Permutation and weighted-permutation entropy analysis for the complexity of nonlinear time series

Commun Nonlinear Sci Numer Simulat 31 (2016) 60–68

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Permutation and weighted-permutation entropy analysis for the complexity of nonlinear time series Jianan Xia∗, Pengjian Shang, Jing Wang, Wenbin Shi Department of Mathematics, School of Science, Beijing Jiaotong University, Beijing, 100044, PR China

a r t i c l e

i n f o

Article history: Received 28 December 2014 Revised 12 June 2015 Accepted 25 July 2015 Available online 30 July 2015 Keywords: Complexity Multiscale permutation entropy Multiscale weighted-permutation entropy Traffic series

a b s t r a c t Permutation entropy (PE) has been recently suggested as a relative measure of complexity in nonlinear systems, such as traffic system and physiology system. A weighted-permutation entropy (WPE) analysis based on the weight assigned to each vector was proposed to consider the amplitude information. We introduce PE/WPE technique to multiple time scales, called multiscale permutation entropy (MSPE)/multiscale weighted-permutation entropy (MSWPE), which are applied to investigate complexities of different traffic series. Both approaches successfully detect the temporal structures of traffic signals and distinguish the differences between workday and weekend time series. © 2015 Elsevier B.V. All rights reserved.

1. Introduction In the era of big data, enormous records can be collected and be badly in need of analysis. Especially, the study on the complexity of real data which is regulated by its environment or mechanism in both spatial and temporal domains, such as traffic series, has drawn considerable attention. Various information-theoretic methods are developed to estimate the complexity, e.g., entropies [1–9], fractal dimensions [10], Lyapunov exponents [11,12], detrended fluctuation analysis [13,14], statistical complexity measures [15], etc. One of the most useful and easily implemented tools is the permutation entropy (PE) [6] which is also applied in this study. Permutation analysis has been applied to time series data for quantifying the degree of complexity based on the appearance of ordinal patterns [16–18]. Ordinal patterns refer to the transformation in which data values are replaced by their rank in data. Due to the ranking, these sequences of ordinal patterns reflect the comparison of neighboring values rather than the actual ones. The advantages of PE are its simplicity, fast calculation, and invariance with respect to nonlinear monotonous transformations. Then, a recent publication describes a modification to generate weighted permutation entropy (WPE) [19] by incorporating amplitude information from the relative order structure, which is not considered in PE. The weighted approach makes it possible to detect the abrupt changes in the data and assigns more weight to the regular spiky patterns [20]. In other word, it is clearly able to differentiate between small fluctuations (may due to effect of noise) and large fluctuations of time series. When analyzing the complexity of time series, PE/WPE analysis may lead to inaccurate or insufficient descriptions of a system. Since many researches take into account the multiple temporal scales, more interesting results can be obtained and compared. The multiscale entropy (MSE) was introduced by Costa et al. [8,9] to measure the complexity of biologic systems and was extended to analyze other systems [18,21–23]. In this paper, we will extend PE and WPE to multiscale analysis called multiscale

∗

Corresponding author. Tel.: +8615120074546. E-mail address: [email protected] (J. Xia).

http://dx.doi.org/10.1016/j.cnsns.2015.07.011 1007-5704/© 2015 Elsevier B.V. All rights reserved.

J. Xia et al. / Commun Nonlinear Sci Numer Simulat 31 (2016) 60–68

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PE (MSPE) and multiscale WPE (MSWPE). Actually, we use PE/WPE to replace sample entropy (SampEn) in original MSE while calculating for each coarse-grained series. In Section 2, we provide the background for calculating the PE and WPE, and the procedures of MSPE and MSWPE. Then we employ the MSPE and MSWPE methods to analyze binomial multifractal model and traffic congestion index (TCI) series, and obtain the interesting results shown in Section 3. Finally, Section 4 presents the conclusions. 2. Methodologies 2.1. Permutation and weighted-permutation entropy Permutation entropy (PE) was introduced by Bandt and Pompe [6] as a complexity measure for nonlinear time series. Consider where N is the series length and generate a vector X jm = {x j , x j+1 , . . . , x j+m−1 } where m is the embedding the time series {xi }N i=1 dimension. Each sequence X jm is sorted in ascending order with permutation pattern πim and there will be m! possible permutations. PE of order m is then defined as:



H (m) = −

i:π

p(πim ) log p(πim )

(1)



m∈ i

and p(πim ) is calculated as: where  denotes the {πim }m! i=1

p(πim ) =

{ j| j = 1, . . . , N − m + 1; X jm has type πim }

(2)

