What is independent component analysis 🌱

• Assume there exist true independent sources $s_1\left(t\right),s_2\left(t\right)$ that are dependent on time.
• Assume that the observed signals $x_1\left(t\right),x_2\left(t\right)$ are linear combinations of the underlying true sources. $\left[\begin{array}{c}{x}_1\left(t\right)\\ x_2\left(t\right)\end{array}\right]=\left[\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right]\left[\begin{array}{c}{s}_1\left(t\right)\\ s_2\left(t\right)\end{array}\right]⟹X\left(t\right)=AS\left(t\right)$

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 $LX\left(t\right)=S\left(t\right)$, 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