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Обсудить все можно У-Е-Нота Это часть дипломной работы Кеннета Массея. Поскольку он владеет английским немного лучше чем я, то я и воспользовался этим текстом чтобы донести до англоязычных, что же собственно я имел ввиду. Главное, что показал Кеннет - это связь е-рейтинга с Марковскими цепями. Я, честно говоря об этом даже и не думал. Однако теперь, когда установлена такая связь, между моими соображениями исходя из "здравого смысла" и отлаженным аппаратом цепей Маркова, конечно чувствуешь себя более уверенно в среде математиков. Хотя, честно говоря, я не чувствовал неуверенности и ранее. Уж больно проста и естественна логика е-рейтинга. Думаю, что если бы было обнаружена некая нестыковка с Марковскими цепями, то цепи бы просто разорвались. The E Rating MethodEugene Potemkin Introduction (Kenneth Massey)The E- rating model was developed by Dr. Eugene Potemkin, who first applied the techniques to international sporting events such as the Olympics, World Cup Soccer, and the Chess Olympiad. A resident of Moscow, Dr. Potemkin has published his results in the Russian press since the early 1980's under the moniker “Elecs-,” short for “Electronic System.” Recent access to the internet has allowed him to introduce the E- method to American sports. The following descriptions have been adapted from my email correspondence with Dr. Potemkin in relation to our joint effort, the “World Wide Ratings and Rankings” web site. Despite its dependence on mathematical formulas, the E- method was not originally intended to generate ratings that satisfy some strict mathematical criteria or conform to standard statistical procedures. Instead, Dr. Potemkin’s motivation for developing this particular rating system was more of a pragmatic nature, based on intuition and a general knowledge of sports competition. However, with the proper interpretation it can be shown, that the E- model is an application of continuous time Markov chains, a class of probability models. This chapter will focus primarily on Dr. Potemkin’s derivation and its practical connection with economic theory; the equivalence to Markov chains is briefly described at the end of this chapter. The E- ModelLike maximum likelihood ratings, the E- system is designed to model probabilities. In particular, the ratio of two teams’ ratings should approximate, the ratio of games won by A to those won by B in a conceptually infinite series of contests between the two teams. Ties may be assumed to contribute half a win to each team. A pair of “Binary Ratings,” brij and brji, can be chosen to represent the win ratio in games between team i and any one of its opponents j. Notice that binary ratings are not fixed numbers because they represent only a ratio; however the relationship between two binary ratings can be established. We set brijwji = brjiwij brijwji + brijwij = brjiwij + brijwij brijgij = wij(brij + brji) (6.1) where gij = wij + wji equals the total number of games played between i and j. This implies that the strength attributed to a team via a binary rating is directly proportional to the percentage of games it won, and also to the total binary rating for both teams. So far, the binary ratings have no relation to actual team ratings; they are only relevant when describing the relationship between an isolated pair of teams. Each team will have as many binary ratings as it has opponents. The goal will be to calculate a vector of general ratings r from which to estimate binary ratings for teams that have never met. A logical choice for a team’s general rating is the average of its known binary ratings. Therefore we define the rating for each team i, i=1..n, to be where gi is the total number of games played by team i. Substituting equation 6.1 gives us For a particular contest, it is generally impossible to determine whether variation in the outcome should be explained by one team’s above average performance or the other team’s below average performance. To quote Dr. Potemkin, “Each team plays the way her opponent lets it play.” Consequently, an assumption made by the E- model that is not likely to be unrealistic is that ri - brij = -(rj - brji) ri + rj = brij + brji (6.2) This simply states that the binary ratings for a single game will total the sum of the participant’s overall ratings. Although this relationship is not entirely