Predicting golf scores at the shot level

1.1Performance based on aggregate golfer skill factors

There have been a number of efforts to quantify a golfer’s performance. Much of the early work model performance based on aggregate golfer skill factors. These aggregate skills are means, percentages and totals representing different parts of a golfer’s game. Table 1 shows the core aggregate skill factors and the abbreviations we will use in describing previous literature.

Davidson and Templin (1986) performed a regression over the top 119 money golfers of the 1986 PGA season to forecast scoring average and prize money earned based on driving proficiency (composed of DRIVEDIST and DRIVEACC), GIR, TOTALPUTT and SANDSAVE. They found that GIR, TOTALPUTT and driving proficiency explained 86% of a players scoring variance, with GIR being the most important skill factor [1]. Using the same dataset but considering a larger number of features that were min-max normalized, Shmanske (1992) performed a regression to model dollar winnings. His work included DRIVEDIST, DRIVEACC, GIR, SANDSAVE, TOTALPUTT, PUTTGIR and the corresponding amount of time spent in practice for each of the skills. He found that PUTTGIR, DRIVEDIST and GIR (in that order) were the three most important skills that contributed to earnings [2]. Jones (1990) also modeled money earned in 99 of the top 100 money winners of the 1980 PGA season using TOTALPUTT, DRIVEDIST and GIR. Similar to Shmanske, he found that TOTALPUTT and GIR are the two highest correlated features to earnings [3]. As an extension to the findings of Davidson and Templin, Belkin, et al. (1994) analyzed a larger dataset of PGA stats for years 1986 to 1988. Performing stepwise multiple regression (a method that doubles as a means of feature selection), they found that the most important predictors of scoring average are GIR, DRIVEACC and TOTALPUTT [4]. Dorsel and Rotunda arrived at a similar conclusion by also performing multiple regression on the same aggregate features for the top 42 players of the 1990 PGA tour [7]. They also show that GIR was the most important skill. On the contrary, Engelhardt (1995) by performing a rank order correlation between the top 10 ranked money winners and top 10 ranked in each aggregate skill category of the 1993 and 1994 PGA seasons, argued that while GIR was seen to be the largest contributor to success up to this point, the trend has now shifted to DRIVEDIST and DRIVEACC being more important than GIR [5]. Peters (2008) using the same methodology, performed his analysis with DRIVEACC, DRIVEDIST, GIR, PUTTGIR, SANDSAVE and years of experience as his regressors. His dataset comprised of tournaments in the PGA tour from 2002 to 2005. He found that PUTTGIR was the most important skill, while DRIVEDIST, SANDSAVE and years of experience were insignificant factors [6].

The studies up to this point in time made a very reasonable assumption that traditional PGA golfer aggregate skills can be used as determinants for performance. While this is generally correct, one has to question the variability of the coefficients of the regressions and the variability in which skill factors are most important across these studies. While some of this effect can be attributed to the gradual evolution of the game in time, a larger part of this discrepancy is due to the fact that these aggregate features are highly correlated, producing non-stable results. For instance, putting is correlated with GIR in that when a player hits the green in regulation, she will have a longer than average putt. In the same way, golfers who are better drivers are more primed to hit the green in regulation.

1.2Attempts at decreasing correlations between skills

Alexander and Kern (2005) realized the correlated skills problem and attempted to construct a pure measure of each skill. Using traditional aggregate stats from PGA tournaments from 1992 to 2001, they construct pure aggregate statistics for approach shots (IRON) and putting skill. They computed the former by taking the residuals when GIR was regressed from DRIVEDIST and DRIVEACC, and the latter by taking the residuals when PUTTGIR was regressed from IRON. Intuitively, this means the variation in GIR that could not be explained by DRIVEDIST and DRIVEACC can be attributed to iron playing skill, with the same being is true for PUTTGIR and putting skill. They find that pure putting skill is the most important factor that contributes to a golfer’s success (measured in earnings this time) [8]. Baugher, et al. (2014) employed the same featurization and analyzed PGA tournaments from 2006– 2013. They also show that putting skill is the most important determinant of success [9].

With the isolation of pure golfer skill factors, both authors Alexander, et al. (2005) and Baugher (2014) have also studied the change in the importance of these skills in time [8,9]. Alexander, et al. (2005) notes an increase in the marginal value of DRIVEDIST and a decrease in the value of pure putting skill in the PGA from 1992 to 2001. Among other trends, Baugher, et al. (2014) notes the increase in importance of DRIVEDIST and DRIVEACC in the PGA from 2008 to 2013. This increase in importance, as they have mentioned in their work, coincides with the decision of the PGA Tour to increase the length of the courses in time.

1.3Variability caused by course difficulty and field strength

Needless to say, changes in the course and relative golfer field strength affects performance of a golfer. Moy and Liaw (1998) compared datasets from the PGA, Senior Tour and LPGA for 1993 season and found that different leagues have different apparent skill factors for success. They report that the PGA commands a more well rounded game with PUTTGIR and GIR as the two largest factors of success, while the Senior Tour and the LPGA, due to its shorter courses, emphasize iron and putting skills [10]. Pfitzner and Rishel (2005) and Shumanske (2000) also investigate determinants of golfer success in the LPGA and how they compare to the PGA [11]. As expected, the relative importance of golfer skill factors varies across tournaments. Since golf is a ranking based sport, field strength also affects a golfer’s individual performance in terms of money earned. In later sections, we will talk more about how researchers have taken a first crack at solving this problem [11,12].

