ABSTRACT 抽象的
In golf, unlike most other sports, individual performance is not the result of direct interactions between players. Instead decision-making and performance is influenced by numerous constraining factors affecting each shot. This study looked at the performance of PGA TOUR golfers in 2011 in terms of stability and variability on a shot-by-shot basis. Stability and variability were assessed using Recurrence Quantification Analysis (RQA) and standard deviation, respectively. About 10% of all shots comprised short stable phases of performance (3.7 ± 1.1 shots per stable phase). Stable phases tended to consist of shots of typical performance, rather than poor or exceptional shots; this finding was consistent for all shot categories. Overall, stability measures were not correlated with tournament performance. Variability across all shots was not related to tournament performance; however, variability in tee shots and short approach shots was higher than for other shot categories. Furthermore, tee shot variability was related to tournament standing: decreased variability was associated with better tournament ranking. The findings in this study showed that PGA TOUR golf performance is chaotic. Further research on amateur golf performance is required to determine whether the structure of amateur golf performance is universal.
与大多数其他运动不同，高尔夫中的个人表现并不是球员之间直接互动的结果。相反，决策和表现受到影响每个镜头的众多限制因素的影响。这项研究考察了 2011 年美巡赛高尔夫球手在每次击球的稳定性和变化性方面的表现。分别使用递归定量分析 (RQA) 和标准差来评估稳定性和变异性。大约 10% 的射击包括短暂的稳定表现阶段（每个稳定阶段 3.7 ± 1.1 次射击）。稳定阶段往往由典型表现的镜头组成，而不是糟糕或异常的镜头；这一发现对于所有镜头类别都是一致的。总体而言，稳定性指标与锦标赛表现无关。所有击球的变化与锦标赛表现无关；然而，开球和短距离击球的变异性高于其他击球类别。此外，开球的可变性与锦标赛排名相关：可变性的降低与更好的锦标赛排名相关。这项研究的结果表明，美巡赛的高尔夫表现是混乱的。需要对业余高尔夫表现进行进一步的研究，以确定业余高尔夫表现的结构是否具有普适性。
Introduction 介绍
Generally, as sports matches unfold, teams and individuals perform actions that are meant to help achieve certain objectives, given the context of the match. Each sport is made unique by the differing objectives and/or different ways of achieving those objectives. Team sports rely on an interaction between several teammates, opponents and the objectives of the match (e.g. Dutt-Mazumder, Button, Robins, & Bartlett, Citation2011). Individual sports can involve a direct interaction with an opponent, as in squash and tennis; these interactions, although less complex than in team sports, have been modelled as self-organising dynamical systems (McGarry, Anderson, Wallace, Hughes, & Franks, Citation2002; Palut & Zanone, Citation2005). Golf is unique to these sports in that the players do not react directly to the actions of the opponent. Golfers do, however, react to numerous constraints affecting each shot, such as: accessibility of pin locations, lie of the ball, wind conditions, and their position in the tournament. All of these factors influence the golfer’s decision making (see Glazier, Citationin press for a theoretical overview). Subsequently, individual skills of golfers (e.g. Robertson, Burnett, Newton, & Knight, Citation2012), although valuable, are unlikely to predict PGA TOUR tournament outcomes because the situations golfers find themselves in, even following well planned and executed shots, are unpredictable. The ability of the golfer to adapt to the situation is likely to be another important characteristic of the top-level golfer. To understand this characteristic in better context, a better understanding of the structure of golf performance is necessary (Lames & McGarry, Citation2007).
一般来说，随着体育比赛的展开，球队和个人会根据比赛的背景采取旨在帮助实现某些目标的行动。每项运动都因不同的目标和/或实现这些目标的不同方式而变得独特。团队运动依赖于多个队友、对手和比赛目标之间的互动（例如 Dutt-Mazumder、Button、Robins 和 Bartlett， Citation 2011 ）。个人运动可能涉及与对手的直接互动，如壁球和网球；这些相互作用虽然没有团队运动那么复杂，但已被建模为自组织动力系统（McGarry、Anderson、Wallace、Hughes 和 Franks， Citation 2002 ；Palut & Zanone， Citation 2005 ）。高尔夫是这些运动的独特之处，因为球员不会直接对对手的行为做出反应。然而，高尔夫球手确实会对影响每次击球的众多限制做出反应，例如：球瓶位置的可达性、球的位置、风力条件以及他们在锦标赛中的位置。所有这些因素都会影响高尔夫球手的决策（参见 Glazier，新闻引文以获取理论概述）。随后，高尔夫球手的个人技能（例如 Robertson、Burnett、Newton 和 Knight， Citation 2012 ）虽然很有价值，但不太可能预测美巡赛的结果，因为高尔夫球手发现自己所处的情况，即使在精心计划和执行的击球之后，也是不可预测的。 高尔夫球手的随机应变能力很可能是顶级高尔夫球手的另一个重要特征。为了更好地理解这一特征，有必要更好地理解高尔夫表现的结构（Lames & McGarry， Citation 2007 ）。
Until recently, performance patterns shown during professional golf tournaments have mostly been analysed indirectly while investigating the existence of the psychological concepts of momentum or streakiness. Round scores (Clark, Citation2004; James, Citation2009) and hole scores (Clark, Citation2005a, Citation2005b; Livingston, Citation2012; Rees & James, Citation2006) have been looked at on the PGA TOUR to determine whether there was a tendency for good rounds or holes to follow other good rounds or holes. Clark (Citation2004) found that there was a tendency for good as well as bad rounds to cluster together, whereas James (Citation2009) found that round scores were rather independent from each other. With respect to hole scores, the support for the idea of streakiness has been weak (e.g. Clark, Citation2005a, Citation2005b; James, Citation2009; Livingston, Citation2012; Rees & James, Citation2006). Since round scores are comprised of hole scores and hole scores are the result of shot sequences, streaks on a shot-by-shot level – the highest level of behaviour and, thus, of performance – has not yet been investigated.
