of the spot oil price, the following
intermediate conclusions were obtained:
1. As in previous analysis, the only 3 measures out of the 5 studied can be used in
research.
2. The most sensitive to critical phenomena are the measures of L and TT.
In figure 13 presents the results of calculations the measures RR, DET, L, TT, and LAM
for the spot oil price returns database. The calculations were carried out for a moving
window size of 500 days and a step of 1 day. Like for the DJIA returns, the figure
clearly shows approximately the same behavior for all studied indicators. Note the
chaotic behavior, that is, the moments of the beginning of critical phenomena exist both
on the areas of growth of indicators and on the areas of their decline. Therefore, based
on the obtained result, we confirmed the impossibility of using returns as indicators of
critical phenomena for measures based on recurrence plots.
Fig. 13. Window recurrence plot-based measures of complexity for the crashes presented in
table 2. The time series of the spot oil price returns is used. The start point of the crash is
marked.
35
4 Recurrence entropy for crash time series
To study the recurrence entropy properties of time series, including periods of crisis,
the following databases have been prepared. The first database included fragments of
the DJIA for the famous crashes of 1929, 1987, and 2008. In a number of daily values
of the DJIA index of 2000 days long, the actual day of the onset of the crash falls at
point 1000 (figure 14). In this case, the fixed point of crashes can easily observe the
indicator capabilities of entropy measures of complexity.
Fig. 14. Fragments of DJIA index with crash at 1000 days.
The following database contains the same length daily values (from March 3, 1990 to
August 30, 2019) of the US stock market indices (DJIA), Germany (DAX), France
(CAC), used to check the universality of the complexity measure regardless of the
index. The index DJIA is also taken for the period from January 1, 1983 to August 30,
2019 in order to cover the crises of 1987 and 1998.
The third database includes the values of daily Bitcoin prices for the entire
observation period (from July 17, 2010) and for a shorter period of stabilization of the
cryptocurrency market from January 1, 2013 to August 30, 2019.
Calculations of recurrence entropy measures of complexity were carried out within
the framework of a moving (sliding) window algorithm [5]. If this or that measures of
complexity behaves in a definite way for all periods of crisis, for example, decreases or
increases during the pre-crisis period, then it can serve as an indicator or precursor of
such a crisis phenomenon.
In Figures 15 and 16 presents the results of calculations RPDEn and RecEn for the
first database with a length of 2000 days.
36
Fig. 15. Window recurrence measures of complexity for the crashes of 1929, 1987, and 2008
using RPDEn procedure. The start point of the crash is marked.
Fig. 16. Window recurrence measures of complexity for the crashes of 1929, 1987, and 2008.
a) RPDEn, b) RecEn. The start point of the crash is marked.
The calculations were carried out for a moving window size of 250 days and a step of
1 day. It can be seen from the figure that the recurrence entropy in the pre-crisis period
is markedly reduced for all crisis events, which is obviously a precursor of such crisis
phenomena. As for RPDEn, such an unambiguous precursor is not observed. Therefore,
37
further we focus on the use of RecEn, leaving for the future a more complete study of
RPDEn.
In figure 17 shows the RecEn dynamics for the long index DJIA, which includes the
last seven well-known crashes (shown in the figure).
Fig. 17. Comparative dynamics of index DJIA and recurrence entropy RecEn.
Fig. 18. The dynamics of the stock index S&P 500 and its corresponding and also the indices of
the DAX and CAC of recurrence entropy.
Obviously, in this case, RecEn is the precursor of crash events in all these cases. In
order to once again verify the universality of RecEn as an indicator-precursor of
financial crashes, we examined its dynamics for various stock indices. As an example,
38
the selected indices are the stock markets of the USA (S&P 500), Germany (DAX) and
France (CAC) for a comparable period of time (figure 18).
Finally, the analysis of a very volatile cryptocurrency market for BTC/$ data with
small window values (50 days) also allows us to identify the main crisis falls in this
market (figure 19).
Fig. 19. Comparison of the dynamics of the BTC/$ price with the corresponding recurrence
entropy.
The periodization of Bitcoin crises, which we conducted earlier, indicates that
recurrence entropy in this case is also a harbinger of crisis phenomena.
5 Conclusion
We have analyzed key measures based on recurrence plots that can be used as indicators
and precursors of critical phenomena in complex economic systems. Based on the
analysis, several main measures were identified that showed satisfactory results for
their use in tools to indicate critical phenomena. Such characteristics were, first of all,
the delay time (TT) and the average length of the lines on the recurrence plot (L), and
it was determined that only one of them can be used in studies due to the similarity of
the forms of their graphs. A measure of determinism (DET) or a measure of laminarity
(LAM) can also be used to identify critical phenomena. In the future we can focus on
the study of the behavior of these characteristics in the analysis of complex economic
systems of different nature.
