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Multiscale recurrence quantification analysis of order recurrence plots
Accepted Manuscript Multiscale recurrence quantification analysis of order recurrence plots Mengjia Xu, Pengjian Shang, Aijing Lin PII: DOI: Reference:
S0378-4371(16)30864-0 http://dx.doi.org/10.1016/j.physa.2016.11.058 PHYSA 17714
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Physica A
Received date: 30 July 2016 Revised date: 8 October 2016 Please cite this article as: M. Xu, P. Shang, A. Lin, Multiscale recurrence quantification analysis of order recurrence plots, Physica A (2016), http://dx.doi.org/10.1016/j.physa.2016.11.058 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Multiscale recurrence quantification analysis of order recurrence plots Mengjia Xu ∗ , Pengjian Shang, Aijing Lin Department of Mathematics, School of Science, Beijing Jiaotong University, Beijing 100044, PR China
Abstract In this paper, we propose a new method of multiscale recurrence quantification analysis (MSRQA) to analyze the structure of order recurrence plots. The MSRQA is based on order patterns over a range of time scales. Compared with conventional recurrence quantification analysis (RQA), the MSRQA can show richer and more recognizable information on the local characteristics of diverse systems which successfully describes their recurrence properties. Both synthetic series and stock market indexes exhibit their properties of recurrence at large time scales that quite differ from those at a single time scale. Some systems present more accurate recurrence patterns under large time scales. It demonstrates that the new approach is effective for distinguishing three similar stock market systems and showing some inherent differences. Keywords: Multiscale, Recurrence quantification analysis, Order recurrence plot, Dynamical system
∗
Corresponding author. E-mail: [email protected]
1
Multiscale recurrence quantification analysis of order recurrence plots Mengjia Xu ∗ , Pengjian Shang, Aijing Lin Department of Mathematics, School of Science, Beijing Jiaotong University, Beijing 100044, PR China
1. Introduction In recent decades, considerable analysis on nonlinear dynamics have been explored. Recurrence plot (RP), a powerful tool for the visualisation and analysis of recurrence, was introduced in the late 1980s [1, 2] which has been widely used to uncover statistically characteristic properties of complex systems [3, 4]. Recurrence is a fundamental property of dynamical systems, which can be especially exploited to characterize nonlinear behaviors of systems. Later on, recurrence quantification analysis (RQA) was developed to quantify the recurrence point density and the diagonal and vertical line structures of the RP [5–7]. Some examples of their successful application in real-word systems can be found in unemployment rate data [8], EEG [9], stock markets [10, 11], and other areas of research [12–14]. Some modifications also have been proposed to improve the performance of nonlinear dynamic analysis [15–18]. Instead of using the spatial closeness between phase space trajectories, order recurrence plots (ORPs) consider a special symbolic dynamic of the system, which incorporate order patterns with the definition of a recurrence [19]. Referring to this symbolic dynamic, in 2002, Bandt and Pompe introduced a complex measure of permutation entropy, by evaluating the probability distribution of order patterns [20]. The main advantage of a symbolic representation is its robust against non-stationarity. The order patterns are invariant with respect to an arbitrary, increasing transformation of the amplitude [19, 21]. Therefore the ORPs have been applied to divers areas of interest [21–23]. In 2005, Costa et al. proposed the multiscale entropy (MSE) over a range of scales to show the complexity changes of a time series and demonstrated distinguished behaviors of complexity at different time scales [24]. Then this technique has been widely used to various research field [25–29]. In this paper, we propose a new method of the MSRQA of order recurrence plots. Here the MSRQA is proposed to uncover recurrence properties of various dynamical systems through quantifying their diagonal line structures of the ORPs over a range of time scales. In fact, this multiscale method has a better performance than the original one due to its ability to distinguish different systems and also accurately reveal the intrinsic information of some certain systems. To characterise behaviors of a system, the coarse-grained time series are first obtained and then further analyzed by the RQA. ∗
Corresponding author. E-mail: [email protected]
1
The reminder of the paper is organized as follows. In Section 2, we review the order recurrence plots and recurrence quantification analysis, and introduce our new method. In Section 3, we apply the MSRQA to the synthetic series and the stock market indexes and analyze the results. In Section 4, a summary is provided. In Appendix we discuss the selection of parameters. 2. Methods 2.1. Conventional RQA of ORPs Consider a dynamic system denoted by a time series of N observations {xi }N i=1 and its time-
delay embedding representation Xim,τ = {xi , xi+τ , · · · , xi+(m−1)τ } for i = 1, 2, . . . , N − (m − 1)τ ,
where m and τ denote the embedding dimension and time delay, respectively. The phase space
can be reconstructed by mapping time series to a trajectory X = [X1 , X2 , · · · , XN −(m−1)τ ]T with
the reconstruction parameters (m, τ ). The order pattern π of Xim,τ is defined by comparing with
neighboring values. For example, if Xi3,τ = {1.1, 2.3, 1.5}, then its order pattern is πi = {0, 2, 1}
as xi ≤ xi+2τ ≤ xi+τ . The trajectory X is hence transformed into the ordinal pattern matrix
π = [π1 , π2 , . . . , πN −(m−1)τ ]T . Obviously, for a given m at most m! order patterns exist. Formally, the order recurrence matrix R is defined as { 1 : πi = πj R(i, j) = i, j = 1, 2, . . . , N − (m − 1)τ. 0 : otherwise,
(1)
Ri,j = 1 (recurrence) is usually represented by a black dot, whereas Ri,j = 0 (no recurrence) is represented by a white dot. A RP represents those times, when a specific rank order recurs in the system. The main advantage of order patterns is its robustness with respect to non-stationarity. The structure of the black dots in a RP is usually quantified by different RQA measures: (1) Recurrence Rate (RR): the percentage of black dots and a measure of the density of recurrence points in the RP, RR =
L 1 ∑ R(i, j), L2
(2)
i,j=1
where L = N − (m − 1)τ is the size of the RP. It describes the probability that a certain state recurs.
