We have a dataset of I items who have been measured over two different sets of features A, with cardinality N, and B with cardinality M, and N > M. We would like to know in which feature space the items are more similar.

Therefore, for each feature space, we first scaled our measurements to have mean 0 and standard deviation 1 (z-scores) and then evaluated the Euclidean distance among all pairs of items. We finally compared the distribution of the distances in the two feature spaces using a Wilcoxon's test.

However, we are now wondering whether the fact that the cardinality of the two feature spaces is different introduces a bias in the distance evaluation, with a higher cardinality leading to a higher distance.

Could this be possible? If yes, how can we make the two feature spaces comparable? Is there a better distance than the Euclidian distance? Dividing the distance by the cardinality would suffice?

Many thanks in advance, any input will be highly appreciated :)

  • $\begingroup$ I'm afraid this is comparing apples with oranges. If you want to decide which feature set a classificator should base on that identifies items belonging to item set I with high sensitivity, you should have an idea about the behaviour of items not belonging to I in both feature sets. This idea will give you (and me) more hints on which distance to choose. $\endgroup$ – Horst Grünbusch Aug 30 '18 at 12:30
  • $\begingroup$ I am not doing classification here, but comparing the items (actually individuals) on two different and not overlapping sets of features. The first set of features is the gut microbiome composition, the second is their feacel metabolic composition. For the former I have values for 643 bacteria, for the latter, I have values for 850 metabolites (for each person). My goal is to determine whether the individuals have a metabolic composition that is closer/farther than their bacterial composition. Hope this make sense and thanks a lot for answering! $\endgroup$ – alesssia Aug 30 '18 at 13:37
  • $\begingroup$ Yes, that makes sence. And yes, the cardinality will cause a bias. Even worse, it is not only the cardinality but also the dependence structure between all the features. Do you have more than 1493 individuals? $\endgroup$ – Horst Grünbusch Aug 30 '18 at 13:57
  • $\begingroup$ No, I have about 600 (why 1493?). And yes, the correlation between them makes things much worse. Any idea on how to overcome this? $\endgroup$ – alesssia Aug 30 '18 at 14:06

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