Karate katas in high-level competitions:
Statistics, Data analysis and Visualizations

Author: Joël Hamilcaro

Abstract

First of all we looked at what are the most practiced katas. The most practiced kata were Anan Dai (11.56%) , followed by Suparinpei (11.47%) , Papuren (11.42%), Unsu (9.95%) and Anan (8.76%). There is a very significant relationship between the athlete's category and their choice of kata ($\chi^2$(30), p-value : 6.84E-279). Athletes in the individual female category prefer Shitoryu katas. On the contrary, the athletes of the men's team category prefer Shotokan katas. Concerning the athletes of the women's team and men's individual category, the most practiced katas are relatively varied between Shotokan and Shitoryu.

Then, we studied the strategies of choice of katas according to the round. We focused on the two styles of karate practiced by the vast majority of athletes. For shotokan style karatekas, Unsu kata and, to a lesser extent, Gojushiho Sho kata, are widely used in both the first and final rounds. On the other hand, Gojushiho Dai is more used in the first rounds, while Kanku Sho and Gangaku are more used in the last rounds. As well as Unsu and Gojushiho Sho, Anan, Anan Dai and Papuren are katas very much used by shitoryu karatekas whatever the round. Chatanyara Kushanku is a kata that is quite practiced in the later rounds, unlike Suparinpei which is very much practiced in the earlier rounds.

Finally, we compared the scores obtained between shotokan and shitoryu katas. Without taking into account the categories, the shitoryu katas obtain better scores, in a significant way (T-test, p-value = 1.36E-16). Without assuming anything about the probability law followed by the scores, we obtain the same results (Mann-Whitney U-test, p-value = 1.02E-15). Taking into account the categories, we have in each category an advantage of the shitoryu style over the shotokan style (except for the men's team category).

Table of Contents

Introduction

What are the kata competitions?

In martial arts, a kata is a codification of technical sequences. The set of karate katas constitutes a historical heritage. Practitioners use it as a database on which they can work on the essential elements of karate (such as strength, speed, explosiveness, balance, flexibility, concentration, aesthetics, etc.). The goal is to assimilate the techniques of a kata in order to be able to adapt them in different combat situations.

Official karate competitions are divided into two disciplines: Kumite and Kata. The first discipline is a fighting competition between two people, which is common in combat sports. The second is a competition in which, in each round, the participant must perform a kata of his/her choice from the WKF (World Karate Federation) kata list. Depending on their performance, participants either move on to the next round or are eliminated. These competitions can be individual or team competitions.

The following video summarizes the evaluation criteria for kata performance.

Purpose of this notebook

In 2019, the rules of kata competitions have been transformed. In the past, kata competitions consisted of individual confrontations. Now it has been replaced by a system of ranking by score where the best ranked move on to the next rounds, and the others are eliminated.

There are a few studies on katas in high level competition. However, the knowledge on this field is still new and limited.

The objectives of this notebook are then as follows:

Data collecting

The data collected is from various WKF senior (i.e. over 18 years old) competitions that have taken place since the implementation of the new kata competition rules:

The data was collected on the website sportdata.org. For a given kata, we record the name of the kata, the score obtained, the category of the competition, the number of rounds remaining before the final, and, if needed, the particularity of the round (for example, a round for a medal or a round to break a tie). These data are readable on the different pages of the website, but this website does not provide a file gathering all the data. For this reason, such a file will not be provided publicly with this notebook (the data migration has been done privately). Moreover, no names or personal information has been collected.

Load data

The archive kata.csv.zip (not provided with this notebook as explained in the previous section) gathers all the collected data in a table in csv format.

Here is a glimpse of a few lines from this table.

This data will be loaded by our program, to allow us to perform all the data analysis and visualization.

Dataset description

Most popular katas

There are 102 katas that are allowed in official competitions. However, only a part of them are actually practiced in high-level competitions. Indeed, the athletes (and their coaches) choose the katas that will allow them to show off their skills the most. Thus, the choice of kata is essential. The complexity of the techniques present in the kata, the presence of jumps, the intensity of the rhythm and the overall duration of the kata, is taken into account. Many katas are therefore considered "too simple" and are never practiced in high-level competitions.

