Independent Component Analysis for dummies
Introduction
ICA is a quite powerful technique and is able (in principle) to separate independent sources linearly mixed in several sensors. For instance, when recording electroencephalograms (EEG) on the scalp, ICA can separate out artifacts embedded in the data (since they are usually independent of each other).
This page intends to explain ICA to researchers that want to understand it but only have a weak background both in matrix computation and information theory. My explanations will be as intuitive as possible and based on practical examples from Matlab.
What is ICA ?
ICA is a technique to separate linearly mixed sources. For instance, let’s try to mix and then separate two sources. Let’s define the time courses of 2 independent sources A(top) and B(bottom) (see the Matlab code)
Whitening the data
We will now explain the preprocessing performed by most ICA algorithms before actually applying ICA.
A first step in many ICA algorithms is to whiten (or sphere) the data. This means that we remove any correlations in the data, i.e. the different channels (matrix Q) are forced to be uncorrelated.
Why do that? A geometrical interpretation is that it restores the initial “shape” of the data and that then ICA must only rotate the resulting matrix (see below). Once more, let’s mix two random sources A and B. At each time, in the following graph, the value of A is the abscissa of the data point and the value of B is their ordinates.
The whitening process is simply a linear change of coordinate of the mixed data. Once the ICA solution is found in this “whitened” coordinate frame, we can easily reproject the ICA solution back into the original coordinate frame.
The ICA algorithm
Intuitively you can imagine that ICA rotates the whitened matrix back to the original (A,B) space (first scatter plot above). It performs the rotation by minimizing the Gaussianity of the data projected on both axes (fixed point ICA). For instance, in the example above,
The projection on both axis is quite Gaussian (i.e., it looks like a bell shape curve). By contrast the projection in the original A, B space far from gaussian.
By rotating the axis and minimizing Gaussianity of the projection in the first scatter plot, ICA is able to recover the original sources which are statistically independent (this property comes from the central limit theorem which states that any linear mixture of 2 independent random variables is more Gaussian than the original variables). In Matlab, the function kurtosis (kurt() in the EEGLAB toolbox; kurtosis() in the Matlab statistical toolbox) gives an indication of the gaussianity of a distribution (but the fixedpoint ICA algorithm uses a slightly different measure called negentropy).
The Infomax ICA in the EEGLAB toolbox (Infomax ICA) is not as intuitive and involves minimizing the mutual information of the data projected on both axes. However, even if ICA algorithms differ from a numerical point of view, they are all equivalent from a theoretical point of view.
ICA in N dimensions
We dealt with only 2 dimensions. However ICA can deal with an arbitrary high number of dimensions. Let’s consider 128 EEG electrodes for instance. The signal recorded in all electrode at each time point then constitutes a data point in a 128 dimension space. After whitening the data, ICA will “rotate the 128 axis” in order to minimize the Gaussianity of the projection on all axis (note that unlike PCA the axis do not have to remain orthogonal).
What we call ICA components is the matrix that allows projecting the data in the initial space to one of the axis found by ICA. The weight matrix is the full transformation from the original space. When we write
X is the data in the original space. For EEG
Time points  
Electrodes 1 Electrodes 2 Electrodes 3 
[ 0.134 0.424 0.653 0.739 0.932 0.183 0.834 ….] [ 0.314 0.154 0.732 0.932 0.183 0.834 0.134 ….] [ 0.824 0.534 0.314 0.654 0.739 0.932 0.183 ….] 
For fMRI
Voxels  
Time 1 Time 2 Time 3 
[ 0.134 0.424 0.653 0.739 0.932 0.183 0.834 ….] [ 0.314 0.154 0.732 0.932 0.183 0.834 0.134 ….] [ 0.824 0.534 0.314 0.654 0.739 0.932 0.183 ….] 
S is the source activity.
In EEG: An artifact time course or the time course of the one compact domain in the brain
Time points  
Component 1 Component 2 Component 3 
[ 0.824 0.534 0.314 0.654 0.739 0.932 0.183 ….] [ 0.314 0.154 0.732 0.932 0.183 0.834 0.134 ….] [ 0.153 0.734 0.134 0.324 0.654 0.739 0.932 ….] 
In fMRI: An artifact topography or the topography of statistically maximally independent pattern of activation
W is the weight matrix to go from the S space to the X space.
Now the rows of W are the vector with which we can compute the activity of one independent component. To compute, the component activity in the formula S = W X, the weight matrix W is defined as
Component 1 Component 2 Component 3 
elec1 elec2 elec3 elec4 elec5
[ 0.824 0.534 0.314 0.654 0.739 …]
[ 0.314 0.154 0.732 0.932 0.183 …]
[ 0.153 0.734 0.134 0.324 0.654 …]

