Abstract
We consider the modeling of individual batting performance in one-day international (ODI) cricket by using a batsman-specific hidden Markov model (HMM). The batsman-specific number of hidden states allows us to account for the heterogeneous dynamics found in batting performance. Parallel sampling is used to choose the optimal number of hidden states. Using the batsman-specific HMM, we then introduce measures of performance to assess individual players via reliability analysis. By classifying states as either up or down, we compute the availability, reliability, failure rate and mean time to failure for each batsman. By choosing an appropriate classification of states, an overall prediction of batting performance of a batsman can be made. The classification of states can also be modified according to the type of game under consideration. One advantage of this batsman-specific HMM is that it does not require the consideration of unforeseen factors. This is important since cricket has gone through several rule changes in recent years that have further induced unforeseen dynamic factors to the game. We showcase the approach using data from 20 different batsmen having different underlying dynamics and representing different countries.
Acknowledgments
The authors have been partially supported by research grants from the Natural Sciences and Engineering Research Council of Canada. The authors thank the Editor, Associate Editor, and the two anonymous reviewers whose comments led to an improvement in the manuscript.
Appendix A: Gibbs Sampling
Gibbs sampling proceeds by repeated application of the following steps:
Given the current value of Θ=θ, we generate a sample path y(N) of the performance states according to (Zucchini and MacDonald 2009):
(16)
where
(17)
and
(18)
Recall that ϕn(j) are the forward step probabilities (2) of the FB algorithm. We simulate the performance states in reverse order by using (17) to generate yN and (18) to generate yn given
Using the simulated hidden performance states y(N), we decompose the observations x(N) into regime contributions by generating r(N)=(r1, …, rN) as described in Section 2.1. Recall that rn=(r1,n, …, rK,n) with
When not-out scores are present, we need to modify the regime decomposition. Under no censoring, and conditional on Xn=xn and Yn=k, the joint distribution of the regimes R1,n, …, Rk,n is multinomial with total xn. Under right-censoring, this still holds, except that xn is a right-censored observation, so that Xn≥xn. In this case, we sample from the distribution of the regimes R1,n,…,RK,n conditional on Xn≥xn. To overcome this, we require an additional sampling step. First, conditional on Xn≥xn, we draw a score
Finally, we note that
(19)
Given the priors and the current values of y(N) and r(N), we update the parameter θ by drawing μ and P according to
(20)
(21)
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