An introduction to football modelling at Smartodds Oxford ...€¦ · to football modelling at...

Anintroductionto footballmodelling atSmartodds

RobertJohnson

An introduction to football modelling atSmartodds

Oxford SIAM Conference 2011

Robert Johnson

Smartodds Ltd

February 9, 2011


RobertJohnson

Introduction

Introduction to Smartodds

Practical example: building a football model


RobertJohnson

What is Smartodds about?

Smartodds provides statistical research andsports modelling in the betting sector

Quant team research and implement the sportsmodels

Primary focus is on Football, however we alsomodel Basketball, Baseball, American Football,Ice Hockey and Tennis

Wide range of interesting problems to work on

Actively recruiting!


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Building a Football model

Suppose we decide to build a football model forthe English football leagues

Here we model the divisions Premier League,Championship, League 1 and League 2

There are 92 teams in total to model

We want to predict the probability of team Awinning against team B where team A and teamB could be from any of the 4 leagues


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Literature review

Maher (1982) assumed independent Poissondistributions for home and away goals

Means based on each teams’ past performance

Dixon and Coles (1997) took this idea further byaccounting for fluctuations in performance ofindividual teams and estimation between leagues

Dixon and Robinson (1998) modelled the scoresduring a game as a two-dimensional birth process


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Model formulation

Assume that home and away goals follow aPoisson distribution

Pr(x goals) =λxe−λ

x!

Pr(y goals) =µye−µ

y !

To estimate the probabilities of x and y goals weneed λ and µ


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Model 1: Mean goals

Assume that home and away teams are expectedto score the same number of goals

Take average goals scored in a game in Englandas 2.56 and divide by two

λ = 1.28

µ = 1.28

However we may believe that there is someadvantage associated with playing at home


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Model 2: Home Advantage

Include a term to take account of homeadvantage

λ = γ × τ

µ = γ

γ is the common mean and τ represents thehome advantage


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Model 2: Home Advantage (Cont)

Mean goals scored by the away team in the fourleagues we model English Leagues is 1.10 giving

γ = 1.10

This implies mean goals scored by the hometeam are 2.56− 1.10 = 1.46Using the above we can estimate τ as

τ = 1.46/1.10 = 1.33


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Model 3: Team Strengths

Previous attempts assumed all teams of equalstrength

Can add team strength parameters for each team

Better teams score more goals. Give each teaman attack parameter denoted α

Better teams concede fewer goals. Give eachteam a defence parameter denoted β


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Model 3: Team Strengths (Cont)

Write λ and µ in terms of the attack anddefence parameters of the home and away teams,which we denote by i and j , giving

λ = γ × τ × αi × βj

µ = γ × αj × βi

The model is overparameterised, so we apply theconstraints

1

n

n∑i=1

αi = 1,1

n

n∑i=1

βi = 1.


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Model 3: Pseudolikelihood

The pseudolikelihood for this model is:

L(γ, τ, αi , βi ; i = 1, . . . , n) =∏k

{exp(−λk)λxkk exp(−µk)µykk }

φ(t−tk )

φ(·) is an exponential downweighting function,which allows us to place less weight on oldergames

Other downweighting functions could be used


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Estimation techniques

Obtaining the parameter estimates is notstraightforward

In this example we have 186 parameters toestimate

Various optimisation techniques could be used toobtain parameter estimates (numericalmaximisation of the likelihood function, MCMC)

High dimensional problems may also requiremore sophisticated computing solutions (MPI)


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Parameter estimates

These are Smartodds’ current estimates of theattack and defence parameters of the top 6teams in the Premier League

Team Attack Parameter Defence Parameter

Chelsea 3.15 0.34Man Utd 3.08 0.35Arsenal 2.84 0.37

Man City 2.44 0.42Tottenham 2.22 0.44Liverpool 2.12 0.39


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Predicting outcomes

Suppose Man Utd are playing at home to ManCity

Using the parameter estimates we get

λ = 1.10× 1.33× 3.08× 0.42 = 1.89

µ = 1.10× 2.44× 0.35 = 0.94

We can use λ and µ to obtain the probability ofMan Utd winning the match


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Predicting outcomes (Cont)

The probability of a specific score is given asfollows

Pr(x , y) =λxe−λ

x!

µye−µ

y !

So the probability of the score, Man Utd 2 ManCity 1, is

Pr(2, 1) =1.892e−1.89

2!

0.941e−0.94

1!= 0.099


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Obtain the probability matrix of all possiblescores

0 1 2 3 4 . . .

0 0.059 0.112 0.105 0.066 0.031 . . .1 0.055 0.105 0.099 0.062 0.029 . . .2 0.026 0.049 0.047 0.029 0.014 . . .3 0.008 0.015 0.015 0.009 0.004 . . .4 0.002 0.004 0.003 0.002 0.001 . . ....

......

......

.... . .


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Sum over all events where home goals aregreater than away goals

0 1 2 3 4 . . .

0 0.059 0.112 0.105 0.066 0.031 . . .1 0.055 0.105 0.099 0.062 0.029 . . .2 0.026 0.049 0.047 0.029 0.014 . . .3 0.008 0.015 0.015 0.009 0.004 . . .4 0.002 0.004 0.003 0.002 0.001 . . ....

......

......

.... . .


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Giving the probability that Man Utd win at hometo Man City as 59.6%

0 1 2 3 4 . . .

