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Eur. Phys. J. Special Topics 227, 943–957 (2018) c EDP Sciences, Springer-Verlag GmbH Germany,

part of Springer Nature, 2018 https://doi.org/10.1140/epjst/e2018-700098-x

THE EUROPEAN PHYSICAL JOURNAL SPECIAL TOPICS

Regular Article

Fractional fuzzy entropy algorithm and the complexity analysis for nonlinear time series Shaobo He1,2,a , Kehui Sun2,b , and Rixing Wang3 1

2

3

School of Computer Science and Technology, Hunan University of Arts and Science, Changde 415000, P.R. China School of Physics and Electronics, Central South University, Changsha 410083, P.R. China Normal College, Hunan University of Arts and Science, Changde 415000, P.R. China Received 20 October 2017 / Received in final form 26 January 2018 Published online 19 October 2018 Abstract. In this paper, fractional fuzzy entropy (FFuzzyEn) algorithm is designed by combing the concept of fractional information and fuzzy entropy (FuzzyEn) algorithm. Complexity of chaotic systems is analyzed and parameter choice of FFuzzyEn is investigated. It also shows that FFuzzyEn is effective for measuring dynamics of nonlinear time series and has better comparing results for different time series. Moreover, changes in the complexity of EEG signals from normal health persons and epileptic patients are observed. The results show that, compared with normal health persons, epileptic patients have the lowest complexity during seizure activity and relative lower complexity during seizure free intervals. The proposed method may be useful for EEG signal based physiological and biomedical analysis.

1 Introduction The most direct way to analyze the real world is to analyze time series from the real systems by means of entropy or complexity measuring methods. At present, investigating complexity in various nonlinear time series has aroused the concern of scholars, e.g., electroencephalogram (EEG) [1], electrocardiosignal (ECG) [2], electromyography (EMG) [3], traffic flow series [4], fault diagnosis signal [5], earthquake signal [6], etc,. Various information-theoretic methods are developed to measuring the complexity of time series, such as entropy measuring algorithms [11], complexity measuring algorithms [12], Lyapunov characteristic exponents (LCEs) [13] and fractal dimensions [14]. Among them, entropy algorithms are the most useful and easy implementation method. Pincus [15] proposed approximate entropy (ApEn) algorithm as a measure of complexity for short and relatively noisy time series. However, ApEn counts each sequence as matching itself the occurrence of ln(0), and it leads a bias in the calculations. To relieve the bias caused by self-matching, Richman and Moorman [16] investigated the mechanism responsible for the bias and proposed another statistic method, sample entropy (SampEn). SampEn displays relative consistency and less dependence on data length. Nevertheless, the similarity definition of a b

e-mail: heshaobo [email protected] e-mail: [email protected]

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vectors in SampEn is based on the Heaviside function as with ApEn. In 2007, Chen et al. [17] modified the two algorithms and proposed Fuzzy entropy (FuzzyEn). They defined the similarity degree of two neighboring vectors by using fuzzy function which makes the results more credible. It has been widely used to calculate complexity of nonlinear time series [18]. Recently, some modified FuzzyEn algorithms, such as multiscale FuzzyEn [19], multivariate multiscale FuzzyEn [20], multivariate generalized multiscale FuzzyEn [21], are proposed to analyze complexity of nonlinear time series. Although FuzzyEn can obtain more satisfying measure results, it is still an interesting topic to design new complexity measure based on FuzzyEn algorithm. Fractional calculus was introduced by Leibniz in mathematics and now it is widely used in the areas of biology, physics and engineering [22–24]. By introducing a power function which is related with factional Fourier and Z transforms, a distinct fractional approach for information and entropy, namely, fractional-order generalized entropy, was proposed and investigated [25]. Whereafter, Lopes et al. [26] investigated fractional-order entropy of earthquake data series, and Xu et al. [27] proposed weighted fractional permutation entropy and fractional sample entropy to analyze dynamics of nonlinear Potts financial system. Actually, the research of fractionalorder generalized entropy on complexity of nonlinear time series is still in exploratory stage and there are many problems to be solved before applying it in real applications. At present, there are a large number of the people in the world suffering from epilepsy. Measuring complexity of EEG signal is a useful way to detect the epileptic. Today there is a large body of literatures exploring how complexity changes when human suffer from epilepsy by analyzing complexity of EEG signals [7–10]. However, these technologies are still a long way off from being able to detect epilepsy for physiological and biomedical analysis and we think it is still an interesting topic to measuring complexity of EEG signals. In this paper, we focus on designing of a new entropy measure algorithm based on concept of fractional order generalized entropy and apply the proposed algorithm to measure the complexity of nonlinear time series including chaotic time series and EEG signals. The rest of the paper is organized as follows. In Section 2, fractional FuzzyEn (FFuzzyEn) is proposed by introducing the concept of fractional order generalized entropy, and relation between fractional-order and measure result is discussed. In Section 3, complexity of nonlinear time series including chaotic time series and EEG signal is analyzed. Meanwhile, parameter choice and characteristics of FFuzzyEn are investigated. Section 4 is the summery of the whole analysis.