For instance to compute the activity of the second source or second independent component (in a matrix multiplication format), you may simply multiply matrix X (see beginning of paragraph) by the row vector
Component 2 
elec1 elec2 elec3 elec4 elec5
[ 0.314 0.154 0.732 0.932 0.183 …]

Now you have the activity of the second component, but the activity is unitless. If you have heard of inverse modeling, the analogy with EEG/ERP sources in dipole localization software is the easiest to grasp. Each dipole has an activity (which project linearly to all electrodes). The activity of the Brain source (dipole) is unitless unless it is projected to the electrodes. So each dipole create a contribution at each electrode site. ICA components are just the same.
Now we will see how to reproject one component to the electrode space. W^{1} is the inverse matrix to go from the source space S to the data space X.
X = W^{1}S
In Matlab you would just type inv(W) to obtain the inverse of a matrix.
Electrode 1 Electrode 2 Electrode 3 
comp1 comp2 comp3 comp4 comp5
[ 0.184 0.253 0.131 0.364 0.639 …]
[ 0.731 0.854 0.072 0.293 0.513 …]
[ 0.125 0.374 0.914 0.134 0.465 …]

If S is a row vector (for instance the activity of component 2 computed above) and we multiply it by the following column vector from the inverse matrix above
Electrode 1 Electrode 2 Electrode 3 
comp2
[ 0.253 ]
[ 0.854 ]
[ 0.374 ]

We will obtain the projected activity of component 2 (the inverse weights for component 2 (column vector; bottom left below) multiplied by the activity for component 2 (row vector; top right below) leads to the component projection (matrix; bottom right).
(on the right one row of the S matrix (the activity of component 2) 
[ 0.314 0.154 0.732 0.932 0.183 0.834 0.134 ….]


[ 0.253 ] [ 0.854 ] [ 0.374 ] 
[ 0.824 0.534 0.314 0.654 0.739 0.932 0.183 ….] [ 0.314 0.154 0.732 0.932 0.183 0.834 0.134 ….] [ 0.153 0.734 0.134 0.324 0.654 0.739 0.932 ….] (above is the projection of one component activity on all the electrodes (note that the calculus not accurate and that the numbers are meaningless). This matrix will be denoted XC2. 
Now, if one want to remove component number 2 from the data (for instance if component number 2 proved to be an artifact), one can simply subtract the matrix above (XC2) from the original data X.
Note that in the matrix computed above (XC2) all the columns are proportional, which mean that the scalp activity is simply scaled. For this reason, we denote the columns of the W^{1} matrix, the scalp topography of the components. Each column of this matrix is the topography of one component which is scaled in time by the activity of the component. The scalp topography of each component can be used to estimate the equivalent dipole location for this component (assuming the component is not an artifact).
As a conclusion, when we talk about independent components, we usually refer to two concepts
 Rows of the S matrix which are the time course of the component activity
 Columns of the W^{1} matrix which are the scalp projection of the components
ICA properties
From the preceding paragraphs, several properties of ICA becomes obvious
 ICA can only separate linearly mixed sources.
 Since ICA is dealing with clouds of point, changing the order in which the points are plotted (the time points order in EEG) has virtually no effect on the outcome of the algorithm.
 Changing the channel order (for instance swapping electrode locations in EEG) has also no effect on the outcome of the algorithm. For EEG, the algorithm has no a priori about the electrode location and the fact that ICA components can most of the time be resolved to a single equivalent dipole is a proof that ICA is able to isolate compact domains of cortical synchrony.
 Since ICA separates sources by maximizing their nonGaussianity, perfect Gaussian sources can not be separated
 Even when the sources are not independent, ICA finds a space where they are maximally independents.
Annex: the math behind whitening
Putting it in mathematical terms, we seek a linear transformation V of the data D such that when P = V*D we now have Cov(P) = I (I being the identity matrix, zeros everywhere and 1s in the Diagonal; Cov being the covariance). It thus means that all the rows of the transformed matrix are uncorrelated (see covariance matrix).
Let consider “centered” matrix Z of matrix D : Cov(D) = Cov(Z) = (Z*Z’)/(n1), (as defined previously z_i,j = d_i,j – mean(d_i))
Then it is easy to show that Cov(P) = I by setting V = C^{1/2}, where C = Cov(D) is the correlation matrix of the data, since then we have
Cov(P) = Cov( V*D )
= Cov( C^{1/2}*Z )
= C^{1/2}*Z*Z’*C^{1/2} by definition
= C^{1/2}CC^{1/2} = I.
(computing C^{x} is not an easy process: to simplify it is defined as to achive C^{x}*C^{x }= I and involve complex computation such as the determinant of the matrix….)