Multivariate permutation entropy and its application for complexity analysis of chaotic systems

Accepted Manuscript Multivariate permutation entropy and its application for complexity analysis of chaotic systems Shaobo He, Kehui Sun, Huihai Wang PII: DOI: Reference:

S0378-4371(16)30280-1 http://dx.doi.org/10.1016/j.physa.2016.06.012 PHYSA 17205

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Physica A

Received date: 2 July 2015 Revised date: 28 March 2016 Please cite this article as: S. He, K. Sun, H. Wang, Multivariate permutation entropy and its application for complexity analysis of chaotic systems, Physica A (2016), http://dx.doi.org/10.1016/j.physa.2016.06.012 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.

*Highlights (for review)

Highlights 1) Multivariate permutation entropy (MvPE) is proposed for complexity analysis 2) Complexity of different kinds of chaotic systems is analyzed 3) Compared with PE, MvPE is better for analyzing complexity of multivariate systems

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Multivariate permutation entropy and its application for complexity analysis of chaotic systems Shaobo Hea , Kehui Suna,b,∗, Huihai Wanga a School

b School

of Physics and Electronics, Central South University, Changsha 410083, China of Physics Science and Technology, Xinjiang University, Urumqi 830046, China

Abstract To measure the complexity of multivariate systems, the multivariate permutation entropy (MvPE) algorithm is proposed. It is employed to measure complexity of multivariate system in the phase space. As an application, MvPE is applied to analyze the complexity of chaotic systems, including hyperchaotic H´enon map, fractional-order simplified Lorenz system and financial chaotic system. Results show that MvPE algorithm is effective for analyzing the complexity of the multivariate systems. It also shows that fractional-order system does not become more complex with derivative order varying. Compared with PE, MvPE has better robustness for noise and sampling interval, and the results are not affected by different normalization methods. Keywords: permutation entropy, multivariate complexity, simplified Lorenz system, financial chaotic system PACS: 05.45-a, 05.45.Tp 1. Introduction At present, complexity measure is an increasingly important topic [1-21]. There are many practical applications of the complexity measures. For instance, Kaffashi F et al. [2] analyzed complexity of EEG signals during sleep with skin-to-skin contact. More recently, Silva L E V et al. [3] and Mukherjee S et al. [4] measured complexity of heart rate variability and heart failure, respectively. Complexity measure of chaotic systems is another focus of interest since it can elegantly describe dynamics and complexity of chaotic systems. For example, Kolmogorov complexity (KC) of H´enon map was estimated by Grassberger P et al. [5]. Micco L D et al. [6] analyzed complexity of continuous chaotic systems by statistical complexity measure (SCM) [7] and permutation entropy (PE) [8]. Recently, He S B et al. [9] measured complexity of multi-wing chaotic systems by applying SCM and spectral entropy (SE) [10]. There are also several other algorithms which can be used to analyze complexity of chaotic systems or nonlinear time series, such as C0 algorithm [11], Lempel-Ziv algorithm [12], approximate entropy (ApEn) [13]. To our knowledge, these algorithms and reports just focused on measuring complexities of a single time series. However, most of chaotic systems are multivariate and their multivariate complexity is few investigated. To analyze complexity of multivariate time series, multivariate sample entropy (MvSampEn) [14], multivariate neighborhood sample entropy (MN-SampEn) [15], and multivariate fuzzy entropy (MvFuzzyEn) [16] are proposed. But calculation speed of these algorithms is slow. Since PE has faster calculation speed and is more accurate for complexity estimation [17-21], we try to ∗ Corresponding

author, Email: [email protected]

Preprint submitted to Physica A

investigate the multivariate PE (MvPE) algorithm, and apply it to analyze complexity of chaotic systems. The structure of the paper is as follows. In Section 2, MvPE algorithm is designed. In Section 3, complexity of three chaotic systems (a discrete chaotic map and two continuous chaotic systems) is investigated. In Section 4, impacts of sampling interval, noise and normalization methods on MvPE are investigated. Finally, we summarize the conclusions. 2. Design of multivariate complexity algorithm In this section, PE algorithm is presented. MvPE algorithm is proposed. 2.1. Permutation entropy For a given time series {x(n), n = 1, 2, 3, · · · , N} and reconstruction dimension d, the reconstructed series is denoted by X(i) = {x(i), x(i + 1), · · · , x(i + d − 1)},

(1)

xi+r0 ≤ xi+r1 ≤ · · · ≤ xi+rd−1 .

