Sunday, January 04, 2009
Form
Russ in a post here suggests a way of measuring form (go and read that post, it's full of goodies). The broad question is, if there are two batsmen with the same true talent, and one has better recent form, will that batsman tend to average more in his next innings?
To start with, Russ weights more recent innings more heavily than earlier innings. Specifically, the k-th innings for a batsmen with N innings in his career is weighted 0.95N-k. (This is, I believe, similar to what the ICC rankings do.)
Now, one problem with assessing "form" statistically is that a batsman will usually play a series against one team, followed by a series against another, etc. Since one of those two opposition teams can have a much stronger bowling attack than the other, what may appear to be good form and bad form may simply be a result of playing against weak bowlers and then strong bowlers. So, for everything I do in this post, I'll adjust the batting averages by the quality of the attack, as explained here.
So, when I talk about a regular average, I really mean an adjusted average. When I talk about a weighted average, I'll mean Russ's weighting by how recent the innings was (each innings also being adjusted for the quality of the attack).
Before continuing about form, I'll just look at the weighted average as a predictive tool. For all batsmen with at least 50 innings, I calculated career-to-date averages and weighted averages, as well as a 10-innings moving average. Then, from the 11th innings of each batsman's career, I calculated the absolute difference between his next innings and each of those three measures. (If the innings was a not-out, I used the not-out score as the absolute difference for each measure.) Then I averaged these errors. I did the same for all batsmen, and then found the "average average" error. The regular average was the best predictor, about 1% better than the weighted average, and 4% better than the moving average. The weighted average becomes more accurate if the 0.95 in the formula is increased towards 1, but it is always worse than the regular average.
So, as a measure of true talent of a batsman, I'll use the regular average rather than the weighted average.
Now to the question of defining form. Russ does this by taking a weighted log average. Defining Ri as the runs scored in the i-th innings, and wi as the weight of that innings, this weighted log average is:
I've actually modified this a bit. If the i-th innings is a not-out, I didn't include it in the sum in the denominator. I hope this isn't too great a crime against statistics.
The measure of form is then the ratio of the weighted log average to the weighted average. Now, if scores are distributed exponentially, then this ratio is about 0.56 (well, it is with equally weighted innings at least). If a batsman makes the same score every innings (and gets out!), the ratio is 1. If a batsman recently has one big score and a bunch of little scores, the ratio is down towards 0.3. So, good form is a high ratio, bad form is a low ratio.
Because I exclude not-outs in the denominator, it's possible to get ratios greater than 1. I'm not really sure how to interpret these, but let's carry on anyway.
Russ's hypotheses are (I hope I've got this right):
a) If there are two batsmen with a similar average, one with a typical ratio and one with a low ratio, then the one with the low ratio will tend to average more in his next innings. The logic here is that the batsman with the low ratio is capable of larger scores, whereas the other batsman is just not so good.
b) Given two batsmen with the same average, one with a high ratio will tend to do better in his next innings than one with a typical ratio.
Both of these are correct, somewhat to my surprise. I went through all batsmen, and for each innings (after the tenth in their career), calculated the career-to-date average, and the ratio-to-date, and binned them as in the table below. I then calculated the overall average for each bin.
Ratios are down the left-hand side, averages across the top. The figures are the low end of the bin. So, e.g., the '5' means that the bin is for averages 5 to 9.99, the '10' is for averages 10-14.99, etc. Only bins with at least 50 innings are shown; bold is used for at least 100 innings.
When the ratio is very low, the batsman does indeed tend to average much more in his next innings. (Since I've used regular averages to define the true talent, the top row may be full of players early in their career. I'm not sure.) Going down each column, the minimum is usually somewhere around 0.5 to 0.6, which seems to correspond to the 0.56 that you'd expect from the exponential distribution.
Really good recent form seems to give a 20% boost and sometimes more. This is a lot more than I had expected.
(My thinking on this issue seems to have been confused — in my last post I said that Johnson was good because he kept getting starts, which is consistent with this analysis.)
To start with, Russ weights more recent innings more heavily than earlier innings. Specifically, the k-th innings for a batsmen with N innings in his career is weighted 0.95N-k. (This is, I believe, similar to what the ICC rankings do.)
Now, one problem with assessing "form" statistically is that a batsman will usually play a series against one team, followed by a series against another, etc. Since one of those two opposition teams can have a much stronger bowling attack than the other, what may appear to be good form and bad form may simply be a result of playing against weak bowlers and then strong bowlers. So, for everything I do in this post, I'll adjust the batting averages by the quality of the attack, as explained here.
