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Extreme events: blending PCA and ICA [20]

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Bullet points include: Most PCA algorithms calculate all eigenvectors/eigenvalues simultaneously. However, suppose V is an n x n covariance matrix with (sorted) eigenvalues lambda 1, lambda 2, ..., lambda n (largest is lambda 1) and corresponding (normalised) eigenvectors q1, q2, ..., qn. Suppose our importance criterion involves f(a) = aTVa, and |a|=1. Then a can be expressed as a = a1q1 + ... + anqn with a12 + ... + an2 = 1. And f(a) = a12 lambda 1  + ... + an2 lambda n so f(a) is maximised when a = q1. And eigenvectors are orthogonal, so removing one from output signals leaves remainder still to be extracted

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