File: ica tutorial2.tex. James V Stone and John Porrill, Psychology Department, Sheeld University, Tel: Fax:

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1 File: ica tutorial2.tex Independent Component Analysis and Projection Pursuit: A Tutorial Introduction James V Stone and John Porrill, Psychology Department, Sheeld University, Sheeld, S 2UR, England. Tel: Fax: j.porrill, j.v.stone@shef.ac.uk Web: April 2, 998 Abstract Independent component analysis (ICA) and projection pursuit (PP) are two related techniques for separating mixtures of source signals into their individual components. These rapidly evolving techniques are currently nding applications in speech separation, ERP, EEG, fmri, and low-level vision. Their power resides in the simple and realistic assumption that dierent physical processes tend to generate statistically independent signals. We provide an account that is intended as an informal introduction, as well as a mathematical and geometric description of the methods. Introduction Independent component analysis (ICA) [Jutten and Herault, 988] and projection pursuit (PP) [Friedman, 987] are methods for recovering underlying source signals from linear mixtures of these signals. This rather terse description does not capture the deep connection between ICA/PP and the fundamental nature of the physical world. In the following pages, we hope to establish, not only that ICA/PP are powerful and useful tools, but that this power follows naturally from the fact that ICA/PP are based on assumptions which are remarkably attuned to the spatiotemporal structure of the physical world.

2 Most measured quantities are actually mixtures of other quantities. Typical examples are, i) sound signals in a room with several people talking simultaneously, ii) an EEG signal, which contains contributions from many dierent brain regions, and, iii) a person's height, which is determined by contributions from many dierent genetic and environmental factors. Science is, to a large extent, concerned with establishing the precise nature of the component processes responsible for a given set of measurements, whether these involve height, EEG signals, or even IQ. Under certain conditions, the underlying sources of measured quantities can be recovered by making use of methods (PP and ICA) based on two intimately related assumptions. The more intuitively obvious of these assumptions is that dierent physical processes tend to generate signals that are statistically independent of each other. This suggests that one way to recover source signals from signal mixtures is to nd transformations of those mixtures that produce independent signal components. This independence is given much emphasis in the ICA literature, although an apparently subsidiary assumption that source signals have amplitude histograms that are non-gaussian is also required. In (apparent) contrast, the PP method relies on the assumption that any linear mixture of any set of (nite variance) source signals is Gaussian, and that the source signals themselves are not Gaussian. Thus, another method for extracting source signals from linear mixtures of those signals is to nd transformations of the signal mixtures that extract non- Gaussian signals. It can be shown that the assumption of statistical independence is implicit in the assumption that source signals are non-gaussian, and therefore that both PP and ICA are actually based on the same assumptions. Within the literature, PP is used to extract one signal at a time, whereas ICA extracts simultaneously a set of signals. However, like the apparently dierent assumptions of PP and ICA, this dierence is supercial, and reects the underlying histories of the two methods, rather than any fundamental dierence between them. Recent applications of ICA include separation of dierent speech signals [Bell and Sejnowski, 995], analysis of EEG data [Makeig et al., 997], functional magnetic resonance imaging (fmri) data [McKeown et al., 998], image processing [Bell and TJ, 997], and the relation between biologicial image processing and ICA [van Hateren and van der Schaaf, Setting the Scene Before becoming too embroiled in the intricacies of ICA, we need to establish the class of problems they can address. Given N time-varying source signals, we dene the amplitudes of these signals at time t as a column vector s t = fst; :::; s Nt g. These signals can be linearly combined to form a signal mixture x t = as t, where each element of the row vector a species how much of the corresponding source signal s it contributes to the signal mixture x t. Given M signal mixtures x t = fxt; :::; x Mt g T we can dene a mixing matrix A = fa ; :::; a M g T in which each row a i species a unique mixture x it of the signals s t = fst; :::; s Nt g. (Note that the t subscript denotes time, whereas the T superscript denotes the transpose operator). Using this matrix notation, the 2

