Final Report For Undergraduate Research Opportunities Project Name: Biomedical Signal Processing in EEG. Zhang Chuoyao 1 and Xu Jianxin 2

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1 ABSTRACT Final Report For Undergraduate Research Opportunities Project Name: Biomedical Signal Processing in EEG Zhang Chuoyao 1 and Xu Jianxin 2 Department of Electrical and Computer Engineering, National University of Singapore In this paper, we will present application of independent component analysis (ICA) to the problem of source identification, especially artifact identification in multi-channel EEG. We first introduce principle of EEG and analyze its characteristic spectral shapes. Then, reason for adopting ICA, instead of traditional linear methods such as PCA and SVD to process EEG signal is stated. After that, we explain the mathematical model of ICA and propose criteria and algorithms. Finally we consider artifact identification (separation) as an essential application of ICA to EEG signal. EEG INTRODUCTION AND SPECTRAL ANALYSIS EEG, the acronym for Electroencephalography, is the neurophysiologic measurement of the electrical activity of the brain by recording from electrodes placed on the scalp. EEG is used extensively for monitoring the electrical activity within the human brain, both for research and clinical purposes. It is in fact one of the most widespread brain mapping techniques to date. EEG is used both for the measurement of spontaneous activity and for the study of evoked potentials. Evoked potentials are activity triggered by a particular stimulus that may be, for example, auditory. Typical clinical EEG systems use around 20 electrodes, evenly distributed over the head. State-of-the-art EEGs may consist of a couple hundred sensors. The signal-to noise ratio is typically quite low: the background potential distribution is of the order of 100 microvolt, whereas the evoked potentials may be two orders of magnitude weaker. The EEG waveforms can be characterized in frequency domain. The different frequency components of EEG denote different brain activities: Alpha wave This component has frequency ranges from 8 to 12 Hz. This typically infers that the person in test is relaxed while awake. The typical waveform is as follows: 1 Student 2 Supervisor

2 Figure 1. Typical waveform of Alpha wave Beta wave This component has frequency ranging from 12Hz to 30Hz. It is also characterized by low amplitude. In reality, this component arises with active mental activities such as anxiety. The typical waveform is as follows: Figure 2. Typical waveform of Beta wave Gamma wave This wave component usually corresponds to frequency ranges Hz. Gamma can only be observed when object is in thinking process. The typical waveform is as follows: Figure 3. Typical waveform of Gamma wave TRADITIONAL LINEAR SIGNAL PROCESSING METHODS IN EEG AND THEIR DRAWBACKS Principal Component Analysis (PCA) and Singular Value Decomposition (SVD) are two linear processing methods utilizing theory of linear algebra. In brief, the PCA decorrelate the output by computing eigenvectors and project the original signal vector to these directions. The components extracted from PCA/SVD are only uncorrelated, while not necessarily statistically independent (unless components are all gaussian). Thus, the resulting components from PCA/SVD may not carry clear physiological significance, thus reducing their use in scenarios where we need to extract independent components, ones that bear brain-functional and physiological meaning. ICA AND ITS MATHEMETICAL MODELS

3 Figure 4. Linear mixing (A) of the latent source signals s(t), demixing by whitening (V) And orthogonal transform (W) of the observed data X(t) Independent component analysis was originally developed to deal with problems that are closely related to the cocktail-party problem [1]. Since the recent increase of interest in ICA, it has become clear that this principle has a lot of other interesting applications as well, such as that in electrical recordings of brain activity as given by an electroencephalogram (EEG). The EEG data consists of recordings of electrical potentials in many different locations on the scalp. These potentials are presumably generated by mixing some underlying components of brain and muscle activity. This situation is quite similar to the cocktail-party problem: we would like to find the original components of brain activity, but we can only observe mixtures of the components. ICA can reveal interesting information on brain activity by giving access to its independent components. As illustrated in Figure 4, The ICA problem [2], [3] can be formulated as given X(t), the observed data, estimate both the matrix A and the corresponding S(t). There are several limitations to solving ICA problem. If no suppositions are made about the noise, then additive noise term in Eq.4 is omitted. X(t) = AS(t) + N(t) = S i (t)a i + N(t) (5) A practical strategy is to include it in the signals as a supplementary term in the sum Eq.5. Thus the ICA model becomes: X(t) = AS(t) = S i (t)a i (6) ICA consists in updating a demixing matrix B(t) of size M * N, without resorting to any information about the spatial mixing matrix A, so that the output vector y(t ) = B(t) x(t) becomes an estimate of the original independent source signals s(t ). Since ICA deals with higher-order statistics it is justified to normalize in some sense the first- and second-order moments. The effect is that the separating matrix B(t ) is divided in two parts: the first is dealing with dependencies in the first two moments the whitening matrix V(t ), and the second comprises the dependencies in higher-order statistics the orthogonal separating matrix W(t ) B(t) = W(t)V(t) -> WV (7) On the other hand, standard PCA, as described previously in this paper, is often used for whitening in a sense that the covariance matrix of resulting vector after PCA is diagonal, indicating that components are uncorrelated. Having discussed PCA or whitening operation V, we now further our discussion on how to find an orthogonal transformation W. Nongaussianity-maximization Criteria& Gradient Algorithm