correct mathematically because of the nonlinear nature of ratios, it serves as a satisfactory estimate for our purposes. Example 6.1Suppose A’s rating equals 5 and B’s rating equals 2. Then A is expected to win 5/2 games for every 1 game won by B. Now assume that A defeated B in three of the five games played between the two teams. The appropriate values would be brab = 4.2 and brba = 2.8. Notice that brab + brba = 7 and brab / brba equals the correct win ratio, 3 / 2. # Based on equation 6.2, the general E- model can be formulated as This results in a set of n linear equations of the form giri - wiri = 3i.jwijrj liri = 3i.jwijrj (6.3) where wi and li equal the number of wins and losses respectively by team i. Exactly n-1 of these equations will be independent because li = 3j.iwji for all i. This should be expected since for any real number c, cr yields ratios equivalent to those obtained from r. Therefore an arbitrary condition can be imposed on the solution. A logical choice would be to set 3ri = c for some positive number c. Another possibility could be to make the ratings relative to one particular team k by setting rk = 1. The E- ratings exhibit certain characteristics based on the model definition. First we notice that each ri will be nonnegative. Zero ratings are possible, occurring if a team has no wins against any of its opponents. In the opposite case, if a team is undefeated, then the model breaks down. Depending on the conditions set by the nth equation, two results are possible. Either ratings for teams that did lose will become zero, or those that didn’t will be infinite. This is a significant weakness of the E- method and ratio models in general. Consequently, measures should be taken to insure that no li equals zero. Replacing wins with points will usually solve the problem. An alternate proposal would be to weight the results with an adjustable parameter x, 0#x#1. A single win would then be treated by the model as x wins and (1-x) losses. As with least squares ratings, it is assumed that enough games have been played to produce a saturated system in which each team has some connection with every other team in the league. The final system of linear equations in the E- model can be solved with the techniques discussed in chapter three. A solution can usually be obtained more efficiently than for nonlinear ratio models. This advantage may offset the sacrifice of mathematical legitimacy caused by the linear approximation of an inherently nonlinear design. Example 6.2Consider the league referred to in example 4.2 and assume the following results: Beast Squares defeat Gaussian Eliminators 10-6 Likelihood Loggers tie Linear Aggressors 4-4 Linear Aggressors defeat Gaussian Eliminators 9-2 Beast Squares defeat Linear Aggressors 8-6 Gaussian Eliminators defeat Likelihood Loggers 3-2 Replacing wins with points and constructing the system of equations defined by equation 6.3 yields This system is obviously singular, so we introduce the restraint that 3ri = 4. Solving for the unique solution gives r = (1.316, 0.614, 0.864, 1.206)T. These ratings can be used to estimate point ratios for any pair of teams, and hence the percentage scored by one team. For instance, equation 6.1 implies that the Beast Squares should score 1.316 / (1.316 + 0.864) = 0.604 = 60.4% of the total points in a game against the Likelihood Loggers. However, notice that the ratings alone do not distinguish between a 3-2 game and a 24-16 result. # Anti-RatingsGeneral E- ratings depend on two factors: the actual number of losses a team had, and the number of losses that would be acceptable given the strengths of the opponents it defeated. Only indirect consideration is given to the quality of the opponents that a team loses to. Therefore we can reasonably conclude that an E- rating measures a team’s relative strength versus its absolute weakness. This suggests the development of a similar rating method to evaluate the opposite, a team’s relative weakness versus its absolute strength. In contrast to the original, the anti-rating model will give higher ratings to poor teams because they are “better” at losing. This is accomplished by following the basic model, with wins and losses transposed. Letting s be the vector of anti-ratings, gisi - lisi =Sum i.j (lijsj) wisi = Sum i.jlijsj (6.4) Example 6.3Applied to the previous example, the anti-rating