3 formation of M signal mixtures from N source signals can be written as: x t = As t () Both ICA and PP are capable of taking the signal mixtures x and recovering the sources s. That the mixtures can be separated in principle is easily demonstrated now that the problem has been summarised in matrix algebra. An `unmixing' matrix W is dened such that: s t = Wx t (2) Given that each row in W species how the mixtures in x are recombined to produce one source signal, it follows that it must be possible to recover one signal at a time by using a dierent row vector to extract each signal. For example, if only one signal is to be extracted from M signal mixtures then W is a M matrix. Thus, the shape of the unmixing matrix W depends upon how many signals are to be extracted. Usually, ICA is used to extract a number of sources simultaneously, whereas PP is used to extract one source at a time. 3 The Geometry of Source Separation The nature of the linear `unmixing' transformation matrix W can be conveniently explored in terms of two signal mixtures. Consider two signals s = fs; s2g that have been mixed with a 2 2 mixing matrix A to produce two signal mixtures x = As, where x = fx; x2g. If we interpret this in terms of two voices (sources) and two microphones, then elements of the ith row in A specify the proximity of each voice to the jth microphone. Each microphone records a weighted mixture x i of the two sources s and s2, where the weightings for each microphone are given by a column of A. Plots of s versus s2, and of x versus x2 can be seen in Figure. We dene the space in which s exists as S with axes S and S2, and x's space is dened as X with axes X and X2. The amplitudes of the source signals st and s2t at time t are represented as a point with coordinates (st; s2t) in S. The corresponding amplitudes of the signal mixtures x = As at time t are represented as a point x t with coordinates (xt; x2t) in X. The `mixing' matrix A denes a linear transformation, so that the mapping from S to X consists of a rotation and shearing of axes in S. Thus, the orthogonal axes S and S2 in S appear as two skewed lines S and S 2 in X (see Figure b). Note that variation along each axis in S is caused by variation in the amplitude of one source signal. Given that each axis S i in S corresponds to a direction Si in X, variation along the projected axes S and S2 in X are caused by variation in the signal amplitudes dened as s and s2, respectively. If we can extract variations associated with one direction, say, S, in X whilst ignoring variations along all other directions, then we can recover the amplitudes of the signal s. This can be achieved by projecting all points in X onto a line In which the rows of A form the basis vectors in X. 3

4 that is orthogonal to all but one direction S. Such a line is dened by a vector w = (w; w2) (depicted as a dashed line in Figure b), dened so that only components of X that lie along the direction S are transformed to non-zero values of y = w x. This is depicted graphically in Figure b, with the result of unmixing both signals y = Wx depicted in Figure 2. To summarise, the linear transformation y t = Wx t produces a scalar value for each point x t in X, so that a single signal results from the transformation y = Wx. The signal amplitude y t at time t is found by taking the inner product of W with a point x t. As the row vector W is dened to be orthogonal to directions corresponding all but one source signal in X, only that signal will be projected to non-zero values y = Wx. Having demonstrated that an unmixing matrix W exists that can extract one or more source signals from a mixture, the following sections describe how PP and ICA can be used to obtain values for W. 4 Independence and Moments of Non-Gaussian Signals 4. Independence and Correlation Statistical independence lies at the core of the ICA/PP methods. Therefore, in order to understand ICA/PP, it is essential to understand independence. At an intuitive level, if two variables x and y are independent then the value of one variable cannot be predicted if the value of the other variable is known. One simple way to understand independence relies on the more familiar denition of correlation. The correlation between two variables x and y is: (x; y) = Cov(x; y) x y (3) where x and y are the standard deivations of x and y, respectively, and Cov(x; y) is the covariance between x and y: Cov(x; y) = (=n) X i where x and y are the means of x and y, respectively. (x i? x )(y i? y ) (4) Correlation is simply a form of covariance that has been normalised to lie in the range f?; +g. Note that if two variables x and y are uncorrelated then (x; y) = Cov(x; y) =, although (x; y) and Cov(x; y) are not equal in general. The covariance Cov(x; y) can be shown to be: Cov(x; y) = (=n) X i x i y i? (=n) X i x i (=n) X i y i (5) Each term in Equation (5) is a mean, or expected value E, and can be written more succinctly as: Cov(x; y) = E[xy]? E[x]E[y] (6) 4

