T U M W P e ISOPAR Method Michael Stöckl, Peter Lamb & Martin Lames Faculty of Sports and Health Sciences Connollystrasse 32 80809 Munich Germany [email protected] [email protected] June 27, 2011 Abstract e ISOPAR method is a method for characterizing the difficulty of golf holes and allows the per- formance of shots to be analyzed. e method is based on the ball locations provided by ShotLink™and the subsequent number of shots required to hole out from each respective location. ISOPAR values are calculated which represent the number of shots the field would require to hole out. ese ISOPAR values can, a) be visualized on an ISOPAR map and, b) lead to a new performance indicator called Shot ality, which is the difference between the ISOPAR values of the starting position and finishing position, respectively. e Shot ality score can also be used to determine how many shots were saved per shot, or per type of shot, with respect to the performance of the field. 1 Introduction In performance analysis, characteristics of a process which describe how an outcome was achieved are used to assess the performance itself (Hughes & Bartle, 2002) and are referred to as performance indica- tors. Classical performance analysis techniques in golf have focused on classes of golf shots (James, 2007), such as driving distance, approach shot accuracy and puing average (James & Rees, 2008). Measures like greens in regulation, average pus per green and driving distance are intended to describe players’ abilities to perform certain types of shots, yet these abilities are not actually assessed. For example, the beginning position of a pu is the result of the approach shot to the green. So a good puing average describes not only puing ability but also all previous shots on the hole – it is a composite measure. erefore, if independent measures for different types of golf shots existed then strengths and weak- nesses of a player’s game could be assessed (Ketzscher & Ringrose, 2002). Currently, golf performance analysis lacks performance indicators which reflect the influence one shot has on the next. For example, on each hole there is a chain of events which starts on the tee and ends once the ball is holed. Each shot represents an event and the final position of shot n determines the starting position for shot n + 1.A model preserving the playing characteristics of the environment (for example, physical contours, play- ing conditions, etc.) and the stroke sequence is more suitable than simply an analysis of frequencies of discrete events. 2 Baground Cochran and Stobbs (1968) introduced the idea of an independent measure of performance, which they represented by the difference between the distance to the hole before and aer the shot. e number of shots required to hole out from certain distances, for pros, was used to create a model which could be 1 used to determine the value of an individual shot. Landsberger (1994) built on the work of Cochran and Stobbs by refining the approach. Landsberger’s Golf Stroke Value System (GSVS) provided a starting point for more recent work on establishing independent measures of performance. Recent projects have emerged which have looked to further advance the shot value idea put forth by Cochran and Stobbs (Broadie, 2011; Fearing, Acimovic, & Graves, 2011; Minton, 2011)¹. Fearing et al. (2011) applied various regression models to achieve the probability of making a pu and a prediction of the distance remaining aer a missed pu. From this the authors have demonstrated a more valid method for describing the performance of individual shots, called strokes gained, from which they assess performance relative to the field. Broadie (2011) takes a similar approach using distance to the hole but expands the analysis off the green and includes a classification of the ball location. Average performance of PGA TOUR players are used as the benchmark from which comparisons of performance can be made. Strokes gained can then be used to explain the contributions of each shot to the total score. Both models provide very sophisticated models of puing performance with respect to the distance from the hole. In the absence of independent measures of individual shot performance, several studies (Clark III, 2004; James, 2007; James & Rees, 2008; Scheid, 1990) have looked at the temporal variance of consecutive golf scores – both hole scores and round scores. Analyses of round scores showed very low correlations between scores of consecutive rounds when considered with respect to external influences on perfor- mance (i.e. weather conditions and course setup). Analyses of hole scores also showed low correlations between successive holes, again considering external influences like hole par and difficulty. Aside from the obvious fact that good players tend to shoot good