(2)

where i = 1, 2, · · · , N − d + 1. Then X(i) can be arranged in an increasing order π = (r0 , r1 , · · · , rd−1 ), where

Obviously, there are d! possible order patters. Taking d = 3 as an example, there are 6 possible patters: {π1 , x1 ≤ x2 ≤ x3 }, {π2 , x1 ≤ x3 ≤ x2 }, {π3 , x2 ≤ x1 ≤ x3 }, {π4 , x3 ≤ x1 ≤ x2 }, {π5 , x2 ≤ x3 ≤ x1 }, and {π6 , x3 ≤ x2 ≤ x1 }, as shown in Fig.1. If we let π j = j, j = 1, 2, · · · , d!, we can get a patter series {s(i), i = 1, 2, · · · , N − d + 1}. If the order patter of X(i) is π j , then s(i) = j. The Bandt-Pompe probability distribution p(π j ) is denoted by p(π j ) =

# {s |i ≤ N − d + 1; s = j } , N−d+1

(3) March 28, 2016

where the symbol # stands for ”number”. According to the definition of Shannon entropy, the normalized PE is defined as PE(x, d) = S [p]/S max = [−

d! X

p(π j ) ln p(π j )]/S max ,

shown in Fig.3, there are 18 possible patterns (M = 18), and the method for values of each Bot and T op is denoted as (

(4)

j=1

where S max = S [Pe ] = ln(d!), and Pe = {1/d!, · · · , 1/d!}. The range of d is {2, 3, · · · , 7} [7,8]. Generally, larger PE value means the time series is more complex.

Botn = 23 min(X(n)) + 13 max(X(n)) . T opn = 13 min(X(n)) + 23 max(X(n))

(9)

In this paper, sub-patterns π18 will be employed to calculate MvPE of three dimension chaotic systems .

Figure 1: Bandt-Pompe order patterns for embedding dimension d = 3 (π6 )

Figure 2: Ways to obtain vector X(n) for PE and MvPE algorithm (d = 3, σ = 3)

2.2. Multivariate permutation entropy Suppose that sequences generated by a multivariate system are presented as {x j (n), n = 1, 2, 3, · · · , N, j = 1, 2, ..., σ}, where σ is the dimension of the system or the number of chosen time series. As the fluctuation range of each time series is different, we should normalize each series. Here the min-max scaling is employed, which is defined as x˜ j (n) =

x j (n) − min(x j ) , max(x j ) − min(x j )

(5)

where max(·) is the maximum function and min(·) is the minimum function. Then the new reconstructed series is defined as X(n) = { x˜1 (n) , x˜2 (n) , . . . , x˜σ (n)} (6)

for MvPE calculation. Obviously, σ time series of the system are used for complexity measure, and the difference between PE and MvPE to obtain the reconstructed series is visualized in Fig.2 for better understanding of the two algorithms. For each X(n), a certain order patter π can be obtained. So we have the corresponding order patter series {s(n), n = 1, 2, · · · , N} according to the reconstructed series {X(n), n = 1, 2, 3, · · · , N}. The probability distribution p(π j ) is obtained according to p(π j ) =

1 # {s |i ≤ N; s = j } , N

Figure 3: Order patterns for embedding dimension σ = 3 with sub-patters π18

Here, it should be pointed out that complexity of chaotic system refers to the complexity of time series generated by the chaotic system. PE and MvPE algorithms are applied to measure the complexity.

