Modeling joint survival probabilities of runs scored and balls faced in limited overs cricket using copulas

1Introduction

Cricket is one of the popular sports in the world, particularly among commonwealth countries. It is a field game that consists of 11 players on each team, which has three main formats that are categorized based on the length of the game: Test cricket, One Day International (ODI) cricket, and Twenty20 (T20) cricket. In this study, we focus on ODI (in which each team gets 50 overs to bat) and T20 (in which each team gets 20 overs to bat). A typical ODI match lasts about nine hours, while a T20 match lasts about three hours. Due to the shorter time span, T20 format is becoming popular among cricket fans around the world.

In cricket, batsmen play as pairs; hence having longer partnerships is a key factor to build the momentum of the game. The goal of the batsmen is to score runs by consuming a low number of balls while protecting the wicket. Therefore, one can use the two variables, the number of balls faced and the runs scored, to model the performance of batsmen. This approach can be used to model the performance of individual batsmen as well as the performance of the partnerships. While partnership at any stage is important, it is extremely important to have a stronger opening partnership as a team plans to accumulate a competitive total (runs). Accordingly, the focus of this paper is divided into two cases: the first is to model the opening partnership scores and the second is to model the runs scored by individual players. Here we used ODI data for opening partnership modeling and T20 data for individual player batting modeling. For the game of cricket, several studies in the literature have discussed the partnership performance. Negative binomial distribution had been used by Scarf et al. (2011) to fit partnership scores and innings scores in test cricket. Valero & Swartz (2012) investigated the importance of partnerships in Test cricket and ODI cricket by comparing the performance of opening batsmen with their “synergistic” partners to the performance of these batsmen with alternative partners. A logistic regression model was applied by Talukdar (2020) to investigate whether the performance of opening partnership influences the outcome of T20 matches along with several other explanatory variables. In their paper, Bhattacharjee et al. (2018) discussed a measure to quantify the batting performance of partnerships based on 2016 T20 world cup data. Furthermore, Swartz et al. (2009) showed how the ODI cricket scores can be simulated using historical ODI data. Moreover, a generalized class of geometric distributions were proposed by Das (2011) as a model for the runs scored by individual batsmen in cricket. It appears that determining the joint distribution of the performance variables of batsmen is an insightful way to understand the overall performance of batsmen. Copula functions have been widely used in the statistical literature to model multivariate distributions. In sports statistics, McHale & Scarf (2007) applied copula functions and Poisson-related marginal distributions to fit soccer data from the English Premier League. From this approach, they highlighted the negative dependency between discrete pairs of shots-for and shots-against. In their paper, Tavassolipour et al. (2013) developed a method to detect and summarize events such as goal, corner foul, offside, and non-highlights in soccer video data using Bayesian network and Farlie-Gumbel-Morgenstern family of copulas. To detect the abnormal variation of batting averages and earned run averages of Major League Baseball (MLB) from 1998 to 2016 seasons, Kim et al. (2019) applied control charts and explored the directional dependence and tail dependence of average run lengths using Markov statistical process control and copula functions. In addition, Boshnakov et al. (2017) proposed a forecasting model using Weibull inter-arrival time count process and copula to obtain the bivariate distribution of number of goals scored at home and away in soccer.

However, to the best of our knowledge, copula functions have not been applied in cricket yet. In cricket, the runs scored and balls faced are generally correlated (positively). In this study, we have used copula functions to obtain the bivariate distribution of runs scored and balls faced by individual batsmen as well as by the opening partnerships. Moreover, these bivariate distributions were used to evaluate the performance of those batsmen or partnerships. For example, using the bivariate distribution, we were able to evaluate the probability of scoring more than 50 runs and staying in the wicket for more than 50 balls. Thereby, those estimated probabilities were then used for ranking partnerships and individual batsmen where the higher the probability for such cases is better the partnership or batsmen would be.

The rest of this paper is organized as follows. In Section 1, we provide an introduction to copula functions and their properties. In the same section, we discuss the types of copula functions that we considered in this study. The proposed method to obtain bivariate distribution and its parameter estimation procedure is explained in Section 3. In Section 4, we apply the proposed method to cricket batting data related to individual batsmen and partnerships. Section 4 is dedicated for discussions and conclusions.

