This article demonstrates the possibility of constructing indicators of critical and crisis phenomena in the volatile market of cryptocurrency. For this purpose, the methods of the theory of complex systems such as recurrent analysis of dynamic systems and the calculation of permutation entropy are used. It is shown that it is possible to construct dynamic measures of complexity, both recurrent and entropy, which behave in a proper way during actual pre-crisis periods. This fact is used to build predictors of crisis phenomena on the example of the main five crises recorded in the time series of the key cryptocurrency bitcoin, the effectiveness of the proposed indicators-precursors of crises has been identified.
Bitcoin is an important electronic and decentralized cryptographic currency system
proposed by Satoshi Nakamoto as the “greatest technological breakthrough since the
Internet” [
The bitcoin has emerged as a fascinating phenomenon in the financial markets. Without any central authority issuing the currency, the bitcoin has been associated with controversy ever since its popularity, accompanied by increased public interest, reached high levels.
Despite an influx of media buzz and venture capital, digital currencies face an uncertain future amid an ever-changing global landscape. Investment requires careful consideration of the potential use cases and risks associated with various cryptocurrencies.
A look back at bitcoin price swings in the last five years, which include several stomach-churning tumbles of 40% and even 50%, makes it clear the world’s most popular cryptocurrency was—and is—extremely volatile. It is also apparent that most of the bitcoin crashes coincide with speculative run-ups coupled with exogenous shocks, such as a major hack or a government crackdown. Also, in most cases, bitcoin has bounced back from the crashes in months or even weeks—suggesting nervous bitcoin buyers will be okay if they are holding for the long run. On the other hand, the crashes of late 2013 and early 2014 are a cautionary tale—recall it took years for those who first bought bitcoin at $1,000 to see their investment recover.
Bitсoin attracts considerable attention of researchers of different levels, using modern methods and models of analysis of the peculiarities of the dynamics of the popular digital currency.
The authors [
A noncausal autoregressive process with Cauchy errors in application to the
exchange rates of the bitcoin electronic currency introduced in [
Taking Bitcoin as a representative example, the authors [
In the work [
It is analyzed the time-varying behavior of long memory of returns on bitcoin and
volatility 2011 until 2017, using the Hurst exponent [
The research [
The bitcoin price was modeled as a geometric fBm, and price predictions were put
forward through a Monte Carlo approach with 104 realisations [
In the paper [
The model proposed is able to reproduce some of the real statistical properties of the price absolute returns observed in the bitcoin real market. In particular, it is able to reproduce the autocorrelation of the absolute returns, and their cumulative distribution function. The simulator has been implemented using object-oriented technology, and could be considered a valid starting point to study and analyse the cryptocurrency market and its future evolutions.
Authors [
During the last two decades, a number of interesting methods have been proposed
to detect dynamical changes. They include, among others, recurrence plots and
recurrence quantification analysis [
Recurrence plots and recurrence quantification analysis
Recurrence plots (RPs) have been introduced to study the dynamics of complex
systems that is represented in an m-dimensional phase space by its phase space trajectory
Xi ∈ Rm (assuming discrete sampling, i = 1, ..., N) [
Xi = (ui, ui+1, ..., ui+(m–1)τ),
(1)
with m the embedding dimension and τ the embedding delay. Both parameters can be
estimated from the original data using false nearest neighbors and mutual information
[
A Recurrence Plot is a 2-dimensional representation of those times when the phase space trajectory Xi recurs. As soon as a dynamical state at time j comes close to a previous (or future) state at time i, the recurrence matrix R at (i, j) has an entry one: Rij = Θ(ε - x- i x j ), i,=j 1,..., N , where || || is a norm (representing the spatial distance between the states at times i and j), ε is a predefined recurrence threshold, and Θ is the Heaviside function (ensuring a binary R).
The RP has a square form and usually the identity Rij ≡ 1 is included in the
graphical representation, although for calculations it might be useful to remove it [
The simplest measure of RQA is the density of recurrence points in the RP, the recurrence rate: that can be interpreted as the probability that any state of the system will recur.
