I didn’t really pay attention to college basketball this year, so I decided to take a different approach to fill out my group.
I started by downloading the full Division 1 men’s basketball schedule (pulled from rivals.yahoo.com), along with each game’s score, date, and home team.
In the model, I assume that each team has two (unknown) vectors of real numbers that describe how good their offense and defense are in various attributes, respectively. For example, we might want to represent how good the guards on each team are, how good the forwards are, and how good the centers are, both offensively and defensively. We could do this using an offensive and defensive vector:
Insulted: [5, 10, 4]
Defending: [2, 3, 10]
This means that the guards are 5 attack and 2 defense etc. In my model, it will be easier if we assume that high numbers are better for offenses and low numbers are better for defenses.
The score of a game between team i and team j can be generated as the dot product of the offensive vector of team i with the defensive vector of team j, and vice versa. In our example run, if our previous team played a team with vectors:
Insulted: [3, 2, 4]
Defending: [2, 5, 5]
So the score of the first team is predicted to be 5 * 2 + 10 * 5 + 4 * 5 = 80
and the second team’s score is predicted to be 3 * 2 + 2 * 3 + 4 * 10 = 52
What a blowout!
Now, the only problem is that we don’t actually know the vectors that describe each team’s offense and defense. Okay, we’ll learn them from the data.
Formally, the objective is to find the latent matrices O and D that minimize the sum of the squared error between the predicted scores and the observed scores. In mathematics,
sum_g (gi_score – O_i: * D_j:)^2 + (gj_score – O_j: * D_i:)^2
where I use the notation that team i played team j in game g (i and j depend on g, but I remove this dependency in the notation to keep things simple)*.
I won’t go into details, but we can take the derivative of the error function with respect to each latent vector to find changes in the vectors that will make them more closely match the results of all games earlier in the season. I repeat this until there is no change that improves the bug (batch gradient descent, for the detail oriented folks).
Results In the case where I choose the latent vectors to be one-dimensional, I get as output an offensive and defensive rating for each team. Remember, to predict the first team’s score against another team, multiply the first team’s offensive rating (higher is better) by the second team’s defensive rating (lower is better).
Here are the top 10 offenses and defenses, as learned by the 1D version of my model:
offenses
North Carolina (9.79462281797)
Pittsburgh (9.77375501699)
Connecticut (9.74628326851)
Memphis (9.71693872544)
Louisville (9.69785532917)
Duke (9.65866585522)
UCLA (9.59945808934)
West Virginia (9.56811566735)
Arizona Street (9.56282860536)
Missouri (9.55043151623)
fangs
North Carolina (7.02359489844)
Pittsburgh (7.0416251036)
Memphis (7.05499448413)
Connecticut (7.07696194481)
Louisville (7.14778041166)
Duke (7.18950625894)
UCLA (7.21883856723)
Gonzaga (7.22607569868)
Kansas (7.2289767174)
Missouri (7.2395184452)
And here are the results of simulating the entire tournament with a 5-dimensional model. For each game, I report the predicted score, but for the pool I just picked the predicted winner.
