|This article relies largely or entirely upon a single source. (April 2013)|
FastICA is an efficient and popular algorithm for independent component analysis invented by Aapo Hyvärinen at Helsinki University of Technology. The algorithm is based on a fixed-point iteration scheme maximizing non-Gaussianity as a measure of statistical independence. It can also be derived as an approximative Newton iteration.
Preprocess the data
Before the FastICA algorithm can be applied, the input vector data should be centered and whitened.
Centering the data
The input data is centered by computing the mean of each component of and subtracting that mean. This has the effect of making each component have zero mean. Thus:
Whitening the data
Whitening the data involves linearly transforming the data so that the new components are uncorrelated and have variance one. If is the whitened data, then the covariance matrix of the whitened data is the identity matrix:
This can be done using eigenvalue decomposition of the covariance matrix of the data: , where is the matrix of eigenvectors and is the diagonal matrix of eigenvalues. Once eigenvalue decomposition is done, the whitened data is:
Single component extraction
The iterative algorithm finds the direction for the weight vector maximizing the non-Gaussianity of the projection for the data . The function is the derivative of a nonquadratic nonlinearity function . Hyvärinen states that good equations for (shown with their derivatives and second derivatives ) are:
The first equation is a good general-purpose equation, while the second is highly robust.
- Randomize the initial weight vector
- Let , where means averaging over all column-vectors of matrix
- If not converged, go back to 2
Multiple component extraction
The single unit iterative algorithm only estimates one of the independent components, to estimate more the algorithm must repeated, and the projection vectors decorrelated. Although Hyvärinen provides several ways of decorrelating results, the simplest multiple unit algorithm follows. indicates a column vector of 1's with dimension M.
- Input: Number of desired components
- Input: Matrix, where each column represents an N-dimensional sample, where
- Output: Un-mixing matrix where each row projects X onto into independent component.
- Output: Independent components matrix, with M columns representing a sample with C dimensions.
for p in 1 to C: Random vector of length N while changes Output: Output:
- Hyvärinen, A.; Oja, E. (2000). "Independent component analysis: Algorithms and applications". Neural Networks 13 (4–5): 411–430. doi:10.1016/S0893-6080(00)00026-5. PMID 10946390.
- Hyvarinen, A. (1999). "Fast and robust fixed-point algorithms for independent component analysis". IEEE Transactions on Neural Networks 10 (3): 626–634. doi:10.1109/72.761722. PMID 18252563.