Predicting Winners of Competitive Events with Topological Data Analysis
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1 Predicting Winners of Competitive Events with Topological Data Analysis Conrad D Souza Ruben Sanchez-Garcia R.Sanchez-Garcia@soton.ac.uk Tiejun Ma tiejun.ma@soton.ac.uk Johnnie Johnson J.E.Johnson@soton.ac.uk Ming-Chien Sung M.SUNG@soton.ac.uk
2 Research Aims Develop topological tools for data analysis Measure the underlying quality of horses based on past race performances Make profitable predictions from the results
3 Horse Racing Data Data consists of outcome of UK horse races between 2005 and races competed in by horses Various performance indicators including finishing position and beaten lengths
4 Handicapping Handicapping decreases the predictability of races Horses are given extra weight to carry to inhibit their performance Aim of handicappers is that races finish in a dead heat Account for this by estimating the results had all horses carried the same weight
5 Pairwise Scores Each race αα, with performance indicator PP, forms a local pairwise score matrix YY αα with YY αα iiii = PP ii PP jj Information is aggregated, with respect to a reliability measure
6 Indirect Comparisons Horse A 1 B 2 Finishing Position Horse B 1 C 2 Finishing Position Which is better? A or C?
7 Optimisation Problem HodgeRank finds global scores ss minimising the weighted square error between the induced and observed pairwise comparisons Want to solve the optimisation problem: min ss R mm ii,jj WW iiii (ss ii ss jj YY iiii ) 2
8 Flag Complex Representation Horses form the 0-skeleton Pairwise scores form the 1- simplices
9 Cochains kk-cochains are realvalued functions on the kk-simplices Reversing direction negates the value of the cochain Set of all kk-cochains is denoted CC kk
10 Inner Products Choose unweighted Euclidean inner products on CC 0 and CC 2 Equip CC 1 with a weighted Euclidean inner product ff, gg CC 1 = ii EE WW ii ff ii gg(ii)
11 Coboundary Operators kk-th coboundary operator is a linear map δδ kk : CC kk CC kk+1 Adjoint of the kk-th coboundary operator is a linear map δδ kk : CC kk+1 CC kk kk-th combinatorial Laplacian is a map Δ kk = δδ kk δδ kk + δδ kk 1 δδ kk 1 : CC kk CC kk
12 Hodge Decomposition Theorem CC kk admits an orthogonal decomposition CC kk = iiii δδ kk 1 kkkkkk Δ kk iiii δδ kk and kkkkkk Δ kk = kkkkkk(δδ kk ) kkkkkk(δδ kk 1 ) Split YY into orthogonal cochains according to the Hodge Decomposition Theorem
13 Globally Consistent iiii(δδ 0 ) are globally consistent cochains Any ff 1 iiii(δδ 0 ) has the form ff 1 ii, jj = ff 0 jj ff 0 (ii) for some ff 0 CC 0 ff 0 = 5 ff 1 = 2 ff 1 = 3 ff 0 = 3 ff 0 = 2 ff 1 = 4 ff 1 = 1 ff 0 = 1
14 Optimisation Problem Since iiii(δδ 0 ) is the set of all globally consistent pairwise scores, the optimisation problem can be written as: min ss R mm ii,jj WW iiii (ss ii ss jj YY iiii ) 2 = min δδ 2 0ss YY ss R mm 2,WW
15 Optimisation Problem Solved Solutions to the optimisation problem satisfy Δ 0 ss = δδ 0 YY and the minimum norm solution is given by ss = Δ 0 δδ 0 YY ss is unique up to an additive constant
16 Inconsistencies ker(δ 1 ) are partial inconsistencies Every triple of horses are consistent but larger cycles are inconsistent
17 Inconsistencies iiii(δδ 1 ) are complete inconsistencies These are functions which are inconsistent on every level
18 Partial Inconsistency Weights Measure how far a 1-simplex is from local consistency by φφ 1 ii = (δδ 1 δδ 1 YY)(ii) Reweight complex by 1 WW ii = φφ 1 ii + 1
19 Predicting Winners Train and test a conditional logit model using public odds as a predictor Assess impact of adding global score variable to the model Measure goodness-of-fit by McFadden s RR 2 Log Likelihood Ratio Test determines benefit of adding signal variable
20 Conditional Logit Model Generate vector of winning probabilities for each horse in each race pp ii αα = (pp 1 αα,, pp nn αα ) Probabilities based on predictive variables xx ii αα = (xx 1 αα 1,, xx 1 αα mm ) Representative utility for horse ii in race αα mm uu ii αα = kk=1 ββ(kk)xx ii αα (kk) + εε ii αα
21 Conditional Logit Model Assuming error terms are identically and independently distributed via a double exponential distribution, probabilities are given by pp ii αα = nn αα ii=1 exp mm kk=1 eeeeee mm kk=1 ββ(kk)xx ii αα (kk) ββ(kk)xx ii αα (kk)
22 Kelly Wagering Strategy Fractional Kelly wagering strategy employed to assigns bets A fraction of the initial capital is bet, given by ff = pp bb bb where bb is the odds ratio bb: 1 and pp is the estimated probability of winning
23 Results Model trained over 2011 to 2013 and evaluated in 2014 LLR test significant at the 0.1% RR 2 increase of 0.227% over public odds model with RR 2 of
24 Results Betting simulation results (initial capital of 1000) Model Profit ( ) Rate of Return (%) excl. scores incl. scores
25
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