I have a set of climate data (temperature, pressure and moisture for example), $X$, $Y$, $Z$ which are matricies with dimensions $n \times p$ where $n$ is the number of observations and $p$ is the number of spatial points.

Previously, to investigate modes of variability in dataset $X$, I simply performed a empirical orthogonal function (EOF) analysis OR Principle component Analysis (PCA) on $X$. This involved decomposing (via SVD), the matrix $X$.

To investigate the coupling of the modes of variability of $X$ and $Y$, I used maximum covariance analysis (MCA) which involved decomposing a covariance matrix proportional to $XY^{T}$.

However, if I wish to looked at all three datasets, how do I go about doing this? One idea I had was to form a fourth matrix, $\Sigma$, which will be the 'feature' concatenation of the three datasets:

$\Sigma = [X, Y, Z]$

so that my matrix $\Sigma$ will have dimensions $n \times 3p$.

I would then use standard PCA/EOF analysis and use SVD to decompose this matrix $\Sigma$ and then I would obtain modes of variabiilty with size $3p \times 1$ and thus subsequently the mode associated with $X$ is the first $p$ values, the mode associated with $Y$ is the second set of $p$ values and the mode associated with $Z$ is the last $p$ values.

Is this correct? Or can anyone suggest a better way of looking at the coupling of all three (or more) datasets?

Thank you so much!


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