Making of a Test bowler: Time it takes to find your feet

Interesting times are back in Cricket. Debutant bowlers since September 2011 have surprised the batmen and as many as six new entrants (four of those are fast bowlers) have started their careers with a five fours. In fact the 12 debutant bowlers of in 2011 have shared as many as 18 five fors. Although data is not sufficient, but this new breed of bowlers seem to bring a hope that finally balance between the ball and the bat will be restored (see Number Game). What is really encouraging is that six of these new bowlers are less than 22 years of age. So I think there are interesting times ahead for Test Cricket where runs will not be easy, or at least that what I would like to happen.


To appreciate the importance these superb performances by the debutant bowlers we (together with Ajit Padmanabhan) looked at the probability to take a certain number of wickets in first four innings. To this end we looked at the bowlers who took at least 200 wickets and bowled at least 71 inning (otherwise the great Clarrie Grimmett would be left out and we dont want that). This way we have 57 bowlers for this analysis.


So we asked how these obviously successful bowlers performed early on in their career. In the figure below we have the probability of taking a certain number of wickets in first four innings of a bowler who finished his career with at least 200 wickets (left panel). The probability to take a certain number of wickets decreases as we increase the wickets count. This decrease is almost linear. But more data may change the picture. The figure also shows that for some of the very successful bowlers the probability to take five wickets or more is only 7% (right panel). This really reveals how amazing  the success of the class of 2011 has been.

Figure 1. Left: Probability of taking a certain number of wickets in the first, second, third and forth career inning of a bowler. The linear decay of the probability traces shows how difficult it is to take more wickets in a test innings.
Right: Probability of taking 'n' or more wickets in the first four career innings. The dark line in both panels is average of the four innings. It turns out that there is only 7% chance to take five of more wickets in your first four outings as a Test bowler. Data taken from cricinfo. 

How much time you want to invest in a new bowler
In cricket early success or failure is no predictors of long term success. So how much time a team should invest in a bowler if he is not responding. We turned to the  numbers of the bowlers who took at least 200 wickets and extended our previous analysis. This time we averaged the innings by innings wickets of these bowler in the 200+ club. It turns out that, when averaged over 57 bowlers, it takes about 10 innings before the bowlers reach their steady state of taking on average 1.5 wickets per inning. 

Figure 2. Evolution of a Test bowler. The probability distribution shown in the figure 1 is color coded and shown for first 71 innings of 57 bowlers who have taken at least 200 wickets. Dark colors mean less probability and other way around for bright colors. The thick blue line is the mean of the probability distributions. The slow rise of the blue line indicates that test bowlers take some time to find their feet at the highest level of Cricket. After about 10 innings the line hovers around 1.5 indicating that after that bowlers stats dont vary much. It also shows that even the very successful bowlers on average take only about 2 wickets per innings. This consistent with the fact that most teams field 4-5 bowler who fight to take 10 wickets.

So we argue that the teams should look into their bowlers at least for 5 test matches before giving up on them. By the same token if a bowlers has not reached his steady state of 1.5 wickets by the fifth match, there is a little chance for him, statistically speaking.


right arm over
Arvind

PS: I was helped by Ajith Padmanabhan in collection and analysis of the data.

Fall of wicket and the Nelson effect in cricket

In cricket there is a superstition including many others, that at the score of 111 or its multipels there is an increased chance of fall of wicket. The number 111 is called the 'Nelson Figure' in cricket. The legend says that in the World War II there was a army officer  Lord Nelson who lost one arm, one leg and one eye. I am not sure whether this office played cricket or was related to cricket in some way but his situation became an expression in cricket to show score of 111. The famous umpire David Shepherd made this number even more special by his one leg jump when the score reached 111.

In the third test match between India and England on the second day, when Cook and Strauss were trouncing the hapless Indian bowling attack there was a discussion on cricinfo about probability of fall of wicket on 111. So I looked in to the partnership data over last 1950 test matches.

The score at the fall of first wicket seems to follow an exponential distribution. The figure below shows that probability to score a certain score before the loss of first wicket. The different lines show this distribution for different innings (top plot). This figure shows that there is a about 0.6% chance that the wicket will fall at the score of 50.

