Basic ICA

Let the individual voxel time courses be arranged in a matrix, $Y$, such that each column represents a single voxel time course, and each row represents a spatial image (at a fixed time point). That is, $Y$ is a $T \times N$ matrix where $T$ is the number of time points, and $N$ is the number of voxels, where we assume $T < N$ so that spatial ICA is being performed.

The matrix is then preprocessed to:

1. remove the mean spatial map (the average of all the rows of $Y$) from each row of $Y$;
2. (optional) normalise the variance of each individual time course (each column of $Y$ set to have unit variance);
3. remove the mean time course (the average of all the columns of $Y$) from each column of $Y$.

With this data, a general ICA decomposition can be written as

$$Y = A S$$ (1)

where $A$ is a $T \times T$ matrix of time courses and $S$ is a $T \times N$ matrix of spatial maps which are pairwise independent in a statistical sense. Note that the $n$th row of $S$ is a spatial map that is associated with the $n$th column of $A$ (a time course).