Who is the most complete athlete? – An insight with the Mahalanobis distance (sport & data analysis)
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The Olympic Games have finished a couple of days ago. Two entire weeks of complete devotion for sport. Unfortunately I hadn’t got any ticket but I didn’t fail to watch many games on TV and internet. I was looking at decathlon men competition and I was very impressed by the general quality of these athletes. They have to be able to do everything: sprinting (100 m), jumping high, jumping fast (110 m hurdles), long, throw heavy (put shot) and light (javelin) things, running longer (400 m) and even longer (1500 m)… It became obvious in my mind that it was the quintessence of the sport, every athlete has to find the perfect balance between those different performances to compete efficiently. This sport induces all the quality of a strong man: power, endurance, flexibility, sprint…
Is it really true? Is it really the most balanced athlete who win the decathlon competition?
I decided to test this assumption with the results of the previous Olympic Games (Beijing, 2008). I only kept the athletes who have completed all the disciplines so that I can do the study on a data set without any missing values. I used the observations of the scores for each discipline which are calculated according to the time of the distance done by the athlete. If you are interested in those details, you can have a look at the way it is calculated on: http://www.iaaf.org/mm/Document/Competitions/ … _Tables_of_Athletics_2011_23299.pdf.
I have been very surprised to see that the winner, Bryan Clay who has an average of 879 points per discipline, did very poorly in 400 meter (865 points), high jump (794 points) and in the 1500 meters race (522 points). On the contrary, he performed very well in 100 meters, 110 meters hurdle and long jump disciplines. Thus, I started wondering if the decathlon was not about power rather than about my socalled balance capacity in all the different areas.
Sir Prasanta Chandra Mahalanobis answered to this question some decades ago. In 1936 he decided to create a new function to measure the distance separating two observations. The most common distance is the Euclidian distance. However, this distance does not take into account two important elements. The first element is the variance of the different variables. Indeed, let’s consider the high jump discipline and the pole vault, a gap of 30 centimeters between two athletes is huge in high jumping whereas it is a reasonable difference in pole vault. The reason is easy to understand, the variance in pole vaulting discipline is higher than in high jumping. Fortunately, most of the robustness to the variance is taken into account by the international athletic association (the federation who sets the scores) – although we will see that this is not perfectly true. But there is another problem which is even more important. The correlation of the different disciplines. For example the following graphic shows a positive correlation between shot put and disc throw, which, if we think about it, makes sense! Thus, if we look for the most complete athlete, there should be no cumulative rewards – we don’t want to give athletes too many points when they have performed well in two very similar disciplines. On the contrary, if two disciplines are negatively correlated such as 1500 meters and 100 meters we want to give extra points to athletes who perform well in both of the disciplines. The Mahalanobis distance has been created in this purpose.
If S is the matrix of variancecovariance of the data set, we can formally write the Mahalanobis distance between the vectors x and y as:
Once the matrix S is computed, we can calculate the Mahalanobis score for every athlete – say the distance between zero and the scores of the athlete in the different disciplines. It was unexpected to see that the gold medal would be claimed by Oleksiy Kasyanov who has finished 7^{th} during the Olympic Games. On the contrary, Bryan Clay the Olympic champion would now rank 5^{th}. You can find below two tables, the first one is the ranking of the athletes according to the Mahalanobis distance, and the second one is the official decathlon ranking. As you can see they are many differences. Therefore, decathlon is not the ultimate sport of complete athlete.
Mahalanobis Ranking

Athlete

Mahalanobis score

1

Oleksiy Kasyanov

790.60

2

Andrei Krauchanka

789.16

3

Maurice Smith

767.85

4

Leonel Suárez

754.27

5

Bryan Clay

742.40

6

Yordanis Garciá

737.40

7

Michael Shrade

723.31

8

Romain Barras

709.31

9

Aleksandr Pogorelov

701.18

10

Andres Raja

696.00

11

Roman Sebrle

693.79

12

Aleksey Drozdov

690.95

13

André Niklaus

687.12

14

Massimo Bertocchi

681.92

15

Jangy Addy

681.16

16

Mikk Pahapill

677.04

17

Mikalai Shubianok

667.82

18

Hadi Sepehrzad

653.71

19

Damjan Sitar

651.63

20

Eugene Martineau

637.66

21

Haifeng Qi

631.22

22

Aliaksandr Parkhomenka

630.64

23

Slaven Dizdarevic

607.92

24

Daniel Awde

607.78

Decathlon Ranking

Athlete

Decathlon Score

1

Bryan Clay

8791

2

Andrei Krauchanka

8551

3

Leonel Suárez

8527

4

Aleksandr Pogorelov

8328

5

Romain Barras

8253

6

Roman Sebrle

8241

7

Oleksiy Kasyanov

8238

8

André Niklaus

8220

9

Maurice Smith

8205

10

Michael Shrade

8194

11

Mikk Pahapill

8178

12

Aleksey Drozdov

8154

13

Andres Raja

8118

14

Eugene Martineau

8055

15

Yordanis Garciá

7992

16

Mikalai Shubianok

7906

17

Aliaksandr Parkhomenka

7838

18

Haifeng Qi

7835

19

Massimo Bertocchi

7714

20

Jangy Addy

7665

21

Daniel Awde

7516

22

Hadi Sepehrzad

7483

23

Damjan Sitar

7336

24

Slaven Dizdarevic

7021

The code (R):
#data and data3 are randomly generated for the example
a = rnorm(24)
data=data.frame(shotPut=a, discusThrow=0.5*a + 0.5 * rnorm(24))
data3=data.frame(X1=a, X2=0.5*a + 0.5 * rnorm(24), X3 = rnorm(24), X4 = rnorm(24), , X5 = rnorm(24), X6 = rnorm(24))
lm.shotPut = lm(data$shotPut~data$discusThrow)
plot(data$discusThrow, data$shotPut, axes=TRUE, ann=FALSE)
abline(lm.shotPut)
title(ylab=”Score at shot put”, xlab = ‘Score at discus throw’, col.lab=rgb(0,0,0))
Sigma = cov(data3)
distance = mahalanobis(data3,0 , Sigma, inverted = FALSE)
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