Cryptocurrencies refer to a type of digital asset that uses distributed ledger, or blockchain technology to enable a secure transaction. Like other financial assets, they show signs of complex systems built from a large number of nonlinearly interacting constituents, which exhibits collective behavior and, due to an exchange of energy or information with the environment, can easily modify its internal structure and patterns of activity. We review the econophysics analysis methods and models adopted in or invented for ifnancial time series and their subtle properties, which are applicable to time series in other disciplines. Quantitative measures of complexity have been proposed, classified, and adapted to the cryptocurrency market. Their behavior in the face of critical events and known cryptocurrency market crashes has been analyzed. It has been shown that most of these measures behave characteristically in the periods preceding the critical event. Therefore, it is possible to build indicators-precursors of crisis phenomena in the cryptocurrency market.
The instability of global financial systems concerning normal and natural disturbances of the modern market and the presence of poorly foreseeable financial crashes indicate, first of all, the crisis of the methodology of modeling, forecasting, and interpretation of modern socio-economic LGOBE (V. N. Soloviev)
https://www.researchgate.net/profile/Andrii-Bielinskyi (A. O. Bielinskyi); http:
CEUR Workshop Proceedings htp:/ceur-ws.org IS N1613-073
CEUR Workshop Proceedings (CEUR-WS.org) 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. Rapidly evolving coronavirus pandemic brings a devastating efect on the entire world and its economy as a whole [1, 2, 3, 4, 5, 6, 7]. Further instability related to COVID-19 will negatively afect 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 [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21]. Despite the complexity of the problem, the results of recent studies indicate that significant success has been achieved within the framework of interdisciplinary approaches, and the theory of self-organization – synergetics [22, 23]. The modern paradigm of synergetics is a complex paradigm associated with the possibility of direct numerical simulation of the processes of complex systems evolution, most of which have a network structure, or one way or another can be reduced to the network. The theory of complex networks studies the characteristics of networks, taking into account not only their topology but also statistical properties, the distribution of weights of individual nodes and edges, the efects of dissemination of information, robustness, etc. [ 1, 2, 3, 4, 24, 25, 26].
Complex systems consist of a plurality of interacting agents possessing the ability to generate new qualities at the level of macroscopic collective behavior, the manifestation of which is the spontaneous formation of noticeable temporal, spatial, or functional structures [27, 28, 29, 30, 31, 32]. As simulation processes, the application of quantitative methods involves measurement procedures, where importance is given to complexity measures. Prigogine notes that the concepts of simplicity and complexity are relativized in the pluralism of the descriptions of languages, which also determines the plurality of approaches to the quantitative description of the complexity phenomenon [5].
Financial markets have been attracting the attention of many scientists like engineers, mathematicians, physicists, and others for the last two decades. Such vast interest transformed into a branch of statistical mechanics – econophysics [33, 34, 30, 31, 32, 35]. Physics, economics, ifnance, sociology, mathematics, engineering, and computer science are fields which, as a result of cross-fertilization, have created the multi-, cross-, and interdisciplinary areas of science and research such as econophysics and sociophysics, thriving in the last two and a half decades. These mixed research fields use knowledge, methodologies, methods, and tools of physics for modeling, explaining and forecasting economic, social phenomena, and processes. Accordingly, econophysics is an interdisciplinary research field, applying theories and methods originally developed by physicists to solve problems in economics, usually those including uncertainty or stochastic processes, nonlinear dynamics, and evolutionary games.
There are deep relationships (as well as crucial diferences) between physics and finance [36] that have inspired generations of physicists as well as economists. In general, physicists apprehend financial markets as complex systems and, as such, they conducted numerous scientific investigations [ 37].
Though statistical physics cannot get along without quantum-mechanical ideas and notions in its fundamentals, the main sphere of its interest is the macroscopic description of systems with a large number of particles, the dynamic behavior of which can’t be brought to microscopic dynamical equations of quantum mechanics figured out for separate particles without the use of respective statistical postulates [38]. During last years an increasing flow of works was observed, in which detailed models of market process participants interactions and quantum-mechanical analogies, notions, and terminology based on methods of describing socio-economic systems are drawn to explain both particular peculiarities of modern market dynamics and economic functioning in whole [39, 40, 41]. Saptsin and Soloviev [42], Soloviev and Saptsin [43] have suggested a new paradigm of complex systems modeling based on the ideas of quantum as well as relativistic mechanics. It has been revealed that the use of quantum-mechanical analogies (such as the uncertainty principle, the notion of the operator, and quantum measurement interpretation) can be applied for describing socio-economic processes.
