Papers › On the explanatory power of principal components
On the explanatory power of principal components
Daniel A. Diaz-Pachon, J. Sunil Rao, Jean-Eudes Dazard
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We show that if we have an orthogonal base (u₁,…,uₚ) in a p-dimensional vector space, and select p+1 vectors v₁,…, vₚ and w such that the vectors traverse the origin, then the probability of w being to closer to all the vectors in the base than to v₁,…, vₚ is at least 1/2 and converges as p increases to infinity to a normal distribution on the interval [-1,1]; i.e., Φ(1)-Φ(-1)≈0.6826. This result has relevant consequences for Principal Components Analysis in the context of regression and other learning settings, if we take the orthogonal base as the direction of the principal components.
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