4 Criteria: Maximization of kurtosis Kurtosis is defined in the zero-mean case by the equation Kurt(x) = E{x 4 } 3[E{x 2 }] 2 (8) For whitened variable x, E{x 2 } = 1; Kurt(x) = E{x 4 } 3 (9) In fact, 3 is for four order moments for Gaussian random variable. Thus for Gaussian random variable Kurt(x) = 3-3 = 0. Thus, the absolute value of kurtosis determines the likeliness of a zero mean, whitened, symmetric random variable to the standard Gaussian variable. In other words, a variable with large absolute value of kurtosis will have large nongaussianity. In context of ICA, given whitened vector z, we want an orthogonal transform W such that the resulting vector has most nongaussianity. In practice, to maximize the absolute value of kurtosis, we would start from some vector w, compute the direction in which the absolute value of the kurtosis of Y = w T z is growing most strongly, based on the available sample z(1),., z(t) of mixture vector z, and then move the vector w in that direction. This idea is implemented in gradient methods and their extensions. Using the principles in Chapter 3, the gradient of the absolute value of kurtosis of w T z can be simply computed as d kurt(w T z) /dw = 4 sign(kurt(w T z))[e{z(w T z) 3 } 3w w 2 ] (10) since for whitened data we have E{(w T z) 2 } = w 2. Since we are optimizing this function on the unit sphere w 2 = 1, the gradient method must be complemented by projecting w on the unit sphere after every step. This can be done simply by dividing w by its norm. Actually, in many cases one knows in advance the nature of the distributions of the independent components, i.e., whether they are subgaussian or supergaussian. Then one can simply plug the correct sign of kurtosis in the algorithm, and avoid its estimation. Minimization of Mutual Information Criteria and Algorithm Criteria: First we need to define several information-theoretic quantity for later use. Differential entropy: Negentropy: (16)

5 (17) Mutual information: (18) Suppose we have for an invertible linear transformation (demixing matrix B) y = Bx (x be observation vector) (19) Using information-theoretic relation of input x and output y, the result of linear transformation B, we have the following equation. I(y 1,.y n ) = Σ H(y i ) H(x) log detb (20) Now, let us consider what happens if we constrain the y i to be uncorrelated and of unit variance. This means E{yy T } = BE{xx T }B T = I, which implies and this implies that detb must be constant since det E{xx T } does not depend on B. Moreover, for y i of unit variance, entropy and negentropy differ only by a constant and the sign, as can be seen in Eq.17. Thus we obtain where the constant term does not depend on B. This shows the fundamental relation between negentropy and mutual information. We see in Eq.21 that finding an invertible linear transformation B that minimizes the mutual information is roughly equivalent to finding directions in which the negentropy is maximized. We have seen previously that negentropy is a measure of nongaussianity. Thus, Eq.21 shows that ICA estimation by minimization of mutual information is equivalent to maximizing the sum of nongaussianities of the estimates of the independent components, when the estimates are constrained to be uncorrelated. Thus, we see that the formulation of ICA as minimization of mutual information gives another rigorous justification of our more heuristically introduced idea of finding maximally nongaussian directions. In practice, however, there are also some important differences between these two criteria: 1. Negentropy, and other measures of nongaussianity, enable the deflationary, i.e., one-by-one, estimation of the independent components, since we can look for the maxima (21)