In this review, we will continue to study Prigogine’s manifestations of the system complexity,
using the current methods of quantitative analysis to determine the appropriate measures
of complexity. The proposed measures of complexity, depending on the methodology and
construction methods, can be divided into the following classes:
(1) informational,
(2) (multi-)fractal,
(
Econophysics, based on a rich arsenal of research on critical phenomena [44], very successfully copes with the description of similar events in economics and finance. These are crises and crashes that are constantly shaking the world economy. The introduced measures of complexity should, to one degree or another, respond to such phenomena.
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 allowed 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.
The cryptocurrency market is a complex, self-organized system, which in most cases can be considered either as a complex network of market agents or as an integrated output signal of this network – a time series, for example, prices of individual cryptocurrency. The research on cryptocurrency price uctuations being carried out internationally is complicated due to the interplay of many factors – including market supply and demand, the US dollar exchange rate, stock market state, the influence of crime, shadow market, and fiat money regulator pressure that introduces a high level of noise into the cryptocurrency data. Moreover, in the cryptocurrency market, to some extent, blockchain technology is tested in general. Hence, the cryptocurrency prices exhibit such complex volatility characteristics as nonlinearity and uncertainty, which are dificult to forecast, and any obtained results are uncertain. Therefore, cryptocurrency price prediction remains a huge challenge [45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59].
As can be seen, markets have seen significant numbers of investors selling of and rebalancing their portfolios with less risky assets. That has been leading to large losses and high volatility, typical of crisis periods. The economy key for preventing such activity may lie in cryptocurrency and constructing efective indicators of possible critical states that will help investors and traders ifll 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 [60]. As was shown by Kristoufek [61], 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 [62, 63, 64]. 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 review we: • present such measures; • study critical and crash phenomena that have taken place in the cryptocurrency market; • try to understand whether crashes and critical events could be identified and predicted by such informative indicators or not.
This review is dedicated to the construction of such indicators based on the theory of complexity. According to our goals and actions, the paper is structured as follows. In Section 2, we present our classification of Bitcoin’s crises for the period from July 16, 2010 to January 21, 2021. In Section 3, we describe the information measures of complexity. In Section 4, we describe the multifractal analysis methodology and its results for the crypto market. Section 5 defines what is chaos-based measures of complexity. In section 6, we deal with the recurrence quantification analysis of critical and crisis phenomena in the cryptocurrency market. Irreversible measure based on permutation patterns is defined in Section 7. Section 8 presents the theory and empirical results on network and multiplex measures of complexity and their robustness for digital currencies. Section 9 defines quantum complexity measures, the features of their manifestation on the crypto market are discussed. Section 10 contains conclusions and some recommendations for further research.
Bitcoin, being the most capitalized cryptocurrency, as a rule, sets and determines the main trends of the crypto market as a whole. Therefore, except for the part of the work where the study of collective processes in the market is carried out, we will use the time series of Bitcoin [65]. From figure 1 it can be seen that at the beginning of its existence, Bitcoin’s dynamic was determined mainly by the processes of the formation of the market as a whole and characterized by high volatility, which, however, was not associated with critical phenomena. Bariviera et al. [66] find that the Hurt exponent changes significantly during the first years of existence of Bitcoin, and now it is less unstable than before. Moreover, they detect more evidence of information since 2014 [67].
Being historically proven, popular, and widely used cryptocurrency for the whole existence of cryptocurrencies in general, Bitcoin began to produce a lot of news and speculation, which began to determine its future life. Similar discussions began to lead to diferent kinds of crashes, critical events, and bubbles, which professional investors and inexperienced users began to fear. Thus, we advanced into action and set the tasks: • classification of such critical events and crashes; • construction of such indicators that will predict crashes, critical events in order to allow investors and ordinary users to work in this market.