5 A histogram plot with abscissas x and y, and with the ordinate denoting frequency, approximates the probability density function (pdf) of the joint distribution of xy. The quantity E[xy] is known as a second moment of this joint distribution. Similarly, histograms of x and y approximate their respective pdfs, and are known as the marginal distributions of the joint distribution xy. The quantities E[x] and E[y] are the rst moments (respectively) of these marginal distributions. Thus, covariance is dened in terms of moments associated with the joint distribution xy. Just because x and y are uncorrelated, this does not imply that they are independent. To take a simple example, given a variable z = f; :::; 2g, we can dene x = sin(z) and y = cos(z). Intuitively, it can be seen that both x and y depend on z. As can be sen from Figure 3, the variables x and y are highly interdependent. However, the covariance (and therefore the correlation) of x and y is zero: Cov(x; y) = E[xy]? E[x]E[y] (7) = E[cos(z) sin(z)]? (8) = (9) In summary, covariance does not capture all types of dependencies between x and y, whereas measures of statistical independence do. Like covariance, independence is dened in terms of the expected values of the joint distribution xy. We have established that if x and y are uncorrelated then they have zero covariance: E[xy]? E[x]E[y] = () Using a generalised form of covariance involving powers of x and y, if x and y are statistically independent then: E[x p y q ]? E[x p ]E[y q ] = () for all positive integer values of p and q. Whereas covariance uses p = q =, all positive integer values of p and q are implicit in measures of independence. Formally, if x and y are independent then each moment E[x p y q ] is equal to the product of the expected values of the pdf's marginal distributions E[x p ]E[y q ], which leads to the result stated in Equation (). The formal similarity between measures of independence and covariance can be interpreted as follows. Whereas covariance measures the amount of linear covariation between x and y, independence measures the linear covariation between [x raised to powers p] and [y raised to powers q]. Thus, independence can be considered as a generalised form of covariance, which measures the linear covariation between non-linear functions (e.g. cubed power) of two variables. For example, using x = sin(z) and y = cos(z) we know that Cov(x; y) =. However, the measure of linear 5

6 covariation between the variables x p and y q as depicted in Figure (3) for p = q = 2 is: E[x p y q ]? E[x p ]E[y q ] =?:23 (2) This corresponds to a correlation between x 2 and y 2 of?:864 (see Figure 3). Thus, whereas the correlation between x sin(z) and y = cos(z) is zero, the fact that the value of x can be predicted from y is implicit in the non-zero values of the higher order moments of the distribution of xy. 4.2 Moments and non-gaussian pdfs We have shown that both the covariance and interdependence between two variables x and y are dened in terms of the moments of the pdf of their joint distribution. However, any variable with a Gaussian pdf is special in the sense that it is completely specied by its second moment E[xy]. That is, the values of all higher moments are implicit in the value of the second moment of a Gaussian distribution. Thus, if the covariance E[xy]? E[x]E[y] of the joint distribution of two Gaussian variables is zero then it can be shown that the quantity E[x p y q ]? E[x p ]E[y q ] is zero for all positive integer values of p and q. From Equation () we know that such variables are statistically independent, and it therefore follows that uncorrelated Gaussian variables are also independent. However, non-gaussian variables that are uncorrelated are not, in general, independent. As stated above, the non-gaussian variables x = sin(z) and y = cos(z). Here, E[xy]? E[x]E[y] =, but (for example) E[x 2 y 2 ]? E[x 2 ]E[y 2 ] ==?:23, and the correlation between x 2 and y 2 is r =?:86. Thus, for non-gaussian variables, the dependency between x and y only becomes apparent in their high order moments. 5 Using Independence and Non-Gaussian Assumptions for Source Separation 5. Projection Pursuit: Mixtures of Source Signals Are Gaussian A critical feature of a random linear mixture of any signals (with nite variance) is that a histogram of its values is approximately Gaussian; that is, it has a Gaussian probability density function (pdf). This follows from the central limit theorem, and is is illustrated in Figures 5, 6 and 7. Most mixtures of of a set of signals therefore produce a signal mixture with a Gaussian pdf. As methods for separating sources use a set of mixtures as input, and produce a linear weighting of them as output, it follows that arbitrary `unmixing' matrices W also produce Gaussian signals. However, if an `unmixing` matrix exists that produces a non-gaussian signal from the set of mixtures then such a signal is unlikely to be a mixture of signals. 6

