Spatial-dependence recurrence sample entropy

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Spatial-dependence recurrence sample entropy Tuan D. Pham a, *, Hong Yan b a b

Department of Biomedical Engineering, Linköping University, 58183 Linköping, Sweden Department of Electronic Engineering, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong

highlights • Quantifying irregularity with sample entropy is based only on the distance measure. • Sequential ordering is an important criterion for computing sample similarity. • The new method is introduced to consider both sources of information.

article

a b s t r a c t

info

Article history: Received 7 August 2017 Received in revised form 8 October 2017 Available online xxxx

Measuring complexity in terms of the predictability of time series is a major area of research in science and engineering, and its applications are spreading throughout many scientific disciplines, where the analysis of physiological signals is perhaps the most widely reported in literature. Sample entropy is a popular measure for quantifying signal irregularity. However, the sample entropy does not take sequential information, which is inherently useful, into its calculation of sample similarity. Here, we develop a method that is based on the mathematical principle of the sample entropy and enables the capture of sequential information of a time series in the context of spatial dependence provided by the binarylevel co-occurrence matrix of a recurrence plot. Experimental results on time-series data of the Lorenz system, physiological signals of gait maturation in healthy children, and gait dynamics in Huntington’s disease show the potential of the proposed method. © 2017 Elsevier B.V. All rights reserved.

Keywords: Time series Irregularity Sample entropy Recurrence plot Binary-level co-occurrence matrix Spatial dependence

1. Introduction In the combination of statistics and information theory, the approximate entropy, denoted as ApEn, was developed for quantifying the amount of irregularity or predictability of fluctuations in time series data [1,2]. The sample entropy [3], denoted as SampEn, was then introduced as a modified algorithm of ApEn that removes the bias in counting selfmatching patterns included in the ApEn. Both ApEn and SampEn, and their modified versions have been increasingly found useful in many applications, particularly in the analysis of physiological time series. Some recently published works include applications in network theory [4], analyses of heart rate variability and systolic blood pressure variability [5], postural analysis [6], and analysis of traffic signals [7]. However, several applications and studies are reportedly in favor of SampEn [8–10]. Given its popularity as a useful measure for quantifying irregularity of time series, SampEn has several technical shortcomings. Many attempts have been made to improve SampEn mainly to reduce its sensitivity to the selection of values for its model parameters [11–13]. One of the most recent efforts has tried to modify SampEn to overcome the limitation

*

Corresponding author. E-mail addresses: [email protected] (T.D. Pham), [email protected] (H. Yan).

https://doi.org/10.1016/j.physa.2017.12.015 0378-4371/© 2017 Elsevier B.V. All rights reserved.

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about the relationship between the parameters and the length of time series [7]. However, there is little effort in developing the formulation of SampEn that can also capture the sequential ordering of the time series, except for the indirect case of using multiple scales of time series data [10,14–16]. In this paper, we introduce a method for quantifying irregularity in time series based on the formulation of SampEn that computes the probability of sample similarity by incorporating sources of information from the distance measure of sample points of a time-series and the spatial orientation of the binary-level cooccurrence matrix of its recurrence plot, where the latter information is a spatial representation of the sequential ordering. The rest of the paper is organized as follows. Section 2 briefly presents the mathematical formulations of the sample entropy, recurrence plots, and binary-level co-occurrence matrix, which are used as the basis for developing the framework of the proposed spatial-dependence recurrence sample entropy. Experimental results and discussion about the proposed method are presented in Section 3, which includes the testing and comparison of proposed method and the sample entropy using three datasets: the time series of the Lorenz system and their surrogate time series, the complex physiological signals of gait maturation in children, and gait dynamics in Huntington’s disease obtained from publicly accessible PhysioNet databases. 2. Methods 2.1. Sample entropy The sample entropy (SampEn) [3] is a measure of irregularity in time series. The formulation of SampEn is briefly described as follows. Consider a time series X of length N taken at regular intervals: X = (x1 , x2 , . . . , xN ), and a given embedding dimension m, a set of newly reconstructed time series from X , denoted as Y m , can be established as Y m = m m m m (ym 1 , y2 , . . . , yN −m+1 ), where yi = (xi , xi+1 , . . . , xi+m−1 ), i = 1, 2, . . . , N − m + 1. The probability of vector yi being similar m m −1 to vectors yj is computed as (N − m − 1) times the number of vectors yj within a similarity tolerance of ym i , where self-matches are excluded, and mathematically expressed as follows Bm i (r) =