So, when I talk about a regular average, I really mean an adjusted average. When I talk about a weighted average, I'll mean Russ's weighting by how recent the innings was (each innings also being adjusted for the quality of the attack).
Before continuing about form, I'll just look at the weighted average as a predictive tool. For all batsmen with at least 50 innings, I calculated career-to-date averages and weighted averages, as well as a 10-innings moving average. Then, from the 11th innings of each batsman's career, I calculated the absolute difference between his next innings and each of those three measures. (If the innings was a not-out, I used the not-out score as the absolute difference for each measure.) Then I averaged these errors. I did the same for all batsmen, and then found the "average average" error. The regular average was the best predictor, about 1% better than the weighted average, and 4% better than the moving average. The weighted average becomes more accurate if the 0.95 in the formula is increased towards 1, but it is always worse than the regular average.
So, as a measure of true talent of a batsman, I'll use the regular average rather than the weighted average.
Now to the question of defining form. Russ does this by taking a weighted log average. Defining Ri as the runs scored in the i-th innings, and wi as the weight of that innings, this weighted log average is:
/ SUM wi log(Ri) \
exp | ------------- |
\ SUM wi /
I've actually modified this a bit. If the i-th innings is a not-out, I didn't include it in the sum in the denominator. I hope this isn't too great a crime against statistics.
The measure of form is then the ratio of the weighted log average to the weighted average. Now, if scores are distributed exponentially, then this ratio is about 0.56 (well, it is with equally weighted innings at least). If a batsman makes the same score every innings (and gets out!), the ratio is 1. If a batsman recently has one big score and a bunch of little scores, the ratio is down towards 0.3. So, good form is a high ratio, bad form is a low ratio.
Because I exclude not-outs in the denominator, it's possible to get ratios greater than 1. I'm not really sure how to interpret these, but let's carry on anyway.
Russ's hypotheses are (I hope I've got this right):
a) If there are two batsmen with a similar average, one with a typical ratio and one with a low ratio, then the one with the low ratio will tend to average more in his next innings. The logic here is that the batsman with the low ratio is capable of larger scores, whereas the other batsman is just not so good.
b) Given two batsmen with the same average, one with a high ratio will tend to do better in his next innings than one with a typical ratio.
Both of these are correct, somewhat to my surprise. I went through all batsmen, and for each innings (after the tenth in their career), calculated the career-to-date average, and the ratio-to-date, and binned them as in the table below. I then calculated the overall average for each bin.
Ratios are down the left-hand side, averages across the top. The figures are the low end of the bin. So, e.g., the '5' means that the bin is for averages 5 to 9.99, the '10' is for averages 10-14.99, etc. Only bins with at least 50 innings are shown; bold is used for at least 100 innings.
r/a 5 10 15 20 25 30 35 40 45 50 55
0.35 23.4 28.4 28.7 40.8 41.3
0.40 12.6 18.8 21.4 31.7 30.5 34.9 40.3 40.7 55.3
0.45 7.4 12.1 15.9 20.9 26.7 30.1 34.0 46.9 39.9 50.2
0.50 9.6 11.1 17.6 22.2 26.4 31.7 36.3 34.9 43.5 46.0
0.55 8.4 11.9 16.5 21.0 26.1 31.6 35.2 40.1 42.6 53.3
0.60 7.6 12.2 18.3 24.4 27.9 33.5 38.2 42.7 46.6 58.0 40.4
0.65 7.8 12.3 18.1 24.7 28.0 33.4 39.2 43.3 48.7 48.1 46.6
0.70 8.2 12.4 19.3 25.7 27.5 34.4 40.4 46.1 44.0 56.7
0.75 9.3 15.3 17.4 23.1 29.3 38.9 41.0 47.4 51.8 50.1
0.80 9.8 12.0 17.7 24.4 30.5 35.1 48.2 54.2 51.8 58.9
0.85 15.0 22.1 25.8 38.8 40.5 44.6 51.3 61.9 46.8
0.90 15.6 26.7 42.1 42.6 59.2 54.1
0.95 46.5
1.00 16.6 35.9 32.5 44.7 53.9 68.5 73.2 62.9
When the ratio is very low, the batsman does indeed tend to average much more in his next innings. (Since I've used regular averages to define the true talent, the top row may be full of players early in their career. I'm not sure.) Going down each column, the minimum is usually somewhere around 0.5 to 0.6, which seems to correspond to the 0.56 that you'd expect from the exponential distribution.
Really good recent form seems to give a 20% boost and sometimes more. This is a lot more than I had expected.
(My thinking on this issue seems to have been confused — in my last post I said that Johnson was good because he kept getting starts, which is consistent with this analysis.)
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