13.3 More Generally

We want

  • From \(P\) original variables \(X_{1}, \ldots , X_{P}\) get \(P\) principal components \(C_{1}, \ldots , C_{P}\)
  • Where each \(C_{j}\) is a linear combination of the \(X_{i}\)’s: \(C_{j} = a_{j1}X_{1} + a_{j2}X_{2} + \ldots + a_{jP}X_{P}\)
  • The coefficients are chosen such that \(Var(C_{1}) \geq Var(C_{2}) \geq \ldots \geq Var(C_{P})\)
    • Variance is a measure of information. Consider modeling prostate cancer.
      • Gender has 0 variance. No information.
      • Size of tumor: the variance is > 0, it provides useful information.
  • Any two PC’s are uncorrelated: \(Cov(C_{i}, C_{j})=0, \quad \forall i \neq j\)

We have

\[ \left[ \begin{array}{r} C_{1} \\ C_{2} \\ \vdots \\ C_{P} \end{array} \right] = \left[ \begin{array}{cccc} a_{11} & a_{12} & \ldots & a_{1P} \\ a_{21} & a_{22} & \ldots & a_{2P} \\ \vdots & \vdots & \ddots & \vdots \\ a_{P1} & a_{P2} & \ldots & a_{PP} \end{array} \right] \left[ \begin{array}{r} X_{1} \\ X_{2} \\ \vdots \\ X_{P} \end{array} \right] \]

  • Hotelling (1933) showed that the \(a_{ij}\)’s are solutions to \((\mathbf{\Sigma} -\lambda\mathbf{I})\mathbf{a}=\mathbf{0}\).
    • \(\mathbf{\Sigma}\) is the variance-covariance matrix of the \(\mathbf{X}\) variables.
  • This means \(\lambda\) is an eigenvalue and \(\mathbf{a}\) an eigenvector of the covariance matrix \(\mathbf{\Sigma}\).
  • Problem: There are infinite number of possible \(\mathbf{a}\)’s
  • Solution: Choose \(a_{ij}\)’s such that the sum of the squares of the coefficients for any one eigenvector is = 1.
    • \(P\) unique eigenvalues and \(P\) corresponding eigenvectors.

Which gives us

  • Variances of the \(C_{j}\)’s add up to the sum of the variances of the original variables (total variance).
  • Can be thought of as variance decomposition into orthogonal (independet) vectors (variables).
  • With \(Var(C_{1}) \geq Var(C_{2}) \geq \ldots \geq Var(C_{P})\).