2 Fractional Fuzzy Entropy Algorithm 2.1 Fuzzy entropy algorithm For a given time series {xN (n), n = 0, 1, 2, . . . , N − 1}, FuzzyEn [17] is calculated by the following steps. Step 1: preparation of the time series. To make the measure results more reliable, data pre-processing of the target time series should be carried out. The following normalization method is applied x (n) − mean (x) x ˆ (n) = . (1) std (x) Here, the mean value of time series x ˆ (n) is zero and its variance is one. Step 2: form the vector as X(i) = [ˆ x(i), x ˆ(i + 1), . . . , x ˆ(i + m − 1)],

(2)

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where i = 1, . . . , N − m + 1 , and X(i) represents m consecutive x values, commencing with the ith point. Step 3: denote the distance between X(i) and X(j) (j = 1, 2, ..., N − m + 1, j 6= i) as d[X(i), X(j)], which is defined by d[X(i), X(j)] = max(|X(i) − X(j)|). (3) m By introducing a fuzzy function, we can get the similarity degree Dij between X(i) and X(j), and it is defined by  2 ! d[X(i), X(j)] m Dij = exp − ln (2) , (4) r

where parameter r is the similarity tolerance. Step 4: then we get Cim (r) = (N − m)−1

N −m+1 X

m Dij .

(5)

j=1,j6=i

Construct Φm (r) = (N − m + 1)−1

N −m+1 X

Cim (r).

(6)

i=1

Step 5: increase the dimension m to m + 1, repeat the above steps and calculate Cim+1 (r) and Φm+1 (r). In practice, the number of data points N is finite, so FuzzyEn is defined by [17]
Φm+1 (r) . FuzzyEn m, r, xN = − ln m Φ (r)

(7)

Taking advantages of the concept of fuzzy sets, FuzzyEn yields more satisfying results than ApEn and SampEn for complexity measure of different nonlinear signals. 2.2 Fractional fuzzy entropy algorithm Recently a generalized expression of Shannon entropy is proposed by considering the fractional calculus, and it is defined by [25] Sα =

X

 pi −

i

 p−α i [ln pi + ψ (1) − ψ (1 − α)] , Γ (α + 1)

(8)

where α is the fractional derivative order, and Γ (·) and ψ(·) represent the gamma and digamma functions. Moreover, the fractional-order information of order α can be denoted as [25] Iα = −

p−α i [ln pi + ψ (1) − ψ (1 − α)] . Γ (α + 1)

(9)

It should be pointed out that the fractional entropy is a novel expression for entropy inspired in the properties of fractional calculus. In fact, Shannon entropy is a special case of fractional entropy.

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By employing the concept of fractional-order information, the FFuzzyEn is defined as N

FFuzzyEn m, r, α, x




 =−

Φm+1 (r) Φm (r)

−α

m+1

(r) ln ΦΦm (r) + ψ (1) − ψ (1 − α)

Γ (1 + α)

.

