A combined measure to differentiate EEG signals using fractal dimension and MFDFA-Hurst

A combined measure to differentiate EEG signals using fractal dimension and MFDFA-Hurst

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A combined measure to differentiate EEG signals using fractal dimension and MFDFA-Hurst ´ S.A. David, J.A.T. Machado, C.M.C. InacioJr., C.A. ValentimJr. PII: DOI: Reference:

S1007-5704(20)30005-8 https://doi.org/10.1016/j.cnsns.2020.105170 CNSNS 105170

To appear in:

Communications in Nonlinear Science and Numerical Simulation

´ Please cite this article as: S.A. David, J.A.T. Machado, C.M.C. InacioJr., C.A. ValentimJr., A combined measure to differentiate EEG signals using fractal dimension and MFDFA-Hurst, Communications in Nonlinear Science and Numerical Simulation (2020), doi: https://doi.org/10.1016/j.cnsns.2020.105170

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Highlights

The dynamics of the electroencephalogram signals of normal and epileptic patients is analyzed.

Multifractal Detrended Fluctuation Analysis (MFDFA), Hurst exponent (H) and fractal dimension (D) are adopted.

A new combined measure, namely the Combined Index (CI), is proposed.

This procedure can avoid bias when a single index is chosen individually.

The results indicate that the new CI is useful for investigating of the EEG dynamics and improve the efficiency of classification.

A combined measure to differentiate EEG signals using fractal dimension and MFDFA-Hurst S.A. Davida,∗, J. A. T. Machadob , C. M. C. In´ acio Jr.a , C. A. Valentim Jr.a a Department

of Biosystems Engineering, University of S˜ ao Paulo, 13635-900 Pirassununga, SP, Brazil - [email protected] b Institute of Engineering, Polytechnic of Porto, Rua Dr. Ant´ onio B. de Almeida 431 4249-015 Porto - Portugal - [email protected]

Abstract The analysis of the brain electrical activity measured by means of electroencephalographic (EEG) records is a fundamental technique for the understanding and diagnosis of neurological diseases. This paper adopts the Multifractal Detrended Fluctuation Analysis, Hurst exponent (H ) and fractal dimension (D) to analyze the dynamics of the EEG signals of normal and epileptic patients using a set of physiological time series. Furthermore, a new measure, namely the Combined Index, is proposed. The results indicate that the indices H and D are useful for building a combined measure of the EEG both for healthy and epileptic cerebral activity. Keywords: Epilepsy, EEG signals, long memory, fractal dimension

∗ Corresponding author: Tel: +55 19 35656711 E-mail address: [email protected] (S.A. David)

Preprint submitted to Elsevier

January 3, 2020

1. Introduction Normal physiology of biological systems requires an intricate network to implement an efficient control function. These networks incorporate a mix of integration, differentiation, feedback loops, and other regulatory mechanisms that enable an organism to perform multiple activities, typical of complex systems with features adapting in time. The modeling of these phenomena is a challenging task and the non-stationary behavior of such mechanisms is an additional problem. In order to describe and quantify the dynamic characteristics of biological systems, different techniques related to complexity theory have been employed [1–8]. The analysis of complex systems and the assessment of complexity emerged as important facets of the mathematical and physical sciences [9–11]. Complexity can be loosely defined as the difficulties faced when describing a signal or predicting its future behavior [12]. The historical development of the concepts of complexity has centered on measuring regularity using various metrics based on nonlinear time series (TS) analysis. The correlation dimension, entropy and Lyapunov exponents are indices often employed to measure the randomness and predictability of a given TS [13–18]. Another alternative approach is to compute the fractal complexity of the TS. The Hurst exponent (H ) and the fractal dimension (D) are measures of the fractal complexity or persistence of fractal processes such as fractional Gaussian noise or Brownian motion [19]. With age conditions and health disorders, there is a loss of complexity in the dynamics of many integrated physiological processes of an organism [20, 21]. The estimates of complexity can be used to probe different aspects of complex signals, eventually affected by aging and disease, and can be applied to a range of physiological measures, including in respiratory signals [22–24], cardiac measures [25, 26], anesthesia dosage monitoring [27, 28] and electrophysiological signals from the brain.

2

The brain is recognized as one of the most complex dynamic systems due to the intricate structure of the neural network. The synapses allow the information transmission among neurons. Through the axon, electrical signs may be (or not) propagated to other cells. The so-called brain activity is formed by the superposition of all those signals and the correlation study of the electrical activity provides significant information about the system dynamics [29]. The recording of cerebral electrical activity can be obtained by means of electroencephalography. In practice, the electroencephalogram (EEG) is a noninvasive technique since it is performed through disc-shaped electrodes arranged in the scalp of patients. The EEG is a bioelectrical manifestation of the nervous system activity. Often, the EEG seeks to pinpoint some problem as, for example, the location of epileptic foci detected from abnormal electrical activity in certain regions of the brain [30]. Epilepsy [31] is a serious neurological disease that affects people regardless of age, gender, or ethnicity. According to the World Health Organization, epilepsy is a serious brain disorder that affects approximately 50 million people (about 1% of the world’s population), 80% of whom live in underdeveloped, or developing countries. The problem is revealed by the occurrence of epileptic seizures, directly affecting the quality of life of patients, in the most varied aspects. Epilepsy is a disorder of the central nervous system characterized predominantly by spontaneous, recurrent and unpredictable interruptions of the normal functioning of the brain, called epileptic seizures. An epileptic seizure is defined as a transient occurrence of signs and/or symptoms caused by synchronous or excessive abnormal cerebral activity [32, 33]. The so-called epilepsy does not refer to a single disease. In fact, it encompasses a variety of disorders and conditions, that have in common an abnormally predisposition for seizures, reflecting the underlying brain dysfunction resulting from different causes. Often epilepsy does not have an identifiable cause, but in some cases it may be related to issues, such as, genetic factors, congenital abnormalities, birth trauma, metabolic or chemical imbalance in the body, head trauma, brain infection, brain tumors, stroke, cortical dysplasia, among others 3

