CHAPTER 5 QUADRATIC PHASE COUPLING DETECTION

Transcription

1 91 CHAPTER 5 QUADRATIC PHASE COUPLING DETECTION 5.1 INTRODUCTION The bispectrum is a useful tool for identifying a process that is either non-gaussian or is generated by nonlinear mechanisms. This nonlinearity is not surprising since a wide range of physical and biological systems shows nonlinear behavior. Application of the bispectrum has been especially popular in biological systems because of the ubiquity of inherently nonlinear characteristics of biological mechanisms. One such characteristic is the presence of nonlinear interactions that have been detected in neural (Pfurtscheller et al. 1997, Huang et al. 24, Schwab et al. 25, Schack 22), renal (Raghavan et al. 26, Chon 25), and cardiovascular (Jamsek et al. 24, Al-Fahoum and Khadra 25) systems, in particular. Past research has demonstrated amply that oscillatory neural activities in different frequency bands can interact in a variety of ways. For instance, both the phase (φ 1 ) and the amplitude (A 1 ) at one frequency (f 1 ) can affect the phase (φ 2 ) at another frequency (f 2 ). In a phenomena called n: m phase synchrony, n cycles of one frequency can lock to m cycles of the other frequency (Palva et al 25, Tass et al 1998). Specifically, the delta phase modulates theta (4-1Hz) amplitude, and the theta phase modulates gamma (3-5Hz) amplitude. Recently, similar phenomena, in terms of the theta phase modulating the gamma oscillation amplitudes were observed in electrocorticogram (ECoG) recordings from human neocortex (Canolty et al. 26).

2 92 An important class of nonlinear interaction, named quadratic phase coupling (QPC) involves frequency triplet f 1, f 2 and f 1 +f 2. QPC means that the sum of the phases at f 1 (φ 1 ) and f 2 (φ 2 ) is the phase at frequency f 1 +f 2 (i.e.φ 1 + φ 2 ) which is often an indication of second order nonlinearities. Bispectral analysis is a powerful tool to detect QPC and has been applied successfully to evaluate QPC types of nonlinear effects in human electroencephalogram (EEG) recordings (Dumermut et al 1971, Zhou and Giannakis 1995, Shils et al 1996). To quantify the degree of a QPC, the bicoherence index (BCI) (Nikias and Raghuveer 1987, Raghuveer and Nikias 1985) is computed. The theoretical values of the BCI correspond to a range from insignificant to highly significant phase coupled peaks. The BCI is designed to consider only phase coupled components, eliminating bispectral peaks resulting from frequency locking alone that should not be present in a bispectrum but are often represented. However, due to technical considerations such as using insufficient number of segments to compute the bispectrum, frequencycoupled peaks will sometimes appear in the BCI. Further, for finite-length data sets, the high variability present in the bicoherence index will cause theoretically Gaussian processes to have a nonzero value. To avoid making erroneous decisions about the presence of phase coupling based on the BCI, a method was proposed by Elgar and Guza (1988) that is based on modeling the significance level for zero bicoherence. Recently Chon et al. (25) introduced a method, an autoregressive bispectrum combined with surrogate data method to test the statistical significance of the obtained quadratic phase coupled peaks. This approach allows better detection of phase-coupled peaks, even with noise contamination. This method still suffers from low specificity due to a normalization procedure that can allow insignificant bispectral peaks to

3 93 become significant. In addition, in certain instance, the BCI values are greater than one when a small amount of time variance was introduced into the data. This further complicates the interpretation of results, as most physiological data inherently have some degree of time variance. To compensate this drawback Pinhas et al. (24) introduced a method that statistically removes these erroneous peaks. However this method suffers from the fact that it is based on the central limit theorem, which requires large amount of data that may be difficult to obtain with physiological data. In a recent study, Kin et al. (28) proposed to quantify the presence of phase coupling using the bispectrum. However, this method only uses one random number to select a point for the rejection method, while normally two random numbers are required to generate an (x, y) coordinates pair. This turns out to be the Achilles heel of the method, potentially affecting the quality of the generated random numbers. Given the aforesaid problems associated with the detection of phase coupling with the use of BCI, the aim of the present work is to circumvent the limitations of the above methods. In this chapter, we propose a novel method to detect QPC that does not involve the use of the BCI; rather, it uses bispectrum estimation followed by testing the significance of the results against improved surrogate data realizations. The goal of an improved surrogate data transformation is to eliminate the nonlinear dynamics in the data. This leaves a time series with only linear properties, thus no phase coupling should be detected. As a result, only bispectral peaks remaining must arise from harmonic components and are insignificant. The efficacy of our new method, based on the use of the bispectrum estimation followed by the use of a improved surrogate data technique, will be compared to the traditional BCI.