2Copula functions

Copula functions are used to link one-dimensional marginal distributions to a multivariate distribution (see, for example, Nelsen 1999, Balakrishnan & Lai 2009). In copula functions, the marginal distributions are uniform on [0,1]. Since we expect to obtain the joint distribution of runs scored and balls faced by a batsman or a partnership of batsmen, in this study, we focus on two-dimensional copulas which allow to obtain the bivariate joint distribution of two random variables.

Suppose u, v ∈ [0, 1], then the two-dimensional copula is denoted as C (u, v) ∈ [0, 1] ² and it has following properties:

Sklar’s Theorem is one of the fundamental theorems in copula literature which provides the baseline to obtain the joint distribution of random variables using their marginal Cumulative Distribution Functions(CDF).

Theorem 1. [Sklar’s Theorem] Let H be a joint distribution function with marginal distribution functions F and G. Then, there exists a copula C (. , .) such that for all x, y ∈ (- ∞ , ∞) (Sklar 1959, Nelsen 1999)

C (. , .) is unique if F and G are continuous; otherwise, C (. , .) is uniquely determined on the (Range of F × Range of G). If C is a copula, and F and G are marginal distribution functions, then the function H defined by Equation (3) is a joint distribution functions with marginals F and G.

The joint Probability Density Function(PDF) of X and Y, h, can be derived from the joint distribution H as

3Method

As mentioned in the previous sections, in this study, we expect to obtain the joint distributions of runs scored and balls faced by a batsman or within a partnership of two batsmen. Suppose X and Y are random variables that correspond to runs scored and balls faced, respectively. Let F_X (x ; _θx) and f_X (x ; _θx) denote the CDF and PDF of X, and similarly, F_Y (y ; _θy) and f_Y (y ; _θy) denote the CDF and PDF of Y. Here _θx and _θy are the vector of parameters of the corresponding distribution of X and Y, respectively. Moreover, suppose we consider a Frank copula function C (F_X (x ; _θx) , F_Y (y ; _θy)) with a parameter ξ. Let (x_i, y_i) ,  i = 1, 2, …, m are sample bivariate data (i.e., runs scored and balls faced) of a batsman or a partnership for m innings. Thus, by using Equation (4), we can use the maximum likelihood estimation method to estimate the model parameters _θx, _θy, and ξ as follows:

If the number of parameters to be estimated is large, the inference function for margin (IFM) method proposed by Joe & Xu (1996) can be used to obtain the MLEs. However, for this study, we directly optimize the likelihood function without using the IFM because we consider two parameter marginal distributions; thus, there are only five parameters to be estimated.

Suppose the parameter estimates for _θx, _θy, and ξ are _hθx, _hθy, and ξˆ , respectively. Using these parameter estimates, the estimated joint distribution based on the Frank copula function for x′ runs and y′ balls is given by

In general, runs scored and balls faced are right-skewed distributions. After performing Kolmogorov-Smirnov goodness-of-fit test for runs scored and balls faced for partnership data and individual batting performance data with different distributions such as gamma, Weibull and exponential, we found gamma distribution to be the most suitable distribution generally for all the cases. Therefore, in this study, we assume the marginal distributions of both runs scored and balls faced follow gamma distributions (particularly, X ∼ Gamma (α_x,  β_x) and Y ∼ Gamma (α_y,  β_y).) Thus, we estimated parameters {α_x, β_x, α_y, β_y, ξ} from maximum likelihood approach using gamma marginals and Frank copula. Since gamma distribution has the support (0, ∞), we imputed zero runs or zero balls faced with 0.01. It is important to note that copula functions have the flexibility to choose any continuous distribution as marginals depending on the goodness-of-fit of the data.

4Results

In this section, we demonstrate and discuss the results obtained for the ODI opening batting partnership performance and for the T20 individual player batting performance based on the copula approach described in Section 3.

4.1Opening partnerships modeling

As described in the previous sections, for the 17 ODI opening partnerships shown in Table 1, the joint survival distribution of runs scored and balls faced is modeled using gamma marginals and the Frank copula. The parameter estimates of this model are derived by maximizing the likelihood function in Equation (6). For example, the MLEs for the bivariate joint distribution for the opening partnership pair Dhawan S and Rahane A M are {αˆx=0.88, βˆx=90.28, αˆy=1.18, βˆy=71.66, ξˆ=34.40} . The parameter estimates for all 17 partnerships are given in Table 6 of Appendix 6.2. Moreover, for the same partnership pair, a contour plot of the joint survival distribution of runs scored and balls faced is shown in Fig. 1.