(7)

Remark 1: Chaos based encryption and security communication are interesting topics today, thus it is interesting to investigate complexity of chaotic time series [5-9]. Higher complexity means the system has greater security for the practical applications.

where j=1, 2, · · · , M, and M is the number of the possible patterns. Thus, MvPE is defined as MvPE(x) = S [p]/S max = [−

M X

p(π j ) ln p(π j )]/S max .

(8)

Lemma 1: When we measure complexity of periodic series with period T , the period of patter series s(n) is equal to T . When we measure complexity of σ periodic time series with period T 1 , T 2 , · · · and T σ , the period of patter series s(n) is equal to the least common multiple (T LCM ) of T 1 , T 2 , · · · and Tσ.

j=1

where S max = S [Pe ] = ln(M). Since, order patterns as shown in Fig.1 cannot always detect changes in time series sensitively [21], order patterns with subpatterns are designed in this paper. We divide each pattern to more situations to obtain more details of the time series. As

Proof: According to Eq.(1) and Fig.2, the reconstructed se2

ries of PE can also be presented as                                   

much faster estimation speed. It is more specifically suited to monitoring real-time complexity of multivariable systems.

X(1) = [ x˜(1), x˜(2), · · · , x˜(d)] X(2) = [ x˜(2), x˜(3), · · · , x˜(d + 1)] .. .

3. Applications of MvPE for analyzing the complexity of chaotic systems

X(T ) = [ x˜(T ), x˜(T + 1), · · · , x˜(T + d − 1)] . (10) X(T + 1) = [ x˜(T + 1), x˜(T + 2), · · · , x˜(T + d)] X(T + 2) = [ x˜(T + 2), x˜(T + 3), · · · , x˜(T + d + 1)] .. .

With the development of chaos science, chaotic systems are widely applied in the field of secure communication and information encryption are found. Complexity measure is a quantitative indicator to show the extent of chaotic time series close to random sequences. It has aroused the concern of scholars. In this section, complexity of a discrete chaotic system (hyperchaotic H´enon map) and two kinds of continues chaotic systems (fractional-order simplified Lorenz system and a novel financial chaotic system) is analyzed. For better observation and legibility, we just show the first two Lyapunov exponents λ1 and λ2 of Lyapunov characteristic exponents (LCEs) to present different states.

Obviously, as the time series { x˜(n), n = 1, 2, 3, · · · , N} is periodic, we can get a periodic patter series s(n) according to Eq.(10), and its period is T . As T LCM is the least common multiple of T 1 , T 2 , · · · and T σ , there exists a set of positive number {κ1 , κ2 , · · · , κd } for T LCM = κ1 T 1 = κ2 T 2 = · · · = κσ T σ . According to Eq.(6) and Fig.2, the reconstructed series of MvPE can also be denoted as                                   

X(1) = [ x˜1 (1), x˜2 (1), · · · , x˜σ (1)] X(2) = [ x˜1 (2), x˜2 (2), · · · , x˜σ (2)] .. .

