## ICA weights and invweights

## Update 01.08.2016

after further discussion we discovered that the correlation matrix was a bit off. This is because I plotted the correlation of the weights without the sphering (=whitening) step. Sphering removes the correlation between the components. I added a plot with the correlation after sphering and updated the topographies.

## Goal

In EEG blind source separation using ICA, understand the difference between the mixing matrix and the unmixing matrix.

## brief introduction

The blind source separation problem in the context of EEG can be understood as follows:

$$ mixing * sourceactivity = EEGdata $$

In words: We have some sources active and we know their projections on each electrode: We take the dot product (we multiply each source with its projection and add them all up).

More formally:

$$

\begin{align}

A &: mixing\\

s &: sourceactivity\\

x &: EEGdata \\

A * s &= x\\

\end{align}

$$

Because every neuronal source projects to most (or even all) sensors/electrodes, the original brain-source activation can only be recored in a mixed way.

Thus in order to unmix we can use blind source separation (ICA details can be found here)

$$

\begin{align}

W &: unmixing \\

unmixing * EEG.data &= sourceactivity \\

W * x &= s \\

\end{align}

$$

The relation between mixing and unmixing matrix in ICA (and other BSS methods) is:

$$

\begin{align}

A * s & = x\\

A^{-1} * A * s & = A^{-1} * x\\

s & = A^{-1} * x\\

A^{-1}& = W\\

\end{align}

$$

This is only true if $A$ is invertible and square (if non-square you have to use a pseudoinverse, see the Haufe 2014 paper for more information)

The matrix $W$ is also known as the *extraction filter*

The matrix $A$ is also know as the *activity pattern*

## EEG ICAs

In eeglab the default infomax-ICA algorithm uses a sphering (whitening) step before calculating the weights. Thus the following relation is true:

[code lang=text] EEG.icawinv = pinv( EEG.icaweights * EEG.icasphere) % the mixing matrix, channels x sourcesEEG.icaact = EEG.icaweights*EEG.icasphere)*EEG.data % the activation, sources x time

[/code]

This is how ICA components look, on the left the weights after sphering-preprocessing, middle the unmixing matrix and on the right the mixing matrix. Usually we should look at the mixing matrix ($A = W^{-1}$).

The weights after decorrelation (=sphering) seem to be quite similar, but the unmixing ($W$) looks very different and is actually very hard to interprete!

One interesting thing to look at is the correlation between the component-vectors (in this case 32 components). As we can see here:

Note that the correlation of the EEG.icaweight matrix is 0 because it is a pure rotation matrix (which is orthognal, thus correlations of 0). As explained here the whitening step estimates the stretching and thus only a rotation remains for the EEG.icaweight matrix.

## More information

Haufe et. al. 2014(pdf-openaccess-link) have a very nice paper around this issue.

**TLDR; Don’t look at your filter/unmixing matrix.**

### Thanks to

Holger Finger, Anna Lisa Gert, Tim Kietzmann,Peter König, Andrew Melnik ( in alphabetical order) for discussions.

01-2023: Thanks to Fabiano Baroni for noticing I had $$W$$ & $$A$$ switched up throughout the blog post!!