4 THIRD ORDER CUMULANT Bispectra Consider three zero-mean third order stationary random processes x (k), y (k) and z (k). The bispectrum of x, y and z is defined as the 2-D Fourier Transform of the third-order cumulants (Nikias and Petropulu, 1993, Wang et al. 26),, (5.1), where the third order cumulant R xyz (m,n) is defined as, (5.2) As in the case of conventional power spectrum estimation, to get better estimates, suitable windows should be used. Two-dimensional windows for bispectrum estimation have been derived and discussed by Sasaki et al. (1975). The window functions should satisfy the following constraints. a) W(m, n) = W(n, m)=w(-m, n-m)=w(m-n, -n) (Symmetry properties of third moments); b) W(m, n)= outside the region of support of R(m, n); c) W(,)=1 (normalizing condition); (5.3) d) W (,, for all ( ω, ω. A class of functions which satisfies equation (5.3), for W (m, n), is the following (Sasaki et al 1975):

5 95 W (m, n)=d(m) d(n) d(n-m) (5.4) where d (m) = d(-m) (5.5a) d (m) = m>l (5.5b) d ()=1 D ( ), for all (5.5c) (5.5d) Equations (5.4) and (5.5) allow a reconstruction of two dimensional window functions for bispectrum estimation using standard one-dimensional lag windows. However, not all conventional power spectrum windows satisfy equation (5.5d). For example, the Hanning window has negative sidelobes in the frequency domain. There are three windows that satisfy (5.5a) (5.5d) have been reported in (Sasaki et al. (1975) such as Optimum window (minimum bispectrum bias supremum), Parzen window and Uniform window in frequency domain. The above three windows have been evaluated in terms of bispectrum bias spectrum (J) and approximate normalized (by triple product of power spectra) bispectrum variance (V). The Optimum window gives variance which is about 26 percent larger than that of the Parzen window while the former achieves a bias which is about 18 percent smaller that of the later (Sasaki et al. (1975). In bispectrum estimation we used parzen window which has been reported in Sasaki et al. (1975) , 2 1 (5.6),, where L is the length for each realization. There are two ways of estimating bispectra from data: direct and parametric. The direct method for bispectral estimation involves the following steps: 1) dividing data into M segments

6 96 where each segment is of length K, 2) computing the Fourier transform of each segment, and 3) estimating the bispectrum according to (Nikias and Raghuveer 1987)., (5.7) where X m (ω), Y m (ω) Z m (ω) are the Fourier transforms of the m th segment of the signal x(k), y(k) and z(k) and * means complex conjugate. The direct method of estimating the bispectrum is similar to the Welch method for power spectral estimation. Depending on how the three signals are chosen, a number of different bispectral quantities can be derived. For example, Letting x(k), y(k)=x(k) and z(k)=x(k) be the same process, (5.7) gives the auto- bispectrum for x(k), denoted as B xxx (ω 1, ω 2 ), which characterizes the signal relations at frequencies 1, 2 and in process x(k). Letting x(k), y(k) and z(k)=x(k), (5.7) gives the cross- bispectrum B xyx (ω 1, ω 2 ), which characterizes the signal relation at frequency 1 in x(k), 2 in y(k) and in x(k). It can be shown that the direct estimator in (5.7) is asymptotically unbiased and consistent (Nikias and Petropulu 1993). For sufficiently long segment length K and sufficiently large realization M, the direct method provides approximately unbiased estimates with asymptotic variances (Nikias and Petropulu 1993),, (5.8) where < ω 2 < ω 1, Re, and Im are the real and imaginary part of the bispectrum, N 3 is a number related to the window size, and P ( ) is the power

7 97 spectra of the investigated signal at frequency. The estimated bispectrum has approximately complex normal distributions (Vanness 1966) Traditional Statistical Methods Traditionally, the determination of phase-coupled peaks in the bispectrum required the calculation of a normalized bispectrum, i.e.bci, the BCI is calculated by,, (5.9) where P is the power spectrum of the signal. From the BCI, one would then need a method for determining significance in the peaks observed. The method developed by Elgar and Guza (1988) for determining the significance in the BCI based on describing the significance levels for zero bicoherence was used. In our test degree of freedom for Chi Square (χ 2 ) is (k-p-1) where k is the number of class intervals and p is the number of parameters of the distribution estimated. Significance levels for zero bicoherence were calculated as a function of dof for both normalizations, and for various combinations of ensemble averaging and/or frequency merging. Since the distributions of bicoherence do not depend significantly on the smoothing method or normalization, the significance levels for zero bicoherence must be independent of smoothing technique. Haubrich (1965) demonstrated that, for a true bicoherence of zero and the normalization given in (9), 2 should be chi-square distributed in the limit of large dof. Thus, for example, the 95 percent significance level for zero bicoherence is approximately 6. As shown in Figure 5.1, the numerical simulation agrees with Haubrich (1965) result even with different normalization and low dof. The agreement of other significance levels with the chi square distribution is illustrated in Figure 5.2. It is clear that large dof