7 If we assume that source signals have non-gaussian pdfs then, whilst most transformations produce data with Gaussian distributions, a small number of transformations exist that produce data with non-gaussian distributions. Under certain conditions, the non-gaussian signals extracted from signal mixtures by such a transformation are in fact the original source signals. This is the basis of projection pursuit methods [Friedman, 987]. In order to set about nding non-gaussian component signals, it is necessary to dene precisely what is meant by the term `non-gaussian'. Two important classes of signals with non-gaussian pdfs have super-gaussian and sub-gaussian pdfs. These are dened in terms of kurtosis, which is dened as R k = T (s? st ) 4 dt R (s? st ) 2 dt? 3 (3) T where s t is the value of a signal at time t, s is the mean value of s t, and the constant (3) ensures that super- Gaussian signals have positive kurtosis, whereas a sub-gaussian signal have negative kurtosis. This can be written more succinctly in terms of expected values, E[:]': k = E[(s? s)4 ] E[(s? s) 2 ]? 3 (4) A signal with a super-gaussian pdf has most of its values clustered around zero, whereas a signal with a sub- Gaussian pdf does not. As examples, a speech signal has a super-gaussian pdf, and a sine function and white noise have sub-gaussian pdfs (see Figure 4). FIGURE 4 HERE. PP methods tend to make use of high-order moments of distributions such as kurtosis in order to estimate the extent to which a signal is non-gaussian. However, here we will use a more general measure, which is borrowed from ICA. The extent to which a signal's pdf is non-gaussian depends on the following critical observation: If the scalar values of a signal s are transformed by the cumulative density function (cdf) of that signal then the resultant distribution of values is uniform. This is useful because it permits the extent of deviation from a Gaussian pdf to be recast in terms of the uniformity, or equivalently, the entropy of the transformed signal, Y = (s) (one way to think of entropy is as a measure of the uniformity of a given distribution). The question of how to nd the linear transformation capable of recovering a source signal follows from the denition of our measure of Gaussian deviation. We have established that a linear transformation W exists such that a signal s = Wx can be recovered from a set of M signal mixtures x, and that this transformation produces a signal s = W x such that S = (Wx) has maximum entropy. By inverting the ow of logic in this argument, it follows that s can be recovered from x by nding a W that maximises the entropy H(Y ) of S = (W x). It can be shown [Girolami and Fyfe, 996] that if a number of signals are extracted from a mixture x (and these are the most non-gaussian signals components of the mixture) then they are guaranteed to be mutually independent. Thus, even though a measure of independence is not explicitly maximised as part of the PP 7

8 optimisation process, extracting non-gaussian signals produces signals that are mutually independent. contrast, ICA explicitly maximises the mutual independence of extracted signals. In 5.2 ICA: Source Signals Are Statistically Independent The ICA methods described in this section are based on the following simple observation: If a set of N signals are from dierent physical sources (e.g. N dierent speakers) then they tend to be statistically independent of each other. The method of ICA is based on the assumption that if a set of independent signals can be extracted from signal mixtures then these extracted signals are likely to be the original source signals. Like PP, ICA requires assumptions of independence that involve the cdfs of source signals, and this is the link that binds ICA and PP methods together. As in the previous section, this problem can be considered in geometric terms. The amplitudes of N source signals at a given time can be represented as a point in an N-dimensional space, and, considered over all times, they dene a distribution of points in this space. If the signals are from dierent sources (e.g. N dierent speakers) then they tend to be statistically independent of each other. As with PP, a key observation is that if a signal s has a cdf then the distribution of (s) has maximum entropy (i.e. is uniform). Similarly, if N signals each have cdf then the joint distribution of (s) = ((s); : : :; (s N )) T has maximum entropy, and is therefore uniform. For a set of signal mixtures x = As, an `unmixing' matrix exists such that s = Wx, where W. Given that (s) has maximum entropy, it follows that s can be recovered by nding a matrix W that maximises the entropy of Y = (Wx) (where is a vector of cdfs in one-to-one correspondence with transformed signals in y = Wx), at which point (Wx) = (s). In summary, for any distribution x which is a mixture of N independent signals each with cdf, there exists a linear unmixing transformation W followed by a non-linear transformation, such that the resultant distribution Y = (W x) has maximum entropy. This can be used to recover the original sources by dening a plausible cdf, and then nding an unmixing matrix W that maximises the entropy of Y. The explicit assumption of independence upon which ICA is based is less critical than the apparently subsidiary assumption regarding the non-gaussian nature of source signals. It can be shown [Girolami and Fyfe, 996] that, given a set of mixtures of independent non-gaussian signals, the sources can be extracted by nding component signals that have appropriate cdfs, and that these signals are independent. The converse is not true, in general, if the number of extracted signals is less than the number of independent signals in the set of signal mixtures. That is, simply nding a subset of independent signals in a set of independent non-gaussian source signal mixtures is not, in general, equivalent to nding the component sources. This is because linear combinations of disjoint sets of source signals are independent. For example, if a subset of independent signals are combined to form a signal mixture x, and a non-overlapping subset of other signals are combined to 8

