Principal Component Analysis

Principal Component Analysis is equivalent to SVD (Singular Value Decomposition) in matrix algebra,
 
 

$$\mathbf{F}=\mathbf{AD}\Psi ^{\mathrm{T}}$$

where

$$\mathbf{A}=\left[ \begin{array}{ccc}\mathbf{a}_{1} & \cdots......s & \Psi _{N}\\\downarrow & & \downarrow\end{array}\right]$$

and N is the rank of F. A is a KT x N matrix of the left eigenvectors of F, D is an N x N diagonal matrix of eigenvalues (a.k.a. singular values), and  is a P x N matrix of right eigenvectors (eigenimages). In Matlab, this is accomplished by the svd routine, but you might want to use pca.m instead, whose results can be fed directly into the subsequent analysis by ind.m, which calculates the indicator functions.

The syntax of pca.m looks like this,
 
 

$$[\Psi ,\mathbf{A},\mathbf{D}]=\mathbf{pca}(\mathbf{F},n)$$

where  is the number of largest eigenvalues (and corresponding eigenvectors) to calculate. In other words, the full data matrix F is approximated by n most significant principal components. Of course, the full spectrum of N eigenvalues can be calculated, but our experience tells us that, for typical imaging data form our laboratory (intrinsic optical imaging of cat or monkey primary visual cortex), most of the neural signal is captured by the first ~100 principal components. We often choose n ~200 , which is a fairly conservative truncation, yet the subsequent calculations are considerably faster, and the results are almost indistinguishable from the full calculation.

After obtaining the principal components, we recommend users to plot the eigenvalue spectrum (semilogy probably gives the best scale), and to check the 's and a's to see if there is any obvious artifacts or strange behavior in the data. The eigenview.m routine displays the 's as images and plots its associated a's, but some modification may be necessary in order to adjust for a particular pixel dimension of the images.