Fraction to score a certain number of runs before the fall of first wicket.

Because of the nature of data it makes sense to plot these graphs in a log-linear plot (lower-plot). In such plots a straight line indicates an exponential distribution. The pale blue line corresponds to an exponential function. Largely the nature of exponential does not change with inning and the behavior remains similar.

Therefore, there is no data that suggests that there an increased chance for the fall of the first wicket at 111.

Next, we can look for runs scored for each wicket in different innings in the same way. In the figure below I chose the log-linear representation because that more revealing. Different colors belong to different wickets with first wicket being blue and 10th wicket being the red line. Different subplots are for different innings.

Inning by inning fraction of fall of wickets as a function of runs scored.

Naturally the lines have a higher slope for lower order wickets but largely the distribution is exponential. That is that fall of wicket, when looked over all the matches, all the conditions, all bowler and batsmen, turns out to be a random process. Exponential distribution is a simplification and more sophisticated distributions can be used to better describe this phenomenon. Maybe I will do it some other time, when I am sitting in a boring lecture, like now...

see also
http://en.wikipedia.org/wiki/Nelson_%28cricket%29

Duckworth-Lewis Calculation Graphs for Different Duration Matches

I think one the best things to have happened to cricket in 1990s was the development of a rule to update the target score should there be an interruption and team batting cannot have full set of overs. Even more remarkable was that ICC who has a reputation for choosing wrong options always, accepted this rule. This rule was devised by two statisticians Frank Duckworth and Tony Lewis and appropriately named after them.

According to Duckworth and Lewis, cricket is about using resources (overs and wickets) to achieve a target set by an opposition who batted first. Therefore, in case of an interruption, both overs and wickets should be considered. 

Exact D-L calculations often become very cumbersome. Thus, in absence of sophisticated computer, most teams in small league and friendly matches devise some ad-hoc rule. Here I show the graphs that can be used to estimate the modified target after an interruption. Originally the D-L method was developed for 50 overs matches, but usually in small leagues and friendly games, matches are shorter, therefore I estimated the D-L calculations for 20,30,40 and 50 over games.

How to read and use the graphs

x-axis is number of over remaining. The different colors of the lines refer to number of wickets fallen. So estimate the scaling factor to update the target, you need to know how many overs have been bowled (i.e. x-axis), how many wickets of the team batting second have fallen, color of the line. Corresponding to the number of wickets fallen and over bowled, y-axis give a fraction, by which you have multiply the original target to get the updated target.

Imagine, you are playing a 40 over game and set a target of 220, and rain stops the play at 25 over and the team chasing has lost 6 wickets. What should be D-L score of the team batting second to win the game? So we draw a vertical line at 15 over, and find out where does it cross the green line, from that crossing draw a horizontal line and find out what value it corresponds to on the y-axis. In this example we get 43.6 on y-axis. This means that the opposition should have scored 56.4 % runs i.e.  220 *(100 - 43.6) = 123 runs. This number would go up if they had lost one more wicket, to 142 and so on.

Enjoy your own simplified graphical D-L update rule, but I do wish that your games go on uninterrupted...

right arm over

Arvind

The Essence of Good Bowlers -- Twenty20 Cricket WC Bowling Analysis

In this post I will analyze the performance of various bowlers in the just concluded Men's Twenty20 Cricket World Cup.
Ananth reported that in Indian Premier League 2009, it did not matter who the bowlers were. Part-timers performed just the same as the regular bowlers on average.

Being an bowler myself, I could not imagine how could a part-time bowler perform be considered same as a professional bowler. Does it mean that there is nothing in toiling for years to perfect deliveries. At the moment all factors indicate that T20 will become the format of choice for furture. But if the quality of a bowler is not relevant in this form of cricket, in all possibility the art of quality bowling would dies and there wont be any Ashes for the bowlers.

It is clear that there is a big difference between Glenn McGrath and say Ashish Nehra when we compare them in Test or limited over cricket matches. In this post I will try to identify the differences between specialist bowlers and part-timers.