N−m+1

It is clear that PE values between in the range [0, log m!] where a completely increasing or decreasing series has a value of 0 and a completely random system where all m! possible permutations appear with the same probability has the upper bound log m!. It is important to realize that the PE provides means to characterize complexity by the ordinal pattern. However, according to the definition, the permutation entropy neglects the amplitude differences between same ordinal patterns and loses the information about the amplitude of the signal. Recently, a publication develops a modification to generate weighted permutation entropy (WPE) which extracted from a given signal by incorporating amplitude information [19]. The procedure of calculating the WPE is briefly described here. First, each vector X jm is weighted with the weight value wj , instead of being weighted uniformly. The weighted relative frequencies are calculated as follows:



pw (π

m i

j:X jm

)=

has type πim



wj (3)

wj

j≤N−m+1

Note that



wj =

i

pw (πim ) = 1. The weight value wj used in [19] is the variance of each vector X jm as m 1  [x j+k−1 − X¯ jm ]2 m

(4)

k=1

where X¯ jm is the arithmetic mean:

X¯ jm =

m 1  x j+k−1 m

(5)

k=1

Then WPE is computed as:

Hw (m) = −



i:π

pw (πim ) log pw (πim )

(6)



m∈ i

when only w1 = w2 = . . . = wN−m+1 , WPE reduces to PE. 2.2. The multiscale analysis Firstly, we review the theory of multiscale entropy (MSE) analysis briefly. Given a time series {xi }N , we construct consecutive 1 (τ )

coarse-grained time series from the original series with the scale factor τ , {y(τ ) }. Each element y j ) y(τ = 1/τ j

jτ  i=( j−1)τ +1

xi , 1 ≤ j ≤ N/τ

is defined as: (7)

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0.012 −6

2

x 10

0.01 1.5 0.008

1 0.5

0.006

0 0

0.004

100 200 300 400 500

0.002

0 0

1

2

3 4 Time (sample number)

5

6 4

x 10

Fig. 1. The sample of binomial multifractal series with a = 0.75 of length 216 .

For scale one (τ = 1), the time series {y(1) } is the original series. Then, for each given τ , we divide the original series into nonoverlapping segments of length τ and calculate the average of data points in each segment, the length of the corresponding coarse-grained time series is N/τ . Then, we calculate an entropy measure for each coarse-grained time series, and then plot the sample entropies over multiple scale factors. In Costa’s method, the sample entropy (SampEn) was applied in MSE analysis. Therefore, outliers and artifacts of real world time series (such as the traffic time series) affect the SampEn values that deviate from the normal ones. Among other information-theoretic measures, PE is using the ranking information to measure the complexity of time series, and this method was extended to multiple scale analysis. The definition of WPE retains the most properties of PE and also incorporates the influence of amplitude information. Due to its easy calculation, simplicity, robustness and invariance under linear transformations, we extend the WPE to MSWPE to analyze TCI series. 3. Results 3.1. Binomial multifractal series The binomial multifractal time-series is very popular and suitable for study in complex models [24–26], as the structures of this kind of time series are well known. Similar to empirical traffic data, the binomial multi-fractal time-series contains some spiky portions shown in Fig. 1. The given series of N = 2nmax numbers i with i = 1, 2, . . . , N is defined by

xi = an(i−1) (1 − a)nmax −n(i−1)

(8)

where 0.5 < a < 1 is a parameter and n(i) is the number of digits equal to 1 in the binary representation of the index i, e.g., n(13) = 3, since 13 corresponds to binary 1,101. In the following study, we generate binomial multifractal series with a = 0.75 (sample in Fig. 1) and the length of {xi } is set to be 210 and 216 . First, we analyze the behaviors of permutation and weighted-permutation entropy (PE and WPE) for each sliding window of binomial multifractal series (see Fig. 2). Windows of 210 points slide by 29 points were used. It is obvious that no marked change in the results of PE, but WPE drops consistently for the region with fluctuations. Because WPE is applying the variance contributes to magnifying the weight to regular spiky patterns corresponding to a higher amount of information, which results in less complexity. Then we estimate the multiscale PE and WPE (MSPE and MSWPE) of {xi } for different embedding dimension m = 3, 4, 5 which are presented in Fig. 3. Some important observations can be find from subplots and show some resemblance: (i) the curves of MSWPE are a little lower than MSPE (except in Fig. 3(a) at scale τ = 6, 7), but both curves have the similar trend with the increasing scale factor τ . (ii) the values of MSPE with same m for different N are very close. To achieve a reliable statistics, it is necessary to hold N  m! which is the reason why we only consider low orders m = 3, 4, 5. It is noticeable that the values of PE (or WPE) do increase with m. The results are consistent with the ref. [6] that the order m has influence on the outcomes, which means the ranges of complexity measurement increase when factor m increases from 3 to 5. Since WPE is applying the variance contributes to weakening effects of small fluctuations (may be caused by the noise) and magnifying the weight to regular patterns, which lead to more predictability and less complexity. The results basically meet our expectations that the values of WPE are a little lower than PE. Though the trends of MSPE and MSWPE are similar, it is still found that the values of PE change significant when scale τ from 7 to 9, but the behaviors of WPE are not influenced greatly. Besides, the trends of MSPE for different N with various m are no obvious differences. When comparing the subplots with same embedding