3.1. Hyperchaotic H´enon map The difference equations of hyperchaotic H´enon map [22] are

  x = a − y2n − bzn  X(T LCM ) = [ x˜1 (κ1 T 1 ), x˜2 (κ2 T 2 ), · · · , x˜σ (κσ T σ )] .   n+1 yn+1 = xn , (12)   X(T LCM + 1) = [ x˜1 (κ1 T 1 + 1), x˜2 (κ2 T 2 + 1), · · · , x˜d (κσ T σ + 1)]   z =y n+1 n X(T LCM + 2) = [ x˜1 (κ1 T 1 + 2), x˜2 (κ2 T 2 + 2), · · · , x˜d (κσ T σ + 2)] .. where a and b are bifurcation parameters. By fixing a = 1.42 . (11) and varying b from -0.2 to 0.2 with step size of 0.001, LCEs So the period of obtained patter series s(n) is T LCM . Thus Lemof hyperchaotic map are presented in Fig.4. Obviously, it is ma 1 is proved. hyperchaotic when b ∈[-0.2,-0.183)∪(-0.139, -0.039)∪(-0.027, 0.0148). It is periodic when b ∈[-0.183, -0.139] and it is chaotIt should be noted that PE and MvPE algorithm cannot alic for b ∈[-0.039, -0.027]∪[0.0148, 0.2]. PE and MvPE comways detect periodic state effectively. For example, given a plexity of this system is calculated with a=1.42 and b varying, periodic series {1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 1, 2, · · · }, PE aland results are shown in Fig.5. x series is used to compute the gorithm is applied to measure complexity. When d = 3, the permutation entropy and the length of the time series is 105 . Bandt-Pompe probability distribution P=[0.6, 0.2, 0.2]. When It shows in Fig.5(a) that PE measures complexity better when d=4, the probability distribution P=[0.4, 0.2, 0.2, 0.2]. When d = 5. MvPE based on pattern π6 does not show much differd ≥ 5, the probability distribution P=[0.2, 0.2, 0.2, 0.2, 0.2]. ence for different states. But MvPE based on sub-patterns π18 Obviously, an unsatisfying complexity estimation is obtained can better measure complexity of the phase space in the hyperwhen d ≥ 5. For MvPE, we will get similar analysis results for chaotic H´enon map. The phase diagrams of the hyperchaotic multivariate periodic series. H´enon map are shown in Fig.6. It shows that the system is hyWhen applying PE to measure complexity of the periodic seperchaotic when a = 1.42 and b = −0.1, and the system is ries {1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 1, 2, · · · }, the patter series {s(n)} periodic when a = 1.42 and b = −0.16. is {1, 1, 1, 4, 5, 1, 1, 1, 4, 5, 1, · · · } for d = 3. The period is 5, which is consistent with the result of Lemma 1. Remark 2: Compared with PE, MvPE has following advan3.2. Fractional-order simplified Lorenz system tages. The simplified Lorenz system is proposed in Ref.[23], and (1) MvPE has better noise robustness. It loses its efficacy its fractional-order case is investigated in Ref.[24, 25]. The only when the noise is strong enough to affect the phase space fractional-order simplified Lorenz system is denoted as of the system.  ∗ q (2) MvPE is used to analyze complexity of the phase space,  D x1 = 10(x2 − x1 )    ∗ t0 while PE measures complexity of a single time series. Dqt0 x2 = (24 − 4c)x1 − x1 x3 − cx2 , (13)     ∗ (3) MvPE can better measure complexity of multivariate sysDqt0 x3 = x1 x2 − 8x3 /3 tem based on sub-patterns since order pattern with sub-patterns can obtain more details in the phase pace. where c is the parameter and (x1 , x2 , x3 ) are state variables. ∗ Dqt0 Remark 3: Compared with other multivariate complexity is the Caputo fractional-order derivative with order q, and it is algorithms, such as MvFuzzyEn and MvSampEn, MvPE has defined as [24, 26] 3

where m is an integer number. Fractional-order chaotic system can be solved by applying Adomian decomposition method (ADM) [25, 26] with high accuracy. Dynamics of fractionalorder simplified Lorenz system is investigated in [25] based on ADM. Based on the numerical solution as presented in [25], LCEs of this system can be calculated by QR decomposition method [27]. Two cases of the system are analyzed according to the time series obtained by ADM-based numerical solution. x1 series is used to compute PE . Case 1: Fix q = 0.96, and vary c from -2 to 8 with step size of 0.02. LCEs is calculated as shown in Fig.7 (a). It shows that the system is chaotic over the most range of c ∈[-2, 8]. PE and MvPE results are calculated as shown in Fig.7(b). The length of time series is 105 , and d=5 for PE calculation. PE values illustrate a slight decrease trend with the increase of c. When the system is chaotic, MvPE values do not have a decrease trend. To further analyze complexity of this system, phase diagrams of the system are computed as shown in Fig.8. Obviously, when q=0.96 and c=-2, the system is convergent, when q=0.96 and c=1, the system is chaotic, and the system is periodic when q=0.96 and c=6.5.