The fraction of recurrence points that form diagonal lines of minimal length µ is the determinism measure: where
DET (μ ) = ∑ N l=μ l ⋅ D(l) ∑ N i, j Ri, j = ∑ N
l=μ l ⋅ D(l) ∑lN=1 l ⋅ D(l) (2) (3) (4)
N D(l) = ∑ (1- Ri- 1,- j 1⋅) (1i, j
l- 1 R+i 1+, j⋅1 ) ∏ R+i k+, j k k =0 is the histogram of the lengths of the diagonal lines. The understanding of ‘determinism’ in this sense is of heuristic nature. 3
The PEn is conceptually simple, computationally very fast and can be effectively used to detect dynamical changes in complex time series.
The degree of disorder or uncertainty in a system can be quantified by a measure of entropy. The uncertainty associated with a physical process described by the probability distribution is related to the Shannon entropy,
P = {pi, i = 1, ..., M}
M S[P] = - ∑ pi ln pi .
i=1
Constructing probability distributions using ordinal patterns from recorded time
series was proposed by Bandt and Pompe [
To obtain the ordinal pattern distribution on which to calculate entropy, one must first choose an appropriate ordinal pattern length D and ordinal pattern delay τ. There are D! possible permutations for a vector of length D, so in order to obtain reliable statistics the length of the time series N should be much larger than D! [20].
The time scale over which the complexity is quantified can be set by changing the ordinal pattern delay τ. This is the time separation between values used to construct the vector from which the ordinal pattern is determined. Its value corresponds to a multiple of the signal sampling period. For a given time series {ut, t = 1, ..., N}, ordinal pattern length D, and ordinal pattern delay τ, we consider the vector XS → (us- ( D- 1)τ , u-s - (D 2)τ , , , ,- , us τ , us ). (8)
At each time s the ordinal pattern of this vector can be converted to a unique symbol π = (r0, r1, ..., rD–1) defined by us- r0τ ≥ u-s r1π ≥ ... ≥ u-s rD- 2τ ≥- us rD- 1τ .
The ordinal pattern probability distribution P = {p(π), i = 1, ..., D!} required for the entropy calculation is constructed by determining the relative frequency of all the (5) (6) (7) D! possible permutations πi. The normalized permutation entropy is then defined as the normalized Shannon entropy S associated with the permutation probability distribution P
H S [P] =
S[ P]
= Smax ∑iD=!1 p(π i ) ln p(π i ) .
ln D! (9)
This normalized permutation entropy gives values 0 ≤ HS ≤ 1 where a completely predictable time series has a value of 0 and a completely stochastic process with a uniform probability distribution is represented by a value of 1. It is important to realize that the PE is a statistical measure and is not able to distinguish whether the observed complexity (irregularity) arises from stochastic or deterministic chaotic processes. It is also important that the PEn provides means to characterize complexity on different time scales, given by the delay.
Thus HS gives a measure of the departure of the time series under study from a
complete random one: the smaller the value of HS, the more regular the time series is.
It is clear that if D is too small, such as 1 or 2, the scheme will not work, since here
are only very few distinct states. In principle, using a large value of D is fine, as long
as the length of a stationary time series under study can be made proportional to D!.
In their paper [
Experimental testing of the effectiveness of indicatorsprecursors of crisis phenomena We have already reached a point where the crash of the bitcoin will have serious global consequences. The degree of involvement of financial institutions in a transaction with cryptocurrencies is now unclear, and, apparently, it will be fully disclosed after the financial catastrophe. This is very similar to the situation in 2007-2008, when nobody really knew where, ultimately, subprime mortgages are concentrated. Until the crash, everyone was only wondering which financial institutions could be bankrupt. Thus, the identification of possible trends of the cryptocurrency movement, construction and modeling of indicators of stability and possible crisis states is extremely relevant.
