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.


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:

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


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.