(2) Determinism (DET ): the ratio of recurrence points that form diagonal structures (of at least length lmin ) to all recurrence points, ∑L
lP (l) l=l DET = ∑Lmin , l=1 lP (l)
(3)
where P (l) is the number of diagonal lines of length l in the RP, and lmin is the line length threshold. It provides an indication of determinism and predictability in the system. (3) The average diagonal line length (Lav ) of the overall diagonal line structure: ∑L lP (l) l=l . Lav = ∑L min l=lmin P (l) 2
(4)
(4) Entropy (EN T R): the Shannon entropy of the probability distribution of diagonal lines, EN T R = −
L ∑
p(l)log2 p(l),
(5)
l=lmin
where p(l) is the probability of a diagonal line with length l. It reflects the complexity and diversity of the diagonal line structure of the RP. 2.2. MSRQA of ORPs The MSRQA of order recurrence plots consists of two steps: i) coarse-graining the original time series at different time scales. Given a time series {xi }N i=1 , the coarse-grained time series
{y (s) } at time scale s is constructed by (s)
yj
=
j+s−1 1 ∑ xi , 1 ≤ j ≤ N − s + 1. s
(6)
i=j
For s = 1, the time series {y (1) } is the original series. The length of each coarse-grained time
series is N − s + 1 by dividing the original time series into overlapping windows of scale factor s.
ii) Using the RQA to quantify the structures of an ORP. For each coarse-grained time series
at time scale s we obtain an ORP, which is then quantified by the RQA. The RR, DET , Lav and EN T R, the representations of the recurrence point density and the diagonal line structures of the recurrence matrix, are estimated. The values of each measure are plotted as a function of the scale factor s. Compared with conventional RQA, the MSRQA extends the time scale ranges of its measures so that the MSRQA curves consist of richer information which can not be found through the RQA. Unless otherwise specified, the values of the reconstruction parameters, the embedding dimension and time delay, are m = 2 and τ = 1. 3. Results and analysis In this section we apply the MSRQA to synthetic data and real-word system, including the logistic map, the white noise, the 1/f noise, and the daily closing prices of three stock market indexes. 3.1. Logistic map Mathematically, the logistic map, a well-known archetypal example, is written as xi+1 = axi (1 − xi ).
(7)
where xi is a real number between 0 and 1 and a is a control parameter. When a ∈ [3.5, 4], the
logistic map shows rich dynamics, e.g., bifurcations, periodic and chaotic states, and inner and outer crises. Here we set a = 3.5 and a = 4 which represent the logistic map is periodic and chaotic respectively. 3
(a)
(b) 1
0.62 N=500 N=1000 0.6
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N=500 N=1000 0.48
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4
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14
N=500 N=1000 2
s
4
6
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14
s
Fig. 1. RR (a), DET (b), Lav (c) and EN T R (d) curves of logistic map with a = 3.5 (periodic regime), plotted versus scale factor s. The original series with two different lengths, N = 500 (blue) and N = 1000 (red), are tested.
Fig. 1 presents four MSRQA measures of the logistic map with a = 3.5 whose period is four, namely, the population will approach permanent oscillations among four values. We test the series with two different lengths, N = 500 (blue) and N = 1000 (red). It is clear that the RR, DET , Lav and EN T R curves, plotted as a function of scale s, are all periodic. Specially, the periods are all four, which are consistent with the period of the original series. For the RR curves, when s = 4, 8, and 12, they have higher values which means at these time scales it is more probable that some specific order patterns recur. For the DET curves, they look like the letter ’M’. When the scale s is odd, the values of DET are all 1 showing the rich deterministic structures, whereas when s = 4, 8, and 12, the ones are all a little lower than 0.8, and when s = 2, 6, 10, and 14, the ones are all a little lower than 0.5. For the Lav curves, the shapes of them are totally opposite to the ones of the RR curves. For the EN T R curves, they also look like the letter ’M’. On the whole, this periodic system has periodic behaviors of the structure of the ORPs and the MSRQA is able to identify it accurately. Besides, the length of original series has some influences on the MSRQA measures especially for the Lav because it is positively correlated with the length in this system. For other measures, the values of the original series with N = 1000 are generally a litter higher than those with N = 500. 4
(b)
(a) 0.8
0.57 N=500 N=1000
0.56
N=500 N=1000
0.79 0.78
0.55 0.77 0.76
DET
RR
0.54 0.53
0.75 0.74
0.52
0.73 0.51 0.72 0.5 0.49
0.71 2
4
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14
2
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s
s
(c)
(d)
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4 N=500 N=1000
N=500 N=1000
2.7
3.8
2.6 2.5
3.6
ENTR
Lav
2.4 3.4
2.3 2.2
3.2
2.1 2
3
1.9 2.8
2
4
6
8
10
12
1.8
14
s
2
4
6
8
10
12
14
s
Fig. 2. RR (a), DET (b), Lav (c) and EN T R (d) curves of logistic map with a = 4 (chaotic regime), plotted versus scale factor s. The original series with two different lengths, N = 500 (blue) and N = 1000 (red), are tested.