8 98 is required to distinguish low, but nonzero bicoherence values from truly zero values (e.g., dof> 1 if the true bicoherence =.2). Improved surrogate method is a modified form of the original data that eliminates nonlinear properties while retaining linear statistical properties. Randomization of phases accomplishes the elimination of nonlinearity and since randomization can be performed in many ways, we can obtain multiple realizations of the improved surrogate data from a single time series. This is useful in the statistical testing of nonlinearity, as one can then use the generated and improved surrogate data as the null hypothesis to be tested against alternate hypothesis %significant bicoherence 95% significance bicoherence Degrees of Freedom Figure 5.1 Zero bicoherence versus degree of freedom for different normalizations and smoothing. Triangles, frequency merging, Kim and Powers or Haubrich normalization; asterisks, ensemble averaging, Kim and Powers normalization (Kim and Powers, 1979); circles, ensemble averaging, Haubrich normalization (Haubrich 1965). The solid line is the theoretical 95% significance level

9 99 95% significance bicoherence 95% significant bicoherence % 95% 9% 8% Degrees of Freedom Figure 5.2 The Zero bicoherence versus degree of freedom (obtained by frequency merging) 5.3 IMPROVED SURROGATE DATA METHOD: Theoretically, the bispectrum of a linear Gaussian process is always zero. However, due to limited data length, the estimated bispectra cannot reach zero. Generally, the improved surrogate data testing method involves three ingredients: 1) a null hypothesis; 2) a method to generate the improved surrogate data; and 3) testing statistics for significance evaluation. The null hypothesis in the present study is that the investigated data are from a linear Gaussian process and thus, the bispectral amplitudes are zero. If the null hypothesis is rejected, then we conclude that the data are either non-gaussian or come from nonlinear process. The improved surrogate data are generated in such a way that it is Gaussian distributed but has the same second order spectral properties (in the bivariate, auto spectra) as the original data. We now give a Matlab program for the improved ziggurat algorithm that provides Gaussian random numbers (Jurgen Doornik, 25): 1. r = ;

10 1 2. m1= ; 3. dn = ; 4. tn = dn; 5. vn = e-3; 6. q = vn/exp(-.5*dn*dn); 7. k(1) = (dn/q)*m1;k(2)=; 8. w(1)=q/m1; 9. w(128)=dn/m1; 1. f(1)=;f(128)=exp(-.5*dn*dn); 11. for (i=127:-1:3) 12. dn =sqrt(-2.*log(vn/dn+exp(-.5*dn*dn))); 13. k(i+1) = (dn/tn)*m1; 14. tn = dn; 15. f(i) = exp(-.5*dn*dn); 16. w(i) = dn/m1; 17. end 18. jsr = ; 19. for i=1: [jsrr,shr,uni,va] = shrn3(jsr); 21. jsr = double(2^32 * rand()); 22. shrd(i) = double(shr); 23. uni(i) = uni; 24. vat(i) = va; 25. end 26. for i=1: va = vat(i); 28. sh = shrd(i); 29. W = w(va); 3. if (sh< k(va)) 31. val = sh * w(va);

11 else 33. val = rep(va,uni,sh,w); 34. end 35. RX(i) = val; 36. End 37. function [rx] = rep(va,uni,sh,w) 38. r = ; 39. if (va == ) 4. x = -log(rand())*(1/r); 41. y = -log(rand()); 42. while((y+y) > (x*x)) 43. x = -log(rand())*(1/r); 44. y = -log(rand()); 45. end 46. if(x > ) 47. rexp = r+x; 48. else 49. rexp = -(r+x); 5. end 51. rx = rexp; 52. return 53. end 54. if (va == 1) 55. va = 2; 56. else 57. va = va; 58. end 59. X = W*sh; 6. A = f(va) + rand()*(f(va-1)-f(va)); 61. B = exp(-.5*x*x);

12 if (A < B) 63. rexp = X; 64. else 65. rexp = rand(); 66. end 67. rx = rep(va,uni,sh,w); The testing statistics is the amplitude of the bispectrum. We will give the steps for generating the improved surrogate data and the theoretical rationale behind the steps (Xue Wang et al. 27), (Schreiber and Schmitz 1996). follows. For easy implementation, use improved surrogate data method as step 1) Estimate G xx and G yy and G xy for original data x(t) and y(t) according to the following equation 1 2 lim, 1 2 lim, 2 lim,, (5.1) where T is the duration of the data, the expectation taken over multiple realizations, and X k and Y k are the Fourier transform of x(t) and y(t). step 2) Calculate Covariance Matrix according to the following equation