Accordingly, during this period in the Bitcoin market, many crashes and critical events shook it. At the moment, there are various research works on what crashes are and how to classify such risk events in the market. The definition of these events still has been debatable. Nevertheless, the proposals of most authors have common elements that allow us to arrive at a consensus. Generally, the market crash is a sudden drastic decline in the prices of financial assets in a specific market [68]. Additionally, the applied model for a specific market takes an important place in the definition of “drastic decline”. For instance, Wang et al. [68] take into account events with a minimum one-day decrease of 5% in the stock returns. These authors [26] identify financial crashes as a decrease of 25% or less of multi-year financial returns. Lleo and Ziemba [69] define a crash as a specific event of a day, which decreasing closing price exceeds a fall of 10% between the highest and the lowest value of the stock index in a year. Hong and Stein [70] postulate that the market crash is an unexpected event in which appearance was not accompanied by any financial news. Moreover, the price change during this event is rather negative. Also, it is worth mentioning the study of Shu and Zhu [71] where their classification of crashes included almost 50 crashes. It remains a little unclear which factors influence their choice of such an enormous amount of crashes in such a short period. Researchers emphasize these drops as such, with a fall of more than 15% and a duration of fewer than three weeks. Nevertheless, regarding this classification, we are going to emphasize the most relevant, where the complexity of the index started to decrease and whose initial deviation from regular behavior was noticeable in advance. Nowadays some people proclaim Bitcoin as a “digital gold”. Gold as a material has served for jewelry and art as well as electronic or medical components. Limited supply and current acceptance of Bitcoin as a “digital gold” may erect it to the same level as gold. While some people back up Bitcoin‘s advantage, demonstrating its similarities with those of gold and silver [72], others argue that it is the new digital coin [73] due to its high volatility and unclear future development. However, researchers find its potential benefits during extreme market periods and provide a set of stylized factors that claim to be successful long-short strategies that generate sizable and statistically significant excess returns [ 74]. Despite volatile swings and many critics, Bitcoin has emerged and attracted much more confidence. Kristoufek [75], Li and Wang [76] consider that measures of financial and macroeconomic activity can be drivers of Bitcoin returns. Reviewing papers of the researches above, the experience of others and our own [77, 78, 79, 80, 81, 82, 83, 84, 85, 86], we have revised our classification of such leaps and falls, relying on Bitcoin time series during the entire period (01.01.2011–21.01.2021) of verifiable ifxed daily values of the Bitcoin price (BTC) ( https://finance.yahoo.com/cryptocurrencies). We emphasize that • crashes are short and time-localized drops that last approximately two weeks, with the weighty losing of price each day. Their volatility is high. In percentage term, their decline exceeds 30 percent, and normalized returns proceed ±3 or near to it; • critical events are those falls that, during their existence, have not had such massive changes in price as crashes.
Relying on these considerations, we emphasize 29 periods on Bitcoin time series, relying on normalized returns and volatility, where returns are calculated as
() = ln ( + Δ) − ln () ≅ [( + Δ) − ()]/() and normalized (standardized) returns as where ≡ √⟨ 2⟩ − ⟨⟩ 2 is the standard deviation of , Δ is time lag (in our case Δ = 1 ), and ⟨ … ⟩ denotes the average over the time period under study and volatility as
From the mentioned stylized facts on BTC dynamics, it was noticed how considerably it started to change near 2014. To gain a deeper understanding of its existence in the starting g() ≅ [() − ⟨⟩ ]/ , () = 1 +−1
∑ |g( ′)| ′= period, we divided the BTC time series into two periods: (01.01.2011-31.08.2016) and (01.09.201621.01.2021). More detailed information about crises, crashes, and their classification under these definitions is given in table 1 and table 2.
Therefore, according to our classification crisis periods with numbers (
Figure 2 confirms the importance of dividing the BTC time series in order to observe its dynamics in more detail. However, as it can be seen, we could separate time series in much deeper time scales.
In figure 3 output Bitcoin time series for the first and the second periods, their normalized returns g(), and volatility () calculated for the window of size 100 are presented.
From figure 3 we can see that during periods of crashes and critical events normalized returns g increases considerably in some cases beyond the limits ±3 . This indicates deviation from the normal law of distribution, the presence of the “heavy tails” in the distribution g, which are characteristics of abnormal phenomena in the market. At the same time volatility also grows. Such qualities are the foundation of our classification for crashes, as it has been mentioned already. All the rest events are critical. These characteristics serve as indicators of crashes and critical events as they react only at the moment of the above-mentioned phenomena and do not allow identifying the corresponding abnormal phenomena in advance. In contrast, most of the indicators described below will respond to critical changes and crashes in advance. It enables them to be used as indicators – precursors of such phenomena.