Fig. 2 presents the RR, DET , Lav and EN T R curves of logistic map with a = 4, which is chaotic, for two different lengths of series, N = 500 (blue) and N = 1000 (red). For the RR curves, the values of s = 1 are highest, and when s ≥ 2 the curves are almost straight. For the
DET curves, the values of s = 1 are highest, the values of s = 2 are lowest , and when s ≥ 3 the
values are almost invariable. For the Lav curves, when 1 ≤ s ≤ 3 they are monotone decreasing
and when s ≥ 3 they are almost straight. For the EN T R curves, when 1 ≤ s ≤ 3 they are also
monotone decreasing and when s ≥ 3 they are first increasing slowly and then basically keep constant.
Obviously, these measures of chaotic system have different behaviors compared with those of periodic system above. Furthermore, they all contain more information on the structure of the ORPs over a range of time scales. The value of s ≥ 2 is smaller than the one of s = 1. In addition, the results of different lengths N = 500 and N = 1000 are almost the same. 3.2. White noise and pink noise In this part, we use two types of synthetic noise signals, the white noise (mean 0, variance 1) and the 1/f noise (pink noise), i.e., uncorrelated and correlated fluctuations, to estimate the efficiency of the MSRQA of ORPs. Thirty independent noise samples are used in each simulation, and each noise sample contains 500 or 1000 data points. Our results show that the standard deviation of thirty samples for each simulation is zero, which means the values of the RR, DET , Lav and EN T R curves are all the same. The MSRQA of ORPs is robust against 5
uncorrelated and correlated fluctuations. Fig. 3 presents these four measures of the white and pink noises with two different lengths, N = 500 and N = 1000. (b)
(a) 0.88
0.505 White noise N=500 White noise N=1000 Pink noise N=500 Pink noise N=1000
0.504 0.503
White noise N=500 White noise N=1000 Pink noise N=500 Pink noise N=1000
0.86 0.84
0.502
0.82
DET
RR
0.501 0.5
0.8 0.78
0.499 0.76
0.498
0.74
0.497
0.72
0.496 0.495
2
4
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14
2
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s
s
(c)
(d)
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2.8 White noise N=500 White noise N=1000 Pink noise N=500 Pink noise N=1000
3.8
White noise N=500 White noise N=1000 Pink noise N=500 Pink noise N=1000
2.7 2.6 2.5
3.6
ENTR
Lav
2.4 3.4
2.3 2.2
3.2
2.1 2
3
1.9 2.8
2
4
6
8
10
12
1.8
14
2
4
s
6
8
s
Fig. 3. RR (a), DET (b), Lav (c) and EN T R (d) curves of the white and pink noises (i.e., uncorrelated and correlated fluctuations), plotted versus scale factor s. The original series with two different lengths, N = 500 and N = 1000, are tested.
Here we mainly discuss the significant differences of the DET , Lav and EN T R between the white noise and pink noise. When time scale s ≥ 3, for these three measures, the values of the pink noise are much higher than those of the white noise. Furthermore, the curves of the pink noise are fluctuant and ascending, whereas the curves of the white noise are almost invariant. Besides, the results of s = 1 is exactly opposite to the results of s ≥ 3. Because
through conventional RQA, the values of the DET , Lav and EN T R of the white noise are all greater than those of the pink noise. However the white noise and pink noise present opposite features of diagonal line structure within large time scale ranges. It means the results under large scales are more reliable for distinguishing these two systems. The method is able to distinguish the white noise and pink noise effectively, which reflects more accurate information on the local characteristics of different scales. The difference of the RR between the white noise and pink noise is irregular and it is not used to describe the relevant behavior of uncorrelated and correlated fluctuations. However compared with conventional RQA (when s = 1) the white noise and pink noise have the opposite relationship under large scales. This is an important finding. The influence of series length is small and the values of long series are generally greater than those of short series. 3.3. Stock market data 6
In this subsection, we investigate three different real-world stock indices, which are considered as the outputs of complex systems. The data used here are the daily closing values of S&P500, DAX Index and SSE Composite Index, from the US, Germany and China respectively. We obtain the data from Yahoo Finance covering the period from Dec. 1, 2011 to Nov. 30, 2015. For convenience, they are abbreviated to S&P500, DAX and SSECI. The lengths of these indices are 1005, 1017 and 960 respectively due to these stock markets have different opening dates. The daily closing values of three stock markets are presented in Fig. 4. (a)
(b)
2200
13000
2100
12000
daily closing values
daily closing values
2000 1900 1800 1700 1600 1500
11000 10000 9000 8000 7000
1400 6000
1300 1200
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t (c) 5500
daily closing values
5000 4500 4000 3500 3000 2500 2000 1500
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t
Fig. 4. The daily closing values of S&P500 (a), DAX (b) and SSECI (c) from Dec. 1, 2011 to Nov. 30, 2015.