13 13 (5.11) (5.12) Now we consider how to generate two zero-mean linear Gaussian processes and which have the same second order statistical properties as x (t) and y (t). Let and denote the Fourier transforms of and. step 3) Draw values (realizations) of,, and from the Gaussian processes with zero mean and covariance matrix for each frequency. This can be done, for example, by using a Matlab function based on the improved ziggurat algorithm.. step 4) Take the inverse Fourier transform of and to obtain the improved surrogate data and. To ensure that the improved surrogate data are real valued, the negative frequency parts of and are taken as the complex conjugate of the positive frequency parts. step 5) Repeat steps 3) and 4) to generate multiple realizations of the improved surrogate data. X k and Y k could be drawn from Gaussian distribution with zero mean and covariance matrix.. This simplified method is similar to the method proposed by Timmer (1995). A normalized histogram is used to approximate the real probability density function (PDF) of the testing statistic under null hypothesis. To generate a detailed histogram that accurately

14 14 portrays the tail of a distribution (i.e., very small p-values), the computational cost may be significant. 5.4 SIMULATION RESULTS We have performed simulation studies to test the effectiveness of the method proposed before. Matlab Signal processing and Spectral analysis toolbox is used in our analysis Linear Bivariate AR Model Driven by Gaussian White Noise The model is written as x( t) =.8x( t 1).5x( t 2) + ε ( t) y( t) =.6x( t 5) + η( t) (5.13) Where and are uncorrelated Gaussian white noise with zero means and unit variances. The data set consists of M=2 realizations where each realization is of length K=512. For sampling frequency of 128Hz, each realization has the duration of 4s. To perform the test P =15 improved surrogate data sets were generated following the procedure in Section 5.3. The one sided power spectra, and cross spectra for both original (solid curve) and one set of improved surrogate data (dotted curve) are shown in Figure 5.3. It is seen that the improved surrogate data s spectra match well with that of the original data. This is an expected result. The maximum amplitude for x (t) original power spectra is (Figure 5.3 (top panel)) and that for improved surrogate data power spectra is and the maximum amplitude for y (t) original power spectra is (Figure 5.3 (middle panel)) and that for improved surrogate data power spectra is Similarly, the maximum amplitude for x

15 15 (t), y (t) original cross power spectra is (Figure 5.3 (bottom panel)) and that for surrogate data, cross power spectra is The plot for the auto-bispectrum (B xxx ) for the original data x (t) and that for one set of the improved surrogate data are shown in Figure 5.4 (a) and (b). The plots for the cross-bispectrum (B xyx ) and one set of its surrogate counterpart are shown in Figure 5.4 (c) and (d). The black and white regions are specified as a minimum and maximum bispectrum values respectively. The maximum amplitude for the original auto-bispectrum is.8259 (Figure 5.4(a)) and that for the improved surrogate data is.9254 (Figure 5.4 (b)). Similarly, the maximum amplitude for the original crossbispectrum is (Figure 5.4(c)) and that for the improved surrogate data is (Figure 5.4 (d)). Notice that the power spectra amplitude is higher between to 3 Hz than other regions can be explained by (5.8). Namely, the estimation variances are proportional to the power spectral amplitudes at those frequencies. Considering that the improved surrogate data contour plots are computed based on a randomly selected data set among P=15 available, these maximum value comparison suggest the known fact that there are no nonlinear or non-gaussian components in the original data. The histogram and Gaussian fit of input data x (t) and y (t) and improved surrogate data and are shown in Figure 5.5. left side figures represent histogram and Gaussian fit of original data x(t) and y(t) and right side figures represent histogram and Gaussian fit of the improved surrogate data and.