Pie chart

We can see below the katas that are the most popular, those that have been the most performed.
(You can select a specific category with the button on the left)

Legend :
- Shotokan style katas
- Shitoryu style katas
- Shitoryu and Shotokan style katas
- Katas of other styles

Katas preferences by category

We now want to see if the choices of katas are related to the sport categories. For that we will carry out a mathematical test of inferential statistics. We take into account only the katas used at least 5 times in each category so that the test conditions are optimal.

Pearson's chi-squared test

We will perform a Pearson's $\chi^2$ test.

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Details:

Let $X$ and $Y$ be two random variables that represent respectively the kata performed and the sport category of the participant. We want to know if $X$ and $Y$ are independent. The hypotheses of our statistical test are:
  • $H_0$ : $X$ and $Y$ are independent (null hypothesis).
  • $H_1$ : $X$ and $Y$ are related (alternative hypothesis).
We study the realization of the pairs $(X,Y)$ of the total sample (the number of times each kata was used for each of the categories is observed). Let us note $O_{ij}$ the observed number of data for which $X$ takes the value $i$ and $Y$ the value $j$. Here is our contingency table consisting of the $O_{ij}$ values observed for any pair $(i,j)$.
Under the null hypothesis $H_0$, the observed differences in frequencies are only due to sampling fluctuations and not to any link between the kata and the category. We can calculate the expected values under $H_0$. For a given pair $(i,j)$, the expected value is $E_{ij}=\dfrac{ {\sum_{i}{O_{ij}}} \times {\sum_{j}{O_{ij}}} }{\sum_{i}\sum_{j}{O_{ij}}}$ Here is the table of expected values under $H_0$.
We compute the distance between the observed values $O_{ij}$ and the expected values $E_{ij}$ if there was independence : $D = \sum_i \sum_j \dfrac{ ( O{ij} - E_{ij} )^2 }{ E_{ij} } $

Pearson's chi-squared test results

Analysis track

In the Pie chart section we can see which katas are the most practiced according to the category.

The differences in the choice of katas between the categories are significant as we showed in the previous section.

Athletes' strategies

In a competition, the more participants there are, the more rounds there will be. However, a competitor cannot perform the same kata several times in the same competition. Therefore, the choice of the different katas and their order must be meticulously planned.

Thus, different strategies are possible. In general, competitors perform katas of the style of karate they specialize in (few perform katas of different styles). Some prefer to save what they consider their best katas for the last rounds. Others prefer to use them before, to increase their chances of passing certain rounds. The idea here is to see if a pattern seems to emerge: if some katas are frequently used in the first rounds and if others are rather last rounds katas.

Kata per round: visualization

In this section, we will see which katas are the most used according to the round. As the number of rounds varies from one competition to another, so we are interested in the number of rounds left before the final.

We will focus on the most used katas in the two styles over represented in high level competition: Shotokan and Shitoryu styles.

You can select a specific category with the button.

Kata per round: analysis track

For Shotokan katas, we can see that Unsu is the kata of Shotokan finalists. It is a "safe bet", as well used in the first rounds as in the last ones. Gojushiho Sho, although less used, is also a very popular kata in the first and last rounds. It can be noticed that Gojushiho Dai is a kata that is mostly used in the first rounds (it is very used when there are three rounds left before the final). On the other hand, Gangaku and Kanku Sho are kata that are used in the last rounds and not much in the first rounds. These observations are similar in each category.

For Shitoryu katas, we can notice that Anan, Anan Dai and Papuren are very used katas whatever the round. Chatanyara Kushanku is rather a kata practiced in the last rounds, unlike Suparinpei which is very much used in the first rounds. These observations are similar when considering the different categories. It is interesting to note that Paiku is used a lot in the women's team category, regardless of the rounds.

Scores analysis

In this section, we are interested in the scores obtained.

Score by category

First of all, we will see a box plot that gives the "five-number summary" (median, upper and lower quartiles, minimum and maximum) of the scores obtained by the athletes.

This graph provides an overview. We have not taken into account performances with a score of zero (which correspond to disqualifications). We can see that the median scores are very close. There is no overwhelming difference that is visible between the scores obtained in the different categories.

