Principal Component Analysis - Derivation of PCA Using The Covariance Method

Derivation of PCA Using The Covariance Method

Let X be a d-dimensional random vector expressed as column vector. Without loss of generality, assume X has zero mean.

We want to find a orthonormal transformation matrix P so that PX has a diagonal covariant matrix (i.e. PX is a random vector with all its distinct components pairwise uncorrelated).

A quick computation assuming were unitary yields:

\begin{array}{rcl} \operatorname{var}(PX) &= &\mathbb{E}\\ &= &\mathbb{E}\\ &= &P~\mathbb{E}P^{\dagger}\\ &= &P~\operatorname{cov}(X)P^{-1}\\ \end{array}

Hence holds if and only if were diagonalisable by .

This is very constructive, as var(X) is guaranteed to be a non-negative definite matrix and thus is guaranteed to be diagonalisable by some unitary matrix.

Read more about this topic:  Principal Component Analysis

Famous quotes containing the word method:

    “... [a] girl one day flared out and told the principal “the only mission opening before a girl in his school was to marry one of those candidates [for the ministry].” He said he didn’t know but it was. And when at last that same girl announced her desire and intention to go to college it was received with about the same incredulity and dismay as if a brass button on one of those candidate’s coats had propounded a new method for squaring the circle or trisecting the arc.”
    —Anna Julia Cooper (1859–1964)