直到最近，职业高尔夫锦标赛期间表现出的表现模式大多是在调查动量或连续性心理概念的存在时进行间接分析。回合得分（Clark， Citation 2004 ；James， Citation 2009 ）和洞得分（Clark， Citation 2005a ， Citation 2005b ；Livingston， Citation 2012 ；Rees & James， Citation 2006 ）已在 PGA 巡回赛上进行查看，以确定是否存在好的轮次或洞会跟随其他好轮次或洞的趋势。 Clark（ Citation 2004 ）发现，好的和坏的回合都有聚集在一起的趋势，而 James（ Citation 2009 ）发现回合分数彼此相当独立。就洞分数而言，对条纹概念的支持很弱（例如Clark， Citation 2005a 、 Citation 2005b ；James， Citation 2009 ；Livingston， Citation 2012 ；Rees & James， Citation 2006 ）。 由于回合得分由洞得分组成，而洞得分是击球序列的结果，因此逐杆水平上的连续得分（行为的最高水平，因此也是表现的最高水平）尚未得到研究。
Since 2003, nearly every shot taken on the PGA TOUR has been recorded and stored in the ShotLinkTM database. The existence of data describing individual shots led to the development of the ISOPAR Method (Stöckl, Lamb, & Lames, Citation2011), a model which allows the assessment of the quality of individual shots. The ISOPAR Method provides the performance indicator, Shots Saved, which quantifies the quality of a shot by comparing its performance to typical performance by the players in the field (field refers to players in the tournament; see next section for more details). Streakiness or momentum are difficult to define as they depend critically on the choice of an arbitrary threshold, above or below which determines the shot’s classification. Instead we used Recurrence Quantification Analysis (Marwan, Romano, Thiel, & Kurths, Citation2007), a rigorous method for analysing time-series data from dynamical systems, which allows the stability of the sequence to be characterised. This study aims to analyse performance patterns in golf – the pattern of stroke sequences according to their quality – in terms of stability and variability, to better understand the nature of professional golf performance.
自 2003 年以来，美巡赛上几乎每一次击球都被记录并存储在 ShotLink TM数据库中。描述单个镜头的数据的存在导致了 ISOPAR 方法的发展（Stöckl、Lamb 和 Lames，引文2011 ），该模型可以评估单个镜头的质量。 ISOPAR 方法提供了表现指标，即保存的射门数，它通过将射门表现与场上球员的典型表现进行比较来量化射门的质量（场是指锦标赛中的球员；有关更多详细信息，请参阅下一节）。条纹或动量很难定义，因为它们主要取决于任意阈值的选择，高于或低于该阈值决定了镜头的分类。相反，我们使用了递归量化分析（Marwan、Romano、Thiel 和 Kurths，引文2007 ），这是一种用于分析动态系统时间序列数据的严格方法，可以表征序列的稳定性。本研究旨在从稳定性和可变性方面分析高尔夫的表现模式（根据其质量而定的击球序列模式），以更好地了解职业高尔夫表现的本质。
Methods 方法
Dataset 数据集
The current study is based on data from the ShotLinkTM database provided by the PGA TOUR, which contains information describing every shot taken on PGA TOUR sanctioned tournaments (these exclude majors and match play events) since 2003. We analysed every shot played in 32 PGA TOUR tournaments (by players who made the tournament cut) in 2011. We included only four-round stroke play tournaments, resulting in 9,152 rounds played by 308 different golfers. The shots (N = 563,354) were divided into five shot types: Tee shots (n = 164,994), Long approach shots (n = 107,023), Short approach shots (n = 14,658), Around the green shots (n = 64,621), and Putts (n = 212,058). Holed putts less than 0.46 m (1.5 ft) were excluded from the analysis because they precluded exceptional performance and could have the effect of making stable phases of exceptional performance appear more typical. The study was approved by the University of Otago Human Ethics Committee.
当前的研究基于 PGA TOUR 提供的 ShotLink TM数据库中的数据，该数据库包含描述自 2003 年以来在 PGA TOUR 认可的锦标赛（不包括大满贯赛和比洞赛）中的每一次击球的信息。我们分析了 32 场 PGA 比赛中的每一次击球2011 年巡回赛（由晋级锦标赛的球员参加）。我们只包括四轮比杆赛，结果是 308 名不同的高尔夫球手参加了 9,152 轮比赛。击球（ N = 563,354）分为五种击球类型：开球（ n = 164,994）、长距离击球（ n = 107,023）、短距离击球（ n = 14,658）、果岭周围击球（ n = 64,621）、和推杆( n = 212,058)。小于 0.46 m（1.5 ft）的入洞推杆被排除在分析之外，因为它们排除了出色的表现，并且可能具有使出色表现的稳定阶段显得更加典型的效果。该研究得到了奥塔哥大学人类伦理委员会的批准。
The ISOPAR method ISOPAR 方法
The analysis in this paper is based on assigning ISOPAR values to each ball location in the dataset. ISOPAR values approximately represent an average number of shots remaining to hole out from a given x,y location on any hole in the above-mentioned PGA TOUR tournaments. The ISOPAR values are calculated in three main steps. The algorithm is explained in detail in Stöckl, Lamb, and Lames (Citation2012), but a short summary is provided below:
本文的分析基于为数据集中的每个球位置分配ISOPAR值。 ISOPAR 值大约代表上述 PGA 巡回赛锦标赛中任何洞上给定x,y位置的剩余击球平均数。 ISOPAR 值的计算分为三个主要步骤。 Stöckl、Lamb 和 Lames（引文2012 ）详细解释了该算法，但下面提供了简短的摘要：
A two-dimensional grid is assigned to the hole. In this study we used a mesh size of two inches.
二维网格被分配给该孔。在这项研究中，我们使用了两英寸的网格尺寸。At the grid nodes ISOPAR values are calculated using an exponential smoothing algorithm (Hamilton, Citation1994). In this study we used a smoothing parameter of 0.17.
在网格节点处，ISOPAR 值使用指数平滑算法计算（Hamilton， Citation 1994 ）。在本研究中，我们使用的平滑参数为 0.17。Based on the ISOPAR values from the previous step a three-dimensional, continuous ISOPAR surface is generated through a cubic spline interpolation (Fahrmeir, Kneib, & Lang, Citation2009).
