International Journal of Golf Science, 2012, 1, 10-24 © 2012 Human Kinetics, Inc. A Model for Visualizing Difficulty in Golf and Subsequent Performance Rankings on the PGA Tour Michael Stöckl, Peter F. Lamb, and Martin Lames Technische Universität München Conventional performance indicators used in golf tend to rely on classifications of shots based on the distance to the hole. Consequently, these indicators ignore the unique factors which make up the difficulty of each shot. The aim of the project was to develop new performance indicators which are independent from the other shots played on a hole as well as a visualization of unique areas on a hole. Using the ISOPAR method a representative performance of the field is calculated at each hole. This allows visualizing difficult areas with respect to the field’s performance and new performance indicators which are independent of preceding shots. We looked at the ShotLink database for all tournaments measured in 2011 and com- pared the rankings of the PGA Tour to rankings generated using the ISOPAR method. We argue that the players who do well in the ISOPAR based rankings tend to be players who are subjectively and popularly known to be good performers for the respective shot types. The theoretical background and anecdotal evidence are used as support for the ISOPAR method’s ability to analyze the performance of individual shots in golf. Keywords: modeling; golf; performance analysis Performance analysis characterizes processes in sport by describing how an outcome was attained to measure performance itself (Hughes & Bartlett, 2002). In doing so, indicators for performance are defined. In golf, classical performance analysis has focused on analyzing classes of shots (James, 2007) like driving distance, approach shot accuracy or putting average (James & Rees, 2008). Per- formance indicators like average putts per green, driving distance or greens in regulation are designated for analyzing players’ abilities in performing certain types of shots. However, these measures do not only describe a specific ability of the player but many abilities of the player. For example, the starting position of a player’s first putt on the green is determined by the approach shot. Hence, a player who performs approach shots well, as a consequence, tends to experience easier putts. Furthermore, the starting position of the approach shot is the result of The authors are with Lehrstuhl für Traininsgwissenschaft und Sportinformatik, Technische Universität München, Munich, Germany. 10 The ISOPAR Method 11 the previous shot. Therefore, putting average is a composite measure since it not only describes putting ability but also the performance of all previous shots on a hole. For this reason, to identify players’ strengths and weaknesses, independent measures of performance for certain types of shots must be developed (Ketzscher & Ringrose, 2002). Currently, there is only one performance indicator, the newly developed indicator strokes gained (Broadie, 2011; Fearing, Acimovic & Graves, 2011), which accounts for the influence one shot has on the next. As suggested above, each shot on a hole is part of a chain of events which starts at the tee and ends when the ball is holed out. Except for the tee shot, the starting conditions of a shot are the result of the finishing position of the previous shot. Hence, a model which includes environmental conditions and the stroke sequence is more suitable than analyzing frequencies of discrete events. The lack of this sort of performance indicator was also recognized by other research groups (Broadie, 2008; Broadie, 2011; Fearing et al., 2011; Landsberger, 1994). These groups developed statistical models which provide probabilities of holing out with respect to the distance to the pin. The development of such statisti- cal benchmarks is based on an idea of Cochran and Stobbs (1968) who manually collected data, albeit a small dataset, and calculated probabilities of holing out from certain distances on the green and the respective remaining average number of shots until the ball was holed from these distances. At the time of their study they were prevented from developing this idea by lack of modern computer technology to collect and analyze data. Broadie (2008) developed this idea further and defined a measurement for individual strokes, a shot value system strokes gained, based on different statistical benchmarks on and off the green (Broadie, 2011). Fearing et al. (2011) worked out a similar statistical model to predict the number of remaining shots to hole out based on the distance to the hole on the green, difficulty of the green, and the performance of the participating golfers. Their performance indica- tor, Strokes Gained—Putting, is based on Broadie’s idea (2008) and is used as the first official measurement for individual shots by the PGA Tour. Whereas these approaches are statistical and