Rolling Markov Chain Monte Carlo

Transcription

1 Rolling Markov Chain Monte Carlo Din-Houn Lau Imperial College London Joint work with Axel Gandy 4 th July 2013

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3 Predict final ranks of the each team. Updates quick update of predictions. Accuracy control of the prediction variance.

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5 Premier League Model: In-Season Each team has a strength - constant over a season. Match: Home Team strength x H vs Away Team strength x A Home Goals: Poisson(λ H exp (x H x A )) Away Goals: Poisson(λ A exp (x A x H )) λ H and λ A - home and away (dis)advantage.

6 Premier League Model: Between-Season Denote X t as the vector of team strengths in season t. Between seasons: X t+1 N(βCX t, σ 2 si). C centering matrix, β contraction / expansion, σ 2 s variance of noise between seasons.

7 Premier League Model: Between-Season Denote X t as the vector of team strengths in season t. Between seasons: X t+1 N(βCX t, σ 2 si). C centering matrix, β contraction / expansion, σ 2 s variance of noise between seasons. Promotion strengths N(µ p, σ 2 pi).

8 X 1 X 2 X 3 Hidden

9 X 1 X 2 X 3... Hidden θ Hidden (λ H, λ A, β, σ s, µ p, σ p )

10 Y 1 Y 2 Y 3,1 Y 3,2 Y 3,3 Observed X 1 X 2 X 3... Hidden θ Hidden (λ H, λ A, β, σ s, µ p, σ p )

11 Y 1 Y 2 Y 3,1 Y 3,2 Y 3,3 Observed X 1 X 2 X 3... Hidden θ Hidden (λ H, λ A, β, σ s, µ p, σ p ) Hidden Markov Model + no closed form for p(θ, x 1:T y 1:T ) + prior = Markov Chain Monte Carlo methods.

12 RMCMC Control Deletion Reweight Sample Database

13 Initialisation RMCMC Control Deletion Reweight Initial Data Sample Database

14 Initialisation RMCMC on GOOD Accuracy Control Indicator BAD RMCMC Control Deletion Reweight Initial Data Sample Database

15 Initialisation RMCMC on GOOD Accuracy Control Indicator BAD RMCMC Control Deletion Reweight Initial Data Sample Database

16 Initialisation RMCMC paused GOOD Accuracy Control Indicator BAD RMCMC Control Deletion Reweight Initial Data Sample Database

17 Initialisation RMCMC paused GOOD Accuracy Control Indicator BAD RMCMC Control Deletion Reweight Initial Data Sample Database

18 New Data Observed Control Indicator Accuracy GOOD BAD RMCMC Control Deletion Reweight New Data Sample Database

19 New Data Observed Reweight on GOOD Accuracy Control Indicator BAD GOOD Quality BAD RMCMC Control Deletion Reweight New Data Sample Database

20 New Data Observed Decrease Database Size GOOD Accuracy Control Indicator BAD GOOD Quality BAD RMCMC Control Deletion Reweight New Data Sample Database

21 New Data Observed Un-pause RMCMC GOOD Accuracy Control Indicator BAD GOOD Quality BAD RMCMC Control Deletion Reweight New Data Sample Database

22 GOOD increase database size Quality pause RMCMC RMCMC on BAD decrease database size GOOD Accuracy BAD

23 Simulation Initial Data: results from 2005/06 to 2009/10 seasons. Batches: reveal data in weekly batches for the 2010/11 and 2011/12 seasons. Typically 10 results per batch, although between 3 and 20 results. Results

24 Conclusion Summary: Use this to control the accuracy of the predictions as new data are revealed. Efficient RMCMC turns on when necessary. Deletes samples when possible. Works in a high dimensional state space. Particle MCMC? (Andrieu et al., 2010) SMC 2? (Chopin et al., 2013) : Big jumps between season strengths causes problems for particle filters.

25 References Football Website: fdl06/premier_league/main.html Andrieu, C., Doucet, A., and Holenstein, R. (2010). Particle markov chain monte carlo methods. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 72(3): Chopin, N., Jacob, P. E., and Papaspiliopoulos, O. (2013). Smc2: an efficient algorithm for sequential analysis of state space models. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 75(3): Lau, F. D.-H. and Gandy, A. (2013). Controlled Estimation Accuracy with Rolling Markov Chain Monte Carlo. ArXiv e-prints.