[31]. Since epilepsy is caused by abnormal brain activity, epileptic seizures can affect any process coordinated by the brain. Thus, the signs and symptoms of an epileptic seizure vary on a case-by-case basis and may involve changes in behavior, mood, sensations (visual, olfactory, auditory) and other cognitive functions, as well as loss of consciousness and movement disorders. Several studies focused on the detection of epilepsy from EEG signals using non-linear methods that can detect and quantify both linear and nonlinear mechanisms and, thereby, reflect somehow the characteristics of the EEG [34– 41]. Non-linear features may be able to extract the hidden complexities embedded in the EEG TS. One of the earliest studies on this topic was developed by Babloyantz et al. [42], that used non-linear parameters such as the correlation dimension and the largest Lyapunov exponent to study the sleep wave signal. Such studies extended the potential application areas of EEG nonlinear analysis methodologies, but the identification of epilepsy still remains one relevant field of research [39]. The dynamic TS analysis of EEG signals may reveal complex phenomena associated with long-range correlation and distinct classes of non-linear interactions. This type of analysis is capable of providing useful diagnostic and prognostic information. Nonetheless, most of the papers found in the literature about EEG consider a single index. Moreover, when several indices are adopted, the fusion of information is not considered and merely an individual analysis of each index is followed. The analysis based on a single index is not sufficient to capture the EEG properties and, consequently, such aproaches lead to limited results [43]. Bearing these facts in mind, three mathematical tools [44–46], namely the Multifractal Detrended Fluctuation Analysis (MFDFA), Hurst exponent (H ) and fractal dimension (D), are adopted in this paper to investigate the EEG of normal and epileptic humans. A new measure embedding the H and D indices is also designed. This approach eliminates the limitations that can occur when adopting a single index for EEG analysis. The main goal of the paper is to have a deeper insight into the relation be4

tween loss of information processing in the brain and changes in the randomness pattern of the EEG when using some indices of assessment. A direct comparison between the indices for each set is drawn in order to evaluate how they perform when identifying whether there is, or not, a dependence between sets, or, by other words, to determine if the signal can indicate if the patient is healthy or not. The paper is structured as follows. Section 2 describes the data characteristics and the methods of EEG used in the follow-up. Section 3 presents the results of the proposed techniques and section 4 their discussion. Finally, section 5 summarizes the main conclusions.

2. Methods Biological dynamic systems require efficient methods for understanding their intricate behavior. Indeed, the analysis of complex systems includes issues such as non-linearity, long-range effects, random-walk, anomalous diffusion and fractal behavior and requires adequate mathematical and computational tools [47]. In the follow-up, the data and the adopted indices are described. 2.1. Data Characteristics The EEG data used in this study was taken from the artifact free EEG time series available at the Department of Epileptology, at the University of Bonn [48]. The referred open-source database has been widely used to test a variety of methods related to the identification of continuous or intermittent epilepsy patients against healthy ones, being frequently considered as a highquality benchmark in the area. The data consists of five sets: • Set A, that represents an extra-cranial recording of healthy subject with eyes open. • Set B, that corresponds to an extra-cranial recording of a healthy subject with eyes closed.

5

• Set C, which is an intra-cranial recordings from the hippocampal formation of the opposite hemisphere of the brain of patients in seizure free interval. • Set D, that reproduces an intracranial from within the epileptogenic zone of patients in seizure free interval. • Set E, that was obtained from depth electrodes EEG recordings. Each set consists of 30 signals of EEG with duration 23.6 s each, recorded at a sampling frequency of 173.61 Hz. More details regarding EEG records can be found in [48]. 2.2. Detrented fluctuation analysis The Detrended fluctuation analysis (DFA) allows the detection of self-similarity in non-stationary TS, identifying intrinsic autosimilarity of the dynamics, and ruling out possible spurious detection of correlation due to non-stationarity of the signal [49, 50]. The DFA can be used to calculate the H index, introduced by H. Hurst [45], for quantifying features of long-range dependence in TS [51–53]. Moreover, H can be related to the concept of Brownian and fractional Brownian motion (in short Bm and fBm, respectively) [54]. We find several computational algorithms for determining H, such as the classical rescaled range analysis originally developed in [55, 56], Fourier analysis using the FFT algorithm [57, 58], Detrended Moving Average (DMA), and wavelet decomposition [2]. In general, a physiological TS is limited, and does not have arbitrarily large signal amplitudes regardless of its time length, which does not allow its characterization as self-similar. Nonetheless, when integrating a TS, we obtain a new non-limited TS, which exhibits fractal properties that can be quantified. The complexity of EEG signals can be characterized by their multifractality. This topic has been considered in the literature and some works showed evidence of better performance when the MFDFA technique is used to distinguish human EEG signals [4–7, 34]. In this work, the MFDFA is used to extract the generalized Hurst exponent, denoted H (q), where q denotes the q th order of the fluctuation function, with 6

properties [45, 59] that can be summarized as follows: (i) 0 < H (q) < 1, (ii) when H (q) = 1/2, the TS has the characteristics of a random walk (Bm) and has not long memory process, (iii) if H (q) > 1/2, there are persistent (long memory) effects in the behavior of the TS, (iv) for H (q) < 1/2, then the TS exhibits anti-persistent (short-term memory) behavior. The closer is the H (q) value to 1, the higher the probability for the next change to be positive, if the last change was also positive. If some trend in the TS (H(q) 6= 1/2) occurs, a reduced number of interactions is perceived leading to a simpler and predictable system (i.e., closer to the cases H(q) = 1 or H(q) = 0). On the other hand, systems where a large number of effects is expected, lead to a rich, complex and random (H(q) = 1/2) system. In particular, for biological TS, it is also possible to obtain H values higher than H = 1, indicating that correlations still exist, but should not follow a power law behaviour [60–62]. The computation of the H (q) index by means of generalized MFDFA applied to a TS involves several 5 steps, being the first three steps identical to the conventional DFA method. First step: Consists of the following estimation [49, 63] P (N ) =

N X (Pt − P ),

(1)

t=1

where N is the number of observations in the TS, Pt represents the value observed in time instant t and P denotes the arithmetic average of this value. Second step: P (N ) is divided into Ns = N/m non-overlapping segments of equal size m. The same procedure is repeated starting from the opposite end. Thereby, 2Ns segments are obtained. Third step: Calculates F (m) by means of the ordinary least squares value [49, 63]

v u N u1 X F (m) = t [P (i) − Pm (i)]2 , N i=1

(2)

where Pm (i) is subtracted from P (i) for removing trend. Fourth step: Average over all segments to obtain the qth order fluctuation 7

function Fq (m) =

2Nm 1 X [F 2 (m)]q/2 2Ns i=1

!1/q

,

(3)

where, in general, the index variable q can assume any real value except zero. For q = 2, the standard DFA method is retrieved. Fifth step: The process is repeated for each value of q and the slope of the plot of log(Fq (m)) versus log(m) determines the H (q) exponent. A large range of values of q was investigated (−10 ≤ q ≤ 10) using the R package (https://www.r-project.org/). Sources of multifractals are not studied as they are not within the main scope of this work. The exploration of generalized Hurst exponent H (q) for q = −2 demonstrated a good performance. Therefore, the case H = H(−2) is used in the follow-up as a component to calculate the CI index for the EEG TS. 2.3. Fractal Dimension The fractal dimension D can be taken as a measure of the local memory of the TS. One can state that we obtain 1 < D ≤ 2 for univariate series. Also, the fractal dimension is connected to the long-term memory of a TS so that D + H = 2, that is, a perfect reflection from a local behavior (fractal dimension) to a global behavior (long-term memory) [64]. The properties of D [43, 64] in TS can also be summarized in accordance to the conditions: (i) 1 < D ≤ 2, (ii) when D = 1.5, TS is a random walk (Bm) phenomenon, (iii) values of D < 1.5, correspond to a persistence (long memory or correlated) process and lead to the concept of the fBm, (iv) if D > 1.5, then anti-persistent process occurs (short-term memory, anti correlated). In order to obtain D we adopt the Hall-Wood (HW) and Robust Genton (RG) estimators [46]. 2.3.1. Hall-Wood estimator The HW proposed [65] is a box-counting estimator that takes into consideration small scales. The area of the boxes covers the curve instead of just

8

their sum. Formally, let us have a scale l = l/n, where l = 1, 2, 3, ..., n. The aforementioned area is [n/l]

b A(l/n) = (l/n)

X i=1

(xil/n − x(i−1)l/n ),

(4)

where [n/l] is the integer part of n/l. The HW estimator is given by the expression b HW = 2 − D

L X i=1

b (si − s) log(A(l/n))

where L ≥ 2, sl = log(l/n) and s = (1/L)

PL

!