Calculations were carried out within the framework of the algorithm of a rolling (sliding, moving) window. For this purpose, the part of the time series (window), for which there were calculated measures of complexity, was selected, then the window was displaced along with the time series in a predefined value, 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 cryptocurrency. If this or that measure of complexity behaves in a definite way for all periods of crashes, for example, decreases or increases during the pre-crashes or pre-critical period, then it can serve as their indicator or precursor.
Calculations of measures of complexity were carried out both for the entire time series, and for a fragment of the time series localizing some of the emphasized crashes and critical events. In the latter case, fragments of time series of the same length with fixed points of the onset of crashes or critical events were selected and the results of calculations of complexity measures were compared to verify the universality of the indicators. Following some described below procedures such time localization as, example, of length 100 or 200, either won‘t make any sense, or won‘t be possible as some of them are sensitive to time localization, or require a longer length of the time series as it is required by the procedure for better accuracy of further calculations
Complexity is a multifaceted concept, related to the degree of organization of systems. Patterns of complex organization and behavior are identified in all kinds of systems in nature and technology. Essential for the characterization of complexity is its quantification, the introduction of complexity measures, or descriptors [87].
We may speak of the complexity of a structure, meaning the amount of information (number of bits) of the structure; this is the minimum space we need to store enough information about the structure that allows us its reconstruction. We may also speak of the algorithmic complexity of a certain task: this is the minimum time (or other computational resources) needed to carry out this task on a computer. And we may also speak of the communication complexity of tasks involving more than one processor: this is the number of bits that have to be transmitted in solving this task [88, 89, 90].
Historically, the first attempt to quantify complexity was based on Shannon’s information theory [91] and Kolmogorov complexity [92].
Lempel-Ziv complexity (LZC) is a classical measure that, for ergodic sources, relates the concepts of complexity (in the Kolmogorov-Chaitin sense), and entropy rate [93, 94]. For an ergodic dynamical process, the amount of new information gained per unit of time (entropy rate) can be estimated by measuring the capacity of this source to generate new patterns (LZC). Because of the simplicity of the LZC method, the entropy rate can be estimated from a single discrete sequence of measurements with a low computational cost [95].
In this paper, we show that the LZC measure can be just such a measure of complexity, which is an early precursor of crisis phenomena in the cryptocurrency market [96, 97, 2, 80].
Historically, the first LZC measure system studies for financial time series were conducted by Da Silva [98], Da Silva et al. [99], Giglio and Da Silva [100], Giglio et al. [97]. They considered the deviation of LZC from that value for a random time series as a measure of actual market eficiency in absolute [ 96, 99, 100, 97] or relative [98] terms. Using this approach Da Silva et al. [99] were able to detect decreases in eficiency rates of the major stocks listed on the Sao Paulo Stock Exchange in the aftermath of the 2008 financial crisis. Lempel and Ziv [101] have surveyed the principal applications of algorithmic (Kolmogorov) complexity to the problem of financial price motions and showed the relevance of the algorithmic framework to structure tracking in finance. Some empirical results are also provided to illustrate the power of the proposed estimators to take into account patterns in stock returns. Brandouy et al. [102] proposed a generic methodology to estimate the Kolmogorov complexity of financial returns. Examples are given with simulated data that illustrate the advantages of our algorithmic method: among others, some regularities that cannot be detected with statistical methods can be revealed by compression tools. Applying compression algorithms to daily returns of the Dow Jones Industrial Average (DJIA), the authors concluded on an extremely high Kolmogorov complexity and by doing so, proposed another empirical observation supporting the impossibility to outperform the market. The structural complexity of time series describing returns on New York’s and Warsaw’s stock exchanges was studied using two estimates of the Shannon entropy rate based on the Lepel-Ziv and Context Tree Weighting algorithms [103]. Such structural complexity of the time series can be used as a measure of the internal (modelless) predictability of the main pricing processes and testing the hypothesis of an eficient market. Somewhat surprisingly, the results of Gao et al. [104], in which the authors computed the LZC from two composite stock indices, the Shanghai stock exchange composite index (SSE) and the DJIA, for both lowfrequency (daily) and high-frequency (minute-to-minute) stock index data. The calculation results indicate that that the US market is basically fully random and consistent with the eficient market hypothesis (EMH), irrespective of whether low- or high-frequency stock index data are used. The Chinese market is also largely consistent with the EMH when low-frequency data are used. However, a completely diferent picture emerges when the high-frequency stock index data are used. Cao and Li [105] presents a novel method for measuring the complexity of a time series by unraveling a chaotic attractor modeled on complex networks. The complexity index, which can potentially be exploited for prediction, has a similar meaning to the LZC and is an appropriate measure of a series’ complexity. The proposed method is used to research the complexity of the world’s major capital markets. The almost absent sensitivity of the LZC to lfuctuations in the time series indicates most likely errors in the calculation algorithm during the transformation of the time series. The complexity–entropy causality plane is employed in order to explore disorder and complexity in the space of cryptocurrencies [105]. They are found to exist in distinct planar locations in the representation space, ranging from structured to stochastic-like behavior.
