I'm working on an assignment which involves two sets of data, each a matrix whose columns are signals. The signals represent the observed knee motion angles of individuals walking - i.e 0 represents a "straight" leg, and 90 a leg bent at a 90 degree angle. Both sets of data are randomized in a way known only to the instructor. One set contains observations for individuals with arthritis, and one set contains "healthy" individuals - the control group. My task is to determine which is which using principal component analysis (in MATLAB).
I've performed the PCA but I'm wondering if my interpretation of the results is correct. The cumulative variance explained by the "first" five principal components for the first dataset ($X$) is:
$$[0.6582, 0.8151, 0.8927, 0.9480, 0.9802].$$
So the first principal component explains 0.6582 of variance, the first two 0.8151, and so on. For the second dataset ($Y$):
$$[0.7033, 0.8532, 0.9545, 0.9815, 0.9925].$$
My hypothesis is that the control group without arthritis will have greater variance between observations than the group with arthritis, who will have poor range of motion. As such, fewer principal components should capture more variance (from the mean signal), for the "arthritis" dataset, relative to the control dataset. That would make $X$ the control dataset.
I'm not sure if my reasoning is correct or if I've approached this wrong.