基于上一步中的 ISOPAR 值，通过三次样条插值生成三维、连续的 ISOPAR 表面（Fahrmeir、Kneib 和 Lang，引文2009 ）。
The algorithm was programmed with MATLAB 2014a using built-in procedures for some steps. Using the ISOPAR values we have defined a measure of performance for individual shots – Shot Quality (SQ). SQ describes the performance of a shot compared to the performance of the field and is defined as the difference between ISOPAR value (IPV) before and after a shot (Stöckl et al., Citation2012):
该算法是使用 MATLAB 2014a 使用内置程序进行某些步骤的编程。使用 ISOPAR 值，我们定义了单次击球性能的衡量标准——击球质量 ( SQ )。 SQ描述了与现场性能相比的射击性能，并定义为射击前后 ISOPAR 值 ( IPV ) 之间的差异（Stöckl 等人，引文2012 ）：
(1) From SQ the performance indicator Shots Saved is derived which is the difference between the SQ of a shot and the average SQ of the field (Stöckl et al., Citation2012)
从SQ可以得出性能指标Shots Saved ，它是一次射门的SQ与场上平均SQ之间的差异（Stöckl 等人，引文2012 ）
(2) and describes how a player gains performance advantage on the field. In this study we calculated the Shots Saved values for each of the five different shot types mentioned above. All strokes were divided into these categories and the Shots Saved values were calculated with respect to the average SQs of the field for the respective shot type.
并描述了球员如何在场上获得表现优势。在本研究中，我们计算了上述五种不同击球类型中每种击球类型的“节省的击球次数”值。所有击球均分为这些类别，并且根据相应击球类型的场上平均SQ计算保存的击球数。
Recurrence plots 重现图
A golfer’s performance in a tournament is represented by the (number of) shots played. The quality of each shot can be expressed by a Shots Saved value. Since the order in which the shots are played forms a shot sequence a golfer’s performance is represented by a Shots Saved sequence.
高尔夫球手在锦标赛中的表现以击球（次数）来表示。每次射击的质量可以通过“已保存的射击”值来表示。由于击球的顺序形成了击球序列，因此高尔夫球手的表现由击球保存序列来表示。
For this analysis, stability captures the fluctuating changes in performance throughout a tournament. We used Recurrence Plots (RP) to identify performance stability in this study. RPs visualise the similarity of states of a dynamical system using a binary system – two states are either similar or not, based on a predefined threshold. In our case RPs highlight when players perform similarly in terms of Shots Saved. If the shots in a sequence are similar to each other they constitute a stable phase.
对于此分析，稳定性捕捉了整个锦标赛中表现的波动变化。在本研究中，我们使用递归图 ( RP)来确定性能稳定性。 RP使用二元系统可视化动态系统状态的相似性——根据预定义的阈值，两个状态要么相似，要么不相似。在我们的案例中，当球员在保存射门数方面表现相似时， RP会突出显示。如果序列中的镜头彼此相似，则它们构成稳定阶段。
In general, the behaviour of a dynamical system is represented by a trajectory in phase space. In our case the behaviour of the system is represented by the one-dimensional stroke series of a player’s Shots Saved values from the whole tournament or from all shots within a shot category. According to Takens (Citation1981) the phase space needs to be reconstructed by embedding the measured trajectory into a higher dimensional phase space which “guarantee[s] the existence of a diffeomorphism between the original and the reconstructed” (Marwan et al., Citation2007, p. 246) phase space. The calculation of an RP is based on the respective dynamical system’s trajectory in the reconstructed phase space. To achieve the phase space reconstruction we used the time delay embedding method suggested by Marwan et al. (Citation2007).
一般来说，动力系统的行为由相空间中的轨迹表示。在我们的例子中，系统的行为由来自整个锦标赛或击球类别中的所有击球的玩家击球保存值的一维击球系列来表示。根据 Takens（引文1981 ），相空间需要通过将测量轨迹嵌入到更高维的相空间中来重建，这“保证原始和重建之间存在微分同胚”（Marwan 等人，引文2007 ，第 246 页）相空间。 RP的计算基于重构相空间中相应动力系统的轨迹。为了实现相空间重建，我们使用了 Marwan 等人提出的时间延迟嵌入方法。 （引文2007 ）。
This method roughly consists of three steps.
该方法大致包括三个步骤。
A time delay is specified for the embedding; using the method of mutual information (Cao, Citation1997) we determined a time delay of 2.
为嵌入指定了时间延迟；使用互信息方法（Cao， Citation 1997 ）我们确定时间延迟为 2。The embedding dimension of the new phase space is determined based on the time delay from step 1. We used a false nearest neighbour algorithm (Kennel, Brown, & Abarbanel, Citation1992) to find the embedding dimension m = 2.
新相空间的嵌入维数是根据步骤 1 的时间延迟确定的。我们使用虚假最近邻算法（Kennel、Brown 和 Abarbanel，引文1992 ）来找到嵌入维数m = 2。The recurrence matrix is calculated as
递归矩阵计算如下
(3) where ε is the threshold distance for assessing the similarity between two states and 
is the Heaviside function (
if 
, otherwise 
; Marwan et al., Citation2007). 
was determined as suggested by Mindlin and Gilmore (Citation1992); 
should be a few percent of the maximum phase space diameter, but not greater than ten percent of the maximum phase space diameter (Koebbe & Mayer-Kress, Citation1992).
其中 ε 是评估两个状态之间相似性的阈值距离， 是赫维赛德函数（
如果
， 否则
; Marwan 等人，引文2007 ）。
根据 Mindlin 和 Gilmore 的建议确定（ Citation 1992 ）；
应该是最大相空间直径的百分之几，但不大于最大相空间直径的百分之十（Koebbe & Mayer-Kress， Citation 1992 ）。
In order to find an appropriate 
which fulfils these constraints and is valid for all players’ stroke series, we determined 
as the mean of the different thresholds of 74 golfers from THE PLAYERS Championship in 2011. Each player’s threshold was determined as 10% of the diameter of the phase space trajectory excluding extreme performances, which resulted in 
= 0.14. The RP visualises the recurrence matrix colour-coded – a black point represents the similarity between two states and is called a recurrence point, while a white point represents non-similarity between two states.