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2.4

63

PE (m=5)

WPE (m=5)

2.2

Entropy value

2 PE (m=4)

1.8

WPE (m=4)

1.6 PE (m=3)

1.4

WPE (m=3)

1.2 0

20

40

60 Window

80

100

120

Fig. 2. The values of PE and WPE for binomial multifractal series sample in Fig. 1, and the calculation with windows of 210 points slide by 29 points under m = 3, 4, 5. Table 1 The TCI values are divided into five grades which correspond to five traffic congestion levels, respectively. TCI

Congestion level

(0,2.0] (2.0,4.0] (4.0,6.0] (6.0,8.0] (8.0,10.0]

Unimpeded Basically unimpeded Slightly congested Moderately congested Seriously congested

dimension m, it is found that in Fig. 3 ((a) vs. (b), (c) vs. (d), (e) vs. (f)) the values of PE are almost identical at all scale for different length N. In contrast, just the range of values of WPE is identical for different N. 3.2. Traffic series analysis In this section, we employ MSPE method and MSWPE method to investigate complexities of different traffic congestion index (TCI) series. The TCI is defined in [27] and TCI series are collected per 15 min from 1 January 2010 to 1 January 2012 by Beijing Transportation Research Center [28]. The days of festivals and holidays are not considered because they show abnormal traffic condition from that of usual days. The index reflects the congestion or fluent of the traffic status and ranges from 0 to 10. Different index values correspond to different congestion levels, shown in Table 1. Due to the commuter trip, there are traffic peaks in the morning and evening (morning peak from 7 a.m. to 9 a.m. and evening peak from 5 p.m. to 7 p.m.) of workdays (from Monday to Friday). But maybe there is one-peak pattern or two-closing-peak pattern in weekend (Saturday and Sunday). Fig. 4 shows a part of TCI values from 1 May 2011 to 31 May 2011. The results are in accord with our intuition that a week is divided into workday and weekend. Moreover, this classification has been widely used in the study and prediction of TCI series. The basic statistics of the data used are shown in Table 2. Then, we analyze the TCI series by using above-mentioned method, MSPE and MSWPE, respectively. But we should note that the length of coarse-grained time series N/τ and the embedding dimension m must satisfy N/τ  m!. From Monday to Sunday, the total number of time series used here is around 8000 to 9000 and the scale τ is up to 15, that make the shortest coarsegrained series is about 550 points. And 550  5!, so we set m = 3, 4, 5 in this research. Fig. 5 describes the PE and WPE values for each day under m = 3, 4, 5. The results of the MSPE and MSWPE analysis of the seven TCI time series are presented in Fig. 6 (for m = 3), Fig. 7 (for m = 4) and Fig. 8 (for m = 5), respectively. In Figs. 6–8 we also can find that the values of PE are larger than the ones of WPE for every weekday time series with any m, which is demonstrated in Fig. 5 clearly and is caused by weakening effects of small fluctuations in WPE method. However,

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J. Xia et al. / Commun Nonlinear Sci Numer Simulat 31 (2016) 60–68

1 MSPE

0

5

MSWPE

10 (c) N=210, m=4

3 2 1 0

5

10

2

1

15 Entropy value

Entropy value

(b) N=216, m=3 Entropy value

Entropy value

(a) N=210, m=3 2

0

5

0

5

10 (d) N=216, m=4

15

10

15

3 2 1

15

16

10

(f) N=2 , m=5 Entropy value

Entropy value

(e) N=2 , m=5 4 3 2 1 0

5

10 Scale factor τ

15

4 3 2 1 0

5

10 Scale factor τ

15

Fig. 3. It shows the results of MSPE and MSWPE for different embedding dimension and also presents the MSE analysis of two series of distinct lengths. Left (for length N = 1010 ): (a) m = 3, (c) m = 4, and (e) m = 5; right (for length N = 1016 ): (b) m = 3, (d) m = 4, and (f) m = 5.