N −m

1

∑

N −m−1

m H [d(ym i , yj )], i ̸ = j,

(1)

j=1

m where r is a real positive value for the similarity tolerance, and H(d(ym i , yj )) is the Heaviside function, defined as

{ d(ym i

H[

,

ym j )

]=

m 1 : d(ym i , yj ) ≤ r

(2)

m 0 : d(ym i , yj ) > r

The distance between the two vectors is obtained by using the Chebyshev distance or the L∞ metric, where the distance between two vectors is the largest of their differences along any coordinate dimension and mathematically expressed as m d(ym i , yj ) = max(|xi+k−1 − xj+k−1 |), k = 1, 2, . . ., m.

(3)

k

The probability of pairs of vectors or data points of length m having the Chebyshev distance ≤ r, denoted as Bm (r), is expressed as Bm (r) =

1 N −m

N −m

∑

Bm i (r).

(4)

i=1

−1 Similarly, Am times the number of vectors yj m+1 within a similarity tolerance of yi m+1 , i (r) is defined as (N − m − 1) where j = 1, . . . , N − m, j ̸ = i, and setting

Am (r) =

1 N −m

N −m

∑

Am i (r).

(5)

i=1

Finally, SampEn is calculated as SampEn(m, r , N) = − log

[

Am (r) Bm (r)

]

,

(6)

where Am (r) ≤ Bm (r), which is imposed by the Chebyshev distance. 2.2. Recurrence plots In nonlinear dynamics and chaos theory, a recurrence plot (RP) [17] is a visualization method that shows the times at which a phase-space trajectory approximately revisits the same area in the phase space. Let X = {x} be a set of phase-space states, in which xi is the ith state of a dynamical system in m-dimensional space. An RP is constructed as an N × N matrix in which an element (i, j), i = 1, . . . , N, j = 1, . . . , N, is represented with a black dot if xi and xj are considered to be closed to Please cite this article in press as: https://doi.org/10.1016/j.physa.2017.12.015.

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Fig. 1. Example of the construction of a BLCM: (a) 3 × 3 image, (b) Its BLCM, using (c) δ = [0 1] that indicates one pixel to the right of the pixel of interest.

each other. For a symmetrical RP, a threshold, denoted as ϵ , is used to define the similarity of a state pair (xi , xj ) as follows [18] R(i, j) = H [d(xi , xj )],

(7)

where R(i, j) is an element (i, j) of the recurrence matrix R, d(xi , xj ) is a distance function of xi and xj , and H(·) is the Heaviside function expressed as H [d(xi , xj )] =

{

1 : d(xi , xj ) ≤ ϵ

(8)

0 : d(xi , xj ) > ϵ

Interested readers can find a web-based tutorial on the construction of recurrence plots at http://www.recurrenceplot.tk/. 2.3. Binary-level co-occurrence matrix A binary-level co-occurrence matrix (BLCM) of an image I of size N × M can be constructed using the concept of the gray-level co-occurrence matrix [19]. The BLCM is a function of binary values and geometric offset δ = (δ x, δ y), and defined as Bδ (p, q) =

N M ∑ ∑

[I(x, y) = p] ∧ [I(x + δ x, y + δ y) = q],

(9)

