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Weighting rounds by sequence or time?

PUBLISHED — 2022-03-19

When estimating baseline skill in our model (i.e. predicting future performance only using
*total* strokes-gained) we average a golfer's historical performances with decreasing
weight applied to rounds played further into the past. In an early version of our model,
this weight decayed only with the sequence in which rounds were played (i.e. whether a golfer's 5th most
recent round was played 10 days ago or 50 days ago did not affect the weight it was given).
This left something to be desired, so beginning in 2021
we started using another weighting method that varied with time:
the weight applied to a given round
was a function of the number of days into the past it was played. Both of these weighting schemes —
by sequence or time — capture useful information that the other misses.
For a golfer who is playing a consistent
schedule and is in the middle of a season, the difference between the averages created by each weighting method will be relatively small.
But for a golfer who is coming off a long layoff, or has only played a few rounds in recent seasons (e.g.
Wesley Bryan currently), the difference between the two schemes can be large.
Clearly we should be
averaging these two weighted averages, but what's the right mix?

In the current version of our model, the choice of how much weight to apply to a golfer's sequence-weighted average SG versus their time-weighted average SG depends on a few basic variables. For example, golfers for whom we have very little data have most of the weight applied to their sequence-weighted average. For the majority of golfers playing in the events we care most about predicting (PGA Tour and European Tour events), their time-weighted and sequence-weighted averages are similar, and so the choice of weight applied to each is fairly inconsequential. But, every so often an edge case would come along that wouldn't be handled particularly well by the model. This was a bit annoying, but these edge cases were almost never relevant players in their respective tournaments. However this week Brendan Jones popped up on our radar; Jones is playing in the Golf Challenge NSW Open this week, an Australasian Tour event. We provide baseline skill predictions for the fields on these minor tours and allow users to simulate the tournaments. Jones had the best predicted skill in this field by more than a shot (!), while Bet365 had him as just the 20th or so best player, with win odds of 80 to 1. Something was clearly amiss.

Take a closer look at Jones' profile. I had never heard of Brendan Jones, but he's had a solid career: his 50-round MA reached the level of a Top 25 player back in 2011 and he's racked up a number of wins on the Japan Tour (and this is only using the data we have: 2010-onwards). As recently as late 2019 he was consistently ranked inside our Top 400 players, which typically requires a skill level around -0.9. However in each of the 2020 and 2021 seasons Jones played only 1 event, and he played his first event of 2022 a couple weeks ago. Critically, Jones played very well in his one event of 2022, finishing 2nd and averaging +1.3 True SG per round. Weighting his rounds by their sequence yields a predicted skill of -1, while weighting them by time yields a skill of +0.65. (You could argue that even the -1 skill is too high given the sparcity of Jones' recent playing schedule, but that's a different discussion.) For our final prediction it looks like we assigned ~60% weight to the time-weighted average and 40% to the sequence-weighted, giving Jones an overall skill level right around 0!

The problem here, from the model's perspective, is that it can't effectively distinguish between someone like Jones who has only played a few rounds after a long layoff, and Wesley Bryan, who has played 5 events since his last layoff. Clearly Bryan's time-weighted average should receive more weight than Jones'. We had a few crude ways of attempting to do this, like adjusting the weight based on the number of rounds played in the last 3 months, or 1 year, but there will always be unique player histories that can slip through these conditions, and Brendan Jones was one of them. Jones' silly prediction motivated me to revisit this problem, and we now have a very nice solution: the weight applied to a golfer's time-weighted SG average versus their sequence-weighted SG average will depend on the uncertainty of each estimate! Recall that the variance of \( aX_{1} + bX_{2} \) is equal to \((a^2 + b^2) \cdot Var(X) \), assuming that \( X_{1}, X_{2} \) are independent and identically distributed with variance equal to \( Var(X) \). In our context,*a* and *b* are the weights applied to each round (except we would have as many weights as
we have rounds), and \( X_{1}, X_{2} \) are strokes-gained values, which are random variables.
The higher is this variance term for the time-weighted average (relative to the sequence-weighted average)
the *less* weight we apply to the time-weighted average. When will the variance of a weighted
average be high or low?
In the case of only 2 rounds, \((a^2 + b^2) \) is minimized when \( a = b = 0.5 \). This same principle
applies when we have many rounds: the more equal are the weights, the lower the variance will be.

Returning to Brendan Jones, the weight on the three 2022 rounds in his time-weighted average were all in excess of 20% (!), while their weights in his sequence-weighted average were all in the 2-2.5% range. If you think of re-sampling each of Jones' rounds, the random draw you get for the 3 rounds in 2022 will have a huge impact on his time-weighted average, while they will have only a small effect on his sequence-weighted average. That is, there is much more uncertainty around Jones' time-weighted estimate than his sequence-weighted esimate, and so — all else equal — basic stats theory would tell us to decrease the weight we assign to Jones' time-weighted average when using it to make predictions.

