An improvement to OPR

[RyanCahoon]

Originally Posted by Frenchie461

This should yield an OPR matrix containing complex entries, which theoretically should have a least squares average for both teleop and auton.

Originally Posted by SoftwareBug2.0

What advantage does this have over simply calculating independently with just auto scores, and then just teleop scores, and then just climb scores?

Since the OPR calculation boils to down to

P = (A^-1) * S

where P is the OPR, A is the binary matrix denoting teams in each alliance and S is the alliance scores, then Frenchie461 is essentially advocating

Pt + Pa*i = (A^-1) * (St + Sa*i)

and since matrix multiplication is distributive

Pt + Pa*i = (A^-1) * St + ((A^-1) * Sa)*i

So you'll end up with the same result as calculating each OPR component independently. You'll get least-squares best fit for each component (as you would otherwise), but there won't be any additional interaction gained between them. This makes sense, because the least-squares fitting part of the operation happens when taking the inverse of A, and isn't affected by the value of S (whether real or complex) that it is post-multiplied by. Performance-wise, I would guess they would take about the same amount of time, assuming you're not re-calculating the value of A^-1 when doing the calculations independently.



note: the inverse operation written ^-1 above becomes the generalized inverse for non-square cases of A