Table 2 The basic statistics (total number, mean, standard deviation) of the data from Monday to Sunday. We eliminate the data which is collected in bad weather conditions such as torrential rain, and any other extreme events like festivals. Days

Number

Mean

Std

Monday Tuesday Wednesday Thursday Friday Saturday Sunday

8439 8922 8917 9097 9213 8352 8160

2.42 2.51 2.52 2.62 1.79 2.40 1.93

1.93 1.92 1.89 2.00 2.08 1.82 1.27

the broad range of WPE values with multiscale analysis show more details which concern to the complexity of TCI series. The differences between them, PE and WPE, are slowly decrease with scale τ , which means that the more loss of effect of small fluctuations (or detail information) with increasing scale τ . That is to say, the complexities of TCI series at large scale, which measured by PE and WPE, tend to be similar. Because in the coarse-graining procedure, do the smoothing of the fluctuations to original series which result in the loss of small fluctuations. The larger scale is, the less detail information will be. Then, the values of PE and WPE are basically same at larger scale. The maximum difference between PE and WPE usually at scale one for workdays, but may be at scale τ = 2 or 3 for weekend. Also, there are some similarities in MSPE and MSWPE analysis of TCI time series in Figs. 6–8. When comparing these six subplots, it is found that the five workday time series are almost overlapped when scale τ ≤ 8 and the behaviors of weekend series are almost same but different from workday series. The PE (or WPE) value of each workday TCI time series are nearly same at scale one which means the complexity measurements of each workday series are difficult to discern. While the WPE values of weekend series at scale one is larger than these values of workday, which indicate the weekend series have higher complexity. Meanwhile, the differences between entropy values of weekend and weekday series in WPE are larger than that in PE. Apparently, the measurement at scale one just stands for the uncertainty degree of TCI series under specific scale. So the WPE method in multiscale analysis is new way to explore the intrinsic complexity of TCI series. In multiscale entropy analysis, we can observe from left portion of Figs. 6–8 that all seven curves decrease to reach their minimum values at scale factor τ = 3, and then workday series increase with a drop at τ = 12 (in Fig. 6 (left)), increase with a drop at τ = 8 (in Fig. 7 (left)) and fluctuate with

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65

10

8

TCI value

6

4

2

0

0

500

1000 1500 2000 From 2011/5/1 to 2011/5/31

2500

3000

Fig. 4. Describes part of the original TCI series, which are collected per 15 min from 1 May 2011 to 31 May 2011.

Entropy value

4

PE(m=5)

3 PE(m=4)

WPE(m=5)

WPE(m=4)

2 PE(m=3)

WPE(m=3) 1

Mon

Tue

Wed

Thu Days

Fri

Sat

Sun

Fig. 5. The values of PE and WPE for each day under m = 3, 4, 5 at scale one. The values of PE for every weekday time series with m = 3 or 4 or 5 are larger than the ones of WPE, but barely fluctuate. The values of WPE for Saturday and Sunday are larger than other obviously, which means the complexity of weekend (Saturday and Sunday) is higher than that of workday.

scale (in Fig. 8(left)), while weekend series increase and start to decrease from τ = 9, respectively (except there is a tiny drop at τ = 6 and τ = 8 in Fig. 8 (left)). In contrast, as shown in right portion of Figs. 6–8 we just find that all weekend curves and the workdays curves (with m = 3) decrease from scale one to scale two, and then all curves increase with scale, but a drop appears at scale τ = 12. It is obvious that the trends of MSPE analysis for weekend are same with diverse embedding dimension m (first fall, and then increase, at last decrease again). For the workday case, the trends is same as weekend trends when m = 5, but m = 3 or 4 (first fall, and then increase, at last stable). In contrast, the trends of MSWPE curves of each TCI series increase gradually with scale τ

66

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MSPE (m=3)

MSWPE (m=3)

1.85

1.8

1.8

1.7 1.6

1.75

Entropy value

Entropy value

1.5 1.7 1.65 1.6

Mon

1.4 1.3 1.2

Tue Wed

1.55

Thu

1.1

Fri Sat

1.5

1

Sun

0

5

10 Scale factor τ

0.9

15

0

5

10 Scale factor τ

15

Fig. 6. MSPE (left) and MSWPE (right) analysis of seven TCI time series for m = 3. The blue solid lines with different symbols represent workdays (Monday (◦), Tuesday (), Wednesday (♦), Thursday (), and Friday ()), and the red dashed lines represent weekend (Saturday (◦), and Sunday ()). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article).

