# Scatter matrices and independent component analysis (ICA)

## Hannu Oja

Assume that a random `\(p\)`

-vector `\(s\)`

has independent components and that `\(x=As\)`

where `\(A\)`

is a positive definite `\(pxp\)`

matrix. Assume that `\(x_1,...,x_n\)`

is a random sample from the distribution of `\(x\)`

. In the independent component analysis (ICA) one tries to estimate a matrix `\(B\)`

such that `\(Bx\)`

has independent components. On the other hand, let `\(S(x)\)`

be a scatter matrix (functional), that is, `\(S(x)\)`

is a positive definite `\(pxp\)`

matrix with the affine equivariance property: `\(S(Bx+b)=BS(x)B'\)`

. In the talk we discuss the problem of finding independent components using two different scatter matrices.