tag:blogger.com,1999:blog-35971090.post6406905648985229468..comments2023-12-21T10:20:43.344-05:00Comments on The Hoover Street Rag: A golden flash of stupidityGeoffhttp://www.blogger.com/profile/09461267960136260783noreply@blogger.comBlogger7125tag:blogger.com,1999:blog-35971090.post-33617653685732086852012-11-30T07:25:47.768-05:002012-11-30T07:25:47.768-05:00So I thought about this problem some more, and the...So I thought about this problem some more, and the only changes of sets of H2H match-ups for which <b>b</b> is invariant are sets containing only entire circles of death. I think we've already established that having only these cycles is sufficient for invariance, so we just need to show it's necessary.<br /><br />Proof sketch: Represent the set of game outcomes as directed graph, with a vertex V_i for each team, and a directed edge E_{i,j} for each game outcome. A "circle of death" is then a sequence of edges that begins and ends and the same vertex. Changing the outcome of a game is the equivalent of removing E_{i,j} and replacing it with E_{j,i}.<br /><br />Let's choose a set of directed edges to transform that cannot be decomposed into cycles. Then there must exist a path E_{a,b} --> E_{b,c} --> ... --> E_{y,z} that is not a cycle, i.e., V_z ~= V_a. Along this path, the out-degree of V_a (i.e., the number of wins for team a) is decreased by one and the in-degree of V_a (i.e., the number of losses for team a) is decreased by one; the reverse is true for team z. Since the function to calculate <b>b</b>_a and <b>b</b>_z is a function of the difference of the in-degree and the out-degree, and transformation that change the overall degree of the vertex are not allowed, a transformation of outcomes that cannot be decomposed into cycles must change at least two values of <b>b</b>.<br /><br />Since <b>C</b> is unchanged by any such transformation, any non-cyclical transformation must result in a change to the ratings vector <b>r</b> and then possibly to the overall rankings.<br /><br />So the Colley ratings will not change if you replace the scenario (Oregon beat Washington beat Stanford beat Oregon) with the scenario (Stanford beat Washington beat Oregon beat Stanford). They will change if you replace (Kentucky beat Kent State beat Rutgers) with (Rutgers beat Kent State beat Kentucky) because that's not a cycle. So, my interpretation is that the ratings do care whether Kent State beat Rutgers or Kentucky <i>unless</i> you included the rest of the cycle.<br /><br />Dear God, I think I understand the Colley rankings better than Colley now. Their assumptions are flawed for the application to ranking football teams, but the method itself is reasonable in the absence of better information. I'd say they were a good ideas whose time has passed now that we're in a more data-rich age. I'd rather have Ed's rankings making the decisions than Colley's or a roomful of NCAA bureaucrats.Davidhttps://www.blogger.com/profile/03104578852711638389noreply@blogger.comtag:blogger.com,1999:blog-35971090.post-53839648780711077132012-11-29T15:51:02.984-05:002012-11-29T15:51:02.984-05:00@dougj: The values of the vector b are where the g...@dougj: The values of the vector <b>b</b> are where the game-specific information dependence lies. <b>b</b> is determined purely by win-loss record, true, but the values of <b>b</b> are not independent of each other - you can't change one value without changing at least one other, because if you change a win to a loss somewhere, you have to change a loss to a win somewhere else. These dependences are caused by the head-to-head matchups.<br /><br />If you change the outcome of any one game so that A beats B instead of B beats A, it can affect every value of the solution <b>r = inv(C)b</b> provided that the adjacency graph of C is fully-connected, as it is by the end of every CFB season. <b>r</b> is thus indirectly determined using game-specific information through changes in the values of <b>b</b>.<br /><br /><b>b</b> is invariant to the direction of "circles of death," and in that sense only does it not depend on game-specific information. I don't know if the Colley ratings are invariant <i>only</i> to changing the directions of circles of death; if there are other transformations that don't affect the ranking, they'll likely have more problematic interpretations.Davidhttps://www.blogger.com/profile/03104578852711638389noreply@blogger.comtag:blogger.com,1999:blog-35971090.post-19601044714489931702012-11-29T15:28:45.775-05:002012-11-29T15:28:45.775-05:00Other than your article above, I have not read abo...Other than your article above, I have not read about the Colley rankings. But based on your description, it indeed does *not* take into account game-specific information. You note the ranking is determined by Cr = b, where b is determined purely by win-loss record and C is essentially a matrix for who has played who (and does not include any information on game outcomes). <br /><br />So r is entirely determined without using game-specific information.dougjhttps://www.blogger.com/profile/11508572828694551993noreply@blogger.comtag:blogger.com,1999:blog-35971090.post-39274396092165321642012-11-29T14:19:28.814-05:002012-11-29T14:19:28.814-05:00Ed (aka Sam-I-Am-Not :) ),
My second favorite tea...Ed (aka Sam-I-Am-Not :) ),<br /><br />My second favorite team is Oklahoma, so I'm not attacking about of bias here. I think your critique of the Colley Matrix was inaccurate but that you also know far more about the subject than came across in the SI.com article. Plus I love flame wars over statistics.<br /><br />I think Colley's rankings are bad because he willfully ignores margin of victory, but that's as much the BCS's fault as his. Within the constraints of the BCS, you could do worse (Billingsley).Davidhttps://www.blogger.com/profile/03104578852711638389noreply@blogger.comtag:blogger.com,1999:blog-35971090.post-30894040225654498722012-11-29T14:04:52.222-05:002012-11-29T14:04:52.222-05:00David,
Thanks for the updated post. I could have...David,<br /><br />Thanks for the updated post. I could have used a better example. You're absolutely correct about the 3 ring chains. I would have used that in the article, but it was more complicated to explain for an already complicated article.<br /><br />Plus, Stanford, Oregon and Washington were in a 3 ring chain of death, and it makes the Cardinal look bad :)Unknownhttps://www.blogger.com/profile/01709709006511218720noreply@blogger.comtag:blogger.com,1999:blog-35971090.post-77753823456119417482012-11-29T11:22:26.686-05:002012-11-29T11:22:26.686-05:00I didn't claim that the matrix is the Colley M...I didn't claim that the matrix is the Colley Matrix. I state that it can be used as the <i>input</i> to Colley's algorithm. The example matrix I give is equivalent to the table above Eq. 24 in Colley's paper, <i>not</i> Eq. 19. At each stage of the iteration, the fact that two teams have different ratings affects the update. For example, since team <i>a</i> has initial rating .4 and team <i>b</i> has initial rating .5, team <i>e</i>'s win over <i>b</i> counts more than its win over <i>a</i> at the first iteration. By the final iteration, when all teams have different ratings, each individual game outcome affects the ratings differently, as each r_j^i in Eq. (7) is different.<br /><br />I quoted your statement that the Colley Matrix does not use this game-specific information. The game-specific information is encoded in the ratings vector <b>r</b> as indicated by Eq. (7). The inputs to the algorithm are a matrix like the one I showed in my post and an initial rating r_i = 0.5 for each <i>i</i> that assumes no prior knowledge of any team's ability. Thus, your statement in your article is false and requires correction.Davidhttps://www.blogger.com/profile/03104578852711638389noreply@blogger.comtag:blogger.com,1999:blog-35971090.post-45855862002524487462012-11-29T10:48:40.876-05:002012-11-29T10:48:40.876-05:00Hi, I'm the author of the SI article. The mat...Hi, I'm the author of the SI article. The matrix you state is not the Colley matrix. It is not consistent with equation 19 of his paper. Thank you.Unknownhttps://www.blogger.com/profile/01709709006511218720noreply@blogger.com