Score by kata

Now we will see which katas have obtained the best scores. It is important to remember that the score obtained depends above all on the performance of the athlete, especially since minor mistakes or small imbalances greatly affect the score obtained in high level competitions. For this reason we will not calculate the average score but rather the median score of each kata, and in each category.

First, we represent the median scores in a graph. The outliers (katas that have been performed less than one time out of a hundred) are not taken into account.
(You can select a specific category with the button on the left)

In a second step, all the median scores for each kata and for each category are represented in a table. Here all the katas are taken into account (even the outliers).

Legend :
- Shitoryu style katas
- Shotokan style katas
- Shitoryu & Shotokan style katas
- Other style katas

We can see that the kata Kishimoto No Kushanku has the highest median score. However, we could see (Section Pie chart : "Table of the number of uses of the katas") that this kata has been practiced only once. We can then ask ourselves if this score depends above all on the performance of the athlete or if it is also the fact of having presented a kata never presented, which allowed the participant to be distinguished.

Shotokan and Shitoryu: Unequal scores

The kata styles over-represented in high level competition are Shotokan and Shitoryu. We want to know if there is a significant difference in the scores obtained according to the style.

Results:

Differences in variances (F-test)

Differences in average scores (Welch T-test)

Differences by category (F-test,Welch T-test)

Differences in average scores (Mann-Whitney U-test)

Differences by category (Mann-Whitney U-test)

F-test of equality of variances

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Details :
Let $U$ and $V$ be two random variables that represent respectivly the score obtained after a performance on a shotokan kata and on a shitoryu kata. We observe $U_1, ... , U_{n_1}$ which are $n_1$ random realizations of $U$ and $V_1, ... , V_{n_2}$ which are $n_2$ random realizations of $V$. The two samples are independent. We want to know if their respectives variances $\sigma_1^2$ and $\sigma_2^2$ are equal.
  • $H_0$ : $\sigma_1^2=\sigma_2^2$ (null hypothesis).
  • $H_1$ : $\sigma_1^2 \neq \sigma_2^2$ (alternative hypothesis).
Let $\mu_1$ and $\mu_2$ be the respectives overall means of $U$ and $V$. Since $n_1$ and $n_2$ are large enough, we approximate the laws of the means $\displaystyle M_1=\dfrac{1}{n_1}\sum_{i=1}^{n_1} U_i$ and $\displaystyle M_2=\dfrac{1}{n_2}\sum_{i=1}^{n_2} V_i$ by respectively $N\left(\mu_1, \dfrac{\sigma_1^2}{n_1}\right)$ and $N\left(\mu_2,\dfrac{\sigma_2^2}{n_2}\right)$ Gaussian distributions. We denote $\displaystyle S_1^2=\dfrac{1}{n_1-1}\sum_{i=1}^{n_1}{(U_i - M_1)^2}$ the unbiased estimator of the variance of $U$ and $\displaystyle S_2^2=\dfrac{1}{n_2-1}\sum_{i=1}^{n_2}{(V_i - M_2)^2}$ the unbiased estimator of the variance of $V$. According to Student's theorem, $K_1=\dfrac{n_1-1}{\sigma_1^2}S_1^2$ follows a $\chi^2(n_1-1)$ distribution and $K_2=\dfrac{n_2-1}{\sigma_2^2}S_2^2$ follows a $\chi^2(n_2-1)$ distribution. Therefore, $F=\dfrac{K_1 \big/ (n_1-1) }{K_2 \big/ (n_2-2) }=\dfrac{S_1^2 \big/ \sigma_1^2 }{S_2^2 \big/ \sigma_2^2 }$ follows a Fisher Law $F(n_1-1,n_2-2)$. Under $H_0$, $F=\dfrac{S_1^2}{S_2^2}$. We compute the observed variances $s_1^2$ and $s_2^2$ on the respectives samples $U_1,...U_{n_1}$ and $V_1,...V_{n_2}$.
We perform a F-test of equality of variances in order to know if variances are equals or not. Our test statistic is $f= \dfrac{s_1^2}{s_2^2}$.

