How To Win The Ashes A Statistician s Guide

Transcription

1 How To Win The Ashes A Statistician s Guide Kevin Wang Sydney Universtiy Mathematics Society Last Modified: August 21, 2015

2 Motivation How come SUMS doesn t do statistics talk very often?

3 Motivation How come SUMS doesn t do statistics talk very often? Motivation Given the dynamic nature of the cricket game, how can we judge a player s career statistics/abilities in a meaningful way?

4 Motivation How come SUMS doesn t do statistics talk very often? Motivation Given the dynamic nature of the cricket game, how can we judge a player s career statistics/abilities in a meaningful way? How can we classify players with similar styles into some categories? How can we find interesting patterns? How does statistics students operate?

5 Outline of This Talk Theories of Principal Component Analysis (PCA) 1. The Don. The Ashes in England. The younger generation. How I will use statistics for evil and manipulate the results to suit my selfish desire to make Australia win the next Ashes. 2013) 1 An Introductory Application of Principal Components to Cricket Data. (Manage and Scariano,

6 Introduction To Cricket

7 Principal Component Analysis (PCA) Good things comes in matrix form. Say, we have a data matrix of n rows (number of samples) and p columns (number of variables).

8 Principal Component Analysis (PCA) Good things comes in matrix form. Say, we have a data matrix of n rows (number of samples) and p columns (number of variables). We want to perform data reduction to cut down the number of variables. PCA allows us to to take an useful linear combination of all variables.

9 Overview of PCA PCA summarise original variables into principal components scores (PCs). All PCs together accounts for 100 % of the variabilities in the data. But in most practical settings, the first few PCs will be enough to account for most (50 80%) of the variability in the data.

10 Overview of PCA PCA summarise original variables into principal components scores (PCs). All PCs together accounts for 100 % of the variabilities in the data. But in most practical settings, the first few PCs will be enough to account for most (50 80%) of the variability in the data. The idea of PCA is to use eigenvectors of Σ as the linear coefficients in our linear combination construction. And at the same time, the resultant PC s must capture all the original data variability (i.e. the variance of the original data equals to the variance of the transformed data).

11 Variance and Correlation For data vectors of the form x = (x 1,..., x p ) we can define Sample variance: V ar(x) = 1 p p 1 i=1 (x i x) 2. Sample Pearson s Correlation Coefficient, or just correlation: (to confuse you) p i=1 Corr(x, y) = (x i x)(y i ȳ) p i=1 (x i x) 2 p i=1 (y i ȳ). (1) 2

12 Correlation Matrix Every time we collect one sample, of dimension p, we can always write it in a vector form: X i = (X i1,..., X ip ) R p, i = 1,..., n. (2) Then, we [X ij ] is an n p matrix: X 1 X 11 X X 1p X 2 X = [X ij ] =. = X 21 X X 2p X n1 X n2... X np X n (3) We can define a p p matrix, called the correlation matrix, Σ = [Σ ij ] element-wise as: Σ ij = Corr(X i, X j ) (4)

13 The Tricky Life of Correlation Matrix Σ is... square and symmetric, non-negative definite, all of its eigenvalues are non-negative,

14 The Tricky Life of Correlation Matrix Σ is... square and symmetric, non-negative definite, all of its eigenvalues are non-negative, it can be decomposed (spectral decomposition) into Σ = UΛU 1 = UΛU, where U is a square p p matrix whose columns are orthonormal eigenvectors of Σ.

15 Skimming Over Some Mathematical Theorems Like Most Statisticians Do Consider only one row of the data (i.e. one player only): X X p 1 R p : Definition The first PC score of X is a linear combination of every element in X. And it can be defined as Y 1 = X U 1. Second and other PC score vectors can be defined similarly.

16 Skimming Over Some Mathematical Theorems Like Most Statisticians Do Consider only one row of the data (i.e. one player only): X X p 1 R p : Definition The first PC score of X is a linear combination of every element in X. And it can be defined as Y 1 = X U 1. Second and other PC score vectors can be defined similarly. Definition In general, the transformed data is Y n p = X n p U p p.

17 See Diagram on Whiteboard This construction: allows a maximisation of variance of any possible linear combination possible.

18 See Diagram on Whiteboard This construction: allows a maximisation of variance of any possible linear combination possible. For the first PC, p j=1 V ar(y j) = p j=1 V ar(x j) = λ 1 Similar holds for higher order PCs.

19 See Diagram on Whiteboard This construction: allows a maximisation of variance of any possible linear combination possible. For the first PC, p j=1 V ar(y j) = p j=1 V ar(x j) = λ 1 Similar holds for higher order PCs. the p columns of Y are called the score vectors. Each one is uncorrelated to all the PCs before it.

20 See Diagram on Whiteboard This construction: allows a maximisation of variance of any possible linear combination possible. For the first PC, p j=1 V ar(y j) = p j=1 V ar(x j) = λ 1 Similar holds for higher order PCs. the p columns of Y are called the score vectors. Each one is uncorrelated to all the PCs before it. Since each score vector has the same length p as the original variables, we can also consider their correlation! This allows us to meaningfully interpret the PC scores.