L X i=1

i=1 si .

(si − s)

2

!−1

,

(5)

Using L = 2, as suggested

by Hall-Wood to avoid limitations [65], one obtains b b b HW = 2 − log(A(2/n)) − log(A(l/n)) . D log(2)

(6)

b HW is used as a component to calculate the CI The fractal dimension D

index for EEG TS.

2.3.2. Robust Genton estimator The approach described in [66] consists of the so-called RG method of moments estimator of scale. However, it is well known that the method is not robust. To avoid this problem, the robust estimator developed by Genton is adopted [66, 67]. The calculations are given by n

c2 (l/n) = V

X 1 (Xi/n − X(i−l)/n )2 . 2(l − n)

(7)

i=l

Similarly to the HW, one obtains the RG estimator as ! L !−1 L X X 1 2 b RG = 2 − c2 (l/n) D (si − s) log(V (si − s) , 2 i=1 i=1

where L ≥ 2, sl = log(l/n) and s = (1/L) the limitations, one obtains

PL

i=1 si .

(8)

Using L = 2 to overcome

c c b RG = 2 − log(V2 (2/n)) − log(V2 (l/n)) . D 2 log(2)

(9)

b RG is also used as a component to calculate the CI The fractal dimension D

index.

9

2.4. Combined Index The presence of randomness in biological TS is an usual response observed in healthy organisms. The complexity of biologic/physiologic signals of organisms reflects their most adaptive (healthy) states. The more adaptive the organism, the more complex the signals it produces are likely to be. A sustained loss of complexity is usually observed in pathological contexts including advanced aging (frailty) syndrome and disease. Bearing in mind that H and D are capable of indicating randomness and the long-range dependencies for TS, a Combined Index (CI) is proposed [43, 64, 68] as



cH − M ∗ M H CI =  RH

!2

+

cD − M ∗ M D RD

!2 1/2 

,

(10)

∗ ∗ = 1.5 are the expected values for the random-walk = 0.5 and MD where MH

phenomenum related to the Hurst exponent H and to the fractional dimension cH is the estimative of H ¯ and the M cD is obtained D, respectively. We note that M b¯ b¯ cD = (D b b from M HW + D RG )/2, with DHW and DRG calculated by (6) and (9), respectively. We consider RH = 1 for the Hurst exponent and RD = 2 for the

fractal dimension, so that the maximum deviation from H = 1 (long memory process) is identical for all measures. Values of CI near zero imply a Bm, meaning lower distance (deviation) of the measured values to the values of random signals. Further away from random behavior points to a loss of complexity of the recorded signal.

10

3. Results The EEG signals of healthy and epileptic patients, both with seizure and free of seizure epochs, are analyzed. The data consists of five sets, as mentioned in subsection 2.1 [48]. Fig. 1 exhibits the EEG signal for one element in each of the sets. ¯ values are calculated from the EEG data. Fig. 2 shows the The H and D ¯ and CI for healthy subjects (Sets A and B), intervalues of H, DHW , DRG , D ictal (sets C and D) and ictal (Set E). Table 1 presents the minimum, maximum and mean values of the Hurst in¯ respectively, in addition to the standard deviation, dex, Hmin , Hmax and H, b¯ b¯ ¯ σH¯ . Furthermore, Table 1 lists the mean values of the D HW , D RG and D and the standard deviations, σHW , σRG and . The CI index is also included.

The box plot for five sets described by the Hurst and fractal dimension indices are depicted in Fig. 3. The boxes mark the maximum and minimum values, the first and third quartiles and the median. ¯ value of the ictal EEG (set Table 1, Fig. 2a and Fig. 3a reveal that the H ¯ values E) is lower than that of the interictal EEG (sets C and D), while the H of the interictal EEG are higher than those of healthy subjects (sets A and B). b¯ b¯ ¯ Table 1, Figs. 2b-2d and Figs. 3b-3d show that the D HW , D RG and D values of the ictal EEG (set E) are also lower than those of the interictal EEG (sets C and D) and healthy patients (sets A and B). One can also note from Table 1, Fig. 2e and Fig. 3e that the CI values of the unhealthy subjects (sets C, D and E) are higher than those exhibited by the healthy individuals (sets A and B). The results reveal that the values of H and CI from epileptic patients are, in general, clearly larger than the corresponding estimated H and CI from healthy ¯ subjects. However, this behavior is not so clear when observing the index D.

11

Example from set A: healthy

Example from set B: healthy 300

200

200

mV) Amplitude (

Amplitude (

mV)

100

0

100

0

-100

-100 -200

-200

-300 0

1000

2000

3000

4000

0

1000

2000

Data points

3000

4000

Data points

Example from set C: epileptic

Example from set D: epileptic

200

300

200

mV) Amplitude (

0

100

0

-100 -100

-200

-200 0

1000

2000

3000

4000

0

1000

2000

Data points

3000

Data points

Example from set E: epileptic 800

mV)

600

Amplitude (

Amplitude (

mV)

100

400

200

0

-200

-400

-600 0

1000

2000

3000

4000

Data points

Figure 1: Amplitude vs time of the EEG signals for one element in each set A, B, C, D, E.

12

4000

A

B

C

D

E

HM IN

0.8285

0.6012

1.1958

1.051

0.7622

HM AX

1.1727

1.0569

1.5368

1.8321

2.16

¯ H

0.9908

0.8288

1.3762

1.4194

1.3554

σH

0.0902

0.1106

0.0921

0.1685

0.4028

b¯ D HW

1.1490

1.0947

1.1563

1.1220

1.0667

σHW

0.0433

0.0467

0.0580

0.0575

0.0390

b¯ D RG

1.1477

1.0940

1.1527

1.1513

1.0727

σRG

0.0904

0.0588

0.1475

0.0918

0.0603

¯ D

1.1483

1.0943

1.1545

1.1367

1.0697

D

0.0555

0.0477

0.0895

0.0533

0.0467

CI

0.6063

0.5313

0.9464

0.9912

0.9783

CI

0.2358

0.3098

0.3029

0.3968

0.6242

Table 1: Hurst, fractal dimensions and combined index for the EEG signals of the five sets.