为了找到合适的 满足这些约束并且对所有球员的击球系列都有效，我们确定
作为 2011 年球员锦标赛 74 名高尔夫球手的不同阈值的平均值。每个球员的阈值被确定为不包括极端表现的相空间轨迹直径的 10%，这导致
= 0.14。 RP用颜色编码可视化递归矩阵 - 黑点表示两个状态之间的相似性，称为递归点，而白点表示两个状态之间的不相似性。
To find stable phases of performance an RP was calculated for each player in each tournament. The recurrence point structure in an RP allows one to draw inferences on the investigated golfer’s behaviour (see
为了找到稳定的表现阶段，我们计算了每场锦标赛中每位选手的RP 。 RP中的重复点结构允许人们对所调查的高尔夫球手的行为进行推断（参见). Vertical lines indicate that the investigated system remains in similar states for a time period corresponding to the line length. The presence of vertical lines in the current analysis indicates that the Shots Saved values achieved stability for a time period indicated by the length of the line. Therefore, vertical lines were used as the measure of stability. We chose a minimum vertical line length of three to reduce the likelihood of accepting chance recurrences of only two strokes. Intuitively, stable performance over a time period might suggest good performance; however, the RP does not discriminate between good and poor performance. Therefore, determining the quality of the performance constitutes a follow-up step. The following measurements on vertical lines were considered in our analysis:
）。垂直线表示所研究的系统在与线长度相对应的时间段内保持相似状态。当前分析中垂直线的存在表明“已保存镜头”值在直线长度指示的时间段内实现了稳定。因此，垂直线被用作稳定性的衡量标准。我们选择最小垂直线长度为三，以减少接受仅两次笔划的偶然重复的可能性。直观上，一段时间内稳定的性能可能表明性能良好；然而， RP并不区分表现的好坏。因此，确定性能质量构成了后续步骤。我们的分析考虑了以下垂直线上的测量：
Proportion of shots composing vertical lines
构成垂直线的镜头比例Average length of the vertical lines (referred to as trapping time in Marwan et al., Citation2007)
垂直线的平均长度（在 Marwan 等人的引文2007中称为捕获时间）Quality of stable phases was determined as the mean Shots Saved value of the strokes forming the respective vertical line.
稳定阶段的质量被确定为形成相应垂直线的笔画的平均射击保存值。
Figure 1. Recurrence plot for a golfer’s performance at the Arnold Palmer Invitational in 2011.
图 1. 2011 年阿诺·帕尔默邀请赛上高尔夫球手表现的重现图。
Finally to assess the structure of professional golf performance as deterministic or chaotic, we used the determinism (DET) measure. Determinism describes the proportion of recurrence points that form diagonal lines in an RP. By investigating the diagonal lines (with a minimum length of two) and their average length (L) the predictability of the system can be inferred. Webber and Zbilut (Citation2007) state that short diagonal lines with the determinism measure tending to 0% indicate a chaotic system, while determinism of 100% and long diagonal lines would indicate a periodic system, which at any point is predictable. Determinism does not relate to stability, but the underlying structure of golf performance.
最后，为了评估职业高尔夫表现的结构是确定性的还是混乱的，我们使用了确定性（ DET ）测量。确定性描述了RP中形成对角线的重复点的比例。通过研究对角线（最小长度为两条）及其平均长度 ( L )，可以推断出系统的可预测性。 Webber 和 Zbilut（ Citation 2007 ）指出，确定性测度趋于 0% 的短对角线表示混沌系统，而确定性为 100% 和长对角线则表示周期系统，该系统在任何点都是可预测的。决定论与稳定性无关，而是与高尔夫表现的基本结构有关。
For the RP analyses, the shot categories short approach shots and around the green shots were combined to a category called short game, so that there were enough shots for meaningful RPs. All RP calculations were conducted using the CRP toolbox for MATLAB (Marwan et al., Citation2007).
对于RP分析，击球类别短距离击球和果岭附近的击球被合并为一个称为短杆的类别，以便有足够的击球来获得有意义的RP 。所有RP计算均使用 MATLAB 的 CRP 工具箱进行（Marwan 等人， Citation 2007 ）。
Statistical analysis 统计分析
Statistical analyses were conducted with respect to stability as well as the variability of the golfers’ performances. Descriptive statistics, mean and standard deviations, of the RQA measurements and Spearman correlations between the tournament ranks and the RQA measurements were calculated to explain the stability of performance.
对高尔夫球手表现的稳定性和变异性进行了统计分析。计算RQA测量的描述性统计数据、平均值和标准差以及锦标赛排名和RQA测量之间的 Spearman 相关性，以解释表现的稳定性。
Performance variability was measured as the standard deviation of the Shots Saved values. Variability for each golfer was calculated as the average of the standard deviation for each tournament played. Because the proportional odds assumption required for regular ordered logistic regression was violated, a generalised ordered logistic regression was calculated to analyse whether the variability in a certain shot type had significant influence on a player’s ranking in the tournament.
表现变异性以保存镜头值的标准偏差来衡量。每个高尔夫球手的变异性计算为每场锦标赛标准差的平均值。由于违反了常规有序逻辑回归所需的比例赔率假设，因此计算广义有序逻辑回归来分析某种击球类型的变异性是否对选手在锦标赛中的排名有显着影响。
The alpha level was set to 0.05 for all statistical test procedures. Statistical procedures were conducted using SPSS (IBM Corp. Released 2013. IBM SPSS Statistics for Windows, Version 22.0. Armonk, NY: IBM Corp).
所有统计测试程序的 alpha 水平均设置为 0.05。使用 SPSS（IBM Corp. 2013 年发布。IBM SPSS Statistics for Windows，版本 22.0。Armonk，NY：IBM Corp）进行统计过程。
Results 结果
Stability 稳定
shows the RP for a golfer’s performance at the Arnold Palmer Invitational in 2011. The RP is fairly typical for PGA TOUR golfers; it contains many isolated points with very few vertical and diagonal lines. His play from the 10th to 13th holes in round one (starting at around shot 35), however, shows a block with a higher recurrence point density. In this section there is also a greater number of vertical lines, compared to the rest of the plot. He made pars on each of those four holes and showed fairly typical performance (according to the Shots Saved values).