21 All Time Batsmen, Correlation Matrix Figure : /Users/kevinwang/Documents/Kevin/SUMS/Ashes_PCA/PCA_

22 All Time Batsmen, Loading Vector Plot

23 All Time Batsmen, Without Bradman

24 All Time Batsmen, With Bradman

25 All Time Batsmen, Rank Table PC1 PC2 PC3 PC4 DG Bradman RG Pollock GA Headley H Sutcliffe SR Tendulkar RT Ponting JH Kallis R Dravid BB McCullum IR Bell AN Cook MJ Clarke Selected players. PC1: high rank desirable. PC2: low rank desirable.

26 All Time Batsmen, Overall Corr(Vari, PCs)

27 Standardisation Goes A Long Way

28 Figure : It is All About The Ashes

29 Ashes in England: The Data Role Country Start Finish Mat Inns NO Runs RT Ponting A AUS MJ Clarke A AUS SR Waugh A AUS KP Pietersen A ENG IR Bell A ENG AJ Strauss A ENG HS Ave BF SR Cent HalfCent Ducks Fours Sixes

30 Ashes in England: Circle Plot and Corr(PCs, Vari) Figure : /Users/kevinwang/Documents/Kevin/SUMS/Ashes_PCA/PCA_

31 Ashes in England: Random Clustering

32 Ashes in England: Biplot of Role

33 Ashes in England: Biplot of Country

34 Ashes in England: Rank And Score Sums Remember those score vectors? Remember that from our plots, lower the first PC scores, better the batsmen.

35 Ashes in England: Rank And Score Sums Remember those score vectors? Remember that from our plots, lower the first PC scores, better the batsmen. Summing the PC scores according to the Country: AUS: , ENG:

36 Ashes in England: Rank And Score Sums Remember those score vectors? Remember that from our plots, lower the first PC scores, better the batsmen. Summing the PC scores according to the Country: AUS: , ENG: Summing the rank of the PC scores according to the Country: AUS: 2010, ENG: PC1: Scores PC1: Rank RT Ponting MJ Clarke SR Waugh KP Pietersen IR Bell AJ Strauss

37 We won the Ashes!

38 Ashes in England: Younger Generations Well, not really... Sadly, a lot of Australian legends have retired post 2007 Ashes series in Australia. The 2009 and 2013 Ashes are essentially composed of younger generation of Australian cricketers.

39 Ashes in England: Younger Generations Well, not really... Sadly, a lot of Australian legends have retired post 2007 Ashes series in Australia. The 2009 and 2013 Ashes are essentially composed of younger generation of Australian cricketers. This is a great lesson in Statistics: put your data into context. You don t have people like Steve Waugh or Glenn McGrath or Shane Warne (these used to be the players who reserve their best performance against the English). If restricting to younger players, the rank sums are 621 vs 414. Firmly in England s favour.

40 Where is hope?

41 AUS vs ENG vs WI We can perform similar analysis on the recent AUS vs WI series and ENG vs WI series. Australia series: two matches, AUS won 2-0. PC1 rank: 113 vs 212 in Australia s favour.

42 AUS vs ENG vs WI We can perform similar analysis on the recent AUS vs WI series and ENG vs WI series. Australia series: two matches, AUS won 2-0. PC1 rank: 113 vs 212 in Australia s favour. England series: three mathces, England won 2-0. PC1 rank: 154 vs 224 in England s favour. Country Series Role Mat Inns NO Runs HS AC Voges AUS 1 A SPD Smith AUS 1 A MJ Clarke AUS 1 A JE Root ENG 2 A GS Ballance ENG 2 A JC Buttler ENG 2 A Ave BF SR Cent Half.Cent Ducks Fours Sixes AC Voges SPD Smith MJ Clarke JE Root GS Ballance JC Buttler

43 Where is hope?

44 Well, not really... What I have not tell you is that ENG actually performed quite well against WI with adjustments. Good statistics does take time. I couldn t complete WI analysis in time for this talk.

45 What to take away? Statistics is different to mathematics. Uncertainty is the blood of statistics. Some analysis are subject to bias of the analyst. If the analyst is half as evil as Kevin, he/she can probably tell you what you want to hear. But the difference is: biased and non-rigorous analysis are subject to criticisms and often indefensible. Good visualisation goes a long way (As STAT3914 Students can attest) Cricket is fun! I would be extremely disappointed if we can predict the outcome of the game based on such simple analysis.

46 Thank You! Theorem Winning the Ashes is every Australians birthright. Corollary (Also know as The McGrath Hypothesis) Australia will always win the Ashes, 5-0. Proof.

47 References STAT3914 Lecture Notes. An Introductory Application of Principal Components to Cricket Data. (Manage and Scariano, 2013) All images were found on the Internet. No copyright infringement intended.