Abstract
We compare various extensions of the Bradley–Terry model and a hierarchical Poisson log-linear model in terms of their performance in predicting the outcome of soccer matches (win, draw, or loss). The parameters of the Bradley–Terry extensions are estimated by maximizing the log-likelihood, or an appropriately penalized version of it, while the posterior densities of the parameters of the hierarchical Poisson log-linear model are approximated using integrated nested Laplace approximations. The prediction performance of the various modeling approaches is assessed using a novel, context-specific framework for temporal validation that is found to deliver accurate estimates of the test error. The direct modeling of outcomes via the various Bradley–Terry extensions and the modeling of match scores using the hierarchical Poisson log-linear model demonstrate similar behavior in terms of predictive performance.
Highlights
The current paper stems from our participation in the 2017 Machine Learning Journal (Springer) challenge on predicting outcomes of soccer matches from a range of leagues around the world (MLS challenge, in short)
The suffix (t) indicates features with coefficients varying with matches played The model indicated by † is the one we used to compute the probabilities for the submission to the MLS challenge The acronyms are as follows: BL, Baseline; CS, Bradley–Terry with constant strengths; LF, Bradley–Terry with linear features; TVC, Bradley–Terry with time-varying coefficients; AFD, Bradley–Terry with additive feature differences and time interactions; hierarchical Poisson log-linear model (HPL), Hierarchical Poisson log-linear model
The sets of features that were used in the LF, TVC, AFD and HPL specifications in Table 4 resulted from ad-hoc experimentation with different combinations of features in the LF specification
Summary
The current paper stems from our participation in the 2017 Machine Learning Journal (Springer) challenge on predicting outcomes of soccer matches from a range of leagues around the world (MLS challenge, in short). The performance of the various modeling approaches in predicting the outcomes of matches is assessed using a novel, context-specific framework for temporal validation that is found to deliver accurate estimates of the prediction error. The direct modeling of the outcomes using the various Bradley–Terry extensions and the modeling of match scores using the hierarchical Poisson log-linear model deliver similar performance in terms of predicting the outcome.
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