A brief analysis of the problem indicates that so far, the Lempel-Ziv informational measure of the complexity has not been used to study the stability and behavior of the cryptocurrency market in a crisis. In this section, we use the Lempel-Ziv complexity measure to study the cryptocurrency market. Using the example of the most capitalized cryptocurrency – Bitcoin – we demonstrate the ability to identify the dynamics of varying complexity. Particularly relevant is the identification of the characteristic behavior of Bitcoin during the crisis phases of market behavior. By observing the dynamics of the Lempel-Ziv measure, precursors of crisis phenomena can be constructed [106].
Let us begin with the well-known degree of complexity proposed by Kolmogorov [107]. The concept of Kolmogorov complexity (or, as they say, algorithmic entropy) emerged in the 1960s at the intersection of algorithm theory, information theory, and probability theory. A. Kolmogorov’s idea was to measure the amount of information contained in individual finite objects (rather than random variables, as in the Shannon theory of information). It turned out to be possible (though only to a limited extent). A. Kolmogorov proposed to measure the amount of information in finite objects using algorithm theory, defining the complexity of an object as the minimum length of the program that generates that object. This definition is the basis of algorithmic information theory as well as algorithmic probability theory: an object is considered random if its complexity is close to maximum.
What is the Kolmogorov complexity and how to measure it? In practice, we often encounter programs that compress files (to save space in the archive). The most common are called zip, gzip, compress, rar, arj, and others. Applying such a program to some file (with text, data, program), we get its compressed version (which is usually shorter than the original file). After that, you can restore the original file using the paired program “decompressor”. Therefore, approximately, the Kolmogorov complexity of a file can be described as the length of its compressed version. Thus, a file that has a regular structure and is well compressed has a small Kolmogorov complexity (compared to its length). On the contrary, a badly compressed file has a complexity close to its length.
Suppose we have a fixed method of description (decompressor) . For this word , we consider all its descriptions, i.e., all words for which ( ) it is defined and equal to . The length of the shortest of them is called the Kolmogorov complexity of the word in this way of description : () = min{( ) | ( ) = } where ( ) denotes the length of the word. The index emphasizes that the definition depends on the chosen method . It can be shown that there are optimal methods of description. The better the description method, the shorter it is. Therefore, it is natural to make the following definition: the method 1 is no worse than the method 2 if
1 ≤ 2() + for some and all .
Thus, according to Kolmogorov, the complexity of an object (for example, the text is a sequence of characters) is the length of the minimum program that outputs the text, and entropy is the complexity that is divided by the length of the text. Unfortunately, this definition is purely speculative. There is no reliable way of identifying this program uniquely, but there are algorithms that are actually just trying to calculate the Kolmogorov complexity of text and entropy. that gives rise to it.
A universal (in the sense of applicability to diferent language systems) measure of the complexity of the finite character sequence was suggested by Lempel and Ziv [101]. As part of their approach, the complexity of a sequence is estimated by the number of steps in the process
Acceptable (editorial) operations are: a) character generation (required at least for the synthesis of alphabet elements) and b) copying the “finished” fragment from the prehistory (i.e. from the already synthesized part of the text).
Let be Σ a complete alphabet, – text (a sequence of characters) composed of elements Σ; [] – ℎ text symbol; [ ∶ ]
– a snippet of text from the ℎ to ℎ character inclusive ( < ) ; = || – length of text . Then the sequence synthesis scheme can be represented as a concatenation () = [1 ∶ 1][ 1 + 1 ∶ 2] … [ −1 + 1 ∶ ] … [ −1 + 1 ∶ ], where [ −1 + 1 ∶ ]is the fragment generated at the ℎ step, and = () is the number of process steps. Of all the schemes of generation is chosen the minimum number of steps. Thus, the Lempel-Ziv complexity of the sequence is () = min{ ()}.