Fig. 1

Contour plot for the joint survival probability distribution obtained using gamma marginals and Frank copula for the partnership between Dhawan D and Rahane A M.

For the purpose of exploring the performance of partnerships, we define five different stages of the game of which we believe crucial for evaluating partnerships. For each case, we have estimated the joint survival probability for scoring more than specific number of runs and facing more than specific number of balls.

• 1 Case 1: Survive scoring more than 45 runs and facing more than 30 balls (Table 3)

• 2 Case 2: Survive scoring more than 50 runs and facing more than 60 balls (Table 3)

• 3 Case 3: Survive scoring more than 60 runs and facing more than 60 balls (Table 4)

• 4 Case 4: Survive scoring more than 75 runs and facing more than 60 balls (Table 4)

• 5 Case 5: Survive scoring more than 100 runs and facing more than 90 balls (Table 5)

Table 3

Marginal and joint survival probabilities for runs scored and balls faced of partnerships case 1 and case 2

	Case 1				Case 2
Partnership	P(R > 45)	P(B > 30)	P(R > 45, B > 30)	Rank	P(R > 50)	P(B > 60)	P(R > 50, B > 60)	Rank
Amla H M and de Kock Q	0.4117	0.5983	0.4114	5	0.3777	0.3526	0.3359	5
Bairstow J M and Roy J J	0.4522	0.5811	0.4508	3	0.4278	0.3656	0.3584	3
Cook A N and Bell I R	0.3630	0.5749	0.3626	11	0.3300	0.3286	0.2988	8
Dhawan S and Rahane A M	0.5415	0.7369	0.5415	1	0.5090	0.5144	0.4914	1
Dilshan T M and Perera M D K	0.3153	0.4234	0.3094	17	0.2851	0.1857	0.1790	17
Fakhar Z and Imam-ul-haq	0.4629	0.6700	0.4627	2	0.4351	0.4544	0.4180	2
Haddin B J and Watson S R	0.3724	0.5778	0.3713	9	0.3364	0.2904	0.2716	10
Jayawardene D P M and Dilshan T M	0.4389	0.6305	0.4364	4	0.3974	0.3265	0.3102	6
McCullum B B and Ryder J D	0.3800	0.5433	0.3768	8	0.3515	0.2766	0.2630	12
McCullum and Guptill M J	0.3297	0.4332	0.3203	16	0.3001	0.1890	0.1809	16
Sharma R G and Dhawan S	0.4115	0.6220	0.4110	6	0.3784	0.3594	0.3363	4
Smith G C and Amla H M	0.3433	0.5407	0.3429	14	0.3082	0.2927	0.2705	11
Sarkar S and Iqbal T	0.3324	0.5209	0.3316	15	0.3000	0.2684	0.2499	14
Tendulkar S R and Sehwag V	0.3731	0.5075	0.3705	10	0.3413	0.2587	0.2512	13
Tharanga W U and Dilshan T M	0.3600	0.5207	0.3586	13	0.3312	0.2926	0.2761	9
Warner D A and Finch A J	0.3995	0.5618	0.3988	7	0.3673	0.3187	0.3082	7
Shewag V and Gambhir G	0.3634	0.5305	0.3620	12	0.3282	0.2496	0.2413	15

Table 4

Marginal and joint survival probabilities for runs scored and balls faced of partnerships case 3 and case 4