13

(a) 12 SET A SET B

11

(-2) (times 5)

SET C SET D

10

SET E

9

Values of the

H

8

7

6

5

4

3 0

5

10

15

20

25

30

20

25

30

20

25

30

20

25

30

20

25

30

Sample

(b) 7.0 SET A

6.8

SET B SET C

Values of the DHW (times 5)

6.6

SET D SET E

6.4

6.2

6.0

5.8

5.6

5.4

5.2

5.0 0

5

10

15

Sample

(c) 8.2 SET A

8.0

SET B

Values of the

DRG

(times 5)

7.8

SET C

7.6

SET D

7.4

SET E

7.2 7.0 6.8 6.6 6.4 6.2 6.0 5.8 5.6 5.4 5.2 5.0 4.8 0

5

10

15

Sample

(d) 7.6 SET A

7.4

SET B

(times 5)

7.2

SET C SET D

7.0

SET E

6.8 6.6

Values of the

D

6.4 6.2 6.0 5.8 5.6 5.4 5.2 5.0 4.8 0

5

10

15

Sample

(e) Values of the Combined Index (times 5)

10 SET A SET B SET C SET D

8

SET E

6

4

2

0

5

10

15

Sample

14 Figure 2: Values of the indices for the sets A-E: (a) Hurst using MFDFA, (b) Fractal Dimension using HW , (c) Fractal Dimension using RG, (d) Fractal Dimension using D, and (e) Combined Index.

(a)

(b)

MFDFA Method

Hall-Wood Method 1.30

2.2

Maximum

Third Quartile

Third Quartile

Median

1.25

Median First Quartile

First Quartile

Minimum

)

Minimum

DHW

1.8

Fractal Dimension (

Hurst index (H (-2))

2.0

Maximum

1.6

1.4

1.2

1.0

0.8

1.20

1.15

1.10

1.05

0.6 A

B

C

D

1.00

E

A

B

Set

(c)

C

D

E

Set

(d)

Robust Genton Method

Fractal Dimension

1.50 Maximum

1.45

1.35

Third Quartile

Third Quartile Median

D

First Quartile Minimum

Mean Fractal Dimension (

)

DRG Fractal Dimension (

1.35

)

Median

1.40

Maximum

1.30

1.25

1.20

1.15

1.10

1.05

1.30

First Quartile Minimum

1.25

1.20

1.15

1.10

1.05

1.00

0.95

1.00 A

B

C

D

E

A

Set

B

C

D

Set

(e)

Combined Index 2.0 Maximum

1.8

Third Quartile Median

1.6

First Quartile Minimum

1.4

CI

1.2

1.0

0.8

0.6

0.4

0.2 A

B

C

D

E

Set

Figure 3: Box-plots for indices and the sets A-E: (a) Hurst using MFDFA, (b) Fractal Dimension using HW , (c) Fractal Dimension using RG, (d) Fractal Dimension using D, and (e) Combined Index.

15

E

To identify the statistically distinguishable datasets (A- E) for each of the

¯ and CI), we propose the one-way analysis purposed indices (H, DHW , DRG , D, of variance (ANOVA). However, the ANOVA analysis is a parametric test and it is assumed that the datasets present a normal distribution [69–71]. Thus, the Shapiro-Wilk normality test is conducted prior to the variance analysis, verifying the null hypothesis of all datasets coming from a normal distribution [72, 73]. The Shapiro-Wilk test results are shown in Table 2 for a significance level of 0.05. The results indicate that only the datasets from H are normally distributed in their totality, i.e., for all sets. Therefore, for the remaining non¯ and CI), we applied the equivalent normal distributed indexes (DHW , DRG , D, non-parametric test to the one-way ANOVA, that is, the so-called “KruskalWallis test”, which in turn does not assume a normal distribution of the residuals [70, 71, 74, 75]. Since the one-way ANOVA and the Kruskal-Wallis tests exclusively verify whether the datasets are statistically distinguishable from one another, or not, its is required a posthoc analysis to specifically confirm which datasets pairs are significantly different and which of them are not. For this purpose, we adopted the Tukey’s honestly significant difference (HSD) test for the one-way ANOVA analysis [76–79] and the Dunn’s multiple comparison test [73, 75, 80–82] for the Kruskal-Wallis analysis using the BenjaminiHochberg method for the p-value adjustment. In both tests, the significance level was set to 0.05. Table 3 shows the results of the posthoc tests of the variance analysis in the combined pairs for all indices. The Tukey’s HSD test is applied for H and indicates that the index is only capable of identifying unhealthy sets (C − D, E − C, and E − D), since the pairs are statistically equal. Similarly, the posthoc test of Dunn is ¯ applied to the combined pairs for the non-parametric indexes (DHW , DRG , D, and CI). The DHW only revealed similarity with the pairs of sets A − C and B − D. On the other hand, the DRG showed a statistical difference between the

¯ pairs A − E, C − E, and D − E. Also, when both indices are combined in D, the distinction between healthy and unhealthy sets is still unsuccessful, since A − C, A − D, C − D, and B − E have statistical similarities. Finally, for the 16

case of the proposed CI index, one can observe that the index is the only one that entirely succeeds in distinguishing all healthy (A, B) from unhealthy sets (C, D, and E). Shapiro-Wilk normality test H

DHW

¯ D

DRG

CI

W

p-value

W

p-value

W

p-value

W

p-value

W

A

0.9752

0.6899

0.7655

0.1636e − 04∗

0.9410

0.9688e-01

0.9252

0.9650

0.4117

B

0.9729

0.6214

0.8553

0.9302

0.4967e − 01∗

0.3658e − 01∗

0.9243

0.8611

0.8491

0.9537

0.2118

0.1066e − 02∗

0.5938e − 03∗

0.1066e − 02∗

0.9791

0.8001

0.9850

0.9364

0.8157

0.1287e − 03∗

0.9400

0.9069e-01

0.8001e − 03∗

C

0.9650

0.4135

0.9369

0.7496e-01

0.8488

D

0.9880

0.9765

0.9305

0.5074e-01

0.8611

E

0.9549

0.2275

0.6974

0.1430e − 05∗

0.8444

0.5847e − 03∗ 0.4751e − 03∗

0.3462e − 01∗

p-value

Table 2: The normality test for the analyzed sets (A-E) for all indices. Significance level of 95% (p < 0.05) rejects the null hypothesis of normality.

