Consistency and Pythagoras
Pythagorean win share has been one of the fundamental contributions to Sabermetrics. Several hundred articles, both academic and non-academic, have explored variations on Bill James' original formula and its fit to empirical data. This paper considers a variation that is previously unexplored on any systematic level, consistency. After discussing several important contributions to the line of literature, we demonstrate the strong correlation between Pythagorean residuals and several notions of run distribution consistency. Finally, we select the "correct" form of consistency and use it to construct a simple regression estimator, which improves RMSE by 11%.
A Derivation of the Pythagorean Won-Loss Formula in Baseball
It has been noted that in many professional sports leagues a good predictor of a team’s end of season won-loss percentage is Bill James’ Pythagorean Formula RSobs γ RSobs +RAobs γ , where RSobs (resp. RAobs) is the observed average number of runs scored (allowed) per game and γ is a constant for the league; for baseball the best agreement is when γ is about 1.82. This formula is often used in the middle of a season to determine if a team is performing above or below expectations, and estimate their future standings. We provide a theoretical justification for this formula and value of γ by modeling the number of runs scored and allowed in baseball games as independent random variables drawn from Weibull distributions with the same β and γ but different α; the probability density is f(x; α, β, γ) = ( γ α ((x− β)/α)γ−1 e−((x−β)/α)γ if x ≥ β 0 otherwise. This model leads to a predicted won-loss percentage of (RS−β)γ (RS−β)γ+(RA−β)γ ; here RS (resp. RA) is the mean of the Weibull random variable corresponding to runs scored (allowed), and RS−β (resp. RA−β) is an estimator of RSobs (resp. RAobs). An analysis of the 14 American League teams from the 2004 baseball season shows that (1) given that the runs scored and allowed in a game cannot be equal, the runs scored and allowed are statistically independent; (2) the best fit Weibull parameters attained from a least squares analysis and the method of maximum likelihood give good fits. Specifically, least squares yields a mean value of γ of 1.79 (with a standard deviation of .09) and maximum likelihood yields a mean value of γ of 1.74 (with a standard deviation of .06), which agree beautifully with the observed best value of 1.82 attained by fitting RSobs γ RSobs +RAobs γ to the observed winning percentages.
Tradeoffs and limitations in statistically based image reconstruction problems.
TRADEOFFS AND LIMITATIONS IN STATISTICALLY BASED IMAGE RECONSTRUCTION PROBLEMS