Figure 4: LCEs of hyperchaotic H´enon map with parameter b varying

Figure 5: Complexity results of hyperchaotic H´enon map with parameter b varying (a) PE; (b) MvPE

Figure 7: Dynamics of fractional-order simplified Lorenz system with q = 0.96 and c varying (a) LCEs result; (b) Complexity result

Here, MvPE complexity of the fractional-order simplified Lorenz system with derivative order q = 1.0, q = 0.86 and q = 0.76 is also calculated, and the results are shown in Fig.9. It shows that complexity curves with different derivative order q have similar trend, but the effective chaotic range shrinks as the fractional derivative order q decreases. Case 2: Fix c = 5, and vary q from 0.56 to 1 with step size of 0.001. LCEs result is shown in Fig.10(a). It shows that the system is chaotic over the most range of q ∈[0.6, 1], and the maximum Lyapunov exponents decrease with the increase of q. MvPE and PE complexity are calculated with q varying, and

Figure 6: Phase diagrams of hyperchaotic H´enon map with a=1.42 and different b. (a) b = −0.1; (b) b = −0.16

∗

Dqt0 x(t)

    =  

Rt m−q−1 (m) 1 x (τ)dτ, m Γ(m−q) t0 (t − τ) m d dtm x(t), q = m

−1
,

(14) 4

Figure 8: Phase diagrams of fractional-order simplified Lorenz system with q=0.96 and different c. (a) c = −2; (b) c = 1; (c) c = 6.5

Figure 10: Dynamics of fractional-order simplified Lorenz system with c = 5 and q varying (a) LCEs result; (b) Complexity result

Figure 9: Complexity of fractional-order simplified Lorenz system with c varying and q = 1.0, q = 0.86, q = 0.76

the results are shown in Fig.10(b). It shows that PE complexity of the fractional-order simplified Lorenz system decreases with the increase of order q. However, MvPE values keep stable. It also shows that MvPE can detect the periodic state when q ∈[0.56, 0.6]. To observe the dynamic behavior better, phase diagrams are shown in Fig.11. It also shows that the system is convergent when q=0.56 and c=5, the system is periodic when q=0.8487 and c=5, and the system is chaotic for q=0.90 and c=5.

Figure 11: Phase diagrams of fractional-order simplified Lorenz system with c=5 and different q. (a) q = 0.56; (b) q = 0.8487; (c) q = 0.90

3.3. Financial chaotic system A nonlinear financial chaotic system is investigated in Ref.[29], and it is denoted as follows   x˙ = z + (y − a)x    y ˙ = 1 − by − x2 , (15)     z˙ = −x − cz

It is not difficult to find out that the chaotic attractors of the fractional-order simplified Lorenz system are all two-wing attractors. Actually, they are topologically equivalent [28]. At this point of view, complexity of phase space with different c or q should be at the same level. According to Fig.7(b) and Fig.10(b), MvPE values of the fractional-order simplified Lorenz system with q or c varying are all about 0.8 for chaotic attractors. Besides, MvPE can identify periodic states better. Thus MvPE is a good choice for complexity analysis of continues chaotic systems.