F-test results

T-test of comparison of means

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We compute $m_1$ and $m_2$ the average values of the respective realizations of $U$ and $V$.
We want to know if there is a significant difference between the average Shotokan and Shitoryu scores.
  • $H_0$ : $\mu_1=\mu_2$ (null hypothesis).
  • $H_1$ : $\mu_1 \neq \mu_2$ (alternative hypothesis).
As $\sigma_1^2$ and $\sigma_2^2$ are significantly different, it is wise to perform a Welch T-Test for testing statistically the hypothesis of equality of two means with two samples of unequal variances. Under $H_0$, $T=\dfrac{M_1-M_2}{\sqrt{\dfrac{S_1^2}{n_1} + \dfrac{S_2^2}{n_2}}}$ follow a Student law with $\nu$ degrees of freedom (with $\nu$ given by the Welch-Satterthwaite equation). Our test statistic is $t= \dfrac{\mu_1-\mu_2}{\sqrt{\dfrac{s_1^2}{n_1} + \dfrac{s_2^2}{n_2}}}$.

T-test results

Style scores according to the category

We perform the same tests as before, this time for each category.

Non-normality of the data

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For the previous tests, we assumed that the distributions of $M_1$ and $M_2$ were Gaussian because $n_1$ and $n_2$ were large enough. This approximation is relevant according to the Central Limit Theorem. However, it may be interesting to test if the distribution of the data is indeed Gaussian, and, if not, to perform a non-parametric test (of course, less powerful) without assuming anything about the distribution of the data.

Visual check of normality

The distributions appear visually Gaussian on the histogram. However, there are small deviations on the Q-Q plots at the extreme values.

Mathematical test of normality


We will now use a statistical test of normality: the Shapiro-Wilk test. Test 1 : We want to know if $U$ is Gaussian.
  • $H_0$ : $U$ is gaussian (null hypothesis).
  • $H_1$ : $U$ is not gaussian (alternative hypothesis).
Test 2 : We want to know if $V$ is Gaussian.
  • $H_0$ : $V$ is gaussian (null hypothesis).
  • $H_1$ : $V$ is not gaussian (alternative hypothesis).
Shapiro-Wilk test results:

Mann-Whitney U-test

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From now on, we do not assume anything about the distributions of the Shotokan and Shitoryu scores. We perform the non-parametric Mann-Whitney U-test. Our statistic is $J = \sum_{i=1}^{n_1}{\sum_{i=1}^{n_2}{ g(U_i,V_i) }}$ with $g(X,Y)=1$ if $Y < X$, $g(X,Y)=1/2$ if $Y = X$ and $g(X,Y)=0$ if $Y > X$
Mann-Whitney U-test results
Mann-Whitney U-test by category

We perform the same test for each category.

We observe that the result obtained is the same as in the case where we assumed that the distributions are Gaussian, except in the Female Team category where the shitoryu katas significantly get higher scores. The global conclusion remains the same, there is a significant advantage of Shitoryu style over Shotokan style (except in the Male Team category)

Conclusion

The data analysis can give us some strategic ideas for high level kata competitions. We can see from the visualizations that very rarely used katas (such as Kishimoto No Kushanku or Ohan) can get much higher scores than frequently practiced katas. Rare katas seem to attract the attention of the judges. Moreover, as we have shown that there is a significant relationship between the choice of katas and the sport category, we can relate this information in order to establish a strategy of choice of kata and an adapted training. We can also ask ourselves the question of the imbalance between styles. We could see that, in all categories (except the Male Team category), the Shitoryu style had an advantage over the Shotokan style. We could ask ourselves if this disadvantage is linked to strategic mistakes in the choice of kata: for example, in the Individual Male category, the kata Kanku Sho is the Shotokan kata which obtains the best marks, even though it is not a kata which is practiced a lot like Unsu or Gojushiho Sho. One could also wonder if this is not rather a lack of Shotokan kata themselves. In this case, it might be wise for Shotokan specialists to open up to other styles in order to increase their range of kata choices.

However, this analysis is limited to high-level competitions. The data are not representative of local and regional competitions which correspond, in terms of number, to the vast majority of karate kata competitions.

To conclude, thanks to the interactive tables and graphs in this notebook, readers will be able to read the data in detail and do their own analysis.

References

Further information