6 of nongaussianity of a single projection b T x. This is not possible with mutual information or most other criteria, like the likelihood. 2. A smaller difference is that in using nongaussianity, we force the estimates of the independent components to be uncorrelated. This is not necessary when using mutual information, because we could use the form in Eq.20 directly. Thus the optimization space is slightly reduced. Algorithm To use mutual information in practice, we need some method of estimating or approximating it from real data. Earlier, we saw two methods for approximating mutual entropy. The first one was introduced in [4]. The second one was based on sublinear approximation technique in [5]. Using mutual information leads essentially to the same algorithms as used for maximization of nongaussianity. In the scenario of maximization of nongaussianity, the corresponding algorithms are those that use symmetric orthogonalization, since we are maximizing the sum of nongaussianities, so that no order exists between the components. Thus, we do not present any new algorithms in this chapter; the reader is referred to algorithm for maximization of nongaussianity. ARTIFACT IDENTIFICATION As a first application of ICA on EEG signals, we consider separation of artifacts. Artifacts are signals that are not generated by brain activity, but by some external interference, such as muscle activity. A typical example is ocular artifacts, generated by eye muscle activity. A review on artifact identification can be found in [6]. The simplest and most widely used method consists of discarding the portions of the recordings containing attributes (e.g., amplitude peak, frequency contents, variance and slope) that are typical of artifacts and exceed a determined threshold. This may lead to significant loss of data, and to complete inability of studying interesting brain activity occurring near or during strong eye activity, such as in visual tracking experiments. Other methods include the subtraction of a regression portion of one or more additional inputs (e.g., from electrooculograms, electrocardiograms, or electromyograms) from the measured signals. This technique is more likely to be used in EEG recordings. ICA gives a method for artifact removal where we do not need an accurate model of the process that generated the artifacts; this is the blind aspect of the method. Neither do we need specified observation intervals that contain mainly the artifact, nor additional inputs; this is the unsupervised aspect of the method. Thus ICA gives a promising method for artifact identification and removal. It was shown in [8] that artifacts can indeed be estimated by ICA alone. It turns out that the artifacts are quite independent from the rest of the signal, and thus even this requirement of the model is reasonably well fulfilled. In the experiments on EEG artifact removal, the EEG signals were recorded in a shielded room with the 122-channel whole-scalp magnetometer described above. The test person

7 was asked to blink and make horizontal saccades, in order to produce typical ocular (eye) artifacts. Moreover, to produce muscular artifacts, the subject was asked to bite his teeth for as long as 20 seconds. Yet another artifact was created by placing a digital watch one meter away from the helmet into the shielded room. Figure 5 presents a subset of 12 artifact contaminated EEG signals, from a total of 122 used in the experiment. Several artifact structures are evident from this figure, such as eye and muscle activity. The results of artifact extraction using ICA are shown in Fig. 6. Components IC1 and IC2 are clearly the activations of two different muscle sets, whereas IC3 and IC5 are, respectively, horizontal eye movements and blinks. Furthermore, other disturbances with weaker signal-to-noise ratio, such as the heart beat and a digital watch, are extracted as well (IC4 and IC8, respectively). IC9 is probably a faulty sensor. ICs 6 and 7 can be breathing artifacts. Figure 5. part of EEG channel recordings IC1 IC2 IC3 IC4 IC5 IC6 IC7 IC8 IC9 Figure 6. Extracted artifact components

8 CONCLUSION AND FURTHER RESEARCH: We have introduced ICA, an alternative biomedical signal processing method applied in EEG and explained its advantages over more traditional linear methods such PCA and SVD, which only operate on second-order statistics. We ve also covered mathematical models of ICA and stated its rationale, criteria and algorithms. Finally, we ve covered one important ICA application to EEG, artifact identification. We ve seen that using ICA, we ve successfully recovered artifacts that are deliberately introduced in the experiment. Throughout this paper, we ve assumed a linear ICA model. This assumption is valid when the source mixing process is linear in nature or can be linearized without much distortion. However, there are scenarios where EEG signals cannot be simply modeled as linear superposition of independent components. In these cases, we need to consider non-linear ICA method along with corresponding criteria and algorithms. REFERENCE 1. S.-I. Amari. Neural learning in structured parameter spaces natural Riemannian gradient. In Advances in Neural Information Processing Systems 9, pages MIT Press, K. Abed-Meraim and P. Loubaton. A subspace algorithm for certain blind identification problems. IEEE Trans. on Information Theory, 43(2): , L. Almeida. Linear and nonlinear ICA based on mutual information. In Proc. IEEE 2000 Adaptive Systems for Signal Processing, Communications, and Control symposium (AS-SPCC), pages , Lake Louise, Canada, October T. Vilmansen. On an Approximation of Mutual Information. In IEEE Transactions on Computers, Volume 23, Issue 12, pages IEEE Computer Society 5 Sudipto Guha, Andrew McGregor and Suresh Venkatasubramanian, "Streaming and Sublinear Approximation of Entropy and Information Distances", 17th ACM-SIAM Symposium on Discrete Algorithms, C. H. M. Brunia, J. Möcks, andm. Van den Berg-Lennsen. Correcting ocular artifacts a comparison of several methods. J. of Psychophysiology, 3:1 50, R. Vig ario. Extraction of ocular artifacts from EEG using independent component analysis. Electroenceph. Clin. Neurophysiol., 103(3): , T. P. Jung, C. Humphries, T.-W. Lee, S. Makeig, M. J. McKeown, V. Iragui, and T. Sejnowski. Extended ICA removes artifacts from electroencephalographic recordings. In Advances in Neural Information Processing Systems, volume 10. MIT Press, 1998.