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I've noticed that correlation heat maps are a popular method of visualising data, but I'm not sure what insights they provide. I can understand why we're interested in the relationship between different predictors and the target, but what do the correlations between the predictors themselves tell us? How do they help us to choose which variables will be useful for a model?

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As ever it depends on the question at hand but often the interesting variables are those which are not correlated with each other but are both correlated with the target.

Collinearity between variables is often insightful because sometimes surprisingly different things end up related to each other and it teaches you something about the domain.

Collinearity increases the variance of a regression fit, so spotting variables that are correlated with each other gives you cause to do something about it. I.e. pick one of them or some linear combination of them to maximise the information you can get.

Finally, when you're preparing your data and transforming it with something more uncorrelated n mind, a correlation heatmap may be a good way to see how good a job you've done and where there is still work to do.

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The heatmap may also show a clustering structure according to correlations, in which case it may be a good idea to represent the clusters by single variables (this could be a principal component of the variables in the cluster, or one "central" variable to represent them all, or some kind newly defined index that puts together the information in the variables in the cluster in a way meaningful from a subject matter perspective).

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