While the official Association of Tennis Professionals (ATP) computer tennis rankings are used to seed players in tournaments, they are not used to predict a player's chance of winning. However, since the rankings are derived from a points rating, an estimate of each player's chance in a head to head contest can be made from the difference in the players' rating points. Using a year's tournament results, a logistic regression model was fitted to the ATP ratings, to estimate the chance of winning as a function of the difference in rating points. Once the draw for a tournament is available, the resultant probabilities can be used in a simulation to estimate each player's chance of victory. The method was applied to the 1998 Men's Wimbledon, 1998 Men's US Open and the 1999 Men's Australian tennis championships.
In this paper we consider the operation of the move-to-front scheme where the requests form a Markov chain of N states with transition probability matrix P. It is shown that the configurations of items at successive requests form a Markov chain, and its transition probability matrix has eigenvalues that are the eigenvalues of all the principal submatrices of P except those of order N—1. We also show that the multiplicity of the eigenvalues of submatrices of order m is the number of derangements of N — m objects. The last result is shown to be true even if P is not a stochastic matrix.
In this paper we consider the operation of the move-to-front scheme where the requests form a Markov chain of N states with transition probability matrix P
. It is shown that the configurations of items at successive requests form a Markov chain, and its transition probability matrix has eigenvalues that are the eigenvalues of all the principal submatrices of P except those of order N—1. We also show that the multiplicity of the eigenvalues of submatrices of order m is the number of derangements of N — m objects. The last result is shown to be true even if P is not a stochastic matrix.
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