9 form a mixture x2, then x and x2 are mutually independent, even though both consist of mixtures of source signals. Thus, statistical independence of extracted signals is a necessary, but not sucient, condition for source separation. Having established the connection between ICA and PP, and conditions under which they are equivalent, we proceed by describing the `standard' ICA method [Bell and Sejnowski, 995]. 6 The Nuts and Bolts of ICA Suppose that the outputs x = (x; : : :; x M ) T of M measurement devices are a linear mixture of N = M independent signal sources s = (s; : : :; s N ) T, x = As, where A is an N N mixing matrix. We wish to nd a N N unmixing matrix W such that each of the N components recovered by y = Wx is one of the original signals s (i.e. K N). As discussed above, an unmixing matrix W can be found by maximising the entropy H(Y) of the joint distribution Y = fy; : : :; Y N g = f(y); : : :; N (y N )g, where y i = W x i. The correct i have the same form as the cdfs of the input signals x i. However, in many cases it is sucient to approximate these cdfs by sigmoids 2 Y i = tanh y i. The entropy of a signal y with pdf f x (x) is given by: H(x) =?E[ln f x (x)] =? Z f x (x) ln f x (x) dy (5) As might be expected, the transformation of a given data set x aects the entropy of the transformed data Y according to the change in the amount of `spread' introduced by the transformation. Given a multidimensional signal x, if a cluster of points in x is mapped to a large region in Y, then the transformation implicitly maps innitesimal volumes from one space to another. The `volumetric mapping' between spaces is given by the Jacobian of the transformation between spaces. The Jacobian combines the derivative of each axis in x with respect to every axis in y to form a ratio of innitesimal volumes in x and y. The change in entropy induced by the transformation W can be shown to be equal to the expected value of ln jjj, where j:j denotes absolute value. Given that Y = (Wx), the output entropy H(Y) can be shown to be related to the entropy of the input H(x) by H(Y) = H(x) + E [ log jjj ] (6) where jjj is the determinant of the Jacobian matrix J Note that the entropy of the input H(x) is constant. Given that we wish to nd a W that maximises H(Y), any W that maximises H(Y) is unaected by 2 In fact sources s i normalised so that E[s i tanh s i ] = =2 can be separated using tanh sigmoids if and only if the pairwise conditions i j > are satised, where i = 2E[s 2 i ]E[sech2 s i ] [Porrill, 997]. 9

10 H(x), which can therefore be ignored. Using the chain rule, we can evaluate jjj as: @y Y N = i (y i)jw j (7) i= are Jacobian matrices. Substituting Equation (7) in Equation (6) yields H(Y) = H(x) + E " N X log i (y i) i= # + log jw j: (8) As the entropy of the x is unaected by W, it can be ignored in the maximisation of H(Y). E [ P log i(y i )] can be estimated given n samples from the distribution dened by y: E " NX # log i (y i) i= nx NX n j= i= The term log i (y(j) i ) (9) Ignoring H(x), and substituting Equation (9) in (8) yields a new function that diers from H(Y) by a constant equal to H(x): nx NX h(w ) = n j= i= If we dene the cdf i = tanh then this evaluates to nx NX h(w ) = n j= i= log i(y (j) i ) + log jw j (2) log(? y (j)2 i ) + log jw j: (2) This function can be maximised by taking its derivative with respect to the matrix W : r h W = [W T ]?? 2yx T (22) Now an unmixing matrix can be found by taking small steps of size to update W : W = ([W T ]?? 2yx T ) (23) In fact, the matching of the pdf of y to each cdf also requires that each signal y i has zero mean. This is easily accomodated by introducing a `bias' weight w i to ensure that y i = Wx + w i has zero mean. The value of each bias weight is learned like any other weight in W. For a tanh cdf, this evaluates to: w i =? 2y (24) In practice, h(w ) is maximised either, a) by using a `natural gradient' [Amari, 998] which normalises the error surface so that the step-size along each dimenstion is scaled by the local gradient in that direction, and which obviates the need to invert W at each step, or b) a second order technique (such as BFGS or a conjugate gradient Marquardt method) which estimates an optimal search direction and step-size under the assumption that the error surface is locally quadratic.

