What is independent component analysis 🌱

• Assume there exist true independent sources $s_1(t), s_2(t)$ that are dependent on time.
• Assume that the observed signals $x_1(t), x_2(t)$ are linear combinations of the underlying true sources. \(\begin{bmatrix} x_1(t) \\ x_2(t) \end{bmatrix} = \begin{bmatrix} a_{11} & a_{12}\\ a_{21} & a_{22} \end{bmatrix} \begin{bmatrix} s_1(t) \\ s_2(t) \end{bmatrix} \implies X(t) = A S(t)\)

The goal is to find a linear transformation $L$ so that we can recover the original sources. We can acheive this by finding $L$ such that $L X(t) = S(t)$, where $L$ is our best guess at the inverse of the mixing matrix $A$.

The results of ICA compared to PCA are shown here. pca_vs_ica