where x is the interest rate, and y represents the investment demand, and z denotes the price index. a, b, and c are saving amount, cost per investment and demand elasticity of commercial markets, respectively. Xin et al. [30] introduced the market confidence into the system (15) and a novel financial system is 5

depicted as

              

x˙ = z + (y − a)x + m1 w y˙ = 1 − by − x2 + m2 w , z˙ = −x − cz + m3 w w˙ = −xyz

(16)

where w denotes the market confidence, and m1 , m2 , m3 are the impact factors. Control and stabilization of this system is investigated theoretically, but dynamics with m1 , m2 or m3 varying is not studied. So complexity with m2 is analyzed to show effect of market confidence as the representative. LCEs result and complexity result are shown in Fig.12. The length of time series is N = 105 . PE complexity of variable x is estimated. According to LCEs as shown in Fig.12 (a), system (16) is chaotic when a = 2.1, b = 0.01, c = 2.6, m1 = 8.4, m3 = 2.2 and m2 ∈[5.04, 5.31]∪[6.11, 6.41]. It shows in Fig.12 (b) that MvPE result agrees with that of LCEs, but PE values of periodic states are higher than that of chaotic state. That is to say, PE is failed to estimate complexity of financial chaotic system. To further observe the dynamic behavior, phase diagrams are shown in Fig.13. Periodic state and chaos are observed. It shows that the market confidence also affects the financial system. Actually, a government should take some effective measures to avoid chaotic state and to make the system stable [30]. As for complexity analysis, we think it is a good method to monitor states of the financial system and it can provide a reference for government.

Figure 13: Phase diagrams of financial chaotic system with different m2 . (a) m2 = 5.25; (b) m2 = 6.4

al order simplified Lorenz system with q=0.96 and c=1. The probability distribution and entropy under different sampling interval τ are calculated and the results are shown in Fig.14. Meanwhile, phase diagrams and time series with different τ are shown in Fig.15.

Figure 14: The impact of sampling interval to values of PE and MvPE

As for MvPE, structure of the histogram appears to remain qualitatively the same and values of the single peaks are not changing. MvPE value is calculated according to values of these single peaks based on Shannon concept. So MvPE is not affected by the sampling time. As for PE, indeed, the structure of the histogram appears to remain qualitatively the same. There are more pattern π1 and pattern π6 detected. But with the increase of sampling time, difference between values of each single peak is getting closer. So PE value is getting larger. That is to say, PE is affected by the sampling time. Meanwhile, Ref.[6] discussed the effect of the sampling time (sampling period) to the Bandt-Pompe procedure, and Fig.15 also shows the reasons. As for PE, when the sampling time increases, time series seems more complex. The probability of each pattern is getting closer. However, MvPE measures complexity of phase space, and each point represents a pattern. According to Fig.15 (a), (c) and (e), phase diagrams remain about the same with the increase of the sampling time. In conclude, MvPE is more objective for complexity analysis.

Figure 12: Dynamics of financial chaotic system with m2 varying (a) LCEs result; (b) Complexity result

4. Discussion 4.1. The impact of sampling interval (τ) The impact of sampling interval (τ) to values of PE and MvPE is investigated. Time series is generated by the fraction6

Figure 16: Values of MvPE with different proportion of noise

where mean(x j ) is the mean value of time series x j , and std(x j ) is the variance of time series x j . The median scaling function is defined as   x j (n) − 0.5 max(x j ) + min(x j )   , (20) x˜ j (n) = 0.5 max(x j ) − min(x j )

and the arctan scaling function is given by   2 x˜ j (n) = arctan x j (n) . (21) π Obviously, the min-max scaling, z-score scaling and median scaling are linear method while the arctan scaling is a nonlinear method.

Figure 15: Phase diagrams and time series of fractional-order simplified Lorenz system with different τ. (a) phase diagram with τ = 1; (b) time series with τ = 1; (c) phase diagram with τ = 10; (d) time series with τ = 10; (e) phase diagram with τ = 20; (f) time series with τ = 20

4.2. The impact of noise Here, we investigate the influence of noise on values of PE and MvPE. As with above, time series are generated by the fractional-order simplified Lorenz system with q = 0.96 and c = 1. We add noise signals η˜ 1 , η˜ 2 , η˜ 3 to each of chaotic time series x1 , x2 and x3 , respectively. Let η1,2,3 are uniform noise signals with the same length as x1,2,3 and vary within [0, 1]. Then η˜ j ( j = 1, 2, 3) are obtained by     η˜ j = Pnoise · max(x j ) − min(x j ) · η j − mean(η j ) , (17) where Pnoise is the proportion of noise and it varies from 0 to 0.2 with step size of 0.002 in Fig.16. The noise polluted chaotic signal is denoted as x j = x j + η˜ j . (18)