x=1 y=1

where I(x, y) and I(x + δ x, y + δ y) are the pixels at locations (x, y) and (x + δ x, y + δ y), p, q ∈ {0, 1}, and ∧ stands for the logical AND operator. As the result, a BLCM is presented with a 2 × 2 matrix, where BLCM(1,1) indicates the number of co-occurrences of black pixels (0,0), BLCM(2,2) records the number of co-occurrences of white pixels (1,1), and BLCM(1,2) is the number of co-occurrences of the black and white pixels (0,1), and BLCM(2,1) is the number of co-occurrences of the white and black pixels (1,0), offset by δ . Fig. 1 illustrates the construction of a BLCM from a 3 × 3 image, using offset δ = [0 1] (one pixel to the right of the pixel of interest). 2.4. Spatial-dependence recurrence sample entropy The mathematical formulation of SampEn expressed in Eq. (6) does not consider the sequential order of the time-series data points, which is useful for time-series analysis, in validating the sample similarity. This sequential information can be incorporated by firstly using the RP representation, of which the sum of each row or column is the total number of matches with other points within a similarity tolerance of the point represented by the row or column where the diagonal element is excluded, and therefore proportional to Bm i (r) of SampEn. Secondly, the RP is then used as a binary image to obtain its BLCM, which quantifies the sample similarity according to spatial orientations. The probability of the total number of the recurrence pairs of dimension m whose Chebyshev distance is less than ϵ at a spatial offset δ can be determined as S m (ϵ, δ ) =

BLCM(1, 1) N(δ )

,

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Table 1 SampEn (m = 1, r = 0.15σ ) and sdrSampEn (m = 1, ϵ = 0.15σ , δ = [0, 1]) values of the components of the Lorenz system. Method

Lorenz system components

SampEn SdrSampEn

Variance

x

y

z

0.1772 0.2039

0.1878 0.1972

0.3374 0.3771

0.0080 0.0104

where BLCM(1,1) is the total number of the recurrences of state pairs in the phase space reconstructed from the time series, and N(δ ) is the number of valid pixel pairs that can be offset by δ . The spatial-dependence recurrence sample entropy, denoted as sdrSampEn, is then given by sdrSampEn(m, ϵ, δ ) = − log

[

S m+1 (ϵ, δ ) S m (ϵ, δ )

]

,

(11)

where S m+1 (ϵ, δ ) ≤ S m (ϵ, δ ). Now, it can be observed that sdrSampEn measures the irregularity of a time series by using double criteria for the quantification of sample similarity: the first criterion is by the distance measure (the Chebyshev distance) and the second one is by the spatial orientation (geometrical offset specified in BLCM of RP) that is the transformed sequential information of the time series. The procedure for computing sdrSampEn can be summarized as follows. 1. 2. 3. 4. 5.

Given a time series and model parameters m, ϵ , and δ . Use time series, m and ϵ , to compute the RP by using the Chebyshev distance. Use offset δ to construct the BLCM of the RP to obtain S m (ϵ, δ ). Set m = m + 1, repeat steps 2–3 to obtain S m+1 (ϵ, δ ). Finally, sdrSampEn(m, ϵ, δ )= − log[S m+1 (ϵ, δ )/S m (ϵ, δ )].

3. Results and discussion To illustrate the performance of the proposed method, sdrSampEn was tested and compared with SampEn, using the time series of the Lorenz system [20], the physiological time series of stride dynamics in children [21,22], and the gait dynamics in patients with Huntington’s disease and healthy controls [23,24]. For the entropy parameters, m usually takes values either 1 or 2, and r is within the range from 0.1 to 0.25 multiplied with σ , where σ is the standard deviation of the time series [10,14,25]. 3.1. Lorenz system The Lorenz system was derived as a set of three ordinary differential equations expressed as x˙ = a(y − x), y˙ = bx − y − xz, and z˙ = xy − cz, where the overdots denote differentiation with respect to time, a, b, and c are three positive system parameters [20]. The system shows chaotic behavior with standard values a = 10, b = 28, and c = 8/3, and when this threedimensional system is plotted, it resembles a butterfly image [20]. Fig. 2 shows the time series of the x, y, and z components of the Lorenz system, where each component has 4000 samples, and their corresponding RPs. For the computation of SampEn and sdrSampEn, m = 1 and 2, and r = ϵ = 0.15σ . For the construction of the BLCM, the offset parameter δ = [0, 1] provides the co-occurrence of binary levels of one pixel to the right of the pixel of interest. SampEn and sdrSampEn values of the three components of the Lorenz system are shown in Table 1 for m = 1 and Table 2 for m = 2. Both methods give the highest entropy value to the z component, while the x component is slightly smaller than the y component for SampEn, and vice versa for sdrSampEn. The visual similarity between the x and y components and the dissimilarity between the x, y, and z components of the Lorenz system can be also observed by means of their recurrence plots shown in Fig. 2. The variances of the three sdrSampEn values (0.0104 for m = 1 and 0.0031 for m = 2) is larger than the variances of the three SampEn values (0.0080 for m = 1 and 0.0022 for m = 2), indicating the better discriminating power of sdrSampEn by taking the spatial (sequential) dependence of the time series into account. To test and compare further the performance of sdrSampEn, the surrogate time series of the Lorenz system components were generated by a method of shuffling time series [26], which generates new time series by randomly permutating the original time series with the same amplitude distribution as the original series but any correlation is eliminated. Each time series of the three components were randomly reordered into 10 surrogate time series. Tables 3 and 4 show the means and standard deviations of SampEn and sdrSampEn values of the surrogate data for m = 1 and 2, respectively. The mean values obtained from sdrSampEn are approximately twice higher than those obtained from SampEn, while the entropy values of the Lorenz components obtained from the two methods are similar, and the variance of the three components obtained by sdrSampEn (0.1678 for m = 1 and 0.0322 for m = 2) is much larger than by SampEn (0.0046 for m = 1 and 0.0041 for m = 2), implying the former method shows better discriminative ability in measuring the irregularity of chaotic and random time series. Please cite this article in press as: https://doi.org/10.1016/j.physa.2017.12.015.

