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:
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:
“One of the grotesqueries of present-day American life is the amount of reasoning that goes into displaying the wisdom secreted in bad movies while proving that modern art is meaningless.... They have put into practise the notion that a bad art work cleverly interpreted according to some obscure Method is more rewarding than a masterpiece wrapped in silence.”
—Harold Rosenberg (19061978)