16 16 G xx Amplitude Magnitude One sided Power Spectra input data improved surrogate data data Frequency G yy 3 25 One sided Power Spectra input data improved surrogate data Magnitude Amplitude Frequency G XY 35 3 One sided Power Spectra input data improved surrogate data Amplitude Magnitude Frequency Figure 5.3 One sided power spectra of and (top panel), and (middle panel), and cross-spectra between x (t) and y (t) (bottom panel). The solid curve indicates the result from the original data and dotted curve indicates the result from one of the 15 improved surrogate data sets

17 17 (a) (b) 12 Maximum= Maximum= f1(hz ) 6 f1(hz ) f2(hz) f2(hz) Figure 5.4 (c) (d) (a) Auto-bispectrum B xxx for the original data x(t) and (b) for one improved surrogate data set, (c) cross bispectrum B xyx for the original data x(t), y(t), and (d) for one improved surrogate data set and

18 input data x(t) x(t) Gaussian Fit 8 7 improved surrogate data x'(t) x'(t) Gaussian Fit P DF 3 PDF t t (a) 7 6 input data y(t) y(t) Gaussian Fit 8 7 Improved Surrogate data y'(t) y'(t) Gaussian Fit P D F 3 P D F t t (b) Figure 5.5 Histogram and Gaussian fit of the original and the improved surrogate data of (a) x (t) and (b) y (t) and

19 Time Series Containing QPC To illustrate the necessity of complementing bicoherence estimation with statistical analysis, a simple simulation example is provided. The simulation consists of a combination of two test signals, involving three frequencies, as shown next:, sin 2 sin 2 sin 2 (5.14) where f 1 and f 2 are set to.1 Hz and.2 Hz respectively. For both test signals, the third frequency f 3 is set to f 1 +f 2 =.3Hz in order to achieve the frequency coupling phases associated with the first two frequencies (θ 1 and θ 2 ) which are randomly generated between π and π with a uniform distribution. For the first test signal y 1 (t), the third frequency is also phase coupled such that θ 3 = θ 1 + θ 2. The second test signal y 2 (t) is not phase coupled so that θ 3 is also randomly generated to be between π and π. The amplitude A is set to.5 for the phase un-coupled signal where as it is set to 1.5 for the phase coupled signal. The amplitude of the phase uncoupled signal is set to a high value to simulate a condition, where high bispectral values can be obtained from frequency matching components alone. Both test signals were generated at 1Hz sampling rate and contained 248 data points. For both test signals, 32 segments each containing 64 data points, with no overlapping segments, were used to estimate the bispectrum. The resulting bispectral (middle panels) and the BCI (bottom panels) for the phase coupled and uncoupled signals with their respective power spectra (top panels) are shown in the left and right panels of Figure 5.6.

20 11 Coupled Uncoupled 1 power spectrum of phase coupled 1 power spectrum of phase uncoupled Power.6 Power Frequency Frequency Bicoherence estimated via the direct (FFT) method Bicoherence estimated via the direct (FFT) method f f f f1 Figure 5.6 Bispectra (middle panels) and bicoherence (bottom panels) with (left panels) and without (right panels) phase coupling. Note the similar phase coupling magnitudes for both phase uncoupled and

21 111 coupled system and top row shows the power spectra for both phase coupled (left) and uncoupled (right) signals The power spectra of the phase-coupled and uncoupled signals are indistinguishable in terms of frequency information, albeit the amplitude of the phase-coupled spectrum is lower especially at.3hz. Although the phasecoupled signal has lower spectral amplitude than the phase uncoupled spectral peak, we observe a similar bispectral peak magnitude at the normalized frequency pair (.1,.2). In these two systems this example demonstrates an important limitation associated with the bispectrum estimation, it is possible to obtain a bispectral peak for the phase uncoupled signal under certain frequencies and amplitude combinations. That is frequency coupling alone is sufficient to generalize peaks in the bispectrum. Hence, a statistical method to distinguish true peaks results from frequency and phase coupling from erroneous peaks resulting from frequency coupling alone. The bottom panel shows the bi-coherence of the test signals. In phase coupled signals, the calculated value of Chi-square is less than the table value, it means that χ 2 cal lie not in the critical region. Hence, the null hypothesis that the given distribution follows a normal distribution is not rejected. In phase uncoupled signals, the calculated value of Chi-square is greater than the table value, it means that χ 2 cal lie within the critical region. Hence, the null hypothesis that the given distribution follows a normal distribution is rejected. The phase coupled points are collected with respect to their frequency origin. Then, in particular segment these points are tested using chi square test and their significance levels p =.5 less than.2, the QPC is not detected strongly. This method has been tested with other simulated data with different number of frequency components and different relative amplitudes.

