# Suitable metrics for summarizing or visualizing the spatial activation of components after temporal Independent Component Analysis (ICA)

I will first describe the Independent Component Analysis (ICA) steps, so that the question becomes clearer, and more relevant to similar questions.

Assume we have $N$ sensors distributed in space. From each sensor we obtain a time series of length $T$. We collect these time series as rows in an $N \times T$ matrix $X$. After performing ICA we obtain a decomposition of the form $X = DS$, where $D$ is an $N \times N$ matrix of weights, and $S$ is an $N \times T$ matrix whose rows are the independent sources. To find the contribution of the $i$-th component to the observed signal we then multiply the $i$-th column of the matrix $D$ with the $i$-th row of $S$. The result is an $N \times T$ matrix which we will denote as $X^{i} = D_{*,i}S_{i,*}$. Then we can write: $X = X^{1} + ... + X^{i} + ... + X^{N}$.

In many domains I see single-picture visualizations of the spatial activation of the $i$-th component from ICA. However, it is not clear to me what metric to extract from each row of $X^{i}$ or from matrix $D$ in order to do this. Is it a summary statistic of the rows of $X^{i}$, e.g. the energy of the signals (rows) of $X^{i}$, the weighting coefficients from the $i$-th column of $D$, or something different altogether?

Thank you.

• Can you put up such a picture of the spatial activation of an i-th component that you are seeing? Each row of the $X_i$ matrix will have a (spatially) scaled version the $i$th original signal. – Tarin Ziyaee May 24 '14 at 20:04
• For example, this is a picture with the activation per component (32 sensors and components) from ICA-decomposed EEG data: ICA EEG. What does "spatially scaled" mean? – Orestis Tsinalis May 24 '14 at 20:29
• Sorry, can't edit my comment above. By "spatially scaled" do you mean that the $i$-th source signal is scaled (because of its distance to the source) to construct each row of the $X^{i}$ matrix? If so, the single value that is used for the sensor $k$ in the visualizations is the weight from the $i$-th column and $k$-th row of $D$, $D_{k,i}$, right? Then for the visualization of the $i$-th component, for each sensor $k$ we assign at its location in the image the value $D_{k,i}$. Is this what you meant? – Orestis Tsinalis May 24 '14 at 20:40
• Firstly, what units does the heatmap in the image represent? – Tarin Ziyaee May 24 '14 at 20:43
• As far as I know, it is not specified in the tutorial of the toolbox where the image comes from (EEGLab ICA tutorial). The image comes automatically out of the ICA analysis, but it is not clear how it is computed. – Orestis Tsinalis May 24 '14 at 20:47

As we know from the instantaneous mixture model of ICA, a signal mixture $X_{N,T}$ is composed of an unknown mixing matrix $D_{N,N}$, and an unknown signal matrix $S_{N,T}$, where:

$$X = DS$$

In ICA, we solve for $D_{N,N}$ and $S_{N,T}$ via maximization of absolute kurtosis. (As one method).

If we now look at the $i$th column of $D$, each element therein will now give us weight by which the $i$th signal gets weighted by in all $N$ mixtures. That is, each element $D_{n,i}$, (row $n$, column $i$), gives us signal $i$'s contribution to mixture $n$. Those are called 'spatial weights' simply because in many ICA applications, we are de-mixing signals that sample a spatial field at different locations. Thus, each element of a column $D$ corresponds to a weight in space, that weighs signal source $i$.

In this way, we can show those weights in space as the heatmap above shows, and find the weight each signal gets weighted by, for a particular mixture, at different locations in space.