The minimum number of steps is provided by the choice to copy at each step the longest prototype from the prehistory. If you mark by the position number () from which the copying begins in step the length of the copy fragment
≤ −1 () = − −1 − 1 = max { ∶ [ −1 + 1 ∶ −1 + ] = [ ∶ + − 1]} and the ℎ component of these complex decomposition can be written in the form [ −1 + 1 ∶ ] ={ [() ∶ () + () − 1] if () ≠ 0, [ −1 + 1]
if () = 0.
The case () = 0 corresponds to a situation where a symbol is in the position −1 + 1 that has not been encountered previously. In doing so, we use a character generation operation.
Complex text analysis can be performed in two regimes – segmentation and fragmentation. The first regime is discussed above. It gives an integrated view of the structure of the sequence as a whole and reduces it to disjoint but interconnected segments (without spaces). The other regime is to search for individual fragments characterized by an abnormally low complexity which means that they characterized by a suficiently high degree of structure. Such fragments are detected by calculating local complexity within variable-length windows that slide along a sequence. Curves of change of local complexity along a sequence are called complex profiles. A set of profiles for diferent window sizes reveals the boundaries of anomalous fragments and their relationship.
We will find the LZC complexity for the time series, which is, for example, the daily values of the cryptocurrency price () . To investigate the dynamics of LZC and compare it with cryptocurrency prices, we will find this measure of complexity for a fixed length (window) contract. To do this, we calculate the logarithmic returns accordingly to equation (1) and turn them into a sequence of bits.
You can specify the number of states that are diferentiated (calculus system). Yes, for two diferent states we have 0, 1, for three – 0, 1, 2, etc. In the case of three states, unlike the binary coding system, a certain threshold is set and the states g are coded as follows [99, 100, 97]: g = 1 if − ≤ g ≤ ,
0 if g < − , ⎧ ⎨ ⎩2 if g > .
The algorithm performs two operations: (1) adds a new bit to an already existing sequence; (2) copies the already formed sequence. Algorithmic complexity is the number of such operations required to form a given sequence.
For a random sequence of lengths , the algorithmic complexity is calculated by expression
= / log . Then, the relative algorithmic complexity is the ratio of the obtained complexity to the complexity of the random sequence = / .
Obviously, the classical indicators of algorithmic complexity are unacceptable and lead to erroneous conclusions. To overcome such dificulties, multiscale methods are used.
The idea of this group of methods includes two consecutive procedures: 1) coarse-graining
(“granulation”) of the initial time series – the averaging of data on non-intersecting segments,
the size of which (the window of averaging) increased by one when switching to the next
largest scale; 2) computing at each of the scales a definite (still mono scale) complexity indicator.
The process of “rough splitting” consists in the averaging of series sequences in a series of
non-intersecting windows, and the size of which – increases in the transition from scale to scale
[108]. Each element of the “granular” time series follows the expression:
=
1
∑
=(−1)+1
g(), for 1 ≤ ≤ / ,
(
The coarse graining procedure for scales 2 and 3 is shown in figure 4.
To find the LZC measure of the time series, the rolling time windows were considered; the index for every window was calculated, and then the average was obtained.
Obviously, the crisis in the cryptocurrency market responds to noticeable fluctuations in standardized returns. Therefore, it is logical to choose as the value for the threshold value .
Figure 5 shows the dependence of the LZC on the scale. The absence of LZC fluctuations at scales exceeding 40 allows us to confine ourselves to this magnitude of the scale when calculating the multiscale measure.
Calculations of measures of complexity were carried for the two periods of BTC. Figure 6 presents the results of calculations of mono-( 1 ) and multi-( 40 ) scaling LZC measures. The calculations were performed for a rolling window of 100 days and an increment of 1 day.
The data in figure 6 indicate that the LZC measure is noticeably reduced both in the case of mono-scale ( 1) and averaged over the scales from 1 to 40 ( 40) for all 29 crashes and critical events in the immediate vicinity of the crisis point.
As the results of calculations showed, the choice of the size of a rolling window is important: in the case of large windows, points of crises of diferent times can fall into the window, distorting the influence of each of the crises. When choosing small windows, the results fluctuate greatly, which makes it dificult to determine the actual point of the crisis. The used window length of 100 days turned out to be optimal for the separation of crises and fixing the LZC measure as an indicator.
