Principal Component Analysis

Given a cloud of vectors S = \{x_1,\ldots,x_m\} in \bm{R}^n what single vector “best” represents the location of the cloud? If m = 1 then obviously \bar{x}_1 = x_1 is the solution to this. If m = 2 then the solution \bar{x}_1 = (x_1 + x_2)/2 seems to be a likely candidate. In general we might suspect the average \bar{x}_1 = \sum_{j=1}^m x_j/m is the best representation of location.

Define the size of S by $|S|^2 = _i |x_i|^2. Given x\in\bm{R}^n define the projection along x by P_xy = y - x\cdot y/\|x\|^2. Note P_x x = 0 and P_x y = y if x\cdot y = 0. The principal component of S is the value of x that minimizes the size of P_xS. The reduction is $R_x = |S|^2 - |P_xS|^2.

Define \Sigma^2 = (1/m)\sum x_j x_j^*. Since \Sigma^2 is symmetric and positive, it has a spectral decomomposition \Sigma^2 = \sum_j \sigma_j^2 e_j e_j^* where (e_j) are orthonormal. Assume \sigma_j are in non-increasing order.

What are the principle components of \{B_t(\omega)\mid 0\le t\le 1, \omega\in C(0,1)\}?

\Sigma^2\colon L^2[0,1]\to L^2[0,1] by \Sigma^2f(x) = \int_0^2 B_t(x)B_t(y)f(y)\,dy.