The Player Kernel Lucas Maystre , Victor Kristof, Antonio Gonzlez - - PowerPoint PPT Presentation

The Player Kernel

Lucas Maystre, Victor Kristof, Antonio González Ferrer, Matthias Grossglauser School of Computer and Communication Sciences, EPFL MLSA workshop @ ECML-PKDD — September 19th, 2016

Context

Our entry to the EURO 2016 Prediction Competition, Challenge 1 Task: probabilistic prediction of match outcomes

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predictor ⇥0.37 0.31 0.32⇤ Italy Belgium "Italy wins" "Belgium wins" "It's a draw"

Starting point

Key challenges with national teams:

• 1. They play few matches every year: recent data is sparse
• 2. Their squad change frequently: old data is stale

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P(u v) = 1 1 + exp[(su sv)] M national teams. Team u has "strength" su

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= 1 1 + exp(−s>x)    s1 . . . sM                 . . . 1 . . . −1 . . .             

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Source: http://www.estadao.com.br/ infograficos/onde-atuam-os-736- jogadores-da-copa-2010,esportes,280906

Inspiration


Many players play against each other in club competitions Can we transfer information from club matches to international matches?

Club matches and international matches share the same parameters. Embed teams in the space of players.

Main idea

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su = X

i∈Lu

˜ si P(u v) = 1 1 + exp[(su sv)] The number of parameters explodes. Seems like it will lead to statistical and computational issues. = 1 1 + exp(−˜ s>z)       ˜ s1 . . . . . . ˜ sP                            . . . 1 1 1 . . . −1 −1 −1 . . .                     

Bayesian approach

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prior distribution

(e.g. Gaussian)

likelihood posterior distribution

p(˜ s | D) ∝ p(D | ˜ s) × p(˜ s)

Statistical issues solved? not clear Computational issues solved? no Keep a distribution over parameters instead of optimizing the likelihood.

In fine, we are only interested in

Dual viewpoint

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p(˜ s>z | D) "strength" difference Accurate estimation of all parameters is not necessary Inference can be done in the dual space f(z) = ˜ s>z p(f | D) ∝ p(D | f) × p(f) Cov[f(z), f(z0)] = σ2z>z0 f(z) ∼ GP[0, k(z, z0)] The player kernel!

The cube

Linear Regression Logistic Regression Kernel Regression Kernel Classification Bayesian Linear Regression Bayesian Logistic Regression GP Regression GP Classification

Kernel Bayesian Classification

Credit: Zoubin Ghahramani

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Dataset


Collected data on 24 887 matches from main football competitions over the last 10 years. 33 157 distinct players appear in the dataset.

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Results

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Logarithmic loss against competing approaches in 2008, 2012 and 2016. 2008 2012 2016

Rao and Kupper (1967) proposed the following extension.

Ternary outcomes

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A draw is (essentially) equivalent to one win and one loss. P(u v) = 1 1 + exp[(su sv α)] P(u ⌘ v) = 1 P(u v) P(v u) / P(u v) · P(v u)