(10)

Obviously, when α = 0, FFuzzyEn is equivalent to FuzzyEn. Let p = Φm+1 (r) Φm (r), then ( FuzzyEn (p) = − ln p FFuzzyEn (p, α) = −

p−α Γ (α+1)

[ln p + ψ (1) − ψ (1 − α)]

,

(11)

where p is calculated from the time series under m and r, and 0 < p < 1. When time series is a constant, p = 1. Theorem 1: for a given constant time series {xN (n), n = 0, 1, 2, . . . , N − 1}, if p is  m m+1 calculated by p = Φ (r) Φ (r) , we have p = 1. Proof. When the time series is constant, for all i and j, d[X(i), X(j)] = m max(|X(i) − X(j)|) = 0 , then Dij = exp(− ln(2).(d[X(i), X(j)]/r)2 ) = 1. According to equations (5) and (6), the following calculation is carried out

Cim (r) = (N − m)−1

N −m+1 X

m Dij =

j=1,j6=i

Φm (r) = (N − m + 1)−1

N −m+1 X

Cim (r) =

i=1

1 × (N − m) = 1, (N − m)

1 × (N − m + 1) = 1. (N − m + 1)

(12)

(13)

m+1

(r) Similarly, Φm+1 (r) = 1. Thus p = ΦΦm (r) = 1. When p varies from 0.1 to 1 and α varies from −1 to 0.6, the values of FFuzzyEn in the p − α plane are shown in Figure 1a. Obviously, the high complexity region is found when p ∈ [0.1, 0.5) and α ∈ (−0.4, 0.5). Actually, when p varies from zero, the time series has much higher complexity. Moreover, let p takes different values from 0.1 to 1 with step size of 0.1, FFuzzyEn with different S fractional-order α is illustrated in Figure 1b. It shows that when α ∈ (−1, −0.5) (0.55, 0.6), FFuzzyEn cannot distinguish different complexities. However, when α ∈ (0.1, 0.3) and there are larger differences between different time series with small p values (high complexity) by FFuzzyEn comparing with that of FuzzyEn. Since there is another parameter α corresponding to FFuzzyEn which needs to estimate properly and two associated gamma and digamma functions, it is a little bit complicated compared with FuzzyEn. However, according to above analysis, FFuzzyEn complexity measurement is more accurate, especially those time series with high complexity.

3 Applications of FFuzzyEn for nonlinear time series In this section, complexity of chaotic systems and EEG signals is analyzed. Since complexity of chaotic systems can also be analyzed by Lyapunov character exponents

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Fig. 1. FFuzzyEn values versus α and p (a) FFuzzyEn in the p − α plane; (b) FuzzyEn with different α.

(LCEs), we investigate the effectiveness of FFuzzyEn by comparing the results of LCEs and FFuzzyEn. Meanwhile, EEG signal plays an important role in the diagnosis of epilepsy, and fractional complexity EEG signal of epilepsy patients is analyzed as another application of FFuzzyEn. 3.1 Complexity analysis of chaotic systems Chaotic time series generated by two discrete chaotic systems, namely, Logistic map and a 2D Sine iterative chaotic map with infinite collapse (2D-SIMM map) [28,29], and by a continuous chaotic system which is the fractional-order simplified Lorenz hyperchaotic system [30], are used to test FFuzzyEn algorithm. Logistic map is denoted as x (n) = µx (n − 1) (1 − x (n − 1)) ,

(14)

in which x (n) ∈ (0, 1) and µ ∈ (0, 4] is the system parameter. When µ > 3.56994, the system is chaotic. 2D-SIMM map is a high complexity chaotic system and it is given by [28]     x (n + 1) = a sin (ωy (n)) sin b x(n)   , (15)  y (n + 1) = a sin (ωx (n + 1)) sin b y(n) where parameters (a, b, ω, x0 , y0 ) are seeds of the system which decide the states of the system. The fractional-order simplified Lorenz hyperchaotic system [30] is defined as  q D x = 10(y − x)   tq0 Dt0 y = (24 − 4c)x − xz + cy + w , (16) q   Dtq0 z = xy − 8z/3 Dt0 u = −ku where x, y, z, w are the state variables. k and c are the system parameters and q is the fractional order. Numerical solution and dynamics of fractional-order simplified Lorenz hyperchaotic are investigated in reference [30] by employing Adomian decomposition method and it shows that the system has rich dynamical behaviors. When µ = 4, the phase diagram of Logistic map is illustrated in Figure 2a. Let (a, b, ω, x0 , y0 ) = (1, 10, π, 0.45, 0.95), the phase diagram of 2D-SIMM map is shown in

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Fig. 2. Phase diagrams of different chaotic systems (a) Logistic map; (b) 2D-SIMM map; (c) fractional-order simplified Lorenz hyperchaotic system.