显示 2011 年阿诺·帕尔默邀请赛上高尔夫球手表现的RP。RP对于 PGA 巡回赛高尔夫球手来说相当典型；它包含许多孤立点，垂直线和对角线很少。然而，他在第一轮第 10 洞到第 13 洞的比赛（从第 35 杆左右开始）显示出具有更高重复点密度的拦网。与绘图的其余部分相比，此部分中的垂直线数量也更多。他在这四个洞中都打出了标准杆，并表现出了相当典型的表现（根据保存的击球数）。
The mean length of a stable phase was roughly the same for all shot categories (range 3.3–3.7) and only slightly longer than the minimum cut-off, standard deviations were also quite similar across shot categories (range 0.7–1.2; see
所有镜头类别（范围 3.3-3.7）的稳定阶段的平均长度大致相同，仅比最小截止时间稍长，各个镜头类别的标准差也非常相似（范围 0.7-1.2；参见). The longest stable phase was 16 shots, which occurred, both, in the Putts category, as well as the All Shots category. These long stable phases did not represent particularly good or poor performance, with average Shots Saved values of 0.05 and −0.01, respectively. The stable phases described in
）。最长的稳定阶段是 16 杆，这发生在推杆类别和全杆类别中。这些长时间的稳定阶段并不代表表现特别好或特别差，平均节省镜头值分别为 0.05 和 -0.01。中描述的稳定相 account for nearly 10% of all shots played. In general, the length of stable phases was not correlated with the tournament rank (ρ = .00, P = .665).
占所有投篮次数的近 10%。一般来说，稳定阶段的长度与锦标赛排名不相关（ρ = .00， P = .665）。
Table 1. Descriptive statistics for the length of stable phases across all tournaments.
表 1.所有锦标赛稳定阶段长度的描述性统计数据。
also shows the descriptive statistics for the quality of stable phases represented by the mean Shots Saved value of the strokes that form stable phases. On average the quality of stable phases is close to zero for all shot categories. However, the stable phases in Putts and Long approach differ from zero slightly, being positive and negative, respectively. The distribution of Shots Saved values of stable phases (
还显示了稳定阶段质量的描述性统计，由形成稳定阶段的笔划的平均击球保存值表示。平均而言，所有镜头类别的稳定阶段质量都接近于零。然而， Putts和Long 方法中的稳定阶段与零略有不同，分别为正和负。稳定阶段的射击保存值的分布（) shows that most of the stable phases tend to consist of typical performance. The quality of the stable phases is positively correlated with tournament performance (low tournament rank indicates good performance e.g. the leader is rank 1) with the exception of Putts (
）表明大多数稳定阶段往往由典型性能组成。稳定阶段的质量与锦标赛表现呈正相关（锦标赛排名低表示表现良好，例如领先者排名 1），推杆除外（). In particular, for Tee shots (ρ = -.17, P < .001) and Long approach shots (ρ = -.14, P < .001) there is a weak but significant trend indicating that better ranked players had stable phases of higher quality. None of the shot categories show a significant correlation between stable phase length and quality.
）。特别是，对于开球(ρ = -.17， P < .001) 和长距离击球 (ρ = -.14， P < .001)，有一个微弱但显着的趋势，表明排名较好的球员有稳定的阶段更高的质量。没有一个镜头类别显示稳定阶段长度和质量之间存在显着相关性。
Figure 2. Distribution of Shots Saved values for all shots considered in this analysis (black) and the shots included in stable phases (grey).
图 2.镜头分布 此分析中考虑的所有镜头的保存值（黑色）和稳定阶段中包含的镜头（灰色）。
About 9% of the strokes played were part of stable phases for the categories All shots, Tee shots and Putts; fewer strokes comprised stable phases for the categories Long approach (3.0%) and Short game (2.4%) (
大约 9% 的击球属于所有击球、开球和推杆类别的稳定阶段；长距离击球(3.0%) 和短距离击球(2.4%) 类别的稳定阶段的杆数较少（). With the exception of Tee shots (ρ = -.08, P < .001), the proportion of shots that were part of stable phases was not correlated with tournament rank. As more Tee shots were part of stable phases, players tended to rank better in the tournament standing, although the correlation was very weak.
）。除了发球区击球(ρ = -.08, P < .001) 之外，稳定阶段的击球比例与锦标赛排名无关。由于更多的开球是稳定阶段的一部分，球员往往在锦标赛中排名更高，尽管相关性非常弱。
Table 2. Proportion of shots comprising stable phases.
表 2.包含稳定相的注射比例。
The proportion of all recurrence points that form diagonal lines is relatively small (range 9.0%–13.8%;
所有重复点形成对角线的比例相对较小（范围9.0%~13.8%；). Further, the existing diagonal lines are short (range 2.0–2.4), which, according to Webber and Zbilut (Citation2007), indicates that the Shots Saved values from the PGA TOUR in 2011 are quite chaotic.
）。此外，现有的对角线很短（范围2.0-2.4），根据Webber和Zbilut的说法（引文2007 ），这表明2011年美巡赛的保存镜头值相当混乱。
Table 3. Determinism for different shot categories.
表 3.不同镜头类别的确定性。
Variability 变化性
Variability in performance across all shots was of 0.32 (
所有镜头的性能差异为 0.32 (). The Spearman correlation between the tournament rank and the variability value showed that as variability increases tournament rank becomes worse (ρ = .13, P < .001), the correlation, however, is quite weak.
）。锦标赛排名和变异性值之间的斯皮尔曼相关性表明，随着变异性的增加，锦标赛排名变得更差（ρ = .13， P < .001），但是相关性相当弱。
Table 4. Mean, standard deviation, range and generalised ordered logistic regression for variability of shots for each shot category with respect to each player in each tournament (model fitting P < .001, Nagelkerke-R2 = .054).