ANOVA - Tukey’s HSD test

Kruskal Wallis - Dunn’s test

H

DHW

¯ D

DRG

CI

Mean difference

p-value

Z

p-value

Z

p-value

Z

p-value

Z

p-value

A−B

-0.1620

0.0264∗

4.2325

5.7767e − 05∗

2.1368

0.0652

3.4012

0.16775e − 02∗

1.5719

1.6566e-01

A−C

0.3854

0.000∗

-0.0596

9.5248e-01

0.4935

0.6908

0.4058

0.76097

-5.3919

A−D

0.4286

0.000∗

2.2302

-0.3559

0.7219

0.5545

7.2407e-01

-5.9089

A−E

0.3647

0.000∗

3.6759e − 02∗

6.4464

0.0015

5.2786

-4.2921

5.8989e − 05∗

0.13015e − 05∗

0.5474

0.000∗

0.75e − 02∗

B−C

-1.6434

0.1672

-2.9953

5.7273e − 10∗

0.13947e − 06∗

0.86098e − 08∗

-4.7797

2.9257e-06

0.54828e − 02∗

-6.9638

0.16561e − 10∗

B−D

0.5906

0.000∗

-2.0023

5.0281e-02

-2.4927

0.0317

-2.8467

-7.4808

0.5267

0.000∗

2.2139

3.3549e − 02∗

0.73628e − 02∗

B−E

1.0393

0.4267

1.8775

0.86361e-01

-6.3517

0.7100e − 09∗

C−D

0.0432

0.9308

2.2898

3.6718e-02

-0.8494

0.4946

0.1487

0.88183

-0.5170

0.60512

C−E

-0.0207

0.9954

6.5060

7.7188e − 10∗

2.6826

0.243e − 01∗

4.8728

0.55009e − 05∗

0.6121

0.6005

D−E

-0.0639

0.7619

4.2162

1.1292

0.32352

4.9702e − 05∗

3.5320

0.41e − 02∗

4.7242

0.77021e − 05∗

0.73845e − 12∗

Table 3: The variance analysis of the sets pairs for all indices. Significance level of 95% (p < 0.05) rejects the null hypothesis that the pairs are statistically equal.

A normalization procedure is adopted by considering the difference between ¯ D ¯ and CI and those for a random walk phenomenon. the calculated values of H, ¯ = |H ¯ − 0.5|, |∆D| ¯ = |D ¯ − 1.5| and |∆CI| = |CI − 0| Therefore, we adopt |∆H| as shown in Table 4. Moreover, Fig. 4 illustrates the 3-D plots reflecting ¯ |∆D| ¯ and |∆CI|. The aforementioned indices are the behavior of the |∆H|, presented in this normalized form allowing a more clear comparison of the results between healthy (sets A and B) and unhealthy (sets C, D and E) individuals. 17

The normalization procedure adopted as well as the values of the normalized variables are listed in Tables 4 and 5, respectively.

Normalization ¯ = |H ¯ − 0.5| |∆H|

Random Scenario ¯ ¯ =0 H = 0.5 and |∆H|

Not-Random Scenario ¯ ≥ 1 and |∆H| ¯ ≥ 0.5 H

¯ = |D ¯ − 1.5| |∆D|

¯ = 1.5 and |∆D| ¯ =0 D

¯ = 1 or D ¯ = 2 and |∆D| ¯ = 0.5 D

|∆CI| = |CI − 0|

CI = 0 and |∆CI| = 0

CI ≥ 0.5 and |∆CI| ≥ 0.5

Table 4: Variable normalization.

A

B

C

D

E

¯ |∆H|

0.4908

0.32875

0.87617

0.91935

0.8554

¯ |∆D|

0.3517

0.4057

0.3455

0.3633

0.4303

|∆CI|

0.6063

0.5313

0.9464

0.9912

0.9783

Table 5: Values of the normalized variables for the five sets.

18

(a)

|

Set A

D

H|

Set C

(b)

|

Set B

D

H|

0.8762

Set C

0.8762

0.4908

0.3288

0

0

0.3455

0.3455

0.3517

0.4057 0.5313

0.6063

0.9464

|

D

|

CI|

(c)

|

0.9464

D

D|

|

Set A

D

H|

Set D

D

|

CI|

(d)

|

D

D|

Set B

D

H|

0.9194

Set D

0.9194

0.4908

0.3288

0

0

0.3517

0.3633

0.3633

0.4057 0.5313

0.6063

|

D

0.9912

|

CI|

(e)

|

D

D|

|

Set A

D

H|

Set E

D

0.9912

|

CI|

(f)

|

D

D|

Set B

D

H|

0.8554

Set E

0.8554

0.4908

0.3288

0

0

0.3517 0.4303

0.4057 0.4303 0.5313

0.6063

|

D

0.9783

CI|

|

D

D|

|

D

0.9783

CI|

Figure 4: Index comparison for the sets: (a) A and C, (b) B and C, (c) A and D, (d) B and D, (e) A and E, and (f) B and E.

4. Discussion 4.1. Index comparison ¯ for the ictal EEG (set As mentioned and observed in Fig. 3a, the value of H E) is lower than the values for the interictal EEG (sets C and D). Nonetheless, 19

|

D

D|

¯ is not lower than the one for the healthy individuals (sets A the value of H and B). Due to the synchronous discharge of large groups of neurons during an epileptic seizure, the loss of complexity in the ictal EEG, results in the decrease of the values of the fractal dimension, as it is visible in Fig. 3b-3d. In turn, the proposed CI index points out that the dynamical behavior is less random for epileptic patients than for healthy subjects. This results is evinced by the CI values for epileptic patients that deviate more clearly from the random walk behavior (Fig. 3e). This is tantamount to saying that the unhealthy subjects are further away from random behavior when compared to healthy ones, indicating loss of information processing in the brain due to the hyper-synchronization of the EEG signals. Fig. 4 highlight these results in the viewpoint of |∆CI|. It is ¯ is analyzed separately in Fig. 4, a similar conclusion is clear that when |∆D| ¯ is further not so obvious. In some cases it is even contradictory, because |∆D| away from the random walk behavior for healthy individuals. In summary, this paper has threefold contributions. First, the choice for the HW and RG estimators to obtain D more robustness. Second, the choice of the MFDFA method prospecting a better performance for the extraction of H in the spectral signature of the applied EEG signals as H(−2), that is, with q = −2 for H(q), if compared to the monofractal method (q = 2). Third, differently from previous works investigating healthy and unhealthy EEG signals, the application of the new CI measure offers a better insight into the dynamical nature and variability of brain signals. This procedure can avoid the limitations observed when a single index is chosen individually.

4.2. Critical analysis It is important to take into account some limitations regarding the nature of the data analyzed in our work. During our study design, we opted for using the most high-quality data available, that is, intracranial recordings for nonhealthy patients (C-D-E) and, due to the impossibility of obtaining such data from patients out of a pre-surgery state, scalp recordings for healthy (A-B) 20

patients. However, since measures from depth electrodes implanted in critical brain positions, as the hippocampal formation and the epileptogenic zone, are presumably more accurate than scalp data, this design may affect the obtained results, because we are comparing data of inherently different nature. Therefore, when directly drawing comparisons between these different groups of sets, any distinction or peculiarity in their signals can be attributed to the nature of the signal (either a conventional EEG or a intracranial recording) or the pathological condition of the patient (either healthy or non-healthy). In this context, regardless of the index being used there still would be doubt about the real agent behind a determined behavior in the signal. As a solution for such situation, one could propose further exploration on a follow-up study using a different design, where all data would be of the same nature (i.e., originate from scalp recording). Using this composition, all data sets would present the same quality (regarding signal-to-noise ratio, for instance) and be comparable without any risk of affecting obtained results.

