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I have a set of objects. I have calculated two distance matrices: $X$ defining distance between each objects pair using metric $f1$, and $Y$ -- using metric $f2$. Now, I would like to understand if two objects are similar according to metric $f1$, then they are also similar according to metric $f2$. How can I do it?

For instance, $f1$ could say whether two objects have similar color, and $f2$ --- whether two objects have similar size. But metrics can be anything. For instance, we could talk about articles, $f1$ could be Jaccard distance measuring how many tags both articles share, and $f2$ could be euclidian distance measuring distance between word vectors of two articles. Now I would like to understand if two objects of blueish objects tend to be big, or whether articles tagged with "racism" have similar content.

Am I asking about correlation? How can I calculate it between $X$ and $Y$?

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You are basically right. You want to check the degree of dependence of one variable with respect to another one. No matter how you generate each variable, if you want to know how dependent it is from the other one, you normally use correlation to run that evaluation.

Very useful options you should consider to analyze the potential dependence between X and Y are:

  • Correlation: to measure both the strength and direction of the linear relationship between two variables.
  • Covariance: to assess the direction of the linear relationship between variables (not the strength).
  • Pearson's correlation: to obtain a single metric representing the degree (strength) of dependence between the two variables
  • Spearman's correlation: to evaluate a potential non-linear dependence between the two variables.

For more information, please, check this post here.

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