11 7 Spatial and Temporal ICA In `standard' ICA, each of N signal mixtures is measured over T time steps, and N sources are recovered as y = Wx, where each source is independent over time of every other source. However, when using ICA to analyse temporal sequences of images it rapidly becomes apparent that there are two alternative ways to implement ICA. 7. Temporal ICA Normally, independent temporal sequences are extracted by placing the image corresponding to each time step in a column of x. We refer to this as ICAt. This essentially treats each of the N pixels as a separate `microphone' or mixture, so that each mixture consists of T time steps. The (large) N N matrix W then nds temporally independent sources that contribute to each pixel grey-level over time. Having discovered the temporally independent signals for an image sequence, this begs the question: what was it that varied independently over time? Given y = Wx we can derive x = Ay (25) where A = W? is an N N matrix. Therefore, each row (source signal) of y species how the contribution to x of one column (image) of A varies over time. So, whereas each row y i of y species a signal that is independent of all rows in y, each column a i of A consists of an image that varies independently over time according to the amplitude of y i. Note that, in general, the rows of s are constrained to be mutually independent, whereas the relationship between columns of A is completely unconstrained. 7.2 Spatial ICA Instead of placing each image in a column of x we can place each image in a row of x. This is equivalent to treating each time step as a mixture of independent images. We refer to this as ICAs. In this case, each source signal (row of y) is an image, where the pixel values in each image (row) are independent of every other image, so that these images are said to be spatially independent. Each column of the T T matrix A is a temporal sequence. In summary, both ICAs and ICAt produce a set of images and a corresponding set of temporal sequences. However, ICAt produces a set of mutually independent temporal sequences and a corresponding set of unconstrained images, whereas ICAt produces mutually independent images and a corresponding set of unconstrained temporal sequences. If it is known that either temporal or spatial independence cannot be assumed then this rules out ICAt or ICAs,

12 respectively. In practice, ICAt is computationally expensive because it involves a P P matrix W. In contrast, ICAs requires a N N matrix W. ICAs has been used to good eect on fmri images [McKeown et al., 998]. If neither spatial nor temporal independence can be assumed then a form of ICA that requires assumptions of minimal dependence over time and space can be used. 8 Using Princpal Component Analysis to Preprocess Data For many data sets, it is impracticable to nd an N N unmixing matrix W because the number N of rows in x is large. In such cases, principle component analysis (PCA) can be used to reduce the size of W. Each of the T N-dimensional column vectors in x denes a single point in an N-dimensional space. If most of these points lie in a K-dimensional subspace (where K N) then we can use K judiciously chosen basis vectors to represent the T columns of x. (E.g. if all the points in a box lie in a two-dimensional square then we can describe the points in terms of the two basis vectors dened by two sides of that square). Such a set of K N-dimensional eigenvectors U can be obtained using PCA. Just as ICA can be used with data vectors in the rows or columns of x, so PCA can be used to nd a set V of K T -dimensional eigenvectors. More importantly, (and momentarily setting K = N) the two sets of eigenvectors U and V are related to each other by a diagonal matrix: x = UDV T (26) Where the diagonal elements of D contain the ordered eigenvalues of the corresponding eigenvectors in the columns of U and the rows of V. This decomposition is produced by singular value decomposition (SVD). Note that each eigenvalue species the amount of data variance associated with the direction dened by a corresponding eigenvector in U and V. We can therefore discard eigenvectors with small eigenvalues because these account for trivial variations in the data set. Setting K N permits a more econimical representation of x: x ~x = U ~ D ~ V ~ T (27) Note that U ~ is now an N K matrix, V ~ is a K K matrix, and D ~ is a diagonal K K matrix. As with ICAs and ICAt, these can be considered in temporal and spatial terms. If each column of x is an image of N pixels then each column of U is an eigenimage, and each column of V is an eigensequence. Given that we require a small unmixing matrix W, it is desirable to use U ~ instead of X for ICAt, and V ~ instead of X T for ICAs. The basic method consists of performing ICA on U ~ or V ~ to obtain K ICs, and then using the relation X ~ = U ~ D ~ V ~ T to obtain the K corresponding columns of A. 2

