Principal Components
Overview
Principal Components is a dimensionality reduction technique that utilizes the covariance of a dataset. It finds the change of basis
of the dataset vector space such that the greatest variance of the data occurs along the first basis vector in the new basis.
The second largest along the second and so on.
Definitions and Justification
Principal Component analysis can be motivated and derived in two ways.
A description of the algorithm can be found at
Algorithm.
Change of Basis
The net effect of a principal component analysis is to transform the data points using a
Change of Basis
transformation.
{% \vec{v}' = M \vec{v} %}
Under this type of transformationg, the covariance matrix of the transformed vectors are given by
{% cov(A \vec{Z}) = A cov(\vec{Z}) A^T %}
See
derivation
for the reasoning.
Monte Carlo
Simulations
