These rankings are derived totally from the scores of the games of the current season, hence no rankings until after the fourth full week of the football season and December 15 for basketball. No analysis of offensive or defensive statistics. No priming with games from the previous year since players have left, new players have arrived, returning players may have gained weight, lost weight, broke up with their girlfriend or be taking Thermo (that'll affect your performance!). The maximum score difference credited is 28 points for football and 24 points for basketball. So, when Kansas St. beats NW South Dakota A&I 98-0 in football, it counts the same as 28-0 (with an exception noted below). Also, the winning team get 5 points added because I said so, and because winning should have some significance. The home team gets 3 points subtracted for their home field advantage. How were these numbers derived? The tried and true SWAG method.
Now, we have the scores of all the games, courtesy of Peter Wolfe for football and Ken Pomeroy for basketball, what do we do with them? Well, I'm not a statistical guy, I'm an optimization guy. The first thing an optimization guy like me says is, "How can I put this in an LP?" Regression? Variance? No time for that now. The first thing we do is construct an equation for each game that looks like this:
WinTeamRank - LoseTeamRank - sqrt(Win Adjustment) + sqrt(Home Team Adjustment) + GameSlack = sqrt(ScoreDifference)
Why use the sqaure root of the score difference, win adjustment and home team adjustment? It seemed reasonble that even though the score difference is capped there still maybe needs to be an adjustment to diminish the impact of each additional point scored. Is it better than before when I didn't use the square root? I'm not sure, because about the time I changed we started our family and for some reason I just haven't the time to study it in detail. How do you define better anyway?
Now we just minimize the sum of the slack variables and we're done, right? Not so fast, my friend. This leads to wild and wacky rankings since the sum of the slack values for any particular team may be very high or very low. So, we sum the slack variable for each team's games, adding the slack for a win and subtracting for a loss, trying to balance out the slack for each team. These equations look something like this:
GameSlack1 + GameSlack2 - GameSlack3 = 0
How do you know it's going to equal zero? Well, those of you who have been paying attention will notice that there is a variable for each team, a variable for each game, an equation for each game and an equation for each team. This results in a system of linear equations which we all know is easily solved. So there, no need for any slack variables in these equations.
Well, a system of linear equations is easily solved in Algebra class when you have three variable and three equations. Here, by the end of the season, we will have approximately 1400 variables and equations (even more for basketball)! What's a fellow to do? Here's that LP I was so looking forward to. Just formulate it as an LP, add an objective function that minimizes the sum of the slack variables and toss it into an lp solver, bake for 10 minutes at 350 degrees and serve with your favorite beverage.
But wait, that's not all! It's seems kind of unfair to penalize your football team if they have to play Duke in a conference game. With a cap of 28 points it's likely to knock you down a bit if you're any good at all. To help alleviate this I adjust the score difference if the team won by the cap to the diffence in the ranking and run the lp again. Now, go ahead and have that beverage, but drink responsibly.
Other notes: Only Division 1-A teams are ranked for football, Division 1 for basketball. All games are used. The "Non Div One" team is used for any games against non Division 1 teams. Early games are de-emphasized as the season goes on. By the end of the season, the games from the first month only count for 70% of there original value. Void where prohibited. Prices higher in Alaska and Hawaii. Side effects may include disbelief at where your alma mater is ranked. Consult your doctor.
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