22 112 Unless otherwise noted, the test signal used consisted of a phased-coupled as described earlier, at a 1-Hz sampling rate with 248 data points. The bispectra was also calculated as described before with segment size of 64, but 128 points were used for the Fast Fourier Transformation (64 points zero padding), and there were no overlapping segments. To guard against counting the same bispectral peak twice (especially possible if the frequency resolution is not high), each peak was checked against its nearest neighbors. If the magnitude of a peak was higher than any of its neighbors, it was then considered as a peak. We repeated this process for the entire bispectrum. 5.5 RESULTS AND DISCUSSION Application to Human EEG Signal In general longer data is needed for meaningful bispectrum estimation (Nikias and Petropulu 1993). We therefore prefer to group the EEG data over selected time (segments) (stages) for the frequency domain analysis. We take the simplest approach by constructing sets where the set members are frontal, central and parietal electrodes, i.e., Set C = (C Z, C 1, C 2, C 3, C 4, C 5, C 6 ), Set F = (F Z, F 1, F 2, F 3, F 4, F 5, F 6, F 7, F 8, F 9, F 1 ); Set P = (P Z, P 1, P 2, P 3, P 4, P 5, P 6, P 7, P 8, P 9 ); The averaged normalized BCI and power spectra and the improved surrogate data BCI of sets C, F, P from sample points are given in Figures 5.7(a-c) and 5.9 (a-c) respectively. In Figure 5.7 (a) the contour plot of the averaged normalized BCI from the channels of data in set C is

23 113 given, whereas in Figure 5.7 (b) of the same figure the averaged normalized power spectrum exhibits a strong peak in the 11-14Hz range, the normalized bicoherence reveals how tightly the phase values interact quadratically among themselves. For example, as seen in Figure 5.7 (a) the frequency regions at (f 1, f 2 ) where f 1 =12 ~ 16 and f 2 = 1 ~ 14, Hz: f 1 = 13 ~ 15 and f 2 = 4 ~ 7 Hz; f 1 =2 ~ 4 and f 2 = 1 ~ 3, and finally f 1 = 15 and f 2 = 12 Hz show strong (almost unity) quadratical interactions. However, it is quite difficult, if not impossible, to extract this information from Figure 5.7 (b) only. Figure 5.7 (a) clearly indicates more frequency interaction. Figure 5.7 (c) shows only prominent QPC using improved surrogate data. As shown on the Figure 5.7 (c), the BCI was able to correctly eliminate the phase uncoupled peaks as these peaks have values lower than the threshold value of.36 using chi square distribution ( 6/(2*32)). Similar quadratic interactions are observed for the set F data in Figure 5.8 (a). Compared to the result of the previous set, the distribution of frequency values seems to be similar. However the more numerous interactions are shifted toward f 1 +f 2 =14Hz line and the region at (f1, f2) where, f 1 = 12 ~ 14 and f 2 = 1 ~ 3 Hz. The averaged and normalized power spectrum shown in Figure 5.8 (b) reveals additional low frequency activity when compared with the result of the previous set. Figure 5.8 (c) shows prominent QPC at f 1 = 5Hz and f 2 = 4 Hz using improved surrogate data.

24 114 Averaged Normalized Bispectra 35 3 f2 Hz f2 Hz f1 Hz 1 (a) Averaged Power Spectrum Averaged power sepctrum.9.8 Magnitude Magnitude Frequency Hz (b) Averaged Normalized Bispectra 35 3 f2 Hz f2 Hz f1 Hz (c) Figure 5.7 For data set of C Z, C 1, C 2, C 3, C 4, C 5, C 6 from samples (a) Contour plot of the averaged normalized BCI. (b) Averaged and normalized power spectrum. (c) Contour plot of the averaged normalized improved surrogate data BCI

25 115 Averaged Normalized Bispectra 35 3 f2 Hz f2 Hz f1 Hz 1 (a) Averaged Power Spectrum Averaged power sepctrum.9.8 Magnitude Magnitude Frequency Hz (b) Averaged Normalized Bispectra 35 3 f2 Hz f2 Hz f1 Hz (c) Figure 5.8 For data set of F Z, F 1, F 2, F 3, F 4, F 5, F 6, F 7, F 8, F 9, F 1 from samples (a) Contour plot of the averaged normalized BCI. (b) Averaged and normalized power spectrum. (c) Contour plot of the averaged normalized improved surrogate data BCI

26 116 Averaged Normalized Bispectra 35 3 f2 Hz f2 Hz f1 Hz 1 (a) Averaged Power Spectrum Averaged power sepctrum.9.8 Magnitude Magnitude Frequency Hz (b) Averaged Normalized Bispectra f2 Hz f2 Hz f1 Hz (c) Figure 5.9 For data set of P Z, P 1, P 2, P 3, P 4, P 5, P 6, P 7, P 8, P 9 from samples (a) Contour plot of the averaged normalized BCI. (b) Averaged and normalized power spectrum. (c) Contour plot of the averaged normalized improved surrogate data BCI