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(a) x component.

(b) RP of x component.

(c) y component.

(d) RP of y component.

(e) z component.

(f) RP of z component.

Fig. 2. Time series and recurrence plots of the three components of the Lorenz system.

Table 2 SampEn (m = 2, r = 0.1σ ) and sdrSampEn (m = 2, ϵ = 0.1σ , δ = [0, 1]) values of the components of the Lorenz system. Method

Lorenz system components

SampEn SdrSampEn

Please cite this article in press as: https://doi.org/10.1016/j.physa.2017.12.015.

T.D.

Variance

x

y

z

0.2751 0.2608

0.2037 0.1789

0.2927 0.2845

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Table 3 Means and standard deviations of SampEn (m = 1, r = 0.15σ ) and sdrSampEn (m = 1, ϵ = 0.15σ , δ = [0, 1]) of surrogate time series of the components of the Lorenz system. Method

SampEn SdrSampEn

Lorenz system components

Variance

x

y

z

2.4727 ± 0.0041 4.686 ± 0.1452

2.3583 ± 0.0144 3.9055 ± 0.1419

2.4792 ± 0.0047 4.5116 ± 0.2350

0.0046 0.1678

Table 4 Means and standard deviations of SampEn (m = 2, r = 0.15σ ) and sdrSampEn (m = 2, ϵ = 0.15σ , δ = [0, 1]) of surrogate time series of the components of the Lorenz system. Method

SampEn SdrSampEn

Lorenz system components

Variance

x

y

z

2.4735 ± 0.0101 4.1548 ± 0.2765

2.3627 ± 0.0222 3.7969 ± 0.4124

2.4750 ± 0.0166 3.9986 ± 0.3276

0.0041 0.0322

Table 5 Means and standard deviations of SampEn (m = 1, r = 0.1σ ) and sdrSampEn (m = 1, ϵ = 0.1σ , δ = [0, 1]) of time series of gait maturation in healthy children. Method

Young

Middle

Old

Variance

SampEn SdrSampEn

1.6403 ± 0.3446 2.4446 ± 0.5637

1.3253 ± 0.7419 2.0453 ± 1.2414

1.7956 ± 0.3560 2.8390 ± 0.6687

0.0574 0.1575

Table 6 Means and standard deviations of SampEn (m = 2, r = 0.2σ ) and sdrSampEn (m = 2, ϵ = 0.2σ , δ = [0, 1]) of time series of gait maturation in healthy children. Method