	Case 3				Case 4
Partnership	P(R > 60)	P(B > 60)	P(R > 60, B > 60)	Rank	P(R > 75)	P(B > 60)	P(R > 75, B > 60)	Rank
Amla H M and de Kock Q	0.3185	0.3526	0.3047	4	0.2477	0.3526	0.2450	5
Bairstow J M and Roy J J	0.3846	0.3656	0.3470	3	0.3304	0.3656	0.3172	3
Cook A N and Bell I R	0.2736	0.3286	0.2625	8	0.2076	0.3286	0.2049	10
Dhawan S and Rahane A M	0.4502	0.5144	0.4471	1	0.3753	0.5144	0.3750	1
Dilshan T M and Perera M D K	0.2340	0.1857	0.1692	17	0.1753	0.1857	0.1455	17
Fakhar Z and Imam-ul-haq	0.3857	0.4544	0.3805	2	0.3242	0.4544	0.3231	2
Haddin B J and Watson S R	0.2749	0.2904	0.2452	11	0.2037	0.2904	0.1942	12
Jayawardene D P M and Dilshan T M	0.3252	0.3265	0.2840	7	0.2400	0.3265	0.2272	7
McCullum B B and Ryder J D	0.3020	0.2766	0.2484	10	0.2424	0.2766	0.2176	8
McCullum and Guptill M J	0.2496	0.1890	0.1715	16	0.1909	0.1890	0.1502	16
Sharma R G and Dhawan S	0.3209	0.3594	0.3045	5	0.2517	0.3594	0.2476	4
Smith G C and Amla H M	0.2488	0.2927	0.2361	13	0.1811	0.2927	0.1783	14
Sarkar S and Iqbal T	0.2452	0.2684	0.2238	15	0.1822	0.2684	0.1759	15
Tendulkar S R and Sehwag V	0.2863	0.2587	0.2380	12	0.2214	0.2587	0.2040	11
Tharanga W U and Dilshan T M	0.2815	0.2926	0.2549	9	0.2223	0.2926	0.2133	9
Warner D A and Finch A J	0.3114	0.3187	0.2870	6	0.2444	0.3187	0.2385	6
Shewag V and Gambhir G	0.2682	0.2496	0.2251	14	0.1988	0.2496	0.1849	13

Table 5

Marginal and joint survival probabilities for runs scored and balls faced of partnerships case 5

	Case 5
Partnership	P(R > 100)	P(B > 90)	P(R > 100, B > 90)	Rank
Amla H M and de Kock Q	0.1640	0.2071	0.1528	4
Bairstow J M and Roy J J	0.2604	0.2340	0.2181	3
Cook A N and Bell I R	0.1324	0.1876	0.1218	8
Dhawan S and Rahane A M	0.2783	0.3533	0.2762	1
Dilshan T M and Perera M D K	0.1099	0.0820	0.0625	17
Fakhar Z and Imam-ul-haq	0.2456	0.3091	0.2396	2
Haddin B J and Watson S R	0.1241	0.1404	0.0968	13
Jayawardene D P M and Dilshan T M	0.1437	0.1589	0.1114	10
McCullum B B and Ryder J D	0.1705	0.1385	0.1145	9
McCullum and Guptill M J	0.1239	0.0826	0.0638	16
Sharma R G and Dhawan S	0.1693	0.2035	0.1520	5
Smith G C and Amla H M	0.1074	0.1585	0.0971	12
Sarkar S and Iqbal T	0.1124	0.1380	0.0932	14
Tendulkar S R and Sehwag V	0.1458	0.1320	0.1070	11
Tharanga W U and Dilshan T M	0.1522	0.1672	0.1279	7
Warner D A and Finch A J	0.1648	0.1812	0.1446	6
Shewag V and Gambhir G	0.1215	0.1138	0.0865	15

Case 1 represents surviving in the innings to score more than 45 runs and facing more than 30 balls (5 overs), which can be considered as minimum reasonable performance by an opening pair. Results of this case are shown in Table 3. The partnership pair Dhawan S and Rahane A M has been ranked the top in the list in terms of the joint survival probability (0.54).

Case 2, case 3, and case 4 can be considered as well-founded partnerships in ODI cricket as these last more than 60 balls (10 overs) with decent scores of 50 runs, 60 runs, and 75 runs, respectively. Moreover, the strengths of the effectiveness of these partnerships increase as it moves from case 2 through case 4. In ODI cricket, it is a known strategy for the bowling team to put all their efforts to dismiss at least one partner (if not both) of the opening pair during the first 10 overs (powerplay). On the other hand, inability to accomplish that may be considered a weakness of the bowling team, which consequently lowers the bowlers’ confidence. Given that, case 4 provides more confidence to the other top order and to the middle order batsmen as they continue to build the innings to go for a higher total. Case 5 represents scoring more than 100 runs and batting more than 15 overs (90 balls), which gives an impressive foundation to the innings of the batting team.

It is interesting to notice that Dhawan S & Rahane A M have the highest joint survival probability in all five cases. Also note that the second and the third highest joint probabilities for surviving in all five cases are recorded by the same partnership pairs: Fakhar Z & Imam-ul-haq, and Bairstow J M & Roy J J, respectively. These results indicate that these two pairs are consistently better than the other partnership pairs considered in this study. Graphical representation of the above cases for all the partnerships considered in this study is shown in Fig. 2. Note that the probabilities are steadily decreasing as the cases move from case 1 to case 5; this is because of moving from case 1 to case 5 the chance of surviving for an opening pair goes down.