5. Conclusions It is well known that billions of neurons in the human brain are connected together with axons and synapses to construct an intrincate system involving a complex electrical activity. The EEG, reflecting the brain processing, typically exhibits a sophisticated dynamics. Linear methods for the analysis of EEG can only provide simple features and are not able to describe adequately the nonlinear behavior of the neural system. This work employed MFDFA, Hurst exponent and fractal dimension in the study of EEG for characterizing their dynamics. We considered the MFDFA approach for obtaining the H exponent. The HW and the RG estimators were adopted to calculate D. A new combined measure CI, was proposed based on the indices D and H, for assessing the EEG nonlinear TS for five distinct data sets involving epileptic patients and healthy subjects. This procedure can avoid the limitations that have been found when a single index is chosen individually.

21

The results indicate that combined H and D are useful for investigating of the EEG dynamics. Furthermore, this paper demonstrated that the new CI achieves a robust discrimination of classification.

Acknowledgments The authors wish to acknowledge the FAPESP (S˜ ao Paulo Research Foundation), grants 2017/13815-3 and 2017/15517-0, for funding support.

References [1] R. Ant˜ ao, A. Mota, J. T. Machado, Kolmogorov complexity as a data similarity metric: Application in mitochondrial DNA, Nonlinear Dynamics 93 (3) (2018) 1059–1071. [2] M. Pereyra, P. Lamberti, O. Rosso, Wavelet Jensen–Shannon divergence as a tool for studying the dynamics of frequency band components in EEG epileptic seizures, Physica A: Statistical Mechanics and its Applications 379 (1) (2007) 122–132. [3] C.-M. Wang, C.-M. Zhang, J.-Z. Zou, J. Zhang, Performance evaluation for epileptic electroencephalogram (EEG) detection by using Neyman–Pearson criteria and a support vector machine, Physica A: Statistical Mechanics and its Applications 391 (4) (2012) 1602–1609. [4] S. Dutta, D. Ghosh, S. Samanta, S. Dey, Multifractal parameters as an indication of different physiological and pathological states of the human brain, Physica A: Statistical Mechanics and its Applications 396 (2014) 155–163. [5] T. Zorick, M. A. Mandelkern, Multifractal detrended fluctuation analysis of human EEG: Preliminary investigation and comparison with the wavelet transform modulus maxima technique, PloS one 8 (7) (2013) e68360.

22

[6] D. Ghosh, S. Dutta, S. Chakraborty, Multifractal detrended crosscorrelation analysis for epileptic patient in seizure and seizure free status, Chaos, Solitons & Fractals 67 (2014) 1–10. [7] A. He, X. Yang, X. Yang, X. Ning, Multifractal analysis of epilepsy in electroencephalogram, in: 2007 IEEE/ICME International Conference on Complex Medical Engineering, IEEE, 2007, pp. 1417–1420. [8] S. Bhaduri, D. Ghosh, Electroencephalographic data analysis with visibility graph technique for quantitative assessment of brain dysfunction, Clinical EEG and Neuroscience 46 (3) (2015) 218–223. [9] J. T. Machado, S. H. Mehdipour, Multidimensional scaling analysis of the solar system objects, Communications in Nonlinear Science and Numerical Simulation 79 (2019) 104923. [10] J. A. T. Machado, Relativistic time effects in financial dynamics, Nonlinear Dynamics 75 (4) (2014) 735–744. [11] M. Mata, J. Machado, Entropy Analysis of Monetary Unions, Entropy 19 (6) (2017) 245. [12] S. Lu, X. Chen, J. K. Kanters, I. C. Solomon, K. H. Chon, Automatic selection of the threshold value R for approximate entropy, IEEE Transactions on Biomedical Engineering 55 (8) (2008) 1966–1972. [13] W. S. Pritchard, D. W. Duke, K. L. Coburn, N. C. Moore, K. A. Tucker, M. W. Jann, R. M. Hostetler, EEG-based, neural-net predictive classification of Alzheimer’s disease versus control subjects is augmented by nonlinear EEG measures, Electroencephalography and clinical Neurophysiology 91 (2) (1994) 118–130. [14] C. J. Stam, B. Jelles, H. A. M. Achtereekte, S. Rombouts, J. P. J. Slaets, R. W. M. Keunen, Investigation of EEG non-linearity in dementia and Parkinson’s disease, Electroencephalography and Clinical Neurophysiology 95 (5) (1995) 309–317. 23

[15] J. Jeong, J.-H. Chae, S. Y. Kim, S.-H. Han, Nonlinear dynamic analysis of the EEG in patients with Alzheimer’s disease and vascular dementia, Journal of Clinical Neurophysiology 18 (1) (2001) 58–67. [16] J. Jeong, S. Y. Kim, S.-H. Han, Non-linear dynamical analysis of the EEG in Alzheimer’s disease with optimal embedding dimension, Electroencephalography and Clinical Neurophysiology 106 (3) (1998) 220–228. [17] J. S. Richman, J. R. Moorman, Physiological time-series analysis using approximate entropy and sample entropy, American Journal of PhysiologyHeart and Circulatory Physiology 278 (6) (2000) H2039—-H2049. [18] S. David, D. Quintino, C. Inacio Jr, J. Machado, Fractional dynamic behavior in ethanol prices series, Journal of Computational and Applied Mathematics 339 (2018) 85–93. [19] J. Beran, Statistics for long-memory processes, Routledge, 2017. [20] L. A. Lipsitz, A. L. Goldberger, Loss of ‘complexity’ and aging: Potential applications of fractals and chaos theory to senescence, JAMA 267 (13) (1992) 1806–1809. [21] L. A. Lipsitz, Physiological complexity, aging, and the path to frailty, Science of aging knowledge environment: SAGE KE 2004 (16). [22] C. M. Ionescu, The human respiratory system: an analysis of the interplay between anatomy, structure, breathing and fractal dynamics, Springer Science & Business Media, 2013. [23] C. M. Ionescu, D. Copot, Monitoring respiratory impedance by wearable sensor device: Protocol and methodology, Biomedical Signal Processing and Control 36 (2017) 57–62. [24] I. Assadi, A. Charef, D. Copot, R. De Keyser, T. Bensouici, C. Ionescu, Evaluation of respiratory properties by means of fractional order models, Biomedical Signal Processing and Control 34 (2017) 206–213. 24

