I've got two data sets on the same observations: one with 4000 variables, one with 5000 variables. I've calculated the first 30 canonical correlations between these two data sets and I look at which correlations are large (r >= 0.6). But now I'm wondering, what with negative correlations? Because when r <=-0.6 there is even a strong relationship between the two data sets. So should I calculate all the canonical correlations, and then (for example) take the first 5 positive ones, en take the last 3 ones?
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1$\begingroup$ I just looked at the Wikipedia page for canonical correlation and I am a bit confused. I always thought that canonical correlation maximised the squared correlation, but the Wikipedia page suggests that it maximises the correlation directly, which means negative correlations are "ignored". I don't have my book on multivariate data analysis here, but I'll check this when I get home. Maybe someone else knows how the "canonical" canonical correlation analysis is defined? $\endgroup$– Tommy LApr 13, 2015 at 7:31
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$\begingroup$ I've used the book of Johnson and Wichern to understand CCA, and they maximizes Corr(X,Y) $\endgroup$– SilkeApr 13, 2015 at 7:41
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$\begingroup$ Your question seems unclear. Canonical correlation is always positive. CCA takes accout for Pearson correlations of any sign. $\endgroup$– ttnphnsApr 13, 2015 at 7:43
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1$\begingroup$ My suestion is: Is it possible to have negative canonical correlation? If yes, why can this occur and how can you take this into account? If no, why can't this occur? $\endgroup$– SilkeApr 13, 2015 at 7:51
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$\begingroup$ My understanding is also, like @ttnphns, that negative correlations are included. They occur because $\mathrm{Corr}(x,y)$ may be negative, but note that the objective function of canonical correlation maximises $\mathrm{Corr}(x,y)^2$ (in my understanding), so positive and negative correlations are treated the same. $\endgroup$– Tommy LApr 13, 2015 at 7:54
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