Figure 2b, while the strange attractor of the fractional-order simplified Lorenz hyperchaotic system is shown in Figure 2c with q = 0.98, k = 5 and c = −1. Comparing Figures 2a and 2b, we find that 2D-SIMM map has higher complexity than that of Logistic map. Fractional complexity of these chaotic systems will be analyzed in the next section. 3.1.1 Parameter choice of FFuzzyEn There are four parameters in the FFuzzyEn algorithm which are reconstruction fractional order α, dimension m, tolerance r, and sequence length N . FFuzzyEn complexities versus different parameters are shown in Figures 3 and 4. Three pieces of chaotic time series corresponding to the three phase diagrams as shown in Figure 2 are used. According to Figure 2, phase diagram of 2D-SIMM map is more complex and the system under these parameters should have higher complexity. As a result, according to Figures 3 and 4, complexities of different systems, in descending order, are 2D-SIMM map, Logistic map and fractional-order simplified Lorenz hyperchaotic system. In Figure 3a, fix m = 3, r = 0.15, N = 1000, while α varies from −1 to 0.6, and it shows that FFuzzyEn can well distinguish different chaotic signals when α ∈ (−0.1, 0.5). However, when α ∈ (0.1, 0.3), larger difference is found between different chaotic systems comparing other values of α including the case α = 0. Time series with high complexity should choose relative larger α, while the time series with low complexity should choose relative smaller values in α ∈ (−0.1, 0.3). Another method to choose a proper fractional order α is given by the following method. According to equation (11), FFuzzyEn measuring results depend on the ratio p. For each p, we increase α from −1 to 0.6 with step size of 0.001, and calculate the complexity of each case. Then we can find the fractional-order α corresponding to the maximum measuring result. The result is shown in Figure 3b, where the red line represents the original relationship and the blue line is the fitting result. The fitting result is given by αmax = −0.9895p3 + 2.3482p2 − 2.4366p + 0.6112.

(17)

In real calculation, when ratio p is obtained, the fractional order α can be calculated according to equation (17), automatically. FFuzzyEn measuring results with different reconstruction dimension m, tolerance r, and sequence length N are shown in Figure 4, where the value of fractional order α in Figures 2a–2c is 0.2 and the value of fractional order α in Figures 2d–2f is αmax . In Figures 4a and 4d, r = 0.15, N = 1000, m varies from 1 to 10, and it shows that when m=1 to 10, the measure results are all correct. It means that FFuzzyEn is robust with dimension m. In this paper, we set m = 3. In Figures 4b and 4e, m = 3, N = 1000,

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Fig. 3. Choice of fractional order α (a) FFuzzyEn versus different fractional order α; (b) relationship between fractional order α and ratio p.

Fig. 4. FFuzzyEn of chaotic systems with different algorithm parameters (a) FFuzzyEn with α = 0.2 and reconstruction dimension m varying; (b) FFuzzyEn with α = 0.2 and tolerance r varying; (c) FFuzzyEn with α = 0.2 and sequence length N varying (d) FFuzzyEn with αmax and reconstruction dimension m varying; (e) FFuzzyEn with αmax and tolerance r varying; (f) FFuzzyEn with αmax and sequence length N varying.

r varies from 0.1 to 1. It indicates that r should be takes values between 0.15 and 0.5 for more satisfying measure results. Here, r = 0.15. In Figures 4c and 4f, m = 3, r = 0.15, and N varies from 50 to 1500. It shows that when N is larger than 200, then the measure result is stable. Thus for FFuzzyEn, the length of the time series should be larger than 200. In this paper, we let N = 1000. Moreover, compared with measuring results with α = 0.2, FFuzzyEn measuring results with αmax are relatively larger. However, both cases can be used in the real applications. 3.1.2 Complexity analysis of chaotic systems Complexity of Logistic map is analyzed by varying parameter as shown in Figure 5. It can be seen from Figure 5 that FFuzzyEn measure results agree well with LCEs, and it means that the proposed FFuzzyEn is an effective method for complexity analysis of chaotic systems. When fractional order is αmax , the measure results do not agree well with LCEs as other cases that α = 0, 0.1 and 0.2. The main reason is that the measure results of this case are maximum ones. It also shows that smaller α leads lager measure values, and larger fluctuation is observed when α is larger.

