Ordinary correlation between two multidimensional variables would give similarity between these variables, whereas canonical correlation analysis (CCA) would find two linear transforms to obtain maximum correlation between the projection of these transform.

Goal of both methods is to find relationship between two variables. My question is why ordinary correlation analysis is not optimal to find the correlation between two variables?

  • $\begingroup$ I must be missing something: by "correlation analysis" at the end of course you mean CCA; but given that CCA considers linear combinations of variables and correlations do not, it's difficult to understand what you're trying to get at. Could you explain how "ordinary correlation analysis" would solve the CCA problem? What, to you, constitutes ordinary correlation analysis? $\endgroup$ – whuber Sep 5 '18 at 13:24
  • $\begingroup$ @whuber, I am trying to understand the difference between these two correlation methods. Lets say Pearson correlation between two time series would give a scalar value quantifying the similarity between them. Then, if I use CCA for these two time series, It will also find the relationship between these two time series. Is it because CCA can find the coordinate system of maximum correlation while Pearson correlation cannot? $\endgroup$ – Vendetta Sep 5 '18 at 13:44
  • $\begingroup$ You seem to be using time series as if they were a single variable. In such cases there's nothing CCA can do that goes beyond correlation, because linear combinations of a single variable are just multiples of that variable, and that changes nothing. Do you have a more pertinent situation in mind? $\endgroup$ – whuber Sep 5 '18 at 15:13
  • $\begingroup$ @whuber, Let us consider a blind source separation problem, there are two source signals $\mathbf{T} \in \mathbb{R}^{2 \times N}$ mixed with a spatial component $\mathbf{W} \in \mathbb{R}^{v \times 2}$, to create a dataset $\mathbf{X} = \mathbf{W}\mathbf{T}$, where $\mathbf{X} \in \mathbb{R}^{v\times N}$. Now the task of CCA is to recover $\mathbf{T}$ and $\mathbf{W}$. To apply CCA, you have to create another instance of $\mathbf{X}$ by varying some conditions, then [$\mathbf{Wx}$, ~, ~]= CCA($\mathbf{X}$, $\mathbf{Y}$) . Why in such a case, CCA would work and ordinary CA would not? $\endgroup$ – Vendetta Sep 6 '18 at 1:08

This answer is on my poor understanding of canonical correlations. It is not based on mathematical properties but on general observations I find that make canonical correlations interesting.

Because ordinary correlation has several problems:

  1. It compares two variables without taking into account the other variables

To overcome this one could use partial correlations, but then we run into the second problem:

  1. There are too many to make sense of the data

When canonical correlation analysis is used you have 40 or thousand of variables, and even if you had few variables and you could make sense of the partial correlations of each block there's another problem:

  1. It only considers two variables without taking into account the relationship of the others with their block of variables

If in partial correlations you give all the other variables expect the two you are considering you are assuming all are equally important and you are not considering the relationship between them.
And if you don't give the the variables of one of the block you are not considering their effect on one of the variables you are attempting to correlate.

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