Young

Middle

Old

Variance

SampEn SdrSampEn

1.3358 ± 0.3145 1.8224 ± 0.4309

1.0931 ± 0.7288 1.6110 ± 1.2072

1.6231 ± 0.4873 2.4246 ± 0.9572

0.0704 0.1782

3.2. Gait dynamics in children It is reported that the immature control of posture and gait of very young children result in an unsteady gait, and gait becomes relatively mature when children reach about three years of age [21]. However, it is hypothesized that gait dynamics would continue to develop beyond the age of three years old, because stride dynamics depends on the neural control [21]. The database of gait dynamics in children was established to test the hypothesis. This database consists of the time series of gait cycle durations or stride intervals in healthy children of ages from 3 to 14 years old, recorded using a portable foot-switch device inserted inside of shoes. A stride interval is defined as the time from heel strike to heel strike of the same foot. There are three groups of aging in children, including male and female: young (11 subjects from 40 to 58 months old), middle (20 subjects from 79 to 94 months old), and old (12 subjects from 133 to 163 months old). Fig. 3 shows the stride-interval time series selected from the youngest to oldest ages of each of the three age groups of the children. It can be observed from Fig. 3 that the fluctuations of the stride intervals largely vary within the same groups, where a high degree of irregularity exhibits in the time series of the 163-month-old male (old group) and 58-month-old male (young group), and a relatively lower fluctuation displays in the time series of the middle group. Fig. 4 shows the RPs of the time series shown in Fig. 3, using ϵ = 0.1σ . Table 5 shows the means and standard deviations of SampEn and sdrSampEn of the gait dynamics of the 3 children groups, where m = 1, r = ϵ = 0.1σ , and δ = [0, 1]. Both SampEn and sdrSampEn results provide the same order of irregularity for the three groups: entropy of the old group is the highest, and the entropy of the middle group is the smallest. The variances of the entropy values of the three groups obtained from SampEn and sdrSampEn are 0.0574 and 0.1575, respectively. These variances suggest that sdrSampEn can provide a better indicator of the gait-dynamics patterns in children than SampEn. Table 6 shows other sets of means and standard deviations of SampEn and sdrSampEn of the gait dynamics of the 3 children groups, using m = 2, r = ϵ = 0.2σ , and δ = [0, 1]. Results obtained from both SampEn and sdrSampEn consistently express the same order of irregularity for the three groups as previously discussed. Once again, the variance of the sdrSampEn values of the three groups, which is 0.1782, is higher than that of the SampEn values of the three groups, which is 0.0704, illustrating the better performance of sdrSampEn in separating the measurements of the irregularity of the gait dynamics of the three groups of healthy children. 3.3. Gait dynamics in Huntington’s disease and healthy control This database consists of records of 20 patients with Huntington’s disease (HD) and 16 healthy control (HC) subjects. The raw data were obtained using force-sensitive resistors that produced the output being proportional to the force under the Please cite this article in press as: https://doi.org/10.1016/j.physa.2017.12.015.

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(a) Young: 40-month-old male.

(b) Young: 58-month-old male.

(c) Middle: 79-month-old male.

(d) Middle: 94-month-old female.

(e) Old: 133-month-old female.

(f) Old: 163-month-old male.

Fig. 3. Time series of stride intervals of six healthy children classified into 3 classes: young, middle, and old.

Table 7 Means and standard deviations of SampEn (m = 2, r = 0.15σ ) and sdrSampEn (m = 2, ϵ = 0.35σ , δ = [0, 1]) of time series of gait dynamics in Huntington’s disease (HD) and healthy control (HC). Method

HD

HC

SampEn SdrSampEn

1.9347 ± 0.7538 1.7242 ± 0.8065

1.8462 ± 0.3239 1.4673 ± 0.4789

foot. The stride-to-stride measures of footfall contact times were derived from these signals. Signals obtained from the rightfoot stride intervals (sec) were used in this study. Fig. 5 shows the right-foot stride-to-stride measures and corresponding RPs of an HD patient and an HC subject, in which the time series of the HD patient can be seen to be more irregular than that of the HC subject, and therefore the recurrence states of the RP of the HD patient are more scattering than those of the RP of the HC subject. As shown in Table 7, both SampEn and sdrSampEn produce higher average entropy values of the HD patients than those of the HC subjects, which indicate the fluctuation of the HD stride-to-stride signals is more irregular than the HC stride-to-stride Please cite this article in press as: https://doi.org/10.1016/j.physa.2017.12.015.

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(a) Young: 40-month-old male.

(b) Young: 58-month-old male.

