Handling Ties. Analysis of Ties in Input and Output Data of Rankings
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1 Analysis of Ties in Input and Output Data of Rankings
2 Knowledge Engineering - Seminar Sports Data Mining 1
3 Tied results in the input data Frequency depends on data source tie resolution policy Knowledge Engineering - Seminar Sports Data Mining 2
4 Colley s Method Massey s Method Markov s Method Colley s method does not account for ties Markov s methods depends on voting mechanism used Elo s, Keener s, Massey s and OD method account for ties Elo s, Keener s, OD Method Knowledge Engineering - Seminar Sports Data Mining 3
5 Colley s Method Massey s Method Markov s Method Elo s, Keener s, OD Method To compare methods with ties and methods without we need to derive a Colley s method accounting for ties create a Massey s method ignoring ties choose a Markov s method allowing for both modify Elo s, Keener s and the OD method Knowledge Engineering - Seminar Sports Data Mining 4
6 Colley s Method Colley s Method Massey s Method Markov s Method Elo s, Keener s, OD Method C r = b, with C n n having entries as follows { 2 + t i, i=j C ij = n ij, i j r n 1 being the unknown Colley rating vector b n 1 defined as b i = (w i l i ) Knowledge Engineering - Seminar Sports Data Mining 5
7 Colley s Method Colley s Method Massey s Method Markov s Method Elo s, Keener s, OD Method Ties in Colley s method are ignored represent an equal chance for either team winning - or losing can be emulated by creating two artificial games do not alter vector b increment C ij by 1 and decrement C ji by 1 thus preserving the Colley property n r i = n 2 i= Knowledge Engineering - Seminar Sports Data Mining 6
8 Massey s Method Colley s Method Massey s Method Markov s Method Elo s, Keener s, OD Method M r = p, with M n n having entries as follows { t i, M ij = n ij, i=j i j r n 1 being the unknown Massey rating vector p n 1 being the vector of all teams point differentials Knowledge Engineering - Seminar Sports Data Mining 7
9 Massey s Method Colley s Method Massey s Method Markov s Method Elo s, Keener s, OD Method Ties in Massey s method are naturally accounted for increment M ij and M ji do not change p can be ignored when forming M to create a No-Ties Method Knowledge Engineering - Seminar Sports Data Mining 8
10 Markov s Method Colley s Method Massey s Method Markov s Method Elo s, Keener s, OD Method Standard Markov voting procedures: 1. Loser casts one vote for each team lost against. 2. Loser casts one vote for each point lost to the other team. 3. Loser and winner cast one vote for each point lost to one another Knowledge Engineering - Seminar Sports Data Mining 9
11 Markov s Method Colley s Method Massey s Method Markov s Method Elo s, Keener s, OD Method In the context of ties: 1. Tied teams cast half a vote each for the other team. 2. Loser casts one vote for each point lost on average. 3. Ignore tied events for a no-ties variant Knowledge Engineering - Seminar Sports Data Mining 10
12 Elo s, Keener s, OD Method Colley s Method Massey s Method Markov s Method Ties in Elo s Method are explicitly taken care of can be ignored for a no-ties method Elo s, Keener s, OD Method Knowledge Engineering - Seminar Sports Data Mining 11
13 Elo s, Keener s, OD Method Colley s Method Massey s Method Markov s Method Ties in Keener s and the OD Method are naturally accounted for can be excluded by setting tied scores to 0 Elo s, Keener s, OD Method Knowledge Engineering - Seminar Sports Data Mining 12
14 Apply variants with and without ties to a data set devoid of ties Introduce a single tie into data set Compare the rankings produced by the variant methods Knowledge Engineering - Seminar Sports Data Mining 13
15 Let C be the Colley matrix for the no-ties variant C denote the Colley matrix for the variant including ties e denote a tied input r be the rating vector related to C r be the rating vector related to C Knowledge Engineering - Seminar Sports Data Mining 14
16 C = C + (e i e j )(e i e j ) T r i r j r = r ( )C 1 (e 1 + [C 1 ] ii 2[C 1 ] ij + [C 1 i e j ) ] jj Knowledge Engineering - Seminar Sports Data Mining 15
17 With ɛ = r i r i+1, ɛ > 0 denoting the difference in pre-disturbance ratings this implies r i < r i+1 Let r i < r j : ɛ < ( (r i r j )([C 1 ] ii [C 1 ] ij [C 1 ] i+1,i + [C 1 ] i+1,j ) 1 + [C 1 ] ii 2[C 1 ] ij + [C 1 ] jj ) if r i r j teams i and j are unlikely to change in rank if r i r j team i is likely to drop in rank C 1 (e i e j ) Knowledge Engineering - Seminar Sports Data Mining 16
18 Movies NHL Hockey Teams Knowledge Engineering - Seminar Sports Data Mining 17
19 Movies Movies NHL Hockey Teams Knowledge Engineering - Seminar Sports Data Mining 18
20 NHL Hockey Teams Movies NHL Hockey Teams Knowledge Engineering - Seminar Sports Data Mining 19
21 Modifications of input data Change non-tied events to a tie Motivation Methods Analysis Knowledge Engineering - Seminar Sports Data Mining 20
22 Motivation Ties are often broken seemingly at random irrespective of the teams actual performance after regular match time Methods Analysis Knowledge Engineering - Seminar Sports Data Mining 21
23 Motivation Methods Analysis Knowledge Engineering - Seminar Sports Data Mining 22
24 Motivation Final results may fail to portray the teams actual strength create a false sense of precision skew the ranking impede a ranking s predictive capabilities Methods Analysis Knowledge Engineering - Seminar Sports Data Mining 23
25 Methods Motivation Induce a tie if a winner is only determined after regular play time points differ only by a small margin match statistics indicate comparable performance Methods Analysis Knowledge Engineering - Seminar Sports Data Mining 24
26 Analysis Motivation Methods Analysis Knowledge Engineering - Seminar Sports Data Mining 25
27 Recapitulation Resolution Methods Ramifications Knowledge Engineering - Seminar Sports Data Mining 26
28 Recapitulation Rankings are total preorders, i.e. relations on a set S that are total ( x, y S : x y y x) transitive ( x, y, z S : x y y z x z) but not antisymmetric ( x, y S : x y y x x y) Recapitulation Resolution Methods Ramifications Knowledge Engineering - Seminar Sports Data Mining 27
29 Resolution Methods Recapitulation Standard Competition Ranking (" ") Modified Competition Ranking (" ") Dense Ranking (" ") Ordinal Ranking (" ") Fractional Ranking (" ") Resolution Methods Ramifications Knowledge Engineering - Seminar Sports Data Mining 28
30 Ramifications Psychological Effects Statistical Effects Recapitulation Resolution Methods Ramifications Knowledge Engineering - Seminar Sports Data Mining 29
31 Statistical Effects - Rank Sum Test Recapitulation Resolution Methods Ramifications Knowledge Engineering - Seminar Sports Data Mining 30
32 Statistical Effects - Rank Sum Test Recapitulation Resolution Methods Ramifications Knowledge Engineering - Seminar Sports Data Mining 31
33 Statistical Effects - Rank Sum Test Recapitulation Resolution Methods Ramifications Knowledge Engineering - Seminar Sports Data Mining 32
34 Statistical Effects - Rank Sum Test Recapitulation Resolution Methods Ramifications Knowledge Engineering - Seminar Sports Data Mining 33
35 Accounting for ties is easy Input ties influence ranking order Inducing ties is beneficial Output ties require special care Knowledge Engineering - Seminar Sports Data Mining 34
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