This is not to say that the variance of the weighted average should be the sole determinant of our time-average vs sequence-average weights. The time-weighted averages will tend to be more variable than the sequence-weighted averages, but that doesn't mean they will necessarily be worse for predicting future performance. In golf, the recency of the data determines how useful it is to a large degree, which gives us this tradeoff between recency and variance. As a reference point, for players playing normal schedules, the typical weight our model applies to their time-weighted average is around 70%, which reflects the fact that it is the "better" weighting scheme when data is abundant. The variance of the time-weighted average relative to the sequence weighted average provides us with a single parameter that can tell us when to adjust the time-average weight away from the default of 70%. This is a much more parsimonious (and also objectively better) solution than using cutoffs for rounds played over different time horizons to adjust this weight. In the case of Brendan Jones, our new method that uses the variance as an adjustment parameter puts 90% of the weight on his sequence-weighted average and just 10% on his time-weighted. This yields a final prediction of -0.81 — still high relative to the betting markets, but not ridiculously so. In the case of Wesley Bryan, 43% of the weight is given to his time-weighted average.

In the current version of our model, the choice of how much weight to apply to a golfer's sequence-weighted average SG versus their time-weighted average SG depends on a few basic variables. For example, golfers for whom we have very little data have most of the weight applied to their sequence-weighted average. For the majority of golfers playing in the events we care most about predicting (PGA Tour and European Tour events), their time-weighted and sequence-weighted averages are similar, and so the choice of weight applied to each is fairly inconsequential. But, every so often an edge case would come along that wouldn't be handled particularly well by the model. This was a bit annoying, but these edge cases were almost never relevant players in their respective tournaments. However this week Brendan Jones popped up on our radar; Jones is playing in the Golf Challenge NSW Open this week, an Australasian Tour event. We provide baseline skill predictions for the fields on these minor tours and allow users to simulate the tournaments. Jones had the best predicted skill in this field by more than a shot (!), while Bet365 had him as just the 20th or so best player, with win odds of 80 to 1. Something was clearly amiss.

Take a closer look at Jones' profile. I had never heard of Brendan Jones, but he's had a solid career: his 50-round MA reached the level of a Top 25 player back in 2011 and he's racked up a number of wins on the Japan Tour (and this is only using the data we have: 2010-onwards). As recently as late 2019 he was consistently ranked inside our Top 400 players, which typically requires a skill level around -0.9. However in each of the 2020 and 2021 seasons Jones played only 1 event, and he played his first event of 2022 a couple weeks ago. Critically, Jones played very well in his one event of 2022, finishing 2nd and averaging +1.3 True SG per round. Weighting his rounds by their sequence yields a predicted skill of -1, while weighting them by time yields a skill of +0.65. (You could argue that even the -1 skill is too high given the sparcity of Jones' recent playing schedule, but that's a different discussion.) For our final prediction it looks like we assigned ~60% weight to the time-weighted average and 40% to the sequence-weighted, giving Jones an overall skill level right around 0!

The problem here, from the model's perspective, is that it can't effectively distinguish between someone like Jones who has only played a few rounds after a long layoff, and Wesley Bryan, who has played 5 events since his last layoff. Clearly Bryan's time-weighted average should receive more weight than Jones'. We had a few crude ways of attempting to do this, like adjusting the weight based on the number of rounds played in the last 3 months, or 1 year, but there will always be unique player histories that can slip through these conditions, and Brendan Jones was one of them. Jones' silly prediction motivated me to revisit this problem, and we now have a very nice solution: the weight applied to a golfer's time-weighted SG average versus their sequence-weighted SG average will depend on the uncertainty of each estimate! Recall that the variance of \( aX_{1} + bX_{2} \) is equal to \((a^2 + b^2) \cdot Var(X) \), assuming that \( X_{1}, X_{2} \) are independent and identically distributed with variance equal to \( Var(X) \). In our context,

Returning to Brendan Jones, the weight on the three 2022 rounds in his time-weighted average were all in excess of 20% (!), while their weights in his sequence-weighted average were all in the 2-2.5% range. If you think of re-sampling each of Jones' rounds, the random draw you get for the 3 rounds in 2022 will have a huge impact on his time-weighted average, while they will have only a small effect on his sequence-weighted average. That is, there is much more uncertainty around Jones' time-weighted estimate than his sequence-weighted esimate, and so — all else equal — basic stats theory would tell us to decrease the weight we assign to Jones' time-weighted average when using it to make predictions.

This is not to say that the variance of the weighted average should be the sole determinant of our time-average vs sequence-average weights. The time-weighted averages will tend to be more variable than the sequence-weighted averages, but that doesn't mean they will necessarily be worse for predicting future performance. In golf, the recency of the data determines how useful it is to a large degree, which gives us this tradeoff between recency and variance. As a reference point, for players playing normal schedules, the typical weight our model applies to their time-weighted average is around 70%, which reflects the fact that it is the "better" weighting scheme when data is abundant. The variance of the time-weighted average relative to the sequence weighted average provides us with a single parameter that can tell us when to adjust the time-average weight away from the default of 70%. This is a much more parsimonious (and also objectively better) solution than using cutoffs for rounds played over different time horizons to adjust this weight. In the case of Brendan Jones, our new method that uses the variance as an adjustment parameter puts 90% of the weight on his sequence-weighted average and just 10% on his time-weighted. This yields a final prediction of -0.81 — still high relative to the betting markets, but not ridiculously so. In the case of Wesley Bryan, 43% of the weight is given to his time-weighted average.

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