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Fig. 5. Complexity analysis results of Logistic map with parameter µ varying.

Fig. 6. Complexity analysis results of 2D-SIMM map with parameter a varying (a) LCEs; (b) FFuzzyEn.

Complexity analysis results of 2D-SIMM map with parameter a varying are illustrated in Figure 6. The system has higher complexity when it is chaotic or hyperchaotic, and periodic windows are observed via both LCEs and FuzzyEn. However, according to Figure 6, the system does not have higher complexity when it is hyperchaotic. Being different from measuring results of Logistic map, smaller α leads smaller FFuzzyEn measure values. According to Figure 1b, FFuzzyEn increases with the increase of fractional order α, and FFuzzyEn increases with the increase of fractional order after a certain value of α. The inflection value increases with the decrease of ratio p or the increase of complexity. Since 2D-SIMM map is more complex than Logistic map, FFuzzyEn increase with the increase of fractional order (α < 0.2). This is the reason why FFuzzyEn complexity of 2D-SIMM map increase with the increase of fractional order α. Moreover, FFuzzyEn with αmax has good measure result with α = 0.2. In conclusion, FFuzzyEn can be used to analyze chaotic system with high complexity and αmax is a good choice for high complexity time series. LCEs and FFuzzyEn complexity of the fractional-order simplified Lorenz hyperchaotic system with derivative order q varying is calculated and the results are illustrated in Figure 7. It shows in Figure 7a that when q > 0.75, the system is hyperchaotic and the maximum LCEs (MLCEs) decrease with the increase of fractional order q. Meanwhile, FFuzzyEn complexity of fractional-order simplified Lorenz hyperchaotic system has the same trend as MLCEs when derivative order increases. Thus FFuzzyEn can be applied to analyze complexity of continuous chaotic systems.

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Fig. 7. Complexity analysis results of fractional-order simplified Lorenz hyperchaotic system (a) LCEs versus derivative order q; (b) FFuzzyEn versus derivative order q. Table 1. Correlation coefficient between MLCEs and FuzzyEn of different systems.

FFuzzyEn(αmax ) FFuzzyEn (α = 0) FFuzzyEn (α = 0.2) FFuzzyEn (α = 0.3)

MLCEs (µ)

MLCEs(a)

MLCEs(q)

0.8930 0.9058 0.9018 0.8973

0.2271 0.3427 0.3131 0.2872

0.8369 0.8348 0.8344 0.8341

3.1.3 Relationship between MLCEs and FFuzzyEn As shown above, FFuzzyEn measuring results agree well with that of MLCEs. The relationship between MLCEs and FFuzzyEn is investigated by calculating correlation coefficients between them as shown in Table 1, and by plotting their diagrams as shown in Figure 8. It should be pointed out that FFuzzyEn and LCEs values of Logistic map are chosen when µ > 3.6 while that of fractional-order simplified Lorenz hyperchaotic system are chosen when q > 0.75. Correlation coefficient is an index for measuring statistical relationships between FFuzzyEn measuring results and MLCEs. Positive liner relation is observed when the value of correlation coefficient is close to one. As shown in Table 1, the first column is the correlation coefficient between MLCEs and FFuzzyEn of Logistic map with different value of α, while the second and third columns are the correlation coefficient between MLCEs and FFuzzyEn of 2D-SIMM map and fractional-order simplified Lorenz hyperchaotic system, respectively. For Logistic map, its correlation coefficient is about 0.9, and correlation coefficient of the fractional-order simplified Lorenz hyperchaotic system is about 0.8. FuzzyEn measuring results and MLCEs have strong positive liner relation which is also verified by Figure 8. However, it is shown in Table 1 that correlation coefficient in 2D-SIMM map is about 0.3 which means that there is no positive liner relation between MLCEs and FFuzzyEn. According to Figure 6, FFuzzyEn decrease with the increase of LCEs when 2D-SIMM map is hyperchaotic, which results the low correlation coefficients. Meanwhile, the corresponding FFuzzyEn-MLCEs plot is not presented. All in all, it further shows that FFuzzyEn is an effective method to analyze complexity of nonlinear time series. 3.2 Complexity analysis of EEG signal As an application of FFuzzyEn, complexity analysis of EEG signal is carried out to diagnose epileptic patients who are during free intervals and during seizure activity intervals.