13 8. ICAt Using SVD Replacing x with ~ V T in y = Wx produces y = W ~ V T (28) where each row of the K T matrix ~ V T is an `eigensequence', and W is a K K matrix. In this case, ICA recovers K mutually independent sequences, each of length T. The set of images corresponding to the K temporal ICs can be obtained as follows. Given ~V T = Ay = W? y (29) and x = ~ U ~ D ~ V T (3) we have x = ~ U ~ DW? y (3) = Ay (32) From which it follows that A = ~ U ~ DW? (33) where A is a N K matrix in which each column is an image. Thus, we have extracted K independent T -dimensional sequences and their corresponding N-dimensional images using a K K unmixing matrix W. 8.2 ICAs Using SVD A similar method can be used to nd K independent images and their corresponding time courses. Replacing x T with ~ U T in y = Wx T produces y = W ~ U t (34) where each row of the K N matrix U ~ T is an `eigenimage', and W is a K K matrix. In this case, ICA recovers K mutually independent images, each of length N. The set of images corresponding to the K spatial ICs can be obtained as follows. Given ~U T = Ay = W? y (35) and x T = ~ V ~ D ~ U T (36) 3

14 we have x = ~ V ~ DW? y (37) = Ay (38) From which it follows that A = ~ V ~ D ~ W? (39) where A is a T K matrix in which each column is a time course. Thus, we have extracted K independent N-dimensional images and their corresponding T -dimensional time courses using a K K unmixing matrix W. Note that, using SVD in this manner requires an assumption that the ICs are not distributed amongst the smaller eigenvectors which are usually discarded. The validity of this assumption is by no means guaranteed. 9 Emulating Singular Value Decomposition We have shown how ICA can be made tractable by using principal components (PCs) obtained from SVD. However, for large dimensional data, U or V can be too large for most computers. For instance, if the data matrix x is an N T matrix then U is N T and V is T T. If each column in x contains an image with N pixels then neither x nor U may be small enough to t into the RAM available on a computer. It is possible to compute U D and V in an iterative manner using techniques that do not rely on SVD. Throughout the following we assume that V is samller than U. For convenience, we assume that x consists of one image per column, and that each column corresponds to one of N time steps. We proceed by nding V and D, from which U can be obtained by combining V with x. The matrix V contains one eigensequence per column. This can be obtained from the temporal N N covariance matrix of x, which is dened by the outer product: C = x T x (4) This covariance matrix is the starting point of many standard PCA algorithms. After PCA we have V and a corresponding set of ordered eigenvalues. The matrix D which is normally obtained with SVD can be constructed by setting each diagonal element to the square of each corresponding eigenvalue. Given that: x = UDV T (4) it follows that: U = xv D? (42) 4

15 So, given V and D from a PCA of the covariance matrix of x, we can obtain the eigenimages U. Note that we can compute as many eigenimages as required by simply omitting corresponding eigensequences and eigenvalues from V and D, respectively. SVD in relation to ICAs and ICAt If we distribute the eigenvalues between U and V by multiplying each column in U and V by the square root of its corresponding eigenvalue then we have: x = UV T (43) which has a similar form to the ICA decomposition: x = As (44) The main dierence between SVD and ICA is as follows. Each matrix produced by SVD has orthogonal columns. That is, the variation in each column is uncorrelated with variations in every other column within U and V. In contrast, ICA produces two matrices with quite dierent properties. Rather than being uncorrelated, the rows of s are independent. This stringent requirement on the rows of s suggest that the columns of A cannot also be independent, in general, and ICA actually places no constraints on the relationships between columns of A. Acknowledgements J Stone is supported by a Mathematical Biology Wellcome Fellowship (Grant number 44823). References [Amari, 998] Amari, A. (998). Natural gradient works eciently in learning. Neural Computation, (2):25{ 276. [Bell and Sejnowski, 995] Bell, A. and Sejnowski, T. (995). An information-maximization approach to blind separation and blind deconvolution. Neural Computation, 7:29{59. [Bell and TJ, 997] Bell, A. and TJ, S. (997). The `independent components' of natural scenes are edge lters. Vision Research, 37(23):3327{3338. [Friedman, 987] Friedman, J. (987). Exploratory projection pursuit. J Amer. Statistical Association, 82(397):249{266. [Girolami and Fyfe, 996] Girolami, M. and Fyfe, C. (996). Negentropy and kurtosis as projection pursuit indices provide generalised ica algorithms. NIPS96 Blind Signal Separation Workshop. 5

