How do I find the directions of the independent components if I have already found the mixing matrix?

Let's say that I have this mixing matrix:

$$\mathcal W = \begin{bmatrix} 2 & -2 \\ 2 & 4 \end{bmatrix}$$

so that $\boldsymbol{x}=W\boldsymbol{h}$, where $\boldsymbol{h}$ are the hidden sources: $p(\boldsymbol{h}) = \frac{1}{4}\prod_{i=1}^2exp(|-h_i|)$

I'd like to sketch the independent components and the contours of the distribution in $\boldsymbol{x}$-space.


1 Answer 1


I have an answer. A simple way to sketch the distribution is to project the x and y axis first so that it is easier to "see" what happens to the projected contours of the distribution. In this case for example projecting the x axis gives the direction of one of the two independent components, which is the line passing from $(0,0)$ and $(2,2)$:

$$ \begin{bmatrix} 2 & -2 \\ 2 & 4 \end{bmatrix} \begin{bmatrix} 1 \\ 0\end{bmatrix} = \begin{bmatrix} 2 \\ 2\end{bmatrix}$$

And similarly the other direction is found by projecting the y axis:

$$ \begin{bmatrix} 2 & -2 \\ 2 & 4 \end{bmatrix} \begin{bmatrix} 0 \\ 1\end{bmatrix} = \begin{bmatrix} -2 \\ 4\end{bmatrix}$$


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