Abstract
The famous Bradley-Terry model for pairwise comparisons is widely used for ranking objects and is often applied to sports data. In this paper we extend the Bradley-Terry model by allowing time-varying latent strengths of compared objects. The time component is modelled with barycentric rational interpolation and Gaussian processes. We also allow for the inclusion of additional information in the form of outcome probabilities. Our models are evaluated and compared on toy data set and real sports data from ATP tennis matches and NBA games. We demonstrated that using Gaussian processes is advantageous compared to barycentric rational interpolation as they are more flexible to model discontinuities and are less sensitive to initial parameters settings. However, all investigated models proved to be robust to over-fitting and perform well with situations of volatile and of constant latent strengths. When using barycentric rational interpolation it has turned out that applying Bayesian approach gives better results than by using MLE. Performance of the models is further improved by incorporating the outcome probabilities.
Highlights
Modelling pairwise comparisons is an important practical problem and well established in research literature [1, 2]
In this paper we extend the Bradley-Terry model to allow for time-varying strengths by combining it with barycentric rational interpolants (BRI) [23] or Gaussian processes (GP) [24]
In this paper we extended the Bradley-Terry model using barycentric rational interpolation (BRI) and GPs to model latent strengths as the time-varying components of the model
Summary
Modelling pairwise comparisons is an important practical problem and well established in research literature [1, 2]. The classical approach is the Bradley-Terry model [3]. The Bradley-Terry model has been extended in several ways: handling ties [6], ranking individual players in multi-player competitions [7, 8], and stochastic non-transitivity of comparisons [9]. A very recent example of such treatment demonstrates a pairwise comparison model where the Weibull distribution is applied [10]. Another common generalization is to allow for the latent strengths to vary with time and it is the focus of our work. The quintessential application domain for time-varying strength models is sports, where ranking is important both for seeding competitions and for fan engagement.
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