1.4Shot level analysis

In 2008, Brodie presents his work on shot level analysis, currently known as “Strokes Gained” as an alternative to the use of the traditional aggregate skill factors [14]. The idea of shot level analysis, wherein the value of a shot is measured by the difference in “state” between the location of the ball before and after the shot was first introduced by Cochran and Stobbs (1986) and Landsberger (1994), but never gained popularity due to the lack of rich data [13].

Shot level analysis provides a more granular way to evaluate performance compared to the aggregate skill factors described in early sections in that it achieves isolation of individual shots (problem discussed in section 1.b) and provides a way to distinguish large and small errors (e.g. missing the green by 1 yard or 30 yards). Equation (1) shows the base strokes gained equation for a shot.

In 2011, Brodie published an extension to his work where he computes benchmarks for average strokes to hole out not only based on distance but also based on shot type (tee, putt, recovery, etc) [15].

The tee shot function u (dist, shot _ type = tee _ shot) was modeled as a piecewise polynomial function for par-3, par-4 and par-5. Shots within 50 yard (sand, rough or fairway) were simply means over each shot type. Putting is a probabilistic model that is a combination of a one-putt and three-putt model. The one-putt model was based on an intuitive physical model developed by Holmes (1991), where a normal distribution was fit for both distance the ball will travel and the angle of putt error. The three putt model was simply selected by Brodie based on statistical data fit and is defined in equation (6) of his paper. The resulting putting model assumes four putts and above have zero probability, hence the two-putt probability is simply the probability that the golfer neither one-putts or three-putts. u (dist, shot _ type = putt _ shot) would then be the expected value of the said discrete probability distribution [15].

1.5Adjusted shot level analysis

The latter half of Brodie’s (2011) paper adjusted strokes gained based on round and course difficulty similar to the method performed by Connolly and Rendleman (2008). The golfer’s 18-hole round score was expressed as the sum of player skill (that is allowed to vary over time), course level difficulty and round level difficulty by iteratively fitting a smoothing cubic spline on time varying individual player skill and then course and then round level difficulty [16]. This allowed Brodie to rank players based on their skills in different types of shots while accounting for course and round difficulty.

4.1Golfer features

The most important aspects to consider when constructing features that describe a golfer’s skill factors is weighing past data to measure current skill levels and normalizing observed skill based on course difficulty. The golfer skills that we ultimately want to produce are Scoring Average, Driving Distance, Driving Spray, SSG(Smoothed Strokes Gained): long approach, SSG: short game, SSG: putting (see Table 2 for more details). The terminology used in this table will be explained in later subsections.

4.1.3Iterative updating

We iteratively update time weighted features and normalized data until the difference of the two mean terms converges. That is to say we compute the value for the feature F as stated in equation (4) and use that value as the skill we assign each player when we compute v_{field-preround} on equation (3). The new skills we derive are fed back to equation (4) to recompute the next iteration of weights and skills.

Table 3

Weights of different weighting methods for different skills

Weighting Method	Scoring Average	Driving Spray	SSG: Iron Play	SSG: Putting
12-week rolling average	0.08	0.16	0.17	0.02
Half-life 50 days	0.17	0.31	0.18	0.11
Half-life 150 days	0.31	0.25	0.29	0.25
Half-life 400 days	0.26	0.14	0.27	0.21
Half-life 800 days	0.15	0.09	0.13	0.2
Intercept	2.16	0.4	– 0.19	0.13

Fig.1

Featurized Scoring Average in Time for Representative Golfers.

4.2Hole features

Computing hole features was an exercise in manipulating spatial data. Hole features built include par (categorical), green size, sand area, normalized penalty strokes, and fairway width. These are shown in more detail in Table 4.

Fairway width and par values were readily available in ShotLink, while we computed the green size, sand area and normalized penalty strokes.

4.2.1Convex hull to estimate green size

Convex hull is a known technique for which given a Euclidean plane, the smallest convex set that contains all points X is computed. Figure 2 shows a visualization of convex hull.

ShotLink gives both the labeled location (i.e. Tee Box, Green) as well as the spatial (x, y, z) coordinates of each shot. To estimate the green size, we can filter for all shots from the Green and compute the convex hull around these points to estimate the size. On average each green has about 700 putts on it per standard tournament, giving us enough data points for a decent estimate. Figure 3 shows an example of hole 1 at Silverado CC in Napa, California.

Fig.2

Convex Hull of a set of points.

Fig.3

Putts on the Green of Hole 1 in the Safeway Open.

The reason convex hull is ideal for putts on the green is because there is only one green per hole. This is not the case for sand bunkers, hence we employ K-nearest neighbors to classify points in the space. It is no coincidence that the points above form four clusters. Each cluster corresponds to a round of the tournament; the course superintendent changes the location of the hole after each round. Note that this method will slightly underestimate the size of each green, as there will be locations on the green outside the hull boundary on which no shots are recorded.

4.3Course features

Computing for course features was more straightforward as most features came directly from ShotLink and only needed minor validation. Course features that we computed are shown in Table 5. They include difficulty, fairway height, rough height, stimp, wind speed, fairway firmness and green firmness. The only feature that we derived was the difficulty feature for the course. This was computed by taking the residual of when actual scoring average is regressed with the scoring average feature; that is, the difference of mean terms from 4.1.2.

4.4State features

The last set of features used were game state features. These features are purely descriptive in nature, and just spell out the state of the ball when predictions are made. The two features that describe state are distance from hole and lie. Table 6 explains these features.