Figure 17: MvPE complexity with different scaling methods

MvPE complexity of the fractional-order simplified Lorenz system with different scaling methods is calculated, and the results are illustrated in Fig.17. It shows that MvPE results based on z-score scaling and median scaling are consistent with results shown in Fig.7 (b), while MvPE result with arctan scaling just have similar trend. As the linear transformation methods do not change dynamics of the time series, the min-max scaling, z-score scaling or median scaling can be chosen for normalization.

Complexity results with different Pnoise show that PE starts to lose efficacy when the proportion of noise increase to 15%, while values of MvPE are not affected. Thus, compared with PE, MvPE has better noise robustness. 4.3. The impact of normalization method As the scales of different time series are different, it needs to adjust values measured on different scales to a notionally common scale. There are various normalization functions in statistics, such as min-max scaling (denoted as Eq.(5)), z-score scaling, median scaling and arctan scaling. The z-score scaling function is denoted as x j (n) − mean(x j ) x˜ j (n) = , std(x j )

5. Conclusion In this paper, we focus on multivariate complexity analysis of chaotic systems. MvPE algorithm is proposed to measure the complexity. Conclusions are drawn as follows.

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(1) MvPE algorithm is an effective method for complexity analysis of chaotic systems. PE cannot always measure complexity of chaotic systems effectively, but MvPE based on order patterns with sub-patterns can estimate complexity of chaotic systems better. (2) MvPE of hyperchaotic H´enon map is consistent with the maximum Lyapunov exponent. MvPE complexity of the fractional-order simplified Lorenz system does not increase with the decrease of order q. Complexity analysis results show that the market confidence has influence on the state of the financial system. (3) Compared with PE, MvPE has better noise robustness and its results are not affected by sampling interval. Meanwhile, MvPE does not depend on choice of different linear normalization methods. It is a objective complexity estimation algorithm.

17. Li J, Yan J, Liu X, et al. Using permutation entropy to measure the changes in EEG signals during absence seizures [J]. Entropy, 2014, 16(6): 30493061. 18. Unakafova V A, Unakafov A M, Keller K. An approach to comparing Kolmogorov-Sinai and permutation entropy [J]. Eur. Phys. J. Special Topics, 2013, 222(2): 353-361. 19. Li D, Li X, Liang Z, et al. Multiscale permutation entropy analysis of EEG recordings during sevoflurane anesthesia [J]. J. Neural Engine., 2010, 7(4): 046010. 20. Ouyang G, Dang C, Li X. Complexity analysis of EEG data with multiscale permutation entropy[J]. Adv. Cogn. Neurodyn., 2011: 741-745. 21. Yin Y, Shang P. Weighted multiscale permutation entropy of financial time series[J]. Nonl. Dyn., 2014, 78(4): 2921-2939. 22. Baier G, Klein M. Maximum hyperchaos in generalized H´enon maps [J]. Phys. Lett. A, 1990, 151(7): 281-284. 23. Sun K H, Sprott J C. Dynamics of a simplified Lorenz system [J]. Inter. J. Bifur. Chaos, 2011, 19(4): 1357-1366. 24. Sun K H, Wang X, Sprott J C. Bifurcations and chaos in fractional-order simplified Lorenz system [J]. Inter. J. Bifur. Chaos, 2012, 20(4): 12091219. 25. Wang H H, Sun K H, He S B. Characteristic analysis and DSP realization of fractional-order simplified Lorenz system based on Adomian decomposition method [J]. Int. J. Bifur. Chaos, 2015, 25(6): 1550085. 26. Cafagna D, Grassi G. Bifurcation and chaos in the fractional-order Chen system via a time-domain approach [J]. Int. J. Bifur. Chaos, 2008, 18(7): 1845-1863. 27. Caponetto R, Fazzino S. An application of Adomian decomposition for analysis of fractional-order chaotic systems [J]. Inter. J. Bifur. Chaos, 2013, 23(3): 1350050. 28. Letellier C, Aguirre L A. Dynamical analysis of fractional-order R¨ossler and modified Lorenz systems [J]. Phys. Lett. A, 2013, 377(28): 1707-1719. 29. Gao Q, Ma J. Chaos and Hopf bifurcation of a finance system [J]. Nonl. Dyn., 2009, 58(2): 209-216. 30. Xin B, Zhang J. Finite-time stabilizing a fractional-order chaotic financial system with market confidence [J]. Nonl. Dyn., 2015, 79(2): 1399-1409.