Fig. 2

Joint probabilities of partnerships for different cases.

4.2Individual T20 batting performance

Similar to the partnership analysis, the runs scored and balls faced by T20 batsmen in Table 2 were modeled using gamma marginals and Frank copula. The parameters of the model were estimated by maximizing the likelihood function in Equation (6). For example, the MLEs for the Pakistan top order batsman, Babar Azam were {αˆx=1.06, βˆx=41.33, αˆy=1.63, βˆy=20.32, ξˆ=22.16} . Parameter estimates for all T20 batsmen considered in this study are given in Table 7 of Appendix 6.2. Moreover, for Babar Azam, a contour plot of the joint survival distribution of runs scored and balls faced is shown in Fig. 3.

Fig. 3

Contour plot for the joint survival probability distribution obtained using gamma marginals and Frank copula for T20 batsman Babar Azam.

To evaluate the individual batting performance using the joint survival distribution, we considered 13 different cases based on different stages of the innings. We used the notation (r, b) to represent the case scoring more than r runs by surviving more than b balls. For example, the notation (20,30) indicates the case of scoring more than 20 runs while surviving more than 30 balls. The thirteen different cases for individual batting performance and their respective joint probabilities are shown in Fig. 4.

Fig. 4

Individual player performance.

Pakistan top order batsman Babar Azam had the highest joint survival probability in all thirteen categories. Interestingly, according to the ICC T20 batsmen ranking as of July 2020, Babar Azam was also the top ranking T20 batsman. Another interesting observation is that Virat Kohli who was the 10^th ranking (based on ICC ranking) batsman has been occupying the number 2 rank in 9 cases and number 3 rank in 4 cases in the joint survival probability list for the thirteen cases. Lokesh Rahul who was the second ranking (based on ICC ranking) batsman in our data set, has been occupying ranks number 2 in 4 cases, rank number 3 in 6 cases, and rank number 4 in 3 cases in our joint survival probability ranking. In this study, we particularly focused on the batsman’s ability to survive in different conditions in the game than evaluating the aggressiveness of scoring runs. Nevertheless, the results in the study fairly match with the ICC T20 batsmen ranking, which indicates that the ICC batsmen ranking system is an effective tool to evaluate the performance of batsmen in cricket.

5Conclusions

Copula applications are common in the fields such as finance and reliability engineering; nevertheless, in sports literature, copula applications are limited. In this article, we have shown the effectiveness of copula methods as an application to the game of cricket. As the main contribution of this study, we were able to model the bivariate joint distribution of the number of runs scored and the number of balls faced using copula functions for the partnerships in ODI cricket and individual batsmen in T20 cricket. Furthermore, these bivariate distributions were used to rank the batting performance of the opening partnerships as well as the batting performance of individual T20 batsmen for different stages in the game.

Based on the joint survival probability estimates, the partnership pair, Dhawan S & Rahane A M, have been ranked the top in all the five partnership cases considered in this study. The second and the third highest joint survival probabilities in all five cases were recorded by the partnership pairs: Fakhar Z & Imam-ul-haq, and Bairstow J M & Roy J J, respectively. These results indicated that above three partnership pairs were consistently better than the other partnership pairs considered in this study.

Babar Azam, who is number one in the ICC T20 batsmen ranking, has been ranked first based on the joint survival probability approach. Furthermore, Virat Kohli and Lokesh Rahul were ranked as the next two highest ranking T20 batsmen with respect to the thirteen joint survival probability cases we considered. It is important to note that the probability based ranking method proposed in this study closely complies with the ICC batting ranking. Nevertheless, there are some noticeable deviations between the two ranking approaches, for example, Virat Kohli is in the tenth place in the ICC ranking; however, he has been ranked as the second or the third from joint survival probability approach.

We believe that the results of this paper would be useful for team managers. In particular, they can use the proposed method to select the best opening pair or individual batsmen from a pool of players for a given match. Furthermore, in this study, copula applications are shown to be a useful tool to evaluate player performance in the game of cricket, and thus, this article may bring attention to the area and open up opportunities for future research. As a future extension, we are expecting to apply a similar approach to evaluate the bowling performance in cricket. In addition, another interesting future research would be to expand for multivariate approach considering other performance indicators such as the number of fours, number of sixes, and number of dot balls.