27 117 The results related to set P is given in Figure 5.9. It is clear that for this time period, this region of brain is highly dominated by the spindle activity as the bispectrum shown in Figure 5. 9(a) and the power spectrum shown in Figure 5.9(b). Moreover the bispectrum suggest that it would be more realistic to think that the sleep spindle activity has atleast some types of second order nonlinearity due to the appearance of the strong interactions on the f 1 =13-15 and f 2 = Hz region. The two methods are compared and the results are summarized in Table 5.1. These results show that a strong consistent phase coupling only exists for sleep spindles region (stage 2) and however, no consistent phase coupling was observed in slow wave sleep (-4Hz region). The problem here is that it would be difficult to distinguish between two signals with high coupling strengths with the real data and improved surrogate data, as they will show up with similar values. This suggests that both the real and improved surrogate statistics are less able to distinguish relative degrees of coupling when signals are strongly coupled. Compared with other methods improved surrogate data method efficiently detect QPC articulately. Table 5.1 Comparisons of the QPC With real data and improved surrogate data Data set Set C Set F Peak freq (Hz) (4,3) (14,7) (15,12) (12,3) (14,12) (5,4) Real data BCI Improved surrogate data Peak freq (Hz) BCI (15,12).8132 (6,5).7932 Set P (11,14).5734 (12,14).9462

28 Test of amount of phase coupling Here our statistical threshold based on the normality assumption was verified using the chi-square goodness of fit test. The calculated p value was.5, which confirms that this set is drawn from a normally distributed population. In testing whether certain observations are consistent with an assumed probability density function, the chi-square goodness of fit test is often used. The amount of phase coupling varied from % to 1% at an increment of 1%. For each level of phase coupling, 1 realizations of the EEG signals were extracted from data base using open and read EDF Matlab files. Each realization of the test signal was corrupted by -3dB AGWN. For sensitivity testing, the calculated value for each method was recorded at the known phase-coupled frequency. For specificity, the total number of significant detected peaks in the entire bispectrum was recorded and the median between the realizations is reported. As shown in the top panels of Figure 5.1, the percent of coupling that each method was sensitive to was approximately 4%, and 18% for the real data and the improved surrogated data respectively. The bottom panels show the specificity of the two tests. As shown in Figure 5.1 (bottom panel) the specificity for the BCI and improved surrogate data method is very high, using the chi-square method. This example provides evidence that the improved surrogate data method has suitable combination of sensitivity and specificity in detecting low levels of phase coupling. Further, it should be noted that the calculated value for the improved surrogate data method linearly increases with increasing coupling percent for values above 18%, thereby suggesting that the method provides a good quantification of the actual amount of phase coupling percent in the system. The bicoherence index and the chi-square statistics, however, show a more sigmoidal relationship with only a small window of linearity from approximately 3% to 8% coupling. The problem here is that

29 119 it would be difficult to distinguish between two signals with high coupling strengths with the BCI and the chi-square statistics, as they will show up with similar values. Varying the number of segments is tested, as it has been demonstrated that only by having a sufficient number of segments will one detect the presence of phase coupling, if it exists (Nikias and Petropulu 1993). The size of each segment was kept constant at 64 data points. The number of segments was varied from 1 to 32 at an increment of one. For example, 1 segment means correspond to 248 data points in total, still 64 in each segment. For each segment, 1 realizations of the test signal were generated. Each realization was corrupted by -3dB AGWN. Similar to previous simulations, the calculated value of the three sets and the number of detected peaks will be recorded for sensitivity and specificity, respectively. The simulation examples presented generally shows that our proposed approach, improved surrogate data method, offers the suitable combination of sensitivity and specificity under all of the tested conditions. In this paper, a statistical method based on improved surrogate data was introduced to analyze bispectral data. Our approach completely bypasses the use of the bicoherence index. As shown in our results, the normalization factor in the computation of the bicoherence index is the main culprit in providing less sensitivity and less specific results. It should be noted that the bispectrum detects not only quadratic phase coupled phenomenon but also provides information regarding nonlinearity, and deviation from Gaussian process. Therefore with the improved surrogate data method, one can obtain statistical quantification regarding the phase coupling, nonlinearity and deviation from normality.

30 12 Figure 5.1 Comparison between the real and improved surrogate data with varying amounts of coupling. Plotted on the top panels are the calculated bicoherence index and bottom panels are detected events The improved surrogate data, BCI and chi-square statistics were all based on nonparametric bispectral estimation. However, all of the methods are also applicable to parametric bispectral estimation. The advantages of using the parametric over the nonparametric bispectrum are higher frequency resolution and its ability to retain the accuracy for data with short data