Fig. 5 shows the four measures of the MSRQA of ORPs about three real-world stock markets. Compared with the results of simulated series, the RR, DET , Lav and EN T R curves of stock markets are all rising as a whole. In regard to the measure RR, it makes these stock markets quite different. Using conventional RQA of ORPs, the RR values of stock markets are all greater than 0.5 and less than 0.51 so that it is not able to analyze them well. However, using the MSRQA, the results totally uncover more overall properties and local characteristics of the stock markets. As we have seen, when scale factor s increases, the RR curve of S&P500 is highest, the one of DAX is middle, and the one of SSECI is lowest. When s > 1 these three curve are completely disjoint. Therefore, we conclude that the recurrences of some certain states in S&P500 are more frequent than those of the other two stock markets at large time scales which is not consistent with the results of conventional RQA. As for the measures DET , Lav and EN T R, although the curves of three stock markets are crossed, they also have a better performance than original method as when s = 1 the three 7
values of each measure are almost the same. In the range 10 ≤ s ≤ 12 the values of S&P500
are lowest, the ones of DAX are middle, and the ones of SSECI is hightest. It is just the reverse about the situation of RR. Therefore, we also conclude that the MARQA of ORPs can well characterise the local properties of diagonal structures at some scales. (a)
(b)
0.56
1 S&P500 DAX SSECI
0.55
0.95
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RR
DET
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s
S&P500 DAX SSECI 2
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s
Fig. 5. RR (a), DET (b), Lav (c) and EN T R (d) curves of S&P500 (circle), DAX (square) and SSECI (diamond), plotted versus scale factor s.
4. Conclusion In summary, we propose the MSRQA of order recurrence plots, to uncover underlying properties of dynamical systems. Simulations are conducted over the logistic map, the white noise, the pink noise, and the stock market data to provide comparative study and show the power of this new method. Experiments with the logistic map show that the RR, DET , Lav and EN T R curves of the MSRQA with periodic and chaotic systems have different behaviors about recurrence. The periodic system has the corresponding periodic measures, and the chaotic system contains rich information on the structure of the ORPs over a range of time scales. Compared with conventional RQA, the MSRQA of ORPs with the white noise and pink noise show that this method distinguishes uncorrelated and correlated fluctuations effectively, and reflects more reliable information on the local characteristics of different scales. The influence of series length is also tested. In addition, in order to estimate the efficiency of the MSRQA of ORPs, series of three stock markets, S&P500, DAX and SSECI, are used as outputs of real-world complex system. 8
We find the MSRQA can uncover overall properties of the recurrence of stock markets and characterise the local properties. In Appendix we discuss the situation about the selection of embedding dimension m. Conventional RQA provides some measures to quantify the properties of the recurrence. However, it does not work well for comparing some similar systems, like three stock markets above. The MSRQA provides a new way to discover the intrinsic nature of recurrence under large time scales. It turns out that it does play an important role in analyzing stock markets. However, there are still some areas that need further study. Parameter selection can be further discussed especially for the parameter τ . In this paper, we simply discuss the selection of parameter m in Appendix. The MSRQA can also be applied to the RPs, and in this case there are more problems of parameter selection need to be discussed. And it can be applied to various fields, like machine learning [30], etc.. Unfortunately, the main shortcoming of this method is that it can only handle short sequence as the complexity of the algorithm is high.
Acknowledgement The financial support by the Fundamental Research Funds for the Central Universities (S16JB00030) are gratefully acknowledged.
Appendix Here we discuss the selection of embedding dimension m. Simulated Gaussian distributed (mean 0, variance 1) white noise and 1/f noise (pink noise) time series with length N = 500 are used to illustrate the effects on the MSRQA of ORPs of the choice of parameter m. Thirty independent noise samples are used in each simulation, and the values of RR, DET , Lav and EN T R curves for each sample of one simulation are also all the same.
(a)
(b)
0.6
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0.7 m=2 m=3 m=4 m=5 m=6
0.3
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0
m=2 m=3 m=4 m=5 m=6
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s
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(c)
(d)
3.6
3 m=2 m=3 m=4 m=5 m=6
3.4 3.2 3
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ENTR
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m=2 m=3 m=4 m=5 m=6
2.5
2.8
1.5
2.6 2.4
1 2.2 2
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s
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s
Fig. 6. RR (a), DET (b), Lav (c) and EN T R (d) curves of the white noise with different embedding dimension m, plotted versus scale factor s. (a)
(b)
0.6
1
0.9
0.5 m=2 m=3 m=4 m=5 m=6
0.3
0.8
DET
RR
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s
Fig. 7. RR (a), DET (b), Lav (c) and EN T R (d) curves of the pink noise with different embedding dimension m, plotted versus scale factor s.
Fig. 6 and Fig. 7 show the RR, DET , Lav and EN T R curves of the white noise and pink noise with different parameter m respectively. Regarding to the results of the white noise, for larger m, the values of the corresponding curves are generally smaller. For the measure RR, the shapes of curves with different m are similar, but for the measures DET , Lav and EN T R, they have a little change. As for the results of pink noise, the situations are slightly more complicated. For the RR curves, the one with larger m is lower. For the DET , Lav and EN T R curves, when only scale s is small, like s ≤ 3 for DET , the situation of RR happens. The local characteristics of these three measures for pink noise are complex.