Since the LZC measure begins to decrease even before the actual crisis point, it can be called an indicator-precursor of crisis phenomena in the cryptocurrency market.
Nowadays, the most important quantity that allows us to parameterize complexity in deterministic or random processes is entropy. Originally, it was introduced by R. Clausius [109], 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 [110], Gibbs [111], linking it to molecular disorder and chaos to make it suitable for statistical mechanics, where they combined the notion of entropy and probability [112].
After the fundamental paper of Shannon [91] in the context of information theory, where entropy denoted the average amount of information contained in the message, its notion was significantly redefined. After this, it has been evolved along with diferent ways and successful enough used for the research of economic systems [113, 114, 115, 116].
A huge amount of diferent methods, as an example, from the theory of complexity, the purpose 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.
The existence of patterns within the series is the core in the definition of randomness, so it is appropriate to establish such methods that will be based on the diferent patterns and their repetition [117]. In this regard, Pincus [118] described the methodology Approximate entropy (ApEn) to gain more detail analysis of relatively short and noisy time series, particularly, of clinical and psychological. Its development was motivated by the length constraints of biological data. Since then it has been used in diferent fields such as psychology [ 119], psychiatry [120], and finance [ 121, 122, 123, 124, 125]. Pincus and Kalman [125] considering both empirical data and models, including composite indices, individual stock prices, the random-walk hypothesis, Black-Sholes, and fractional Brownian motion models to demonstrate the benefits of ApEn 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. Duan and Stanley [126] showed that it is possible to efectively distinguish the real-world financial time series from random-walk processes by examining changing patterns of volatility, approximate entropy, and the Hurst exponent. The empirical results prove that financial time series are predictable to some extent and ApEn is a good indicator to characterize the predictable degree of financial time series. Delgado-Bonal [127] gives evidence of the usefulness of ApEn. The researcher quantifies the existence of patterns in evolving data series. In general, his results present that degree of predictability increases in times of crisis.
Permutation entropy (PEn), according to the previous approach, is a complexity measure that is related to the original Shannon entropy (ShEn) that applied to the distribution of ordinal patterns in time series. Such a tool was proposed by Bandt and Pompe [128], 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 [129, 130] and invariance to nonlinear monotonous transformations [131]. 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.
As an example, Henry and Judge [132] applied PEn to the Dow Jones Industrial Average (DJIA) 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. Sigaki et al. [133] applied PEn and statistical complexity over sliding time-window of daily closing price log-returns to quantify the dynamic eficiency of more than four hundred cryptocurrencies. Authors address to the eficient market hypothesis when the values of two statistical measures within a time-window cannot be distinguished from those obtained by chance. They found that 37% of the cryptocurrencies in their study stayed eficient over 80% of the time, whereas 20% were informationally ineficient in less than 20% of the time. Moreover, the market capitalization was not correlated with their eficiency. Performed analysis of information eficiency over time reveals that diferent currencies with similar temporal patterns formed four clusters, and it was seen that more young currencies tend to follow the trend of the most leading currencies. Sensoy [134] compared the time-varying weak-form eficiency of Bitcoin prices in terms of US dollars (BTC/USD) and euro (BTC/EUR) at a high-frequency level by PEn. He noticed that BTC/USD and BTCEUR have become more informationally useful since the beginning of 2016, namely Bitcoin in the same period. Researcher also found that with higher frequency in the Bitcoin market, we had lower price eficiency. Moreover, cryptocurrency liquidity (volatility) had a significant positive (negative) efect on the informational eficiency of its price.
Also, researh by Metin Karakaş [135] is dedicated both to Bitcoin and Ethereum. Here, the concept of entropy was applied for characterizing the nonlinear properties of the cryptocurrencies. For her goal, Shannon, Tsallis, Rényi, and Approximate entropies were estimated. From her empirical results, it was obtained that all entropies were positive. Of great interest was the results of ApEn which demonstrated larger value for Ethereum than for Bitcoin. In this case, it concluded that Ethereum had higher volatility.
Pele and Mazurencu-Marinescu-Pele [136] investigated 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 estimated 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 showed that entropy had strong explanatory power for the quantiles of the distribution of the daily returns. They confirmed the hypothesis that there was a strong correlation between the daily logarithmic price of Bitcoin and the entropy of intraday returns based on Christofersen’s tests for Value at Risk (VaR) backtesting, they concluded that the VaR forecast built upon the entropy of intraday returns was the best, compared to the forecasts provided by the classical GARCH models.