MSPE (m=4)

MSWPE (m=4)

3

3

2.9

2.8 2.6

2.8

Entropy value

Entropy value

2.4 2.7 2.6 2.5

Mon

2.2 2 1.8

Tue Wed

2.4

Thu

1.6

Fri Sat

2.3

1.4

Sun

2.2

0

5

10 Scale factor τ

15

0

5

10 Scale factor τ

15

Fig. 7. MSPE (left) and MSWPE (right) analysis of seven TCI time series for m = 4. Monday (◦), Tuesday (), Wednesday (♦), Thursday (), and Friday () with blue solid lines, Saturday (◦), and Sunday () with red dashed lines. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article).

under the condition of different m. Both results are helpful to characterize the multiscale properties of each TCI time series and distinguish the workday and weekend series. But the MSPE method will be influenced by the parameter m, which in other word means the possible permutations m! increase may lead to important vector weight reduction. This problem can be overcome by MSWPE method which assigns more weight to important vectors and less weight to unimportant ones. It’s reasonable to believe that MSWPE approach is steady and reliable in TCI series analysis.

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MSPE (m=5)

67

MSWPE (m=5)

4.6

4 Mon Tue

4.4

Wed

3.5

Thu

4.2

Fri Sun

Entropy value

Entropy value

Sat

4

3.8

3

2.5

3.6 2 3.4

3.2

0

5

10 Scale factor τ

15

1.5

0

5

10 Scale factor τ

15

Fig. 8. MSPE (left) and MSWPE (right) analysis of seven TCI time series for m = 5. Monday (◦), Tuesday (), Wednesday (♦), Thursday (), and Friday () with blue solid lines, Saturday (◦), and Sunday () with red dashed lines. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article).

What’s more, the weekend series obtain higher values of PE than the workday series when τ < 9, while in the case of WPE, it just happens at scale τ ≤ 2. These indicate that the local-order structure of weekend series are more complex than that of the workday series when τ < 9, in the MSPE analysis, but results are reversed when τ ≥ 9. Nevertheless, the MSWPE analysis show that the complexities of pattern structure of workday series are more than that of the weekend series, which may be due to the busy and various daily traffic mode in workday and random or simple traffic plans in weekend. These results are quite consistent with our earlier search of MSE analysis [18]. We believed that MSWPE is better capable of reflecting the complexity changing in each scale and distinguishing the workday and weekend series. The structure of weekend series is fairly less complex than that of the workday series in our common sense, which can be found in MSWPE analysis. Also, we find that the MSWPE approach shows some distinction in workday series at large scales, which can help to do deep research on characterizing workday series (difference from Monday to Friday) and studying the complexity of each day. 4. Conclusions In this paper, we investigate the complexities of multifractal binomial series and TCI time series by applying the MSPE and MSWPE methods, with comparison between the results of different embedding dimension m. In multifractal binomial series analysis, we find that WPE is capable of reflecting the significant fluctuations of signal, e.g. spiky regions. The multiscale analyses results show that the range of entropy values of both methods do increase with embedding dimension obviously. In addition, the values of MSWPE curves are relatively lower than MSPE curves since MSWPE method incorporates amplitude information to weaken the effect of noise. While analyzing TCI time series, the results from the MSPE and MSWPE methods are a little different, we can still get that the weekend series have different behaviors from the workday series, which means that we could divide the TCI series into workday group (from Monday to Friday) and weekend group (Saturday and Sunday). The complexities of TCI series at scale one, which measured by PE and WPE, deviate greatly from each other, but tend to be similar at large scale. Even so, the large range of WPE values show more details which concern to the complexity of TCI series. However, the MSWPE approach is much stable with changing m, and shows some distinction in workday series at large scales. In brief, permutation entropy as a simple and fast calculation measure for time series complexity has been confirmed on real data, while WPE retains most of PE’s properties and shows more robustness to noise. But, further work is still required to focus on the limitation and potential of PE in multiscale analysis, such as MSPE changes with embedding dimension. And the study of information interaction between systems based on PE is developing. According to the better performance of MSWPE method, it can be used to measure the complexity and dependence of two systems, such as traffic volume vs. speed. Acknowledgments Financial support by China National Science (61371130, 61304145), and Beijing National Science (4122059) are gratefully acknowledged.

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