What is the primary objective of Principal Component Analysis (PCA)?
PCA finds linear orthogonal directions (principal components) that maximize projected data variance or equivalently minimize reconstruction error.
The first Principal Component (PC1) in PCA corresponds to the eigenvector of the covariance matrix associated with:
PC1 corresponds to the eigenvector with the maximum eigenvalue lambda_max, representing maximum variance.
In PCA, what property do the principal component directions (eigenvectors) possess relative to each other?
Eigenvectors of a symmetric real covariance matrix corresponding to distinct eigenvalues are mutually orthogonal.
Before applying PCA to a dataset X, what mandatory preprocessing step must be performed?
Data must be mean-centered (zero mean) so the covariance matrix X^{T} X / (n-1) correctly computes sample variances.
If a dataset has d features, how many principal components can PCA generate at most?
A d-dimensional feature space yields at most d principal components.