We note that the larger m is, the smaller RR is, which means in this situation more information about the recurrence of system lose. When m = 4, the RR curve of the white noise is 10
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HIGHLIGHTS We propose a new method of multiscale recurrence quantification analysis to analyze order recurrence plots. The proposed method is applied to synthetic series and stock market indexes to show their local characteristics. Compared with conventional method, the new method can much reflect richer and more recognizable information of diverse systems.
S0378-4371(16)30864-0 http://dx.doi.org/10.1016/j.physa.2016.11.058 PHYSA 17714
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Physica A
Received date: 30 July 2016 Revised date: 8 October 2016 Please cite this article as: M. Xu, P. Shang, A. Lin, Multiscale recurrence quantification analysis of order recurrence plots, Physica A (2016), http://dx.doi.org/10.1016/j.physa.2016.11.058 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Multiscale recurrence quantification analysis of order recurrence plots Mengjia Xu ∗ , Pengjian Shang, Aijing Lin Department of Mathematics, School of Science, Beijing Jiaotong University, Beijing 100044, PR China
Abstract In this paper, we propose a new method of multiscale recurrence quantification analysis (MSRQA) to analyze the structure of order recurrence plots. The MSRQA is based on order patterns over a range of time scales. Compared with conventional recurrence quantification analysis (RQA), the MSRQA can show richer and more recognizable information on the local characteristics of diverse systems which successfully describes their recurrence properties. Both synthetic series and stock market indexes exhibit their properties of recurrence at large time scales that quite differ from those at a single time scale. Some systems present more accurate recurrence patterns under large time scales. It demonstrates that the new approach is effective for distinguishing three similar stock market systems and showing some inherent differences. Keywords: Multiscale, Recurrence quantification analysis, Order recurrence plot, Dynamical system
∗
Corresponding author. E-mail: [email protected]
1
Multiscale recurrence quantification analysis of order recurrence plots Mengjia Xu ∗ , Pengjian Shang, Aijing Lin Department of Mathematics, School of Science, Beijing Jiaotong University, Beijing 100044, PR China
1. Introduction In recent decades, considerable analysis on nonlinear dynamics have been explored. Recurrence plot (RP), a powerful tool for the visualisation and analysis of recurrence, was introduced in the late 1980s [1, 2] which has been widely used to uncover statistically characteristic properties of complex systems [3, 4]. Recurrence is a fundamental property of dynamical systems, which can be especially exploited to characterize nonlinear behaviors of systems. Later on, recurrence quantification analysis (RQA) was developed to quantify the recurrence point density and the diagonal and vertical line structures of the RP [5–7]. Some examples of their successful application in real-word systems can be found in unemployment rate data [8], EEG [9], stock markets [10, 11], and other areas of research [12–14]. Some modifications also have been proposed to improve the performance of nonlinear dynamic analysis [15–18]. Instead of using the spatial closeness between phase space trajectories, order recurrence plots (ORPs) consider a special symbolic dynamic of the system, which incorporate order patterns with the definition of a recurrence [19]. Referring to this symbolic dynamic, in 2002, Bandt and Pompe introduced a complex measure of permutation entropy, by evaluating the probability distribution of order patterns [20]. The main advantage of a symbolic representation is its robust against non-stationarity. The order patterns are invariant with respect to an arbitrary, increasing transformation of the amplitude [19, 21]. Therefore the ORPs have been applied to divers areas of interest [21–23]. In 2005, Costa et al. proposed the multiscale entropy (MSE) over a range of scales to show the complexity changes of a time series and demonstrated distinguished behaviors of complexity at different time scales [24]. Then this technique has been widely used to various research field [25–29]. In this paper, we propose a new method of the MSRQA of order recurrence plots. Here the MSRQA is proposed to uncover recurrence properties of various dynamical systems through quantifying their diagonal line structures of the ORPs over a range of time scales. In fact, this multiscale method has a better performance than the original one due to its ability to distinguish different systems and also accurately reveal the intrinsic information of some certain systems. To characterise behaviors of a system, the coarse-grained time series are first obtained and then further analyzed by the RQA. ∗
Corresponding author. E-mail: [email protected]
1
The reminder of the paper is organized as follows. In Section 2, we review the order recurrence plots and recurrence quantification analysis, and introduce our new method. In Section 3, we apply the MSRQA to the synthetic series and the stock market indexes and analyze the results. In Section 4, a summary is provided. In Appendix we discuss the selection of parameters. 2. Methods 2.1. Conventional RQA of ORPs Consider a dynamic system denoted by a time series of N observations {xi }N i=1 and its time-
delay embedding representation Xim,τ = {xi , xi+τ , · · · , xi+(m−1)τ } for i = 1, 2, . . . , N − (m − 1)τ ,
where m and τ denote the embedding dimension and time delay, respectively. The phase space
can be reconstructed by mapping time series to a trajectory X = [X1 , X2 , · · · , XN −(m−1)τ ]T with
the reconstruction parameters (m, τ ). The order pattern π of Xim,τ is defined by comparing with
neighboring values. For example, if Xi3,τ = {1.1, 2.3, 1.5}, then its order pattern is πi = {0, 2, 1}
as xi ≤ xi+2τ ≤ xi+τ . The trajectory X is hence transformed into the ordinal pattern matrix
π = [π1 , π2 , . . . , πN −(m−1)τ ]T . Obviously, for a given m at most m! order patterns exist. Formally, the order recurrence matrix R is defined as { 1 : πi = πj R(i, j) = i, j = 1, 2, . . . , N − (m − 1)τ. 0 : otherwise,
(1)
Ri,j = 1 (recurrence) is usually represented by a black dot, whereas Ri,j = 0 (no recurrence) is represented by a white dot. A RP represents those times, when a specific rank order recurs in the system. The main advantage of order patterns is its robustness with respect to non-stationarity. The structure of the black dots in a RP is usually quantified by different RQA measures: (1) Recurrence Rate (RR): the percentage of black dots and a measure of the density of recurrence points in the RP, RR =
L 1 ∑ R(i, j), L2
(2)
i,j=1
where L = N − (m − 1)τ is the size of the RP. It describes the probability that a certain state recurs.
