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Principle component analysis (PCA) is a coordinate transformation of the original data, so that the projection of the data onto the first new coordinate has the greatest variance. I want to know whether there are situations where it is advantageous to split the variables into disjoint subsets, and doing PCA on each subset, instead of doing one PCA on all the data.

In matrix notation, suppose we have a data matrix $M$, of size $n_{\text{obs}} \times p$. That is, the number of rows in $M$ is equal to the number of observations, and the number of columns equal to the number of variables. The new coordinates for each observation is given by $MT$, where $T$ is a $p \times p$ matrix. If the variables are split into subsets, and arranged appropriately, $T$ will be block diagonal.

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Perhaps you believe that there are two different sets of correlated variables that are independent, i.e. there two sets are uncorrelated with each other. So you run PCA on one of the sets and then on the other and then you can check whether they return a similar projection or a very different one and you could check how far the two projections are from each other.

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