[25] A. E. Aubert, S. Vandeput, F. Beckers, J. Liu, B. Verheyden, S. Van Huffel, Complexity of cardiovascular regulation in small animals, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 367 (1892) (2009) 1239–1250. [26] M. A. K. A. Braeken, A. Jones, R. A. Otte, D. Widjaja, S. Van Huffel, G. J. Y. J. Monsieur, C. M. Van Oirschot, B. R. H. den Bergh, Anxious women do not show the expected decrease in cardiovascular stress responsiveness as pregnancy advances, Biological Psychology 111 (2015) 83–89. [27] E. Mortier, M. Struys, T. De Smet, L. Versichelen, G. Rolly, Closed-loop controlled administration of propofol using bispectral analysis, Anaesthesia 53 (8) (1998) 749–754. [28] L. C. Jameson, T. B. Sloan, Using EEG to monitor anesthesia drug effects during surgery, Journal of Clinical Monitoring and Computing 20 (6) (2006) 445–472. [29] R. S. Frackowiak, Human brain function, Elsevier, 2004. [30] J. Praline, J. Grujic, P. Corcia, B. Lucas, C. Hommet, A. Autret, B. De Toffol, Emergent EEG in clinical practice, Clinical Neurophysiology 118 (10) (2007) 2149–2155. [31] W. A. Hauser, D. C. Hesdorffer, et al., Epilepsy: Frequency, causes, and consequences, Epilepsy Foundation of America Landover, MD, 1990. [32] R. S. Fisher, C. Acevedo, A. Arzimanoglou, A. Bogacz, J. H. Cross, C. E. Elger, J. Engel Jr, L. Forsgren, J. A. French, M. Glynn, et al., ILAE official report: A practical clinical definition of epilepsy, Epilepsia 55 (4) (2014) 475–482. [33] R. S. Fisher, W. V. E. Boas, W. Blume, C. Elger, P. Genton, P. Lee, J. Engel Jr, Epileptic seizures and epilepsy: Definitions proposed by the international league against epilepsy (ILAE) and the international bureau for epilepsy (IBE), Epilepsia 46 (4) (2005) 470–472. 25

[34] S. Lahmiri, Generalized Hurst exponent estimates differentiate EEG signals of healthy and epileptic patients, Physica A: Statistical Mechanics and its Applications 490 (2018) 378–385. [35] U. R. Acharya, F. Molinari, S. V. Sree, S. Chattopadhyay, K.-H. Ng, J. S. Suri, Automated diagnosis of epileptic EEG using entropies, Biomedical Signal Processing and Control 7 (4) (2012) 401–408. [36] Q. Yuan, W. Zhou, S. Li, D. Cai, Epileptic EEG classification based on extreme learning machine and nonlinear features, Epilepsy Research 96 (12) (2011) 29–38. [37] R. A. S. Ruiz, R. Ranta, V. Louis-Dorr, EEG montage analysis in the blind source separation framework, Biomedical Signal Processing and Control 6 (1) (2011) 77–84. [38] D. Coyle, T. M. McGinnity, G. Prasad, Improving the separability of multiple EEG features for a BCI by neural-time-series-prediction-preprocessing, Biomedical Signal Processing and Control 5 (3) (2010) 196–204. [39] N. F. Ince, F. Goksu, A. H. Tewfik, S. Arica, Adapting subject specific motor imagery EEG patterns in space–time–frequency for a brain computer interface, Biomedical Signal Processing and Control 4 (3) (2009) 236–246. [40] A. Accardo, M. Affinito, M. Carrozzi, F. Bouquet, Use of the fractal dimension for the analysis of electroencephalographic time series, Biological Cybernetics 77 (5) (1997) 339–350. [41] S. A. David, C. C. Inacio Jr, Detrended fluctuation analysis and Hurst exponent as a measure to differentiate EEG signals, in: Biomath Communications Supplement V, BIOMATH, 2018. [42] A. Babloyantz, J. Salazar, C. Nicolis, Evidence of chaotic dynamics of brain activity during the sleep cycle, Physics Letters A 111 (3) (1985) 152–156.

26

[43] L. Kristoufek, How are rescaled range analyses affected by different memory and distributional properties? A Monte Carlo study, Physica A: Statistical Mechanics and its Applications 391 (17) (2012) 4252–4260. [44] R. Bryce, K. Sprague, Revisiting detrended fluctuation analysis, Scientific Reports 2 (2012) 315. [45] H. Hurst, The long-term dependence in stock returns, Transactions of the American Society of Civil Engineers 116 (1951) 770–99. ˇ c´ıkov´ [46] T. Gneiting, H. Sevˇ a, D. B. Percival, Estimators of fractal dimension: Assessing the roughness of time series and spatial data, Statistical Science (2012) 247–277. [47] J. Machado, Fractional order generalized information, Entropy 16 (4) (2014) 2350–2361. [48] R. G. Andrzejak, K. Lehnertz, F. Mormann, C. Rieke, P. David, C. E. Elger, Indications of nonlinear deterministic and finite-dimensional structures in time series of brain electrical activity: Dependence on recording region and brain state, Phys. Rev. E 64 (2001) 061907. doi:10.1103/PhysRevE.64. 061907. [49] C.-K. Peng, S. V. Buldyrev, S. Havlin, M. Simons, H. E. Stanley, A. L. Goldberger, Mosaic organization of DNA nucleotides, Phys. Rev. E 49 (1994) 1685–1689. doi:10.1103/PhysRevE.49.1685. [50] S.-J. Shieh, Long memory and sampling frequencies: stock index futures markets,

Evidence in

International Journal of Theoretical

and Applied Finance 09 (05) (2006) 787–799.

arXiv:http://www.

worldscientific.com/doi/pdf/10.1142/S0219024906003780, doi:10. 1142/S0219024906003780. [51] C. W. Granger, R. Joyeux, An introduction to long-memory time series models and fractional differencing, Journal of Time Series Analysis 1 (1) (1980) 15–29. doi:10.1111/j.1467-9892.1980.tb00297.x. 27

[52] P. Cizeau, Y. Liu, M. Meyer, C.-K. Peng, H. E. Stanley, Volatility distribution in the S&P500 stock index, Physica A: Statistical Mechanics and its Applications 245 (3-4) (1997) 441–445. [53] M. Ausloos, N. Vandewalle, P. Boveroux, A. Minguet, K. Ivanova, Applications of statistical physics to economic and financial topics, Physica A: Statistical Mechanics and its Applications 274 (1) (1999) 229–240. [54] B. B. Mandelbrot, J. W. V. Ness, Fractional Brownian motions, fractional noises and applications, SIAM Review 10 (4) (1968) 422–437. URL http://www.jstor.org/stable/2027184 [55] B. Mandelbrot, Statistical methodology for nonperiodic cycles: from the covariance to R/S analysis, in: Annals of Economic and Social Measurement, NBER, 1972, pp. 259–290. [56] B. B. Mandelbrot, J. R. Wallis, Robustness of the rescaled range R/S in the measurement of noncyclic long run statistical dependence, Water Resources Research 5 (5) (1969) 967–988. doi:10.1029/WR005i005p00967. [57] P. Welch, The use of fast Fourier transform for the estimation of power spectra: A method based on time averaging over short, modified periodograms, IEEE Transactions on Audio and Electroacoustics 15 (2) (1967) 70–73. doi:10.1109/TAU.1967.1161901. [58] G. J. Roerink, M. Menenti, W. Verhoef, Reconstructing cloudfree NDVI composites using Fourier analysis of time series, International Journal of Remote Sensing 21 (9) (2000) 1911–1917. doi:10.1080/014311600209814. [59] M. Tarnopolski, On the relationship between the Hurst exponent, the ratio of the mean square successive difference to the variance, and the number of turning points, Physica A: Statistical Mechanics and its Applications 461 (2016) 662 – 673. doi:https://doi.org/10.1016/j.physa.2016.06.004. [60] Z. Chen, P. C. Ivanov, K. Hu, H. E. Stanley, Effect of nonstationarities on detrended fluctuation analysis, Physical Review E 65 (4) (2002) 041107. 28