During the entire period (16.07.2010 - 10.02.2018) of verifiable fixed daily values of the bitcoin price (BTC) (https://finance.yahoo.com/cryptocurrencies) in relative units, five crisis phenomena were recorded and marked with arrows on Fig. 1. Fig. 1. Dynamics of price fluctuations of bitcoin over time. The arrows indicate the beginning of one of the five known crises In order to study the possibility of constructing indicators of crisis phenomena in the market of cryptocurrency, the price range of bitcoin was divided into five parts in accordance with the periodization of crises [21]: 1). From 19.02.2013 to 31.05.2013. 2). From 10.10.2013 to 31.12.2013. 3). From 18.12.2013 to 02.03.2014. 4). From 22.04.2017 to 31.07.2017. 5). From 15.07.2017 to 02.10.2018.
For each of the time series phase portraits, recurrent diagrams were constructed, their quantitative analysis was carried out, and there were entropies of permutations estimated. 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 measures of complexity (RR, DET, PEn), 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 cryptocurrency. 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.
We expect that the variation of RR, DET, PEn as a function of time or certain timevarying parameter can accurately indicate interesting dynamical changes in a time series.
The simulation results are quite sensitive to the window width selection. Indeed, if the window is too large, several crisis or shock (critical states) may enter it. As a result, we get an average case where it is impossible to reliably divide one crisis from another. On the contrary, when over a small window, the measures of complexity is not that exact, it fluctuates noticeably and requires smoothing.
In Fig. 2 for the first crisis the phase portrait, the recurrent diagram and the measures of complexity calculated for the window in 15 days in a one-day increments are given.
a) c) b) d) Fig. 2. Phase portrait (a), recurrence diagram (b) and, respectively, the measure of recurrence
RR and determinism DET (c) and the permutation entropy PEn (d) Unlike, for example, the stock markets, the cryptocurrencies market is more volatile, and critical phenomena are separated by a smaller time lag. This justifies the choice of the size of the window of a few days. We have proved calculations for windows in 15, 25 and 35 days. The best way is to share critical events in time when choosing a window in 15 days.
At the phase portrait there are no attractive areas, although fluctuations during the first crisis are visible both on the phase portrait and on the recurrence diagram. But the measures of complexity look interesting: before the crisis, both recurrent and entropy measures are noticeably diminished, thus signaling the oncoming crisis.
For the second crisis, the indicators-precursors produce dynamics, which is depicted in Fig. 3.
a) b)
Fig. 3. Dynamics of RR and DET (a) and permutation entropy PEn (b) for the second crisis For the third crisis, the behavior of indicators-precursors has the form, presented in Fig. 4. A fourth crisis could also have been predicted using the indicators introduced (Fig. 5). Finally, the last crisis is preceded by shock states, which are identified by the introduced indicator measures. But most clearly, they "prevent" the rapid fall of the main phase of the crisis of the end of 2017 beginning of 2018 (Fig. 6).
a) b) It should be noted that other of the most capitalized cryptocurrencies, such as Ethereum, Ripple, Bitcoin Cash have coefficients of pair correlation with bitcoin at the level of 0.6-0.8 and similarly react to crisis phenomena. 5
Consequently, in this paper, we have shown that monitoring and prediction of possible critical changes on cryptocurrency is of paramount importance. As it has been shown by us, the theory of complex systems has a powerful toolkit of methods and models for creating effective indicators- precursors of crisis phenomena. In this paper, we have explored the possibility of using the recurrent and entropy measures of complexity to detect dynamical changes in a complex time series. We have shown that the measures that have been used can indeed be effectively used to detect crisis phenomena for the time series of bitcoin. Certainly there is no reason to expect that the RR, DET or PE is universally and indiscriminately applicable. It is most likely that no such measure exists; instead, various measures would have to be used in a complementary fashion, to take best advantage of their respective merits within their ranges of applicability. We have concluded though by emphasizing that the most attractive features of the RR, DET and PE, namely its conceptual simplicity and computational efficiency make it an excellent candidate for a fast, robust, and useful screener and detector of unusual patterns in complex time series.
As for the prospects for further research, we plan to investigate the fractal and network properties of cryptomarket, as well as its correlation with other sectors of the global financial market.