表 4.每次锦标赛中每个球员的每个击球类别的击球变异性的平均值、标准差、范围和广义有序逻辑回归（模型拟合P < .001，Nagelkerke-R 2 = .054）。
Performance variability across different shot types was similar (
不同击球类型的表现差异相似（). Variability was roughly the same across all shot types (~0.3 Shots Saved) and very close to the variability based on All shots. The generalised ordered logistic regression model was not able to explain much of the variance in the tournament ranks because the Nagelkerke-R2 = .054 was so small, according to Backhaus, Erichson, Plinke, and Weiber (Citation2006). Only the variability of Tee shots significantly correlated with the tournament rank; its regression parameter was very large (
, P < .001), thus the probability to be ranked worse was very large if the variability unit was increased by 1. The latter is not realistic compared to the range of possible Shots Saved values (
）。所有镜头类型的变异性大致相同（已保存约 0.3 次镜头），并且非常接近基于所有镜头的变异性。根据 Backhaus、Erichson、Plinke 和 Weiber 的说法，广义有序逻辑回归模型无法解释锦标赛排名中的大部分方差，因为 Nagelkerke-R 2 = .054 非常小（引文2006 ）。只有开球的可变性与锦标赛排名显着相关；它的回归参数非常大（ ， P < .001），因此如果变异性单位增加 1，则排名更差的概率非常大。与可能的已保存镜头值的范围相比，后者是不现实的（); however, the magnitude of the regression parameter suggests that a significant change in Tee shot variability will influence a player’s tournament ranking.
）；然而，回归参数的大小表明开球变异性的显着变化将影响球员的锦标赛排名。
Discussion 讨论
Stability 稳定
Human movement systems have been treated as dynamical systems whose behaviour emerges from the synergy of many components. Golfers can also be treated as dynamical systems whose observable performance is the result of the interaction of many components under the influence of numerous constraining factors. Although the golfer does not interact with action(s) of opponent(s), each shot the golfer plays presents a changing array of constraints, which affects performance. The dynamic nature of those factors makes golf a dynamic sport. Given the adaptive qualities of golf performance, conventional performance indicators may not be informative as each situation experienced in a tournament round is unique. We looked at the recurrence of states of golf performance throughout single golf rounds played on the PGA TOUR to see if professional golf performance is stable to the perturbations imposed by the various constraints affecting each shot. We were also interested to see whether top performing golfers are more resilient than lower ranking golfers. RQA analysis allowed us to describe the nature of golf performance in terms of being chaotic or predictable.
人类运动系统被视为动力系统，其行为是由许多组件的协同作用产生的。高尔夫球手也可以被视为动态系统，其可观察到的表现是许多组件在众多约束因素的影响下相互作用的结果。尽管高尔夫球手不与对手的动作互动，但高尔夫球手的每次击球都呈现出一系列不断变化的约束，这会影响表现。这些因素的动态性质使高尔夫成为一项动态运动。考虑到高尔夫表现的适应性，传统的表现指标可能无法提供信息，因为锦标赛回合中经历的每种情况都是独特的。我们观察了 PGA 巡回赛中单轮高尔夫球比赛中高尔夫表现状态的重复情况，以了解职业高尔夫表现对于影响每次击球的各种限制所施加的扰动是否稳定。我们也有兴趣了解表现最好的高尔夫球手是否比排名较低的高尔夫球手更有弹性。 RQA 分析使我们能够用混乱或可预测的方式来描述高尔夫表现的本质。
The analysis of stability showed that relatively short, stable phases of performance occurred fairly infrequently. Stable phases refer to a series of shots, which is relatively robust to perturbations and, thus, do not necessarily imply good performance. The mean Shots Saved values for shots that occurred during stable phases was close to zero, meaning stable phases tended to consist of fairly typical performance, although the Shots Saved distribution is skewed slightly toward negative values. This could suggest that players are more likely to find themselves in stable phases of poor performance than good performance. Although the negative Shots Saved values are still very close to zero, we speculate that a professional golfer’s expectations of performance are much higher than ‘average performance’ and therefore, a stable phase of slightly lower than average performance could be perceived as a bout of very poor performance. Data on golfers’ perceptions of their performance would be helpful in substantiating these interpretations.
稳定性分析表明，相对较短、稳定的性能阶段很少出现。稳定阶段是指一系列镜头，其对扰动相对稳健，因此并不一定意味着良好的性能。稳定阶段发生的投篮平均保存次数值接近于零，这意味着稳定阶段往往包含相当典型的表现，尽管保存投篮次数分布稍微偏向负值。这可能表明，玩家更有可能发现自己处于表现不佳的稳定阶段，而不是表现良好的阶段。尽管负的“节省击球数”值仍然非常接近于零，但我们推测职业高尔夫球手对表现的期望远高于“平均表现”，因此，略低于平均表现的稳定阶段可以被视为一场非常激烈的比赛。表现不佳。有关高尔夫球手对其表现的看法的数据将有助于证实这些解释。
We should emphasise that stability analysis is not the same as hot hand, momentum or streakiness, as found in the literature (Clark, Citation2005a, Citation2005b; Hughes et al., Citation2015; James, Citation2009; Livingston, Citation2012; Rees & James, Citation2006); although extended stable phases of exceptional performance could be compared to the so-called hot hand phenomenon. With this in mind we found very little evidence of extended stable phases of exceptional performance from any players in any tournaments in the 2011 PGA TOUR season. For example, considering only stable phases lasting at least nine shots (there were 87 in 2011) average Shots Saved was 0.01 ± 0.03, with maximum and minimum mean Shots Saved value for the stable phase of 0.1 and −0.06, respectively. Here the best performance was by a player at the Valero Texas Open in round 2; the stable phase included shots on holes 11–14 and lasted for ten shots. He went par – par – par – birdie on those holes, which, as the best performance in a stable phase, does not suggest that “hot streaks” on a shot-by-shot basis are common on the PGA TOUR.