(c) Middle: 79-month-old male.

(d) Middle: 94-month-old female.

(e) Old: 133-month-old female.

(f) Old: 163-month-old male.

Fig. 4. Recurrence plots of time series of stride intervals of six healthy children classified into 3 classes: young, middle, and old.

signals. However, sdrSampEn shows better discriminative ability in measuring the irregularity of the HD (1.7242) and HC (1.4673) signals than SampEn (1.9347 for HD and 1.8462 for HC). 4. Conclusion A new method for quantifying irregularity in time series has been presented and discussed. The proposed method can take both distance measure and spatial dependence of sequential samples into account for measuring the irregularity of the Please cite this article in press as: https://doi.org/10.1016/j.physa.2017.12.015.

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(a) Right-foot stride-to-stride measure of a HD patient.

(b) RP of right-foot stride-to-stride measure of the HD patient.

(c) Right-foot stride-to-stride measure of an HC subject.

(d) RP of right-foot stride-to-stride measure of the HC subject.

Fig. 5. Right-foot stride-to-stride measures and RPs of an HD patient and an HC subject.

time series, where conventional SampEn considers only the criterion of distance measure. The new method has the potential to be utilized as a computational tool for identifying physiologic markers in complex time series data. Optimal selections of sdrSampEn parameters m and ϵ , which are used for constructing the RP, and δ , which is required for computing the BLCM, were not explored in this study. A small value of ϵ results in a small number of recurrence points and therefore may not adequately reveal the recurrence structure of the time series. On the other hand, a large value of ϵ allows many points to become similar with other points, leading to artifacts [18]. For the computation of the BLCM, different spatial orientations for δ specify different distances between the pixel of interest and its neighbor, and would result in different matrices of spatial relationships. Guidelines for choosing these parameters remain open issues for research in recurrence plots and gray-level co-occurrence matrix, respectively, and are certainly worth studying in future investigation. Finally, as an alternative to SampEn, sdrSampEn can be applied for multiscale entropy analysis [10,14–16], and analysis of other types of signals [27,28] to gain more insights into the quantification of time-series irregularity. Acknowledgments This work was carried out during the visiting period of the first author (TDP) with the Signal Processing and Biocomputing Lab in the Department of Electronic Engineering at City University of Hong Kong (CityU). The project was partially supported by a CityU research grant (9610308). References [1] S.M. Pincus, Approximate entropy as a measure of system complexity, Proc. Natl. Acad. Sci. USA 88 (1991) 2297–2301. [2] S.M. Pincus, Approximate entropy (ApEn) as a complexity measure, Chaos 5 (1995) 110–117.