We have demonstrated also that the entropy analysis of financial time series in phase
space reveals the characteristic recurrent properties of complex systems. It turned out
that recurrence entropy, unlike other entropy indicators of complexity, is an indicator
and an early harbinger of crisis phenomena. The recurrence entropy methodology has
39
several advantages compared to the traditional recurrence entropy defined in the
literature, namely, the correct evaluation of the chaoticity level of the signal, the weak
dependence on parameters. In the future, a thorough comparative analysis of the
possibilities of recurrence entropy with other promising types of entropy indicators of
complexity should be carried out [4; 8; 10; 14; 19].
References
1. Arunachalam, S.P., Kapa, S., Mulpuru, S.K., Friedman, P.A., Tolkacheva, E.G.: Rotor pivot
point identification using recurrence period density entropy. In: 54th Annual Rocky
Mountain Bioengineering Symposium, RMBS 2017 and 54th International ISA Biomedical
Sciences Instrumentation Symposium 2017, 2017-March
2. Bielinskyi, A., Soloviev, V., Semerikov, S., Solovieva, V.: Detecting stock crashes using
Levy distribution. CEUR Workshop Proceedings 2422, 420–433 (2019)
3. Corso, G., Prado, T., Lima, G., Lopes, S.: A novel entropy recurrence quantification
analysis. arXiv:1707.00944v1 [stat.OT] (2017)
4. Danylchuk, H., Derbentsev, V., Soloviev, V., Sharapov, A.: Entropy analysis of dynamics
properties of regional stock market. Science and Education a New Dimension. Economics
4(2), 15–19 (2016_
5. Derbentsev, V., Semerikov, S., Serdyuk, O., Solovieva, V., Soloviev, V.: Recurrence based
entropies for sustainability indices. E3S Web of Conferences 166, 13031 (2020).
doi:10.1051/e3sconf/202016613031
6. Diaz, J.F.T.: Evidence of Noisy Chaotic Dynamics in the Returns of Four Dow Jones Stock
Indices. Annual Review of Chaos Theory, Bifurcation and Dynamical System 4, 1–15
(2013)
7. Faure P., Lesne, A.: Estimating Kolmogorov entropy from recurrence plots. In: Webber,
G.C.L., Marwan, N. (eds.) Recurrence Quantification Analysis, pp. 45–63. Springer
International Publishing, Cham (2015)
8. Gu, R.: Multiscale Shannon Entropy and its application in the stock market. Physica A:
Statistical Mechanics and its Applications 484, 215–224 (2017)
9. Kantz, H., Shreiber, T.: Nonlinear time series analysis, 2nd edn. Cambridge University
Press, Cambridge (2004)
10. Lim, J.R.: Rapid Evaluation of Permutation Entropy for Financial Volatility Analysis – A
Novel Hash Function using Feature-Bias Divergence”. Department of Computer Science,
Imperial College of London, London. https://www.doc.ic.ac.uk/teaching/distinguished-
projects/2014/j.lim.pdf (2014). Accessed 17 Aug 2020
11. Little, M.A., McSharry, P.E., Roberts, S.J., Costello, D.A.E., Moroz, I.M.: Exploiting
nonlinear recurrence and fractal scaling properties for voice disorder detection. BioMedical
Engineering OnLine 6, 23 (2007)
12. Lopes, S., Lima, T., Corso, G., Lima, G., Kurths, J.: Parameter-free quantification of
stochastic and chaotic signals. arXiv: 1905.02284v1 [physics.data-au] (2019)
13. Marwan, N., Carmen Romano, M., Thiel, M., Kurths, J.: Recurrence plots for the analysis
of complex systems. Phys. Rep. 438(5–6), 237–329 (2007)
14. Pele, T., Lazar, E., Dufour, A.: Information Entropy and Measures of Market Risk. Entropy
19(5), 1–19 (2017)
15. Rabarimanantsoa, H., Achour, L., Letellier, C., Cuvelier, A., Muir, J.-F.: Recurrence plots
and Shannon entropy for a dynamical analysis of asynchronisms in noninvasive mechanical
ventilation. Chaos 17, 013115 (2007)
40
16. Soloviev, V., Bielinskyi, A., Solovieva, V.: Entropy analysis of crisis phenomena for DJIA
index. CEUR Workshop Proceedings 2393, 434–449 (2019)
17. Soloviev, V.N., Belinskiy, A.: Complex Systems Theory and Crashes of Cryptocurrency
Market. Communications in Computer and Information Science 1007, 276–297 (2019)
18. Sornette, D.: Why Stock Markets Crash: Critical Events in Complex Systems. Princeton
University Press, Princeton (2003)
19. Wang, G.-J., Xie, C., Han, F.: Multi-Scale Approximate Entropy Analysis of Foreign
Exchange Markets Efficiency. Systems Engineering Procedia 3, 201–208 (2012)
20. Webber, C.L., Zbilut, J.P.: Recurrence quantification analysis of nonlinear dynamical
systems. In: Riley, A., Van Orden. G.C. (eds.) Contemporary Nonlinear Methods for the
Behavioral Sciences, pp. 26–94. http://www.nsf.gov/sbe/bcs/pac/nmbs/nmbs.jsp (2005).
Accessed 21 Mar 2017
21. Zhou, R., Cai, R., Tong, G.: Applications of entropy in finance: A review. Entropy 15(11),
4909–4931 (2013)