model produces this system of equations: After applying the restraint and solving, we find that s = (0.706, 1.424, 1.175, 0.695)T. Using these results we expect the Beast Squares to score 1 - 0.706 / (0.706 + 1.175) = 0.625 = 62.5% of the total points in a contest with the Likelihood Loggers. # Based on their derivations, we would expect si to be approximately equal to 1 / ri. Indeed this relationship is nearly satisfied, especially as the number of game observations increases. However the previous examples illustrate that an exact match is not necessarily guaranteed. In particular, notice the slight difference in 60.4% and 62.5% that are the listed predictions of r and s respectively. In fact, even the rankings disagree; r implies that the Beast Squares are superior to the Linear Aggressors while the anti-rating vector s concludes otherwise. An obvious question is how to join the two sets of ratings, which essentially measure two distinct but related aspects of sports performance. Wishing to maintain the ability to estimate ratios, we seek a combined set of ratings t such that the expected ratio of wins between A and B is equal to . Furthermore each ti should be an appropriate function of ri and si. The following approximations are derived from r and s.Hence we can establish the relationship Setting ti = (ri / si). the following condition is met. Example 6.4Combining the results of examples 6.2 and 6.3 gives the following table: Team Rating (r) Anti-Rating (s) Overall (t) Beast Squares 1.316 0.706 1.365 Gaussian Eliminators 0.614 1.424 0.657 Likelihood Loggers 0.864 1.175 0.858 Linear Aggressors 1.206 0.695 1.317 The Beast Squares should therefore win 1.365 / (1.365 + 0.858) = 0.614 = 61.4% of games against the Likelihood Loggers, a proper compromise between the two prior estimates. # Economics ApproachThe E- model is quite similar to the economic principles of supply and demand. We can consider our league as a marketplace in which teams exchange products, which correspond to games. If a team wins then it in effect purchases a game from the loser, who receives compensation equal to the price for that game. The primary goal is to calculate equilibrium prices for each team’s games. Corresponding to the basic E- model is a sellers’ market, meaning that each team sets the price for its own games. Ratings will coincide with prices, so the total expenses for a particular team i is expressed as 3j.iwijrj. Equilibrium will occur when each team’s income equals its expenses so liri = 3j.iwijrj, which is identical to the system defined by equation 6.3. From this model we can conclude that a team creates demand for its games by winning, especially against “wealthy” opponents. Price is proportional to the ratio of demand and supply, so from the league’s perspective a total of 3j.iwijrj losses are demanded from team i; however only li are supplied. Consequently as supply decreases or demand increases, a win against team i becomes more valuable and expensive to purchase. The anti-rating model parallels a buyers’ market in which the purchaser of a game determines how much it is willing or able to pay. Total income is then dependent on what the customers can afford, expressed as 3j.ilijsj. Accordingly, equilibrium will occur when wisi = 3j.ilijsj, which matches equation 6.4, is satisfied for each team. From team i’s perspective, it desired a total of 3j.ilijsj wins and was able to purchase wi. Therefore, as supply increases or demand decreases, each of i’s wins becomes worth less. This indicates that i is a strong team with an abundance of “wealth” such that success is taken for granted. Markov Chain EquivalenceIt was mentioned earlier that the E- rating system is actually an application of continuous time Markov chains. To see this, imagine that each team represents a possible state of the system, which corresponds to the league as a whole. Furthermore, if the system is in state i, then this indicates that team i recorded the most recent success, in terms of either winning a game or scoring a point. Now if some opponent j comes along and defeats i, then the system transfers to state j. This process continues indefinitely as teams fight for control of the system. Eventually equilibrium will be reached, and it will be possible to state the exact probability that the system will be in a particular state at any instant in time. With