# PCA with on data with unequal length

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.

• Excuse me, what does this mean data is T1Xn and T2Xn? And did by "unequal length" you just mean that there are missing values? – ttnphns Jul 5 '12 at 19:24
• So the data is $TXn$, but you could break it into the respective panels that would be of different length. The first $T_{2}-T_{1}Xn$ block would be missing in the second panel. – John Jul 5 '12 at 19:45
• I'm sorry if this comes a bit too late... I'm currently looking for a solution to the same problem. You mention that 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... would you mind kindly giving me some references? Thank you in advance! – Imaco Mar 18 at 15:56
• This is a common one: ideas.repec.org/p/fth/pennfi/05-96.html – John Mar 18 at 22:10