Acknowledgements This work supported by the National Natural Science Foundation of China (Grant Nos. 61161006 and 61573383), and the authors would like to thank the editor and the referees for their valuable suggestions. References 1. Horgan J. From complexity to perplexity [J]. Sci. Am., 1995, 272(6): 104109. 2. Kaffashi F, Scher M S, Ludington-Hoe S M, et al. An analysis of the kangaroo care intervention using neonatal EEG complexity: a preliminary study [J]. Clin. Neurophysiol., 2013, 124(2): 238-246. 3. Silva L E V, Cabella B C T, Neves U P D C, et al. Multiscale entropy-based methods for heart rate variability complexity analysis [J]. Phys. A, 2015, 422(422): 143-152. 4. Mukherjee S, Palit S K, Banerjee S, et al. Can complexity decrease in congestive heart failure? [J]. Phys. A, 2015, 439: 93-102. 5. Grassberger P, Procaccia I. Estimation of the Kolmogorov entropy from a chaotic signal [J]. Phys. Rev. A, 1983, 28(4): 2591-2593. 6. Micco L D, Fern´andez J G, Larrondo H A, et al. Sampling period, statistical complexity, and chaotic attractors [J]. Phys. A, 2012, 391(8): 25642575. 7. Lopez-Ruiz R, Mancini H L, Calbe X. A statistical measure of complexity[J]. Phys. Lett. A, 2010, 209(5-6): 321-326. 8. Bandt C, Pompe B. Permutation Entropy: a natural complexity measure for time series [J]. Phys. Rev. Lett., 2002, 88(17): 174102. 9. He S B, Sun K H, Zhu C X. Complexity analyses of multi-wing chaotic systems [J]. Chin. Phys. B, 2013, 22(5): 220-225. 10. Staniczenko P P, Lee C F, Ns. J. Rapidly detecting disorder in rhythmic biological signals: a spectral entropy measure to identify cardiac arrhythmias [J]. Phys. Rev. E, 2009, 79(1): 100-115. 11. Shen E H, Cai Z J, Gu F J. Mathematical foundation of a new complexity measure[J]. Appl. Math. Mech., 2005, 26(9): 1188-1196. 12. Ziv J. and Lempel A. A universal algorithm for sequential data compression [J]. IEEE Trans. Infor. Theory, 1977, 23(3): 337-343, . 13. Pincus S M. Approximate entropy as a measure of system complexity [J]. Proc. Natl. Acad. Sci. USA, 1991, 88(6): 2297-2301. 14. Ahmed M U, Mandic D P. Multivariate multiscale entropy: a tool for complexity analysis of multichannel data [J]. Phys. Rev. E., 2011, 84(6): 30673076. 15. Richman J S. Multivariate neighborhood sample entropy: a method for data reduction and prediction of complex data [J]. Meth. Enzymol, 2011, 487: 397-408. 16. Li P, Liu C Y, Li L P, et al. Multiscale multivariate fuzzy entropy analysis [J]. Acta Phys. Sinica, 2013, 62(62): 120512.

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