Data
I took the bowling data from cricket.org. I extracted runs conceded and wickets taken by a bowler. Subsequently I divided all the bowlers used in the T20 world-cup in four groups according to the number of overs each bowler bowled. The four groups consists of bowlers who bowled (1) 1-5 overs (red), (2) 6-12 overs (orange) (3) 13-21 overs (pale blue) (4) 22-28 overs (blue). It is possible that some regular bowlers ended up in the 6-12 over group because their team made an early exist. I did not correct for this. However, in principle we can compare the performance of bowlers with 1-5 over with rest of the bowlers.

Hypothesis
Very naively I hypothesize (and you will agree with that) that a good bowler is more likely to bowl a good over and a bad over is a less likely event. On the other hand a part-time bowler is equally likely to bowl a good over and bad over. These differences may not be captured by average economy rate or strike rate of the bowlers, because the distribution of runs-per-over and wickets-per-over are likely to be skewed for specialist bowlers and, symmetric and wide for part-time bowlers.

Bowling performance distribution
In panel A below I show the mean economy rate of the bowlers in four different groups. The average values are indeed different but the standard deviation is too big and the differences are not significant. Similarly in panel B I show the probability of a wicket. Once again the group-4 is likely to take more wickets but the standard deviation is rather high and differences with respect to the first group (part-timers) are not significant.


So panels A and B show that first and second order moment (mean and standard deviation) of the wickets and run distribution are not informative in differentiating the quality of various bowlers.

A detailed look at the data in terms of the full distribution give a complete picture. In panel D I plotted the distribution of runs given in over by specialist bowlers (blue) and part-times (red). Now the differences are much more stark.
This figure also provides support for my hypothesis that a good bowler is more likely to bowl a good over than a bad one, and the distribution of the specialists is skewed with a fat-tail. As the quality of the bowlers declines (as indicated by their use in the tournament), the distribution becomes symmetric and broader such that bowers who bowed 1-5 overs are equally likely to bowl an maiden over and an over with 16 runs.


The differences between the distributions of the specialists and part-times bowlers' economy rate is very satisfying not just because it supports my hypothesis, but also because it gives a more quantitative way to differentiate the quality of bowlers.
There are sophisticated tools to differentiate distributions shown in panel D, but at the moment I have too few data. Later when I will have a bigger database I will be able to put numbers on quality of bowlers.
Cricket.org only recently has started to put the commentary of full matches, else it would have been indeed possible to provide a quantitative difference between Ashish Nehra and Glenn McGrath beyond the averages.

right arm over
Arvind

Why Garry Sobers is better than Jacque Kallis -- Analysis of Test Cricketers

In my previous post I gave hints how one can estimate the quality of a player as a batsman, bowler or all-rounder. The idea is rather straightforward. Good batsmen will increase their cumulative runs at a much faster pace as a function of matches played, while bowler will increase their cumulative wickets at much faster pace. The Criterion So I looked into the data of about 400 players, for their total runs and wickets over their career. The plot is shown below. In panels A runs are plotted as a function of matches played. It turns out that the top 20% batsmen in the history of test cricket increased their runs at nearly 72 runs per match. Top 20% bowlers on the other hand increased their runs at about 10 runs per match. Similar trend is bowling data, where top 20% bowlers increase their wickets at 4.73 per match. So 72 runs per match and 4.7 wickets is a criteria to be a good bowler or batsman, respectively.  

