Monday, December 22, 2014

Kentucky is So Good They Broke My Model

The Matrix model uses a series of n x n matrices involving only wins and losses to calculate a "true" win percentage that factors in the strength of each team's opponents. There is a matrix for wins and a matrix for losses, which are then inverted, factoring in the total number of games each team has played, and then multiplied by an n x 1 vector that is each team's net wins (Note: n = number of teams (in NCAAB, 351)):


net wins = wins - losses

Each game simply is denoted by a "1". Therefore, the outputs of the win and loss matrices are bound between -1 and 1. If a team has a winning "adjusted" record, their rating is above 0; otherwise it's negative. I then standardize this rating to be analogous to win percentage, bound between 0 and 1:

i = initial Matrix rating; -1 <= i <= 1
i + 1 = i'
i' / 2 = f
f = final Matrix rating; 0 <= f <= 1

Kentucky has a rating of 1.005. As seen above, the final rating has an upper bound of 1. So the issue must be within the win and loss matrices themselves.

Kentucky's loss matrix rating is 1.000; this checks out, since they're undefeated and thus have 0 losses. The issue then is pinpointed to their win matrix: their rating is 1.010.

In their win matrix, they have 12 wins, and 12 total games (Note: only Division 1 games are included, but in this case all of Kentucky's opponents have been Division 1). This checks out too. Their strength of schedule is very high: 0.645 (0.500 is average). This is the only explanation I can offer as to why Kentucky's rating is so high: they've played a very tough slate of teams and beaten them all. Even so, it shouldn't be above 1. 

A full rundown of the Matrix method can be found here on page 31, "Least Squares Ratings", written by Kenneth Massey. 

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