=================== ROUND 1 ======================
Louisville 75.8969699266, Morehead St. 54.31731649
Ohio St. 74.9907105909, Siena 69.6702059811
Utah 69.7205426091, Arizona 69.2592708246
Wake Forest 72.3264784371, Cleveland St. 64.3143396939
West Virginia 66.7025939102, Dayton 57.550404701
Kansas 84.0565034675, North Dakota St. 71.281863854
Boston College. 65.0669174572, USC 68.7027018576
Michigan Street 77.3858437718, Robert Morris 59.6407479
Connecticut 91.9763662649, Chattanooga 63.9941388666
BYU 74.7464520646, Texas A&M 70.5677646712
Purdue 69.8634461612, Northern Iowa 59.4892887466
Washington 81.8475059935, Mississippi St. 74.6374151171
Marquette 73.4307446299, Utah Street 69.1796188404
Missouri 83.8888903275, Cornell 68.1053984941
California 74.9638076999, Maryland 71.2565877894
Memphis 78.3145709447, CSU Northridge 59.0206289492
Pittsburgh 85.5983991252, E. Tennessee St. 64.8099546261
Oklahoma St. 81.6131739754, Tennessee 81.8021658489
Florida Street 59.994769086, Wisconsin 60.9139371828
Javier 77.3537694, Portland St. 63.8161558802
UCLA 76.790261041 VCU 65.2726887151
Villanova 72.9957948506, American 58.6863439306
Texas 64.5805075558, Minnesota 62.3595994418
Duke 85.084666484, Binghamton 61.1984347353
North Carolina 99.2788271609, Radford 69.7291392149
LSU 65.0807263343, Butler 64.9895028812
Illinois 70.6250577544, West. Kentucky 57.6646396014
Gonzaga 75.0447785407, Akron 61.0678281691
Arizona Street 64.7151394863, Temple 58.0578420156
Syracuse 74.7825424779, Stephen F. Austin 60.5056731732
Clemson 74.4054903161, Michigan 70.8395522274
Oklahoma 78.5992492855, Morgan Street 59.7587888038
=================== ROUND 2 ======================
Louisville 67.3059313968, Ohio St. 60.5835683909
Utah 71.3007847464, Wake Forest 73.2895225467
West Virginia 67.9574088476, Kansas 67.4869037187
USC 62.1192840465, Michigan Street 64.56295945
Connecticut 76.8719158147, BYU 71.8412099454
Purdue 74.245343296, Washington 73.6100911982
Marquette 76.4607554812, Missouri 80.5497967091
California 64.7143532135, Memphis 70.9373235427
Pittsburgh 79.1278381289, Tennessee 70.6786108051
Wisconsin 63.0943233452, Xavier 63.5379857382
UCLA 74.1282015782, Villanova 71.4919550735
Texas 66.3817261194, Duke 70.9875941571
North Carolina 86.2296333847, LSU 73.8695973309
Illinois 62.6218220536, Gonzaga 65.6078661776
Arizona Street 74.0588194422, Syracuse 71.254787147
Clemson 76.9943827197, Oklahoma 78.9108038697
==================== SWEET 16 ====================
Louisville 72.8097088102, Wake Forest 68.2411945982
West Virginia 66.1905929215, Michigan St. 65.2198396254
Connecticut 70.4975234274, Purdue 67.014115714
Missouri 66.6046145365, Memphis 69.9964130636
Pittsburgh 72.8975484716, Xavier 64.848615134
UCLA 72.3676109557, Duke 73.1522519556
North Carolina 84.6606149747, Gonzaga 80.3910425893
Arizona Street 67.8668018941, Oklahoma 67.0441371239
================== ELITE EIGHT =====================
Louisville 64.0822047092, West Virginia 61.7652102534
Connecticut 64.875382557, Memphis 65.9485921907
Pittsburgh 72.8027424093, Duke 70.5222034022
North Carolina 76.2640153058, Arizona St. 72.3363504426
==================== FINAL FOUR =====================
Louisville 60.7832463768, Memphis 61.4830569498
Pittsburgh 80.3421788636, North Carolina 81.0056716364
==================== END GAME ======================
Memphis 73.8935857273, North Carolina 74.259537592
In the end, these predictions were enough to win my key. Obviously, everything has to be taken with a grain of salt, but being a PhD student in Machine Learning [http://www.machinelearningphdstudent.com/]it was fun to put my money where my mouth was and have a little fun.
Oh, and let me know if you want the data I collected or the code I wrote to make this work. I am pleased to share it.
* I also regularize the latent vectors by adding independent zero-mean Gaussian priors (or, equivalently, a linear penalty on the squared L2 norm of the latent vectors). This is known to improve these matrix factorization-like models by encouraging them to be simpler and less willing to detect spurious features of the data.