What is an all-rounder These two numbers (72 for batting and 4.7 for bowling) also can allow us to suggest how close someone is close to be a good bowler and batsman simultaneously i.e. all-rounder. How do we check whether a player has been a more of a bowler or a batsman or both. In fact, by following careers of some players, like Steve Waugh who was considered as an all-rounder in his early days but ended his career as a batsman. Similarly Ravi Shastri. That is a player with good abilities with bat and ball, can switch roles very dynamically. So I thought to see the runs per match (run slope) and wickets per match (wicket slope), over a period of 10 matches. This window of 10 matches was slided by 3 matches to get another estimate. The sliding can also be done with 5 matches as many Test series are of that length, but then I consider the fact that an allrounder may not play full series due to injuries or such factors. With a window 10 matches which was slided by 3 matches, I calculated the run-slope and wicket-slope of 28 players. Average run-slope and average wicket-slope are plotted across each other. The players name is indicated in the figure. It is clear that Brian Lara, Ricky Ponting, Rahul Dravid, Sachin Tendulkar are right on the top, together with Sunny Gavaskar. This kind of plot also reveal similarity between Viv Richards, Alan Border and Javed Miandad. On the bowling side, that is in left-bottom corner, all usual suspects (Glen McGrath,Ambrose, Warne, Murali) are placed. Now the allrounders are in the middle of this plot, the likes of Imaran, Kapil, Botham, Daniel Vettori, Flintoff. I also added Vinno Mankad and Keith Miller to get a historical perspective. So indeed the good allrounders are those who maintain a good run-slope and wicket slope. What wait, what about Gary Sobers and Jacque Kallis?

 

The curious case of Sobers and Kallis 

In figure above Gary Sobers is placed slightly higher than Jacque Kallis. Both are great players and simple analysis indicates that Kallis is slightly better than Sobers. But in the figure above, subtle differences are accounted for and we see that Sobers was slightly better, both as a bowler and a batsman. To get a detailed look on this, I plotted the progression their cumulative wickets and cumulative runs in figure below. The red trace(Sobers) in both panels (A,B) is above Kallis (blue trace). Panels C and D show the run-slope and wicket slope for both in 10 match segments. An interesting picture emerges. Sobers leads Kallis on batting scale very early. Actually its the inning of 365 runs that gave Sobers an early surge in run-slope. Kallis was a bit slow in scoring runs early on. Both players did good with ball only in their mid-career i.e. around 40-60th matches and that improved their image as all-rounders.  

 From panels C and D in figure above we can safely conclude that in their mid-career both Kallis and Sobers were highly comparable, in fact, tended to be more like bowling all-rounders . Sobers started to be more of a bowler towards the end of his career as see by dip in red trace in panel D. Kallis is having a very good time with bat but struggling with ball of late. In general both Sobers and Kallis more or less all the time were in top 20% batter group, but only occasionally made it to 20% bowlers. These dynamic changes in the nature of performances over short periods of 10 matches in case of Kallis and Sobers prompted me to do make similar figures for Ian Botham, Imran Khan, Kapil Dev and Richard Hadlee. In the beginning of their respective career the four celebrated all-rounders were more or less alike, but then around 50th match Richard Hadlee outperforms his compatriots in batting!!. Just like the beginning all four all-rounders had similar decline in their batting and bowling towards the end. Kapil Dev observed a slower decline, which perhaps is the reason of his longevity. Again these four all-rounders were nearly always in top 20% bowlers but only occasionally in top 20% batsmen.

 

From an all-rounder to batsmen Further, I wanted to look at the career progression of Steve Waugh, Ravi Shastri and Sanath Jayasuriya. Steve Waugh was regarded as an all-rounder in the beginning. The analysis show that he had a start like that of Kallis and Sobers but he was far from the class of Imran/Botham/Kapil. His decline as a bowler and rise as a batsman is clearly visible in the red traces in panels C,D below. Towards the end he was a batsman, but you don't need to to bowl when Glen McGrath and Shane Warne are on your side. Similarly Ravi Shastri was a bowler of about average class in the beginning and ended as an average batsman and a poor bowler. Sanath Jayasuriya has interesting spell in his career when he was a good bowler, or a good batsman or both. Around the time of his 50th test match he was in the class of Sobers/Kallis, currently he is just a good batsmen. They way he is showing no signs of retirement, who know he may again return to his all-roudner status for a short time.

 

In all I think estimation of run-slope and wicket-slope in short duration is a very good indicator of the abilities of a player as an allrounder or batsman or a bowler. This analysis reveals that the four celebrated all rounders (Kapil/Imran/Botham/Hadlee) were in fact more bowlers who were very effective with bat. On the other hand Sobers and Kallis are more of batsmen who are really good with ball. right arm over Arvie PS: In the plots above I should have added the error-bars but I just wanted to keep the figures uncluttered, but if someone needs I can provide those.