16 [Jutten and Herault, 988] Jutten, C. and Herault, J. (988). Independent component analysis versus pca. In Proc. EUSIPCO, pages 643 { 646. [Makeig et al., 997] Makeig, S., Jung, T., Bell, A., Ghahremani, D., and Sejnowski, T. (997). Blind separation of auditory event-related brain responses into independent components. Proc. Natl. Acad. Sci, 94:979{984. [McKeown et al., 998] McKeown, M., Makeig, S., Brown, G., Jung, T., Kindermann, S., and Sejnowski, T. (998). Spatially independent activity patterns in functional magnetic resonance imaging data during the stroop color-naming task. Proceedings of the National Academy of Sciences USA (In Press). [Porrill, 997] Porrill, J. (997). Independent component analysis: Conditions for a local maximum. Technical Report 23, Psychology Department, Sheeld University, England. [van Hateren and van der Schaaf, 998] van Hateren, J. and van der Schaaf, A. (998). Independent component lters of natural images compared with simple cells in primary visual cortex. Prc Royal Soc London (B), 265(7):359{366. 6

17 Figure : The geometry of source separation. a) Plot of signal s versus s2. Each point s t in S represents the amplitudes of the source signals st and s2t at time t. These signals are plotted separately in Figure 2. b) Plot of signal mixture x versus x2. Each point x t = As t in X represents the amplitudes of the signal mixtures xt and x2t at time t. These signal mixtures are plotted separately in Figure 2. The orthogonal axes S and S2 in S (solid lines in Figure a) are transformed by the mixing matrix A to form the skewed axes S and S2 in X (solid lines in Figure b). An `unmixing' matrix W consists of two row vectors, each of which `selects' a direction associated with a dierent signal in X. The dashed line in Figure b species one row vector w of an `unmixing' matrix W which is (in general) orthogonal to every transformed axis Si except one (S, in this case). Variations in signal amplitude associated with directions (such as S2) that are orthogonal to w have no eect on the inner product y = w x. Therefore, y only reects amplitude changes associated with the direction S, so that y = ks where k is a constant that equals unity if S and w are co-linear. 7

18 Figure 2: Separation of two signals. Original signals s = s; s2 are displayed in the left hand graphs. Two signal mixtures x = As are displayed in middle graphs. The results of applying an unmixing matrix W = A? to the mixtures x = Wx are displayed in the right hand graphs. 8

19 Figure 3: The interdependence of x = sin(z) and y = cos(z) is only apparent in the higher order moments of the joint distribution of xy. a) Plot of x = sin(z) versus y = cos(z). Even though the value of x is highly predictable given the corresponding value of y (and vice versa), the correlation between x and y is r =. For display purposes, noise has been added in order to make the set of points visible. b) Plot of sin 2 (z) versus cos 2 (z). The correlation between sin 2 (z) and cos 2 (z) is r =?:864. Whereas x and y are uncorrelated if the correlation between x and y is zero, they are statistically independent only if the correlation between x p and y q is zero for all positive integer values of p and q. Therefore, sin(z) and cos(z) are uncorrelated, but not independent. 9

20 Figure 4: Histograms of signals with dierent probability density functions. From left to right, histograms of super-guassian, Guassian, and sub-guassian signal. The left hand histogram is derived from a portion of Handel's Messiah, the middle histogram is derived from Gaussian noise, and the right hand histogram is derived from a sine wave. 2

21 Figure 5: Six sound signals and their pdfs. Each signal consists of ten thousand samples. From top to bottom: chirping, gong, Handel's Messiah, people laughing, whistle-plop, steam train. 2

22 Figure 6: The outputs of six microphones, each of which receives input from six sound sources according to its proximity to each source. Each microphone receives a dierent mixture of the six non-gaussian signals displayed in Figure 5. Note that the pdf of each signal mixture shown on the rhs is approximately Gaussian. 22

23 Figure 7: A typical signal produced by applying a random `unmixing' matrix to the six signal mixtures displayed in 6. The resultant signal has a pdf that is approximately Gaussian. From top to bottom: a single mixture of the six signals shown in Figure 5, the mixture's pdf, pdf of a Gaussian signal. Note that correct unmixing matrix would produce each of the original source signals displayed in Figure 5. 23