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Fig. 8. FFuzzyEn-MLCEs plots of different systems (a) Logistic map; (b) fractional-order simplified Lorenz hyperchaotic system. Table 2. Descriptions about the chosen data sets. Data sets

Data description

Set A

Signals from five health volunteers relaxed in a wake state with eye open, data is presented as A001–A100 Signals from epileptic patients during seizure free intervals, data is presented as C001–C100 Signals from epileptic patients during seizure activity, data is presented as E001–E100

Set B Set C

3.2.1 Datasets In this study, we use data from the Department of Epileptology at the University Hospital of Bonn, freely available at http://epileptologie-bonn.de/ and more details about these data were given in reference [31]. There are five sets data which are denoted as A–E, and each data contains 100 single-channel EEG segments with a segment duration of 23.6 s. These data segments are selected from the continuous multichannel EEG recordings after visual inspection for artifacts, e.g., due to muscle activity or eye movements. Set A and Set B are taken from five health and relaxed in an awake state and volunteers, with eye open and closed, respectively. In this study, Set A is chosen as the control sample. Set C and Set D are originated from the epileptic patients during seizure free intervals, and Set C is used as the example of seizure free state. Set E is recorded only when the patients is at the state of epileptic seizure activity. Moreover, descriptions about the chosen data sets are illustrated in Table 2. Exemplary EEG time series taking from A001, C001 and E001 are depicted in Figure 9. 3.2.2 FFuzzyEn analysis of the datasets Complexity of each EEG data segment is calculated by introducing the sliding window method, where FFuzzyEn of each window is calculated. The final complexity measuring result of the data segment is the mean values of all windows. The flow chart of the sliding window method is presented in Figure 10. In the first step, length of windows N , sliding step L and total number of windows are set. Here, we choose N = 1000, L = 30 and M = 100. Then FFuzzyEn complexity of each window is calculated and saved until complexity values of all windows are obtained. Finally, the mean value of the complexities is calculated.

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Fig. 9. Exemplary specimens of the EEG signals (a) A001; (b) C001; (c) E001.

Fig. 10. Flow diagram of the sliding window method.

FFuzzyEn complexity of EEG signals as shown in Figure 9 is calculated and the results are presented in Figure 11 where parameters of FFuzzyEn are m = 2, r = 0.15, N = 1000 and α = αmax , 0, 0.1, 0.2. Meanwhile, mean and variance values of each case are presented in Table 3, in which the results are shown as form of MeanStd. As

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Fig. 11. FFuzzyEn complexity analysis results of A001, C001 and E001 with different parameter α (a) αmax ; (b) α = 0; (c) α = 0.1; (d) α = 0.2. Table 3. Mean ± Std result of A001, C001 and E001.

αmax α=0 α = 0.1 α = 0.2

A001

C001

E001

0.7902 ± 0.0149 0.7571 ± 0.0194 0.6569 ± 0.0233 0.4680 ± 0.0264

0.5910 ± 0.0155 0.5232 ± 0.0113 0.4626 ± 0.0259 0.3449 ± 0.0205 0.3138 ± 0.0293 0.1819 ± 0.0227 0.0896 ± 0.0314 −0.0500 ± 0.0237

shown in Figure 11, when α takes different values, FFuzzyEn complexity shows similar trends. It means FFuzzyEn measures complexity effectively when α = αmax , 0, 0.1 and 0.2. However, when α is larger, the difference between different states becomes larger, and this can also be verified by both Figure 11 and Table 3. Moreover, it shows that health EEG signals have much higher complexity than that of EEG signals recorded from epileptic patients, and it means epileptic leads to less complexity of EEG signal. For epileptic patients, EEG signals show higher complexity during seizure free intervals. FFuzzyEn complexity of other EEG segments including channel 025, 060, 080 and 098 is calculated and results are illustrated in Figure 12. It is shown in Figure 12 that Set A has the highest complexity and while complexity of Set E is the lowest and is much lower than that of Set A. Thus it also shows that normal health persons have more complex EEG signals than epileptic patients during seizure activity. Complexity of Set C lies in the middle. However, it is shown in Figure 12c that it may increase to the level of Set A. It means that epileptic patients during seizure free intervals could have high complexity EEG as normal health persons, but the complexity of their EEG signals are lower in most cases. There are 100 single-channel data segments for each Set A, Set C and Set E, and the complexity of all channels are calculated. Measuring results are shown in