我们应该强调，稳定性分析与文献中发现的热手、动量或条纹不同（Clark， Citation 2005a ， Citation 2005b ；Hughes 等人， Citation 2015 ；James， Citation 2009 ；Livingston，Citation 2012；Livingston， Citation 2012 ）。 Rees 和 James，引用2006 年）；尽管卓越性能的延长稳定阶段可以与所谓的热手现象进行比较。考虑到这一点，我们发现很少有证据表明任何球员在 2011 年 PGA 巡回赛赛季的任何锦标赛中都具有长期稳定的出色表现。例如，仅考虑持续至少 9 次射击的稳定阶段（2011 年有 87 次），平均节省镜头数为 0.01 ± 0.03，稳定阶段的最大和最小平均节省镜头值分别为 0.1 和 -0.06。最佳表现是来自瓦莱罗德克萨斯公开赛第二轮的选手；稳定阶段包括11-14洞的击球，持续10杆。他在这些洞中打出了标准杆-标准杆-标准杆-小鸟球，这是稳定阶段的最佳表现，但这并不意味着逐杆“热连胜”在美巡赛上很常见。
This analysis also shows that shot series’ performed by PGA TOUR players are not particularly robust to the perturbations imposed by the changing constraints from shot to shot. This finding is not unique to players low on the leaderboard or players at the peak of their performance; rather the lack of stability found should be considered part of the structure of golf performance. The diagonal lines on the recurrence plots, quantified by their proportion of points (determinism) and their length, shows us that golf performance is quite chaotic as opposed to being predictable. This can potentially have important implications for players and golf psychologists – if a player finds him or herself in what they consider a stretch of poor performance they should feel assured that it may be that their perceptions are incorrect since stable phases of poor performance are quite rare. The shots they consider to be poor are not likely as poor as they perceive, relative to the field. By improving their outlook they may be able to improve their performance on average, accepting the fact that the stability of their performance is likely to be unaffected.
该分析还表明，美巡赛球员进行的一系列击球对于因击球之间不断变化的约束而施加的扰动并不是特别稳健。这一发现并不是排行榜上排名较低的玩家或处于巅峰状态的玩家所独有的。相反，缺乏稳定性应该被视为高尔夫表现结构的一部分。递归图上的对角线通过点的比例（确定性）和长度进行量化，向我们表明高尔夫表现是相当混乱的，而不是可预测的。这可能对球员和高尔夫心理学家产生重要影响——如果球员发现自己处于他们认为的一段表现不佳的状态，他们应该放心，他们的看法可能是错误的，因为表现不佳的稳定阶段非常罕见。相对于球场而言，他们认为较差的投篮可能并不像他们认为的那么差。通过改善他们的前景，他们也许能够提高他们的平均表现，并接受他们的表现的稳定性可能不受影响的事实。
The chaotic nature of golf performance may be unique to golf because of all the factors unique to the sport. For example, McGarry, Khan, and Franks (Citation1999) suggested that in squash a steady state of performance could be perturbed by exceptional or poor performance by one of the competitors. Golfers do not react to the actions of their opponents in nearly the same way as in net/wall games such as squash. Accordingly, golfers are often instructed to play their own game (e.g. Stockton & Rudy, Citation2014) and not take notice of the performance of their opponents. In golf there is also more time in between shots, compared to squash, for the player to recover from the excitement of a good shot or the disappointment of a poor shot. Furthermore, the ISOPAR method of quantifying golf performance helps standardize performance, so that, a difficult shot that gives the player little chance to get the ball close to the hole does not preclude exceptional performance. Rather, relative to the average, a difficult shot does not have to end up as close to the hole as an easy shot to be given the same quality. In other sports where the outcome depends on other factors than the performance itself, it can be difficult to determine whether an outcome was the result of a player’s good performance or the poor performance of the opponent.
由于这项运动独有的所有因素，高尔夫表现的混乱性质可能是高尔夫独有的。例如，McGarry、Khan 和 Franks（ Citation 1999 ）指出，在壁球比赛中，一名参赛者的出色或糟糕表现可能会扰乱稳定的表现状态。高尔夫球手对对手的行为的反应与壁球等网/墙游戏中的反应方式几乎不同。因此，高尔夫球手经常被指示打自己的比赛（例如Stockton & Rudy， Citation 2014 ）而不注意对手的表现。与壁球相比，高尔夫运动中击球之间的时间也更长，可以让球员从好击球的兴奋或糟糕击球的失望中恢复过来。此外，量化高尔夫表现的 ISOPAR 方法有助于标准化表现，因此，即使球员几乎没有机会让球接近球洞，但高难度的击球并不妨碍出色的表现。相反，相对于平均水平，困难的击球不必像简单的击球一样接近球洞，也能获得相同的质量。在其他运动中，结果取决于表现本身以外的其他因素，因此很难确定结果是运动员良好表现还是对手表现不佳的结果。
Variability 变化性
The player who wins a tournament (assuming no play-off) will have the lowest average round score and, therefore, the highest average Shots Saved values calculated using the ISOPAR method (Stöckl et al., Citation2011). We were interested to see whether players who do well in tournaments are also more or less variable than players who finish lower and whether the performance of certain shots were more or less variable.
赢得锦标赛（假设没有附加赛）的选手将获得最低的平均回合得分，因此，使用 ISOPAR 方法计算出的平均节省射门数最高（Stöckl 等人，引文2011 ）。我们有兴趣了解在锦标赛中表现出色的球员是否比成绩较差的球员或多或少存在变化，以及某些击球的表现是否或多或少存在变化。
The average variability for each player’s Shots Saved values for each tournament was fairly consistent across all shots and between the shot types. However, variability was slightly more variable for Tee shots and much more variable for Short approach shots, which indicates a greater spread around the mean variability score for Tee shots and Short approach shots. Importantly, variable performance for the Short approach category did not correlate with tournament ranking. Variability in Tee shots performance, on the other hand, was strongly related to tournament rank, showing that less variable tee shot performance tends to be rewarded with better tournament standing. In other words the leaders are less variable, shot to shot, on Tee shots than players who finish lower in the rankings, whereas variability for Short approach shots is consistently higher compared to other shot types for all players, regardless of their position in the tournament. For instance, an exceptional tee shot can gain the player about 0.5 shots on a risk-reward hole (see 18th hole at Pebble Beach Golf Links in Stöckl et al., Citation2012); however, a poor drive finding a hazard or out of bounds costs the player much more. The best players tend to make small gains on the field for most tee shots and according the variability analysis here, they limit the number of very poor shots, compared to players who finish further down in the standings. Moreover, we have previously shown that driving performance, according to mean Shots Saved, was most highly correlated with money earnings in 2011 on the PGA TOUR (ρ = .61) and in THE PLAYERS tournament (ρ = .57; Stöckl, Lamb, & Lames, Citation2013) compared to other shot types. Similarly, Broadie (Citation2012) also showed that long-game performance, which consists of shots longer than 90 m to the hole, explained most of the variability in PGA TOUR scoring from 2003 to 2010. Again, the implications of these findings relate to how players perceive their performance. A player may perceive his or her performance on short approaches as being inconsistent; however, inconsistency for this shot type is typical and has a minor effect on their place in the tournament.