(2) Determinism (DET ): the ratio of recurrence points that form diagonal structures (of at least length lmin ) to all recurrence points, ∑L
lP (l) l=l DET = ∑Lmin , l=1 lP (l)
(3)
where P (l) is the number of diagonal lines of length l in the RP, and lmin is the line length threshold. It provides an indication of determinism and predictability in the system. (3) The average diagonal line length (Lav ) of the overall diagonal line structure: ∑L lP (l) l=l . Lav = ∑L min l=lmin P (l) 2
(4)
(4) Entropy (EN T R): the Shannon entropy of the probability distribution of diagonal lines, EN T R = −
L ∑
p(l)log2 p(l),
(5)
l=lmin
where p(l) is the probability of a diagonal line with length l. It reflects the complexity and diversity of the diagonal line structure of the RP. 2.2. MSRQA of ORPs The MSRQA of order recurrence plots consists of two steps: i) coarse-graining the original time series at different time scales. Given a time series {xi }N i=1 , the coarse-grained time series
{y (s) } at time scale s is constructed by (s)
yj
=
j+s−1 1 ∑ xi , 1 ≤ j ≤ N − s + 1. s
(6)
i=j
For s = 1, the time series {y (1) } is the original series. The length of each coarse-grained time
series is N − s + 1 by dividing the original time series into overlapping windows of scale factor s.
ii) Using the RQA to quantify the structures of an ORP. For each coarse-grained time series
at time scale s we obtain an ORP, which is then quantified by the RQA. The RR, DET , Lav and EN T R, the representations of the recurrence point density and the diagonal line structures of the recurrence matrix, are estimated. The values of each measure are plotted as a function of the scale factor s. Compared with conventional RQA, the MSRQA extends the time scale ranges of its measures so that the MSRQA curves consist of richer information which can not be found through the RQA. Unless otherwise specified, the values of the reconstruction parameters, the embedding dimension and time delay, are m = 2 and τ = 1. 3. Results and analysis In this section we apply the MSRQA to synthetic data and real-word system, including the logistic map, the white noise, the 1/f noise, and the daily closing prices of three stock market indexes. 3.1. Logistic map Mathematically, the logistic map, a well-known archetypal example, is written as xi+1 = axi (1 − xi ).
(7)
where xi is a real number between 0 and 1 and a is a control parameter. When a ∈ [3.5, 4], the
logistic map shows rich dynamics, e.g., bifurcations, periodic and chaotic states, and inner and outer crises. Here we set a = 3.5 and a = 4 which represent the logistic map is periodic and chaotic respectively. 3
(a)
(b) 1
0.62 N=500 N=1000 0.6
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14
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s
4
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14
s
Fig. 1. RR (a), DET (b), Lav (c) and EN T R (d) curves of logistic map with a = 3.5 (periodic regime), plotted versus scale factor s. The original series with two different lengths, N = 500 (blue) and N = 1000 (red), are tested.
Fig. 1 presents four MSRQA measures of the logistic map with a = 3.5 whose period is four, namely, the population will approach permanent oscillations among four values. We test the series with two different lengths, N = 500 (blue) and N = 1000 (red). It is clear that the RR, DET , Lav and EN T R curves, plotted as a function of scale s, are all periodic. Specially, the periods are all four, which are consistent with the period of the original series. For the RR curves, when s = 4, 8, and 12, they have higher values which means at these time scales it is more probable that some specific order patterns recur. For the DET curves, they look like the letter ’M’. When the scale s is odd, the values of DET are all 1 showing the rich deterministic structures, whereas when s = 4, 8, and 12, the ones are all a little lower than 0.8, and when s = 2, 6, 10, and 14, the ones are all a little lower than 0.5. For the Lav curves, the shapes of them are totally opposite to the ones of the RR curves. For the EN T R curves, they also look like the letter ’M’. On the whole, this periodic system has periodic behaviors of the structure of the ORPs and the MSRQA is able to identify it accurately. Besides, the length of original series has some influences on the MSRQA measures especially for the Lav because it is positively correlated with the length in this system. For other measures, the values of the original series with N = 1000 are generally a litter higher than those with N = 500. 4
(b)
(a) 0.8
0.57 N=500 N=1000
0.56
N=500 N=1000
0.79 0.78
0.55 0.77 0.76
DET
RR
0.54 0.53
0.75 0.74
0.52
0.73 0.51 0.72 0.5 0.49
0.71 2
4
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s
s
(c)
(d)
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4 N=500 N=1000
N=500 N=1000
2.7
3.8
2.6 2.5
3.6
ENTR
Lav
2.4 3.4
2.3 2.2
3.2
2.1 2
3
1.9 2.8
2
4
6
8
10
12
1.8
14
s
2
4
6
8
10
12
14
s
Fig. 2. RR (a), DET (b), Lav (c) and EN T R (d) curves of logistic map with a = 4 (chaotic regime), plotted versus scale factor s. The original series with two different lengths, N = 500 (blue) and N = 1000 (red), are tested.