[61] H. Stanley, L. N. Amaral, A. Goldberger, S. Havlin, P. C. Ivanov, C.K. Peng, Statistical physics and physiology: Monofractal and multifractal approaches, Physica A: Statistical Mechanics and its Applications 270 (1-2) (1999) 309–324. [62] M. N. Flynn, W. Pereira, Ecological diagnosis from biotic data by Hurst exponent and the R/S analysis adaptation to short time series, Biomatematica 23 (2013) 1–14. [63] F. Serinaldi, Use and misuse of some Hurst parameter estimators applied to stationary and non-stationary financial time series, Physica A: Statistical Mechanics and its Applications 389 (14) (2010) 2770 – 2781. doi:https: //doi.org/10.1016/j.physa.2010.02.044. [64] L. Kristoufek, M. Vosvrda, Measuring capital market efficiency: Global and local correlations structure, Physica A: Statistical Mechanics and its Applications 392 (1) (2013) 184–193. [65] P. Hall, A. Wood, On the performance of box-counting estimators of fractal dimension, Biometrika 80 (1) (1993) 246–251. [66] M. G. Genton, Variogram fitting by generalized least squares using an explicit formula for the covariance structure, Mathematical Geology 30 (4) (1998) 323–345. [67] Y. Ma, M. G. Genton, Highly robust estimation of the autocovariance function, Journal of Time Series Analysis 21 (6) (2000) 663–684. doi: 10.1111/1467-9892.00203. [68] L. Kristoufek, M. Vosvrda, Commodity futures and market efficiency, Energy Economics 42 (2014) 50–57. [69] D. C. Howell, Statistical Methods for Psychology, Duxbury, 2002. doi: 10.2307/2348956.

29

[70] F. Aporti, R. Borsato, G. Calderini, R. Rubini, G. Toffano, A. Zanotti, L. Valzelli, L. Goldstein, Age-dependent spontaneous EEG bursts in rats: Effects of brain phosphatidylserine, Neurobiology of Aging 7 (2) (1986) 115 – 120. doi:https://doi.org/10.1016/0197-4580(86)90149-1. URL

http://www.sciencedirect.com/science/article/pii/

0197458086901491 [71] J. Pripfl, S. Robinson, U. Leodolter, E. Moser, H. Bauer, EEG reveals the effect of fmri scanner noise on noise-sensitive subjects, NeuroImage 31 (1) (2006) 332 – 341. doi:https://doi.org/10.1016/j.neuroimage.2005. 11.031. URL

http://www.sciencedirect.com/science/article/pii/

S105381190502495X [72] S. S. Shapiro, normality 611.

M. B. Wilk,

(complete

samples),

An analysis of variance test for Biometrika

52

(3-4)

(1965)

591–

arXiv:http://oup.prod.sis.lan/biomet/article-pdf/52/3-4/

591/962907/52-3-4-591.pdf, doi:10.1093/biomet/52.3-4.591. URL https://doi.org/10.1093/biomet/52.3-4.591 [73] J. T. Oliva, J. L. Garcia Rosa, How an epileptic EEG segment, used as reference, can influence a cross-correlation classifier?, Applied Intelligence 47 (1) (2017) 178–196. doi:10.1007/s10489-016-0891-y. URL https://doi.org/10.1007/s10489-016-0891-y [74] W. H. Kruskal, W. A. Wallis, Use of ranks in one-criterion variance analysis, Journal of the American Statistical Association 47 (260) (1952) 583–621.

arXiv:https://www.tandfonline.com/doi/pdf/10.1080/

01621459.1952.10483441, doi:10.1080/01621459.1952.10483441. URL

https://www.tandfonline.com/doi/abs/10.1080/01621459.

1952.10483441 [75] M. Eva Gonzlez-Trujano, E. Tapia, L. Lpez-Meraz, A. Navarrete, A. Reyes-Ramrez, A. Martnez, Anticonvulsant effect of annona di30

versifolia saff and palmitone on penicillin-induced convulsive activity. A behavioral and EEG study in rats, Epilepsia 47 (11) (2006) 1810– 1817. arXiv:https://onlinelibrary.wiley.com/doi/pdf/10.1111/j. 1528-1167.2006.00827.x, doi:10.1111/j.1528-1167.2006.00827.x. URL

https://onlinelibrary.wiley.com/doi/abs/10.1111/j.

1528-1167.2006.00827.x [76] W. Haynes, Tukey’s Test, Springer New York, New York, NY, 2013, pp. 2303–2304. doi:10.1007/978-1-4419-9863-7_1212. URL https://doi.org/10.1007/978-1-4419-9863-7_1212 [77] P. Hans, P.-Y. Dewandre, J. F. Brichant, V. Bonhomme, Comparative effects of ketamine on Bispectral Index and spectral entropy of the electroencephalogram under sevoflurane anaesthesia, BJA: British Journal of Anaesthesia 94 (3) (2004) 336–340. doi:10.1093/bja/aei047. URL https://doi.org/10.1093/bja/aei047 [78] V. Nikulin, T. Brismar, Phase synchronization between alpha and beta oscillations in the human electroencephalogram, Neuroscience 137 (2) (2006) 647 – 657.

doi:https://doi.org/10.1016/j.neuroscience.

2005.10.031. URL

http://www.sciencedirect.com/science/article/pii/

S0306452205011711 [79] P. Chang, L. Arendt-Nielsen, T. Graven-Nielsen, P. Svensson, A. C. Chen, Different EEG topographic effects of painful and non-painful intramuscular stimulation in man, Experimental Brain Research 141 (2) (2001) 195–203. doi:10.1007/s002210100864. URL https://doi.org/10.1007/s002210100864 [80] O. J. Dunn, Multiple comparisons using rank sums, Technometrics 6 (3) (1964) 241–252.

arXiv:https://www.tandfonline.com/doi/

pdf/10.1080/00401706.1964.10490181, doi:10.1080/00401706.1964. 10490181. 31

URL

https://www.tandfonline.com/doi/abs/10.1080/00401706.

1964.10490181 [81] M. Fraschini, M. Demuru, A. Crobe, F. Marrosu, C. J. Stam, A. Hillebrand, The effect of epoch length on estimated EEG functional connectivity and brain network organisation, Journal of Neural Engineering 13 (3) (2016) 036015. doi:10.1088/1741-2560/13/3/036015. URL https://doi.org/10.1088%2F1741-2560%2F13%2F3%2F036015 [82] S. Hensel, B. Rockstroh, P. Berg, T. Elbert, P. W. Schnle, Lefthemispheric abnormal EEG activity in relation to impairment and recovery in aphasic patients, Psychophysiology 41 (3) (2004) 394– 400.

arXiv:https://onlinelibrary.wiley.com/doi/pdf/10.1111/j.

1469-8986.2004.00164x, doi:10.1111/j.1469-8986.2004.00164x. URL

https://onlinelibrary.wiley.com/doi/abs/10.1111/j.

1469-8986.2004.00164x

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