developed to predict performance and eventually compare performance to expected benchmarks, our working group took a different approach, which is specifically aimed at characterizing the perfor- mance of the participating golfers, rather than predicting it. We call the model the ISOPAR method (Stöckl, Lamb & Lames, 2011). The approach of the ISOPAR project comes from a systems perspective which has been empirically applied to many levels of analysis of human movement and performance (e.g., Davids et al. (2003); Kelso (1995); Mayer-Kress et al. (2006)). The central concept is that neurobiological systems behave as complex systems and theories from physical sciences, e.g., dynamical systems theory, are appropriate for understanding and mod- eling human performance. We extended this perspective to golf performance on the PGA Tour measured by ShotLink. The underlying assumption is that each shot a player faces, represents a new set of constraints and the player must adapt to the constraints associated with the shot, which can be divided into three main categories: environment, organism, task (Newell, 1986). We have already shown in a previous study that the ISOPAR method is suitable to visualize constraints in putting (Stöckl & Lames, 2011). This idea can be extended to entire holes to illustrate difficulty on a hole represented by the number of remaining shots, since each shot is part of a player’s shot sequence. Hence, a representation of the performance of the participating golfers is calculated, 12 Stöckl, Lamb, and Lames according to the number of remaining shots, for a hole and a specific pin position by the ISOPAR method. Based on that, the ISOPAR method provides a) a visualization of unique areas on a hole with respect to the participating players and a certain pin position and b) a performance indicator for individual shots, Shot Quality, which describes the quality of a shot with respect to the difficulty of its starting location and its finishing location. Using the ISOPAR method we analyzed the PGA Tour ShotLink data from all measured tournaments in 2011. Methods In this section the concept on which the ISOPAR method is based will be explained. Furthermore, the different steps of the algorithm of the ISOPAR method will be described. The Concept The idea of the ISOPAR method is based on and can be explained well by an anal- ogy to isobar maps used in meteorology. In meteorology, isobar maps visualize barometric pressure by plotting lines of equal pressure, the isobar lines (iso—means equal; bar—means pressure), on geographical maps. Minima and maxima, which are areas of low and high air pressure respectively, can be identified by closed lines with usually a small diameter. Closely packed lines represent a large change in pressure and spread out lines represent a small change in pressure. Similarly, the ISOPAR method was developed to calculate and visualize difficulty on golf holes and the changes in difficulty. By plotting the values on a map of a golf hole unique areas can be identified. We call these maps ISOPAR maps. Data and the ISOPAR Algorithm The calculation of ISOPAR values and ISOPAR maps is based on information about shots. The ISOPAR method needs as input arguments the ball locations and the number of remaining shots from each ball location. In this study we used information about all shots (n = 1,009,362) measured by ShotLink from 38 PGA Tour tournaments in 2011 to calculate ISOPAR values and ISOPAR maps for 2,754 holes played in 153 rounds. The ISOPAR values and ISOPAR maps are calculated in three and four steps, respectively. The ISOPAR method was programmed with MATLAB R2011b using common MATLAB procedures for some of the steps. More details of the method are explained in Stöckl et al. (2011). Although the algorithm is described only with respect to the green in Stöckl et al. (2011), the analysis in the current study is based on the same algorithm for calculating ISOPAR values and the corresponding maps for entire holes; therefore, only a summary of the steps is provided. 1. A two dimensional grid is assigned to the hole. This gives us a set of grid nodes represented by pairs (xij, yij), i = 1, ..., m, j = 1, ..., n. In this study a grid size of two inches was used. 2. At the grid nodes ISOPAR values zij are calculated. This calculation is based on the ball locations and the respective number of remaining shots at the ball The ISOPAR Method 13 locations. The ISOPAR value calculation at any grid node considers all ball locations which are in an area 60 degrees left and 60 degrees right of the straight line between hole and this grid node (vertex being the hole). This angle was decided upon so that ball locations beyond the hole do not influence the calcu- lation. The included ball locations are then sorted in ascending order by their Euclidean distanced dijp, p = 1, ..., q (q = number of included ball locations) to the respective grid node.