0 0.059 0.112 0.105 0.066 0.031 . . .1 0.055 0.105 0.099 0.062 0.029 . . .2 0.026 0.049 0.047 0.029 0.014 . . .3 0.008 0.015 0.015 0.009 0.004 . . .4 0.002 0.004 0.003 0.002 0.001 . . ....

......

......

.... . .


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Practical issues

Betfair’s odds imply Man Utd has a 63% chanceof winning the game, potentially leaving value fora bet on Man City. However, should we bet?

These models take into account no externalinformation about match circumstances

InjuriesMotivationFatigueNewly signed players

So betting off a mathematical model would bedangerous!


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Shortcomings of the model

If we compare the expected full-time scoresunder the model with the observed scores, wefind our modelling assumptions don’t hold

Goals don’t have a Poisson distributionGoals scored by the home and away teams aren’tindependent

Dixon and Coles corrected for this by modifyingthe predicted distribution to increase probabilityof draws and 0-1 and 1-0 scores

However this isn’t entirely satisfactory — wouldbe better to model what is happening directly


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Goal time distribution

0.00

0.01

0.02

0.03

0.04

0.05

Goal time (mins)

Den

sity

0 10 20 30 40 50 60 70 80 90

Goals in injury time at the end of each half arerecorded as 45 / 90 min goals

Goal rate steadily increases over the course ofthe game

Notice the spikes every 5 minutes in the secondhalf - due to rounding?


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Dixon and Robinsons’ model

If we assume that the goal scoring processes forthe home and away teams are independenthomogeneous Poisson processes then our modelreduces to the full time model discussedpreviously.

For match k between teams i and j

λk(t) = λk = γ × τ × αi × βj

µk(t) = µk = γ × αj × βi


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Dixon and Robinsons’ model (continued)

Three changes:

1 Goal-scoring rate dependent on the current score

2 Modelling of injury time

3 Increasing goal-scoring intensity through thegame (due to tiredness of players)


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(1) Goal-scoring rate dependent on current score

Assume that home and away scoring processesare independent Poisson processes

Scoring rates are piecewise constant

Home and away intensities are constant until agoal is scored and only change at these times

Denote λxy and µxy as parameters determiningthe scoring rates when the score is (x ,y)

Scoring rates are now

λk(t) = λxyλk

andµk(t) = µxyµk


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Estimates of λ(x , y) and µ(x , y)

λ̂(0, 0) = 1µ̂(0, 0) = 1

λ̂(1, 0) = 0.88µ̂(1, 0) = 1.35

λ̂(0, 1) = 1.10µ̂(0, 1) = 1.07


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(2) Increase the scoring rate during injury time

Goals scored during injury time are recorded ashaving occurred at either 45 or 90 minutes.

Define two new parameters ρ1 and ρ2 to modelinjury time.

The adjusted scoring rates are

λk(t) =

ρ1λxyλk t ∈ (44, 45]mins,ρ2λxyλk t ∈ (89, 90]mins,λxyλk otherwise

and similarly for µk(t)


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(3) Increasing goal-scoring intensity

Allow the scoring intensities to increase over time

Model scoring rates as time inhomogeneousPoisson processes with a linear rate of increase

Replace λk(t) and µk(t) with

λ∗k(t) = λk(t) + ξ1t,

µ∗k(t) = µk(t) + ξ2t

ξ1 and ξ2 could be constrained to be positive toensure that the hazard functions above areconstrained to always be positive, but in practicethis is not neccessary

Scoring rates are estimated to be about 75%higher at the end of the game then at the startof the game.


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Model usage

This ‘in-running’ model can be useful in its ownright (for deriving in-running prices)

Also explains the home/away dependencies andnon-Poisson pdfs observed in the data


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Summary

The Dixon-Coles model is a simple and robustfull-time score model, but not all of itsassumptions are met

A continuous time model such as theDixon-Robinson model can model dependenciesbetween home and away scoring rates

Mathematical models cannot model team news(unless this is incorporated into the modelsomehow)

These models can be extended to other sports bychanging the distributions, eg

Normal distribution for American FootballNegative binomial for baseball


RobertJohnson

References

M.J. Maher, 1982, Modelling association footballscores, Statist. Neerland., 36, 109-1188

M. Dixon and S.G. Coles, 1997. ModellingAssociation Football Scores and Inefficiencies inthe Football Betting Market. Applied Statistics,46(2), 265-280

M. Dixon and M. Robinson, 1998. A birthprocess model for association football matches.JRSS D, 47(3), 523-538


RobertJohnson

Interested?

If you are interested in sports modelling andpossess the following skills:

Post graduate qualification (at least MMath /MSc, PhD. preferred) in mathematics, statisticsor another subject with considerablemathematical contentExperience in developing and implementingmathematical / statistical modelsExperience of computer programming(preferably in C++, C, R or Python)Enthusiasm, self-motivation and the ability towork under pressure to strict deadlines

Then email us at [email protected]

For more information see our website:http://www.smartodds.co.uk

[email protected]://www.smartodds.co.uk

HomeIntroductionWhat is SmartOdds about?Building a Football modelLiterature review

Model formulationModel 1: Mean goalsModel 2: Home AdvantageModel 3: Team StrengthsModel 3: PsedolikelihoodEstimation techniquesParameter estimatesPredicting outcomesPractical issuesShortcomings of the modelGoal time distribution

Dixon and Robinson's model(1) Goal-scoring rate dependent on current scoreEstimates of λ(x,y) and μ(x,y)

(2) Increase the scoring rate during injury time(3) Increasing goal-scoring intensityModel usage

SummaryReferencesInterested?

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