31 121 records. However, the main disadvantage of the parametric approach is the determination of model order, which can be complex. A. Comparison with the asymptotic methods One method that is derived from the asymptotic theory is the Hinch method (1982). The Hinch (1982) test statistics is computed by summing over the average skewness for all frequency combinations in the bispectral principal domain. Since the bisepctral estimates are approximately independent across the frequency plane, this sum will be approximately distributed as a chi-square distribution. This method has been shown to be effective and conservative (Patterson and Ashley 2); however, the confidence threshold from the asymptotic method is based on the argument of an infinite sample size and one needs a larger number of averages for the argument to become applicable. An improved surrogate data method approach has the advantage of providing a more exacting test for finite data sets (Hinch 25). B. Effect of smoothing technique at low dof For the case of no frequency merging (i.e., statistical stability is obtained solely by ensemble averaging), it can be shown that the Kim and Powers (Kim and Powers 1979) normalization leads to bicoherence values between and 1. On the other hand, the Haubrich (Haubrich 1965) normalization is not bounded above by 1. Furthermore, it can be shown that with frequency merging, the Kim and Powers normalized bicoherences also become unbounded above 1. In the present numerical simulation with low dof (< 32) the bicoherence values greater than 1 were occasionally obtained for the Haubrich normalization and for the Kim and Powers normalization when frequency merging was used (Kim and Powers1979). This slightly raises the upper tail of the bicoherence distributions. However, the

32 122 corresponding increases of the significance levels at very low dof is small (13 percent for 8 dof, Figure 5.1), and negligible at higher dof. C. Comparison with Other Surrogate Data Methods Multiple ways are currently in use to generate surrogate data sets to test the significance of the bispectrum. In one method, the Fourier transform (FT) is estimated for each segment and the phase of the Fourier transform is randomized without changing the Fourier amplitude. The result is then inverse Fourier transform to generate a surrogate time series (Schanze and Eckhorn 1997), (Schwilden and Jeleazcov 22). While the phase randomization destroys nonlinearity and leads to linear Gaussian processes (Venema 26), the Fourier amplitudes are random variables for a stationary process as well. By keeping them constant, one loses an important degree of freedom (Timmer 1995). In contrast, our method, in which both amplitudes and phases of the Fourier transform are randomly generated, produces improved surrogate data sets that are explicitly Gaussian, nonlinear, and share the same second order statistics with the original data. 5.6 SUMMARY We compared, on different approaches of surrogate data generation. Such as the AR method, the FT method and the iteratively refined amplitudeadjusted FT (IAAFT). Consistency in the outcome of the nonlinearity test among the three methods was interpreted as an indication of the linearity of the investigated time series, when all approaches did not reject the null hypothesis, or as a marker of presence of nonlinear dynamics, when all approaches rejected the null hypothesis. These situations were successfully reproduced by the simulations (see Figure 4.4 and Figure 5.5), were encountered in a considerable part of the experimental data.

33 123 On the other side, different outcomes of the nonlinearity test observed using the different surrogate approaches may give important information about the nature of the investigated time series. In particular, a disagreement between the AR approaches based surrogate data generation, may suggest of a role of nonstationarity on the detection of nonlinear dynamics. Our simulations showed that, in the presence of nonstarionarity behaviors like quadratic phase coupling imposed on linear time series, FT based surrogates tend to reject the null hypothesis, thus erroneously indicating nonlinearity. While IAAFT based surrogates are correctly consistent with null hypothesis (see Figure 5.1). Moreover, in the presence of weak nonlinearities confined to a short epoch, FT surrogates tend to accept the null hypothesis while only IAFFT surrogates show a high rejection ratio. There are two situations mentioned in the difference between FT and IAAFT surrogate realizations. The first situation might reflect the fact that FT surrogates are forced to have Gaussian marginal distribution, and thus departures from Gaussianity of the distribution of the observed time series may include false rejections when FT surrogates are used (see Figure 4.7). The second situation might be due to the strict adherence of the power spectrum of constrained surrogates to the spectrum of the original series, when can favor false rejections (Dolan and Spano, 21), (Kugiumtzis, 1999), combined with larger variance in power spectrum replication obtained by typical surrogates, which can lead to misleading consistency with the null hypothesis. More generally, it is worth remarking the discrepancies among the outcomes of the nonlinearity test performed with the various surrogate approaches could be caused by subtle differences between the distribution (and / or the linear correlation) of the original series and of the surrogate seires. Hence some care has to be taken in interpreting differences in the detection of nonlinear dynamics among the various surrogate approaches.

34 124 The various approaches for nonlinear detection are Markov models, spline interpolation, radial basis functions and neural networks. In summary, the improved surrogate data method proposed in this chapter is shown to give accurate results when applied to the test significance of bicoherence. It is based on solid statistical principles and overcomes some weakness in previous methods for the same purpose. It is the expected repertoire of nonlinear analysis methods for neuroscience and other biomedical applications.