Fig. 2 presents the RR, DET , Lav and EN T R curves of logistic map with a = 4, which is chaotic, for two different lengths of series, N = 500 (blue) and N = 1000 (red). For the RR curves, the values of s = 1 are highest, and when s ≥ 2 the curves are almost straight. For the
DET curves, the values of s = 1 are highest, the values of s = 2 are lowest , and when s ≥ 3 the
values are almost invariable. For the Lav curves, when 1 ≤ s ≤ 3 they are monotone decreasing
and when s ≥ 3 they are almost straight. For the EN T R curves, when 1 ≤ s ≤ 3 they are also
monotone decreasing and when s ≥ 3 they are first increasing slowly and then basically keep constant.
Obviously, these measures of chaotic system have different behaviors compared with those of periodic system above. Furthermore, they all contain more information on the structure of the ORPs over a range of time scales. The value of s ≥ 2 is smaller than the one of s = 1. In addition, the results of different lengths N = 500 and N = 1000 are almost the same. 3.2. White noise and pink noise In this part, we use two types of synthetic noise signals, the white noise (mean 0, variance 1) and the 1/f noise (pink noise), i.e., uncorrelated and correlated fluctuations, to estimate the efficiency of the MSRQA of ORPs. Thirty independent noise samples are used in each simulation, and each noise sample contains 500 or 1000 data points. Our results show that the standard deviation of thirty samples for each simulation is zero, which means the values of the RR, DET , Lav and EN T R curves are all the same. The MSRQA of ORPs is robust against 5
uncorrelated and correlated fluctuations. Fig. 3 presents these four measures of the white and pink noises with two different lengths, N = 500 and N = 1000. (b)
(a) 0.88
0.505 White noise N=500 White noise N=1000 Pink noise N=500 Pink noise N=1000
0.504 0.503
White noise N=500 White noise N=1000 Pink noise N=500 Pink noise N=1000
0.86 0.84
0.502
0.82
DET
RR
0.501 0.5
0.8 0.78
0.499 0.76
0.498
0.74
0.497
0.72
0.496 0.495
2
4
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14
2
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s
s
(c)
(d)
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2.8 White noise N=500 White noise N=1000 Pink noise N=500 Pink noise N=1000
3.8
White noise N=500 White noise N=1000 Pink noise N=500 Pink noise N=1000
2.7 2.6 2.5
3.6
ENTR
Lav
2.4 3.4
2.3 2.2
3.2
2.1 2
3
1.9 2.8
2
4
6
8
10
12
1.8
14
2
4
s
6
8
s
Fig. 3. RR (a), DET (b), Lav (c) and EN T R (d) curves of the white and pink noises (i.e., uncorrelated and correlated fluctuations), plotted versus scale factor s. The original series with two different lengths, N = 500 and N = 1000, are tested.
Here we mainly discuss the significant differences of the DET , Lav and EN T R between the white noise and pink noise. When time scale s ≥ 3, for these three measures, the values of the pink noise are much higher than those of the white noise. Furthermore, the curves of the pink noise are fluctuant and ascending, whereas the curves of the white noise are almost invariant. Besides, the results of s = 1 is exactly opposite to the results of s ≥ 3. Because
through conventional RQA, the values of the DET , Lav and EN T R of the white noise are all greater than those of the pink noise. However the white noise and pink noise present opposite features of diagonal line structure within large time scale ranges. It means the results under large scales are more reliable for distinguishing these two systems. The method is able to distinguish the white noise and pink noise effectively, which reflects more accurate information on the local characteristics of different scales. The difference of the RR between the white noise and pink noise is irregular and it is not used to describe the relevant behavior of uncorrelated and correlated fluctuations. However compared with conventional RQA (when s = 1) the white noise and pink noise have the opposite relationship under large scales. This is an important finding. The influence of series length is small and the values of long series are generally greater than those of short series. 3.3. Stock market data 6
In this subsection, we investigate three different real-world stock indices, which are considered as the outputs of complex systems. The data used here are the daily closing values of S&P500, DAX Index and SSE Composite Index, from the US, Germany and China respectively. We obtain the data from Yahoo Finance covering the period from Dec. 1, 2011 to Nov. 30, 2015. For convenience, they are abbreviated to S&P500, DAX and SSECI. The lengths of these indices are 1005, 1017 and 960 respectively due to these stock markets have different opening dates. The daily closing values of three stock markets are presented in Fig. 4. (a)
(b)
2200
13000
2100
12000
daily closing values
daily closing values
2000 1900 1800 1700 1600 1500
11000 10000 9000 8000 7000
1400 6000
1300 1200
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t (c) 5500
daily closing values
5000 4500 4000 3500 3000 2500 2000 1500
100
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t
Fig. 4. The daily closing values of S&P500 (a), DAX (b) and SSECI (c) from Dec. 1, 2011 to Nov. 30, 2015.