scores and poor players tend to shoot poor scores, these results suggest that performance in golf is not subject to “streakiness”. In other words, the nature of the performance of individual shots which make up hole and round scores seems not to be well un- derstood. In summary, consecutive round scores do not depend on one another, and consecutive hole scores do not depend on one another. However, individual shots played on the same hole present a different scenario; these shots make up a continuous chain of events so that the finishing position of shot n represents the starting position for shot n + 1. Although shots on the same hole are related, one would expect the same lack of “streakiness” that has been demonstrated in the literature. is means that although a well played shot tends to set up an advantage on the ensuing shot compared to a poorly ¹see PGA TOUR Academic Data Program page, available at: http://www.pgatour.com/stats/academicdata/ for de- tailed explanations of these projects. 2 played one, a well played shot will not likely predict the performance of the ensuing shot. is question has not been properly addressed in the literature, mainly because of the lack of a genuine performance indicator for individual shots. 3 e ISOPAR method 3.1 e concept Here we present two analogies to help explain the following methods. In meteorology, lines of equal barometric pressure are ploed on geographical maps. ese maps are called isobar maps and the lines are isobar lines. e term isobar (iso - meaning equal and bar - meaning pressure) is used appropriately as the isobar map shows lines of equal pressure. Small diameter, closed lines represent minima and maxima by which, areas of low-pressure and high-pressure can be identified. Densely packed isobar lines indicate a steep gradient of air pressure. Meteorologists can therefore make weather predictions using isobar maps. Our second analogy is to contour maps used in geography to show elevation. Similar to isobar lines, lines of equal elevation are ploed on geographical maps. Here, densely packed lines represent steep ascents and descents. In both analogies, lines that are relatively close together represent steep changes in pressure or elevation, respectively. Likewise, lines that are relatively widely spaced represent areas of lile change in pressure or elevation. For golf, we have developed the ISOPAR method for calculating a gradient of difficulty for a golf hole. e output can then be ploed on a map of the golf hole to visualize the difficulty of certain areas. We call these maps ISOPAR maps and a detailed explanation of how they are calculated is provided below. 3.2 Development and testing e three-dimensional spatial coordinates (x;y;z) of the green gives the first of three sets of triplets, (xg;yg;zg), where g represents the number of measuring points. When available, this set of triplets can be used for ploing the physical contour of the green. For each ball position, (x;y), the corresponding number of strokes, z, required for the player to hole out are used in the calculation. is gives our second of three sets of triplets (xp;yp;zp). For example, if a player took four shots on a hole, that player contributed four data points to our dataset: the x;y 3 coordinates from the location of the first shot with a corresponding z value of 4 and the x;y coordinates from the second shot and a corresponding z value of 3 and so on. 3.2.1 Computing ISOPAR values and maps Before explaining the details of the algorithm for computing the ISOPAR values and maps, a rough overview of the steps involved in calculating an ISOPAR map for a green is given (see Stöckl, Lamb, & Lames, 2011): 1. Assign a grid to the green (Figure 1). 2. Calculate the ISOPAR value of every grid point subject to all measuring points with a modified application of the exponential smoothing algorithm. 3. Compute a surface out of the ISOPAR values of the grid points using a smoothing spline interpo- lation (Fahrmeir, Kneib, & Lang, 2009) to finely remove rough edges. 4. Calculate the ISOPAR map which consists of ISOPAR lines. e following explains the steps for computing ISOPAR values and maps in detail. All computations were performed in MATLAB (e Mathworks, Inc.). Assign grid to green: A grid with a specified mesh size is assigned to the green (Figure 1). e ISOPAR values are computed at the grid nodes. For positions which lie between grid nodes the ISOPAR values must be estimated. erefore, a grid with an extremely small mesh size represents the data very well, while a very large mesh size does not. However, there is a trade-off between representational power and computational intensity. A mesh size which optimizes this trade-off should be used. Exponential smoothing algorithm: From Step 1, coordinates (xi j;yi j) were assigned to the grid nodes.