I’ve been familiar with this issue for quite some time as it was misleading some relatively smart people in the context of infectious disease research. My initial take was also to view it as an extreme example of over fitting. But I think it’s more helpful to think of it as some thing inherent to random walks. Actually the phenomena has very little to do with d>>T & persists even with T>>d. The fraction of variance in PC1 tends to be at least 6/π^2≈61% irrespective of d & T. I believe you need multiple independent random walks for PCA to behave as naively expected.
I’ve been familiar with this issue for quite some time as it was misleading some relatively smart people in the context of infectious disease research. My initial take was also to view it as an extreme example of over fitting. But I think it’s more helpful to think of it as some thing inherent to random walks. Actually the phenomena has very little to do with d>>T & persists even with T>>d. The fraction of variance in PC1 tends to be at least 6/π^2≈61% irrespective of d & T. I believe you need multiple independent random walks for PCA to behave as naively expected.