Fig. 5 shows the four measures of the MSRQA of ORPs about three real-world stock markets. Compared with the results of simulated series, the RR, DET , Lav and EN T R curves of stock markets are all rising as a whole. In regard to the measure RR, it makes these stock markets quite different. Using conventional RQA of ORPs, the RR values of stock markets are all greater than 0.5 and less than 0.51 so that it is not able to analyze them well. However, using the MSRQA, the results totally uncover more overall properties and local characteristics of the stock markets. As we have seen, when scale factor s increases, the RR curve of S&P500 is highest, the one of DAX is middle, and the one of SSECI is lowest. When s > 1 these three curve are completely disjoint. Therefore, we conclude that the recurrences of some certain states in S&P500 are more frequent than those of the other two stock markets at large time scales which is not consistent with the results of conventional RQA. As for the measures DET , Lav and EN T R, although the curves of three stock markets are crossed, they also have a better performance than original method as when s = 1 the three 7
values of each measure are almost the same. In the range 10 ≤ s ≤ 12 the values of S&P500
are lowest, the ones of DAX are middle, and the ones of SSECI is hightest. It is just the reverse about the situation of RR. Therefore, we also conclude that the MARQA of ORPs can well characterise the local properties of diagonal structures at some scales. (a)
(b)
0.56
1 S&P500 DAX SSECI
0.55
0.95
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RR
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s
S&P500 DAX SSECI 2
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s
Fig. 5. RR (a), DET (b), Lav (c) and EN T R (d) curves of S&P500 (circle), DAX (square) and SSECI (diamond), plotted versus scale factor s.
4. Conclusion In summary, we propose the MSRQA of order recurrence plots, to uncover underlying properties of dynamical systems. Simulations are conducted over the logistic map, the white noise, the pink noise, and the stock market data to provide comparative study and show the power of this new method. Experiments with the logistic map show that the RR, DET , Lav and EN T R curves of the MSRQA with periodic and chaotic systems have different behaviors about recurrence. The periodic system has the corresponding periodic measures, and the chaotic system contains rich information on the structure of the ORPs over a range of time scales. Compared with conventional RQA, the MSRQA of ORPs with the white noise and pink noise show that this method distinguishes uncorrelated and correlated fluctuations effectively, and reflects more reliable information on the local characteristics of different scales. The influence of series length is also tested. In addition, in order to estimate the efficiency of the MSRQA of ORPs, series of three stock markets, S&P500, DAX and SSECI, are used as outputs of real-world complex system. 8
We find the MSRQA can uncover overall properties of the recurrence of stock markets and characterise the local properties. In Appendix we discuss the situation about the selection of embedding dimension m. Conventional RQA provides some measures to quantify the properties of the recurrence. However, it does not work well for comparing some similar systems, like three stock markets above. The MSRQA provides a new way to discover the intrinsic nature of recurrence under large time scales. It turns out that it does play an important role in analyzing stock markets. However, there are still some areas that need further study. Parameter selection can be further discussed especially for the parameter τ . In this paper, we simply discuss the selection of parameter m in Appendix. The MSRQA can also be applied to the RPs, and in this case there are more problems of parameter selection need to be discussed. And it can be applied to various fields, like machine learning [30], etc.. Unfortunately, the main shortcoming of this method is that it can only handle short sequence as the complexity of the algorithm is high.
Acknowledgement The financial support by the Fundamental Research Funds for the Central Universities (S16JB00030) are gratefully acknowledged.
Appendix Here we discuss the selection of embedding dimension m. Simulated Gaussian distributed (mean 0, variance 1) white noise and 1/f noise (pink noise) time series with length N = 500 are used to illustrate the effects on the MSRQA of ORPs of the choice of parameter m. Thirty independent noise samples are used in each simulation, and the values of RR, DET , Lav and EN T R curves for each sample of one simulation are also all the same.
(a)
(b)
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0.7 m=2 m=3 m=4 m=5 m=6
0.3
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(d)
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ENTR
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m=2 m=3 m=4 m=5 m=6
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Fig. 6. RR (a), DET (b), Lav (c) and EN T R (d) curves of the white noise with different embedding dimension m, plotted versus scale factor s. (a)
(b)
0.6
1
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s
Fig. 7. RR (a), DET (b), Lav (c) and EN T R (d) curves of the pink noise with different embedding dimension m, plotted versus scale factor s.
Fig. 6 and Fig. 7 show the RR, DET , Lav and EN T R curves of the white noise and pink noise with different parameter m respectively. Regarding to the results of the white noise, for larger m, the values of the corresponding curves are generally smaller. For the measure RR, the shapes of curves with different m are similar, but for the measures DET , Lav and EN T R, they have a little change. As for the results of pink noise, the situations are slightly more complicated. For the RR curves, the one with larger m is lower. For the DET , Lav and EN T R curves, when only scale s is small, like s ≤ 3 for DET , the situation of RR happens. The local characteristics of these three measures for pink noise are complex.
We note that the larger m is, the smaller RR is, which means in this situation more information about the recurrence of system lose. When m = 4, the RR curve of the white noise is 10
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HIGHLIGHTS We propose a new method of multiscale recurrence quantification analysis to analyze order recurrence plots. The proposed method is applied to synthetic series and stock market indexes to show their local characteristics. Compared with conventional method, the new method can much reflect richer and more recognizable information of diverse systems.