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## Data Analysis – March Madness Predictions

I didn’t really pay attention to college basketball this year, so I decided to take a different approach to filling out my bracket.

I started by downloading the full Division 1 men’s basketball schedule (taken 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 describing the quality of their offense and defense in several attributes, respectively. For example, we might want to represent the quality of each team’s guards, the quality of forwards, and the quality of crosses, both in attack and defense. We could do this using an offensive and defensive vector:

Offense: [5, 10, 4]

Defense: [2, 3, 10]

This means the guards are a 5 in attack and a 2 in defense, etc. In my model, it will be easier if we assume that high numbers are better for attacks and low numbers are better for defenses.

The score of a match between team i and team j can then be generated as the dot product of team i’s offensive vector with team j’s defensive vector, and vice versa. In our race example, if our forward team was playing against a team with vectors:

Offense: [3, 2, 4]

Defense: [2, 5, 5]

Then the score of the first team should be 5 * 2 + 10 * 5 + 4 * 5 = 80

and the score of the second team should be 3 * 2 + 2 * 3 + 4 * 10 = 52

What an eruption!

Now the only problem is that we don’t actually know the vectors describing each team’s offense and defense. It’s OK – we’ll learn them from the data.

Formally, the goal is to find 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 (score_gi – O_i: * D_j:)^2 + (score_gj – O_j: * D_i:)^2

where I use the notation that team i played against team j in game g (i and j depend on g, but I drop this dependency in notation to keep things simple)*.

I won’t go into detail, but we can take the derivative of the error function with respect to each latent vector in order to find changes in the vectors that will make them closer to the results of all games earlier in the game. season. I repeat this until there are no more changes that will improve the error (batch gradient descent, for the detail minded).

Results In the case where I choose 1-dimensional latent vectors, I get an offensive and defensive rating for each team as output. Remember that 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 attacks 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)

tusks

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 the full tournament simulation with a 5-dimensional model. For each game I report the predicted score, but for the range I just choose the predicted winner.

==================== ROUND 1 ====================

Ohio St. 74.9907105909, Siena 69.6702059811

Utah 69.7205426091, Arizona 69.2592708246

Wake Forest 72.3264784371 Cleveland Street 64.3143396939

West Virginia 66.7025939102, Dayton 57.550404701

Kansas 84.0565034675 North Dakota Street 71.281863854

Coll. from Boston. 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 St. 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 Street 64.8099546261

Oklahoma St. 81.6131739754, Tennessee 81.8021658489

Florida Street 59.994769086, Wisconsin 60.9139371828

Xavier 77.3537694 Portland Street 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

LSU 65.0807263343, Butler 64.9895028812

Illinois 70.6250577544, West. Kentucky 57.6646396014

Gonzaga 75.0447785407, Akron 61.0678281691

Arizona St. 64.7151394863, Temple 58.0578420156

Syracuse 74.7825424779, Stephen F. Austin 60.5056731732

Clemson 74.4054903161, Michigan 70.8395522274

Oklahoma 78.5992492855 Morgan St. 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 St. 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 St. 74.0588194422, Syracuse 71.254787147

Clemson 76.9943827197, Oklahoma 78.9108038697

==================== SOFT 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 St. 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

==================== FINAL GAME ====================

Memphis 73.8935857273, North Carolina 74.259537592

In the end, these predictions were enough to win my slice. Obviously, everything must 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 gathered or the code I wrote to make this work – I’m happy to share it.

* I also regularize the latent vectors by adding independent zero-mean Gaussian priors (or equivalently, a linear penalty on the L2 norm squared of the latent vectors). This is known to improve these matrix factorization type models by encouraging them to be simpler and less prone to picking up spurious features in the data.

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