In imaging, we deal with multivariate data, like in array form with several spectral bands. And trying to come up with interpretation across correlations of its dimensions is very challenging, if not impossible. For example let's recall the number of spectral bands of AVIRIS data we used in the

previous post. There are 152 bands, so in total there are 152$\cdot$152 = 23104 correlations of pairs of random variables. How will you be able to interpret that huge number of correlations?

To engage on this, it might be better if we group these variables into two and study the relationship between these sets of variables. Such statistical procedure can be done using the canonical correlation analysis (CCA). An example of this on health sciences (from Reference 2) is variables related to exercise and health. On one hand you have variables associated with exercise, observations such as the climbing rate on a stair stepper, how fast you can run, the amount of weight lifted on bench press, the number of push-ups per minute, etc. But you also might have health variables such as blood pressure, cholesterol levels, glucose levels, body mass index, etc. So two types of variables are measured and the relationships between the exercise variables and the health variables are to be studied.

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Methodology

Mathematically we have the following procedures:

- Divide the random variables into two groups, and assign these to the following random vectors:
\begin{equation}\nonumber
\mathbf{X} = [X_1,X_2,\cdots, X_p]^T\;\text{and}\;\mathbf{Y} = [Y_1,Y_2,\cdots, Y_q]^T
\end{equation}