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[3] J.S. Richman, J.R. Moorman, Physiological time-series analysis using approximate entropy and sample entropy, Am. J. Physiol Heart Circ. Physiol 278 (2000) H2039–H2049. [4] J. West, L. Lacasa, S. Severini, A. Teschendorff, Approximate entropy of network parameters, Phys. Rev. E 85 (2012) 04611. [5] A. Singh, B.S. Saini, D. Singh, An alternative approach to approximate entropy threshold value (r) selection: Application to heart rate variability and systolic blood pressure variability under postural challenge, Med. Biol. Eng. Comput. 54 (2016) 723–732. [6] H. Clint, W. Qin, S. Jiann-Shing, F. Paul, I. Brice, M. Lina, Sample entropy, univariate, and multivariate multi-scale entropy in comparison with classical postural sway parameters in young healthy adults, Front. Hum. Neurosci. (2017). http://dx.doi.org/10.3389/fnhum.2017.00206. [7] D. Shang, M. Xu, P. Shang, Generalized sample entropy analysis for traffic signals based on similarity measure, Physica A 474 (2017) 1–7. [8] X. Chen, I. Solomon, K. Chon, Comparison of the use of approximate entropy and sample entropy: Applications to neural respiratory signal, in Proceedings of the 27th Annual Conference IEEE Engineering in Medicine and Biology (Shanghai, China, September 1–4, 2005), 2005, pp. 4212–4215. [9] M.O. Sokunbi, Sample entropy reveals high discriminative power between young and elderly adults in short fmri data sets, Front. Neuroinformatics (2014). http://dx.doi.org/10.3389/fninf.2014.00069. [10] T.D. Pham, Time-shift multiscale entropy analysis of physiological signals, Entropy 19 (2017) 257. [11] L. Zhao, S. Wei, C. Zhang, Y. Zhang, X. Jiang, F. Liu, C. Liu, Determination of sample entropy and fuzzy measure entropy parameters for distinguishing congestive heart failure from normal sinus rhythm subjects, Entropy 17 (2015) 6270–6288. [12] C.C. Mayer, M. Bachler, M. Hortenhuber, C. Stocker, A. Holzinger, S. Wassertheurer, Selection of entropy-measure parameters for knowledge discovery in heart rate variability data, BMC Bioinform. 15 (Suppl 6) (2014) S2. http://dx.doi.org/10.1186/1471-2105-15-S6-S2. [13] R. Alcaraz, D. Abasolo, R. Hornero, J.J. Rieta, Optimal parameters study for sample entropy-based atrial fibrillation organization analysis, Comput. Methods Programs Biomed. 99 (2010) 124–132. [14] M. Costa, C.K. Peng, A.L. Goldberger, J.M. Hausdorff, Multiscale entropy analysis of human gait dynamics, Physica A 330 (2003) 53–60. [15] M. Costa, A.L. Goldberger, C.K. Peng, Multiscale entropy analysis of complex physiologic time series, Phys. Rev. Lett. 89 (2002) 068102. [16] M. Costa, A.L. Goldberger, C.K. Peng, Multiscale entropy analysis of biological signals, Phys. Rev. E 71 (2005) 021906. [17] J.P. Eckmann, S.O. Kamphorst, D. Ruelle, Recurrence plots of dynamical systems, Europhys. Lett. 4 (1987) 973–977. [18] N. Marwan, M.C. Romano, M. Thiel, J. Kurths, Recurrence plots for the analysis of complex systems, Phys. Rep. 438 (2007) 237–329. [19] R.M. Haralick, K. Shanmugam, I. Dinstein, Textural features for image classification, IEEE Trans. Syst. Man Cybern. 3 (1973) 610–621. [20] E.N. Lorenz, Deterministic nonperiodic flow, J. Atmos. Sci. 20 (1963) 130–141. [21] J.M. Hausdorff, L. Zemany, C.K. Peng, A.L. Goldberger, Maturation of gait dynamics: stride-to-stride variability and its temporal organization in children, J. Appl. Physiol. 86 (1999) 1040–1047. [22] Gait Maturation Database and Analysis. PhysioNet, http://dx.doi.org/10.13026/C2H014, https://physionet.org/physiobank/database/gait-maturationdb/. [23] J.M. Hausdorff, S.L. Mitchell, R. Firtion, C.K. Peng, M.E. Cudkowicz, J.Y. Wei, A.L. Goldberger, Altered fractal dynamics of gait: reduced stride-interval correlations with aging and huntington’s disease, J. Appl. Physiol. 82 (1997) 262–269. [24] Gait Dynamics in Neuro-Degenerative Disease Data Base. PhysioNet, http://dx.doi.org/10.13026/C27G6C, https://www.physionet.org/physiobank/ database/gaitndd/. [25] S.M. Pincus, I.M. Gladstone, R.A. Ehrenkranz, A regularity statistic for medical data analysis, J. Clin. Monit. 7 (1991) 335–345. [26] J. Theiler, S. Eubank, B. Galdrikian, A. Longtin, D.J. Farmer, Testing for nonlinearity in time series: the method of surrogate data, Physica D 58 (1992) 77–94. [27] P. Casaseca-de-la Higuera, M. Martin-Fernandez, C. Alberola-Lopez, Weaning from mechanical ventilation: A retrospective analysis leading to a multimodal perspective, IEEE Trans Biomed. Eng. 53 (2006) 1330–1345. [28] M.K. Yum, K.Y. Jung, H.C. Kang, H.D. Kim, Y.M. Shon, J.K. Kang, I.K. Lee, K.J. Park, O.Y. Kwon, Effect of a ketogenic diet on eeg: Analysis of sample entropy, Seizure 17 (2008) 561–566.

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