PCA with on data with unequal length

Question

I am wondering how to perform PCA on data with unequal lengths. In the simplest case you could split up the data matrix so that one block would be $T_{1}Xn$ and the rest is $T_{2}Xn$. The second block has effectively missing data in the first $T_{1}-T_{2}Xn$ block.

Many of the applications of PCA on missing data are variants of the EM algorithm methodology (these typically will assume the distribution is multivariate Gaussian) that are more suited to applications where the data is missing at random. Oftentimes, these techniques do not particularly apply to data with uneven lengths. However, there are techniques to estimate the mean and covariance matrix in these cases and it is possible to apply PCA to the resulting covariance matrix. However, when the number of variables becomes large (absolutely or relative to the number of observations), then it may no longer be convenient to estimate the covariance matrix in this fashion.

Answer 1

What I've seen is an interpolation of the data in order to match the length (and even the sampling) of the data. This was particulary used (successfully, I must say) in the classification of variable stars using PCA, where the data where the actual light curves of the stars. For more details, see the paper of Deb & Singh (2009).

I must add, however, that if you are going to do PCA interpolating data, maybe Functional Data Analysis techniques are more suitable (I've been thinking in writing a paper with this method, in fact, and compare it with the PCA approach).

Answer 2

PCA is not the only dimensionality reduction that can be used, there are plenty of other methods, e.g. you may consider feature extraction to extract distinctive features from your data (with different lengths). FFT on a sliding window; overlapped or not, might be a good option, DCT of the results of FFT or without might be another option, it all depends on the Data used.