the proper model definition, these probabilities are proportional to the E- ratings. Without going into a detailed description of Markov chains, we will assume that transitions from team to team are instantaneous. In addition, the rate at which transitions occur from team i to team j is equal to the number of losses i had to j. Letting Q denote the transition rate matrix, we see that qij = lij = wji. It can be shown that under the assumptions associated with continuous time Markov chains, qii = -3j.i qij (Stewart 1994). This is essentially a measure of how often a team is forced to give up control of the system. As expected, this occurs whenever that team loses. Therefore qii = -li. Consider the example from example 6.2. The resulting transition rate matrix is Equilibrium probablities p are a solution to pQ = 0, where 3 pi = 1. You can verify that p = (0.329, 0.154, 0.216, 0.301) satisfies these conditions. Also notice that p = 0.25r where r is the set of E- ratings. Therefore except for a constant factor, the E- rating method is equivalent to the application of a continuous time Markov chain. Of course, the reversal of wins and losses provides a similar correspondence to anti-ratings. BibliographyArnold, Jesse C., and J.S. Milton. Introduction to Probability and Statistics: Principles and Applications for Engineering and the Computing Sciences. Third Edition. New York: McGraw-Hill, 1995. Bassett, Gilbert W. "Predicting the Final Score." Unpublished Manuscript. Bassett, Gilbert W. "Robust Sports Ratings Based on Least Absolute Errors." The American Statistician. May 1997:1-7. Burden, Richard L., and J. Douglas Faires. Numerical analysis. Fifth Edition. Boston: PWS Publishing Company, 1993. Cormen, Thomas H., Charles E. Leiserson, and Ronald L. Rivest. Introduction to Algorithms. Cambridge: MIT Press, 1992. Degroot, Morris H. Probability and Statistics. Second Edition. Reading, MA: Addison-Wesley Publishing Company, 1989. Glickman, Mark, and Hal Stern. "A State-Space Model for National Football League Scores." Unpublished Manuscript, 1996. Harris, Chance. "More WBW Babble." Personal e-mail (Sep. 24, 1996). Harville, David, and Michael H. Smith. "The Home-Court Advantage: How Large Is It, and Does It Vary From Team to Team?" The American Statistician. February 1994: 22-28. Harville, David. "Predictions for National Football League Games Via Linear-Model Methodology." Journal of the American Statistical Association. September 1980: 516- 524. Harville, David. "The Use of Linear-Model Methodology to Rate High School or College Football Teams." Journal of the American Statistical Association. June 1977: 278-289. Leake, Jeffrey R. "A Method for Ranking Teams: With an Application to College Football." Management Science in Sports. Ed. Robert Machol, et.al. New York: North-Holland Publishing Company, 1976. 27-46. Myers, Raymond H. and J.S. Milton. A First Course in the Theory of Linear Statistical Models. Boston: PWS-Kent/Duxbury Press, 1992. Potemkin, Eugene. "Eugene Potemkin's E- Rating." http://www.digiserve.com/wwrr/describe/E-1.htm (Aug. 1996). Potemkin, Eugene. "Eugene Potemkin's E-rating Description." http://www.digiserve.com/wwrr/describe/erating.htm (Nov. 1996). Rabenstein, Albert L. Elementary Differential Equations with Linear Algebra. Fourth Edition. New York: Harcourt Brace Jovanovich, 1992. Stern, Hal. "On the Probability of Winning a Football Game." The American Statistician. August 1991:179-183. Stern, Hal. "Who's Number 1 in College Football?...And How Might We Decide?" Chance. Summer, 1995: 7-14. Stewart, William J. Introduction to the Numerical Solution of Markov Chains. Princeton, NJ: Princeton University Press, 1994. Stefani, Raymond T. "Football and Basketball Predictions Using Least Squares.” IEEE Transactions on Systems, Man, and Cybernetics. February 1977: 117-121. Stefani, Raymond T. "Improved Least Squares Football, Basketball, and Soccer Predictions." IEEE Transactions on Systems, Man, and Cybernetics. February 1980: 116-123. Trotter, Hale F. and Richard E. Williamson. Multivariable Mathematics. Third Edition. Upper Saddle River, NJ: Prentice Hall, 1996. Woolner, Keith. "Baseball." Personal e-mail (June. 3, 1996). Zenor, Michael. "Zenor's College Football Power Ratings." http://www.cae.wisc.edu/`dwilson/ rsfc/rate/zenor.html (Dec. 1995). |
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