[Ether]

Originally Posted by brennonbrimhall

This is going to sound extremely stupid, but what is the formula for OPR?

The "short" answer is, the "formula" for OPR is

[A][OPR]~[SCORE]

...where OPR for a given team is one of the N elements of the [OPR] Nx1 column vector approximate solution to the overdetermined system of linear equations shown above. N is the number of teams, M is the number of matches, and [SCORE] is a (2M)x1 column vector of alliance scores for each match in the database being used to do the computation. [A] is a (2M)xN matrix generated from the database¹.

There are several different methods for finding an approximate solution to an overdetermined system of linear equations¹, but for this particular problem using Normal Equations and Cholesky factorization to find the least squares solution is perhaps the most common.

Normal Equations solution method:

Multiply both sides of [A][OPR]~[SCORE] by the transpose of [A]:

[A]T[A][OPR]=[A]T[SCORE]

which can be re-written as

[P][OPR]=[S]     (see footnote 2 below)

...where [P]=[A]T[A] and [S]=[A]T[SCORE]

then use Cholesky factorization (see footnote 3 below)
to find the lower triangular matrix [L] such that

[L][L]T=[P]

to get

[L][L]T[OPR]=[S]

now substitute [y] for [L]T[OPR] to get

[L][y]=[S]

and use forward substitution to solve for [y]

then, with [y] known, use backward substitution to solve

[L]T[OPR]=[y]

for [OPR]

¹ I can provide the details if anyone's interested

² The matrix [P] and the vector [S] can be constructed directly from the database, rather than constructing [A] and [SCORES]. In fact, doing so is quicker and requires less memory. I show all the steps in order to make it clear that [OPR] is an approximate solution to the overdetermined data in the database, and that approximate solutions other than least squares are possible.

³ [P]=[A]^T[A] will be symmetric positive definite