The state of the system can be described by the set of variables. Its observational state can be expressed through a -dimensional vector or matrix, where each of its components refers to a single variable that represents a property of the system. After a while, the variables change, resulting in diferent system states.
Usually, not all relevant variables can be captured from our observations. Often, only a single variable may be observed. Thakens’ theorem [137] that was mentioned in previous sections ensures that it‘s possible to reconstruct the topological structure of the trajectory formed by the state vectors as the data collected for this single variable contains information about the dynamics of the whole system.
For an approximate reconstruction of the original dynamics of the observed system, we project the time series onto a Reconstructed Phase Space [138, 131, 139] with the commonly used time delay method [131] which relied on the embedding dimension and time delay.
The embedding dimension is being the dimensionality of the reconstructed system (corresponds to the number of relevant variables that may difer from one system to another. The time delay parameter specifies the temporal components of the vector components. As an example, in recurrence analysis that will be described in section 6, Webber, Jr. and Zbilut [140] recommend setting the embedding dimension between 10 and 20. Regarding the analysis of ifnancial systems, values between 1 and 20 for the embedding dimension are considered to be reasonable as well as the time delay.
The general approach can be described as follows. Formally, we represent the underlying dynamic state of the system in probability distribution form and then the Shannon entropy with an arbitrary base (i.e. 2, e, 10) is defined as: (4)
Here, in equation 4, represents the probability that price occurs in the sample’s distribution of the Bitcoin time series, and is the total amount of data in our system. Dealing with continuous probability distributions with a density function () , we can define the entropy as: +∞ ( ) = − ∫ () log (). (5) −∞
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 have the same value, i.e. where all values are equally probable, and [ ] reaches its minimum for more structured time series (events that are more certain). Equation 5 is obeyed to the same rules as 4. In figure 7 are the empirical results for Shannon entropy and Bitcoin time series.
It can be seen from the figure that Shannon’s entropy is rapidly increasing at the very moment of the crisis itself and is an excellent indicator of crisis phenomena.
To gain more detail analysis of the complex financial systems, it is known other entropy methods have become known, particularly, ApEn developed by Pincus [118] for measuring regularity in a time series.
When calculating it, given data points { ( )| = 1, … , } are transformed into subvectors X⃗ ( ) ∈ ℜ , where each of those subvectors has [( ), ( + 1), … , ( + − 1)]for each , 1 ≤ . Correspondingly, for further estnimations, we would like to calculate a probability of finding such patterns whose Chebyshev distance [ X⃗(), X⃗()]does not exceed a positive real ℋ ( − [ X⃗(), X⃗()]) where ℋ (⋅)is the Heviside function which count the number of instances [ X⃗ (), X⃗()] ≤ . Next, we estimate ( ) = ( − + 1)−1 ln( and ApEn of a corresponding time series (for fixed and ) measures the logarithmic likelihood that patterns that are close for adjacent observations remain close on the next comparison: − +1 ∑ =1 − +1 ∑ =1 ( , , ) = ( ) − +1( ), (6) i.e., equation (6) measures the logarithmic likelihood that sequences of patterns that are close for observations will remain close after further comparisons. Therefore, if the patterns in the sequence remain close to each other (high regularity), the ApEn becomes small, and hence, the time series data has a lower degree of randomness. High values of ApEn indicate randomness and unpredictability. But it should be considered that ApEn results are not always consistent, thus it depends on the value of and the length of the data series. However, it remains insensitive to noise of magnitude if the values of and are suficiently good, and it is robust to artifacts and outliers. Although ApEn remains usable without any models, it also fits naturally into a classical probability and statistics frameworks, and, generally, despite its shortcomings, it is still the applicable indicator of system stability, which significantly increased values may prognosticate the upcoming changes in the dynamics of the data.
The empirical results for the corresponding measure of entropy along with two periods of BTC are presented in figure 8.
Long before the crisis, the value of this type of entropy begins to decrease, the complexity of the system decreases. This measure, in our opinion, is one of the earliest precursors of the crisis.
PEn, according to the previous approach, is a complexity measure that is related to the
fundamental Information theory and entropy proposed by Shannon. Such a tool was proposed
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