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Fig. 12. FFuzzyEn complexity of several data segments (a) A025, C025 and E025; (b) A060, C060 and E060; (c) A080, C080 and E080; (d) A098, C098 and E098.

Fig. 13. FFuzzyEn complexity error bar plot of Set A, Set C and Set E.

Figure 13 where complexity values are the averages of FFuzzyEn values of each data segment. It is shown in Figure 13 that in the most majority of cases Set A has higher complexity than Set E, but the fluctuation range of Set C overlaps with Set A or Set E in some channels. It means that some channels may lead unsatisfying results. To compare complexities of different cases, further statistical analysis is needed. For the statistical analysis of complexity of Set A, Set C and Set E, one-way analysis of variance (ANOVA) and post hoc multiple comparison least significant difference (LSD) are carried out. The experiments are carried out by employing the Matlab

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Fig. 14. Boxplots of FFuzzyEn complexity of Set A, Set C and Set E. Table 4. ANOVA result. Source

SS

df

MS

F

p − V alue

Columns Error Total

5.52518 2.53259 8.04877

2 297 299

2.76259 0.0085 –

325.13 – –

0 – –

Table 5. Values of (p, F ) of LSD.

A C E

A

C

E

– (0, 233.53) (0, 563.39)

(0, 233.53) – (0, 96.72)

(0,563.39) (0, 96.72) –

function p = anova1(X). When running function anoval1(·), boxplots of FFuzzyEn complexity of Set A, Set C and Set E are shown in Figure 14, and ANOVA analysis result is illustrated in Table 4. The median in the box is shown as a horizontal line, which extends from 25% to 75%. The crosses represent the outliers. It also shows that Set A has the highest complexity, but complexity measuring results of the three sets overlap with each other to a certain extent. However, according to the ANOVA analysis result as shown in Table 4, p-value is zero and F value is 365.85. Since p-value is smaller than 0.005, we can reject the null hypothesis which means FFuzzyEn complexity analysis results of the three sets are statistically significant. In fact, a health person is a complex system. For instance, high complexity can be found from the EEG ECG and EMG signals of a health person. However, the dynamics of a person’s biological signals could be changed when he or she gets a disease such as epileptic and heart failure. As a result, complexity of signals from those patients is lower. It is shown in the above analysis that, compared with complexity on EEG signals of normal health persons, the complexity of epileptic patients have the lowest complexity during seizure activity and relative lower complexity during seizure free intervals. Moreover, the complexity of ECG signal from normal healthy persons and congestive heart failure patients is investigated in reference [32] by employing the idea of gradient cross recurrence and mean gradient cross recurrence density, and it also shows that higher complexity is found in the ECG signal from the normal healthy persons.

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4 Conclusion In this paper, by introducing the concept of fractional information, the FFuzzyEn algorithm is proposed. Based on chaotic systems, parameter choice and effectiveness of FFuzzyEn are investigated. It shows that the proposed FFuzzyEn algorithm is effective for measuring complexity of nonlinear chaotic systems because the results agree well with dynamical behaviors, and larger difference between different signals is observed when fractional order α is larger than zero. Particularly, there is a strong positive liner relation between FFuzzyEn and maximum Lyapunov exponents. In addition, we studied the nature of complexity of EEG signals by employing sliding window method based FFuzzyEn, and the results are shown in the boxplots. One-way ANOVA and LSD are applied to test the statistical characteristics. It is observed that the dynamics of EEG signal from epileptic patients is less complex compared with that of health persons and lowest complexity is observed during the seizure activity. The FFuzzyEn algorithm is effective for complexity analysis of nonlinear time series. This work was supported by the National Natural Science Foundation of China [Nos. 11747150 and 11704120] and the Startup Foundation for Doctoral Research in Hunan University of Arts and Science [No. E07016048].

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