I am trying to measure the correlation between a probability distribution and a scalar value. For instance, I have the following:
| Vector of values | Corresponding Scalar |
|---|---|
| Vec 1 | Scalar 1 |
| Vec 2 | Scalar 2 |
| ... | ... |
| Vec n | Scalar n |
What I currently do is that, I calculate the wasserstein distance between each pair of vectors and calculate the difference between each pair of corresponding scalars, so the resulting is:
| Wasserstiein Distance (x,y) | Scalar Difference(x,y) |
|---|---|
| wass(vec1, vec2) | scalar1 - scalar2 |
| wass(vec1, vec3) | scalar1 - scalar3 |
| ... | ... |
| wass(vec n-1, vec n) | scalar n-1 - scalar n |
Then I calculate the correlation between these two streams (wass values) and (diff values). I use five different correlation coefficients to catch both linear and non-linear relations:
- Peasron's Coeff.
- Kendall Tau
- Spearman
- Distance Correlation.
- Maximal Information.
The results suggest some correlation between the wasserstein distance and the corresponding scalar vector.
My question is: Is this correlation process legit? Is there any references I can look into utilizing the same concept?
Edit: Regression is not possible in this context because the size of the vectors are not equal, so, this is a deficient data. Additionally, I don't have enough samples (vectors) for training a regression model.
That's why I am trying to avoid regression in this context.