每个锦标赛中每个球员的击球保存值的平均变异性在所有击球和击球类型之间相当一致。然而，开球击球的变异性稍微大一些，而短距离击球的变异性则大得多，这表明开球击球和短距离击球的平均变异性得分分布更大。重要的是，短距离类别的可变表现与锦标赛排名无关。另一方面，开球表现的变化与锦标赛排名密切相关，这表明开球表现变化较小往往会获得更好的锦标赛排名。换句话说，与排名较低的球员相比，领先者在开球时的击球变化较小，而所有球员的短距离击球变化始终高于其他击球类型，无论他们在锦标赛中的位置如何。例如，在风险回报洞中，出色的开球可以使球员获得大约 0.5 次击球机会（参见 Stöckl 等人的 Pebble Beach Golf Links 的第 18 洞，引文2012 ）；然而，如果发现障碍物或出界，开球不佳会让球员付出更大的代价。最好的球员往往会在球场上的大多数开球中获得微小的收益，并且根据此处的变异性分析，与排名靠后的球员相比，他们限制了非常差的击球数量。 此外，我们之前已经表明，根据平均节省击球数，驾驶表现与 2011 年 PGA 巡回赛 (ρ = .61) 和球员锦标赛 (ρ = .57) 的收入高度相关；Stöckl、Lamb、 & Lames，引文2013 ）与其他击球类型的比较。同样，布罗迪（Broadie， 2012 年引文）还表明，长距离比赛表现（包括距离球洞超过 90 m 的击球距离）解释了 2003 年至 2010 年 PGA 巡回赛得分的大部分变化。同样，这些发现的含义与球员如何看待他们的表现。球员可能会认为他或她在短距离进攻中的表现不一致；然而，这种击球类型的不一致是很常见的，并且对他们在锦标赛中的位置影响较小。
Limitations 局限性
We can only generalise our results to professional golfers – specifically, those on the PGA TOUR. We expect amateur golf performance to be much more variable than professional golf performance, however, it remains to be seen whether the chaotic nature of golf performance is inherent to the sport. In future work we look to gain amateur golf performance data, which will allow us to better understand characteristics of amateur golf performance.
我们只能将我们的结果推广给职业高尔夫球手——特别是那些参加美巡赛的高尔夫球手。我们预计业余高尔夫表现比职业高尔夫表现变化更大，然而，高尔夫表现的混乱本质是否是这项运动固有的还有待观察。在未来的工作中，我们希望获得业余高尔夫表现数据，这将使我们更好地了解业余高尔夫表现的特点。
Conclusions 结论
This study is the first to have looked at performance stability in golf on a shot by shot basis. By looking specifically at stable phases of exceptional performance we found very little evidence for the “hot hand” phenomenon in golf. We have also shown further support for the importance of long game performance in PGA TOUR golf; specifically that low variability in tee shot performance is related to better tournament performance. In future research we look to collect performance data from amateur golfers so that comparisons with PGA TOUR golfers can be made. In particular we would like to know whether golf performance, in general, is chaotic or if this is specific to professional golf. Future work should also look at whether golfers’ abilities to adapt to certain constraints can be effectively quantified.
这项研究是第一个以逐杆为基础考察高尔夫运动表现稳定性的研究。通过专门观察卓越表现的稳定阶段，我们发现高尔夫中“热手”现象的证据很少。我们还进一步支持美巡赛高尔夫中长距离比赛表现的重要性；具体来说，开球表现的低变异性与更好的锦标赛表现有关。在未来的研究中，我们希望收集业余高尔夫球手的表现数据，以便与美巡赛高尔夫球手进行比较。我们特别想知道高尔夫表现是否总体上是混乱的，还是职业高尔夫特有的。未来的工作还应该研究高尔夫球手适应某些限制的能力是否可以有效量化。
Acknowledgements 致谢
We thank the PGA TOUR for providing us with access to the ShotLinkTM data used in this study.
我们感谢 PGA TOUR 为我们提供了本研究中使用的 ShotLink TM数据的访问权限。
Disclosure statement 披露声明
No potential conflict of interest was reported by the authors.
作者报告没有潜在的利益冲突。
References 参考
- Backhaus, K., Erichson, B., Plinke, W., & Weiber, R. (2006). Multivariate Analysemethoden. Berlin: Springer.
Backhaus, K.、Erichson, B.、Plinke, W. 和 Weiber, R. (2006)。多变量分析方法。柏林：施普林格。 - Broadie, M. (2012). Assessing golfer performance on the PGA TOUR. Interfaces, 42(2), 146–165. doi:10.1287/inte.1120.0626
布罗迪，M.（2012）。评估高尔夫球手在美巡赛上的表现。接口， 42 ( 2 )，146–165。 doi:10.1287/inte.1120.0626 - Cao, L. (1997). Practical method for determining the minimum embedding dimension of a scalar time series. Physica, D, 110(1–2), 43–50. doi:10.1016/S0167-2789(97)00118-8
曹L.(1997)。确定标量时间序列的最小嵌入维数的实用方法。物理学，D ， 110 （ 1-2 ），43-50。号码：10.1016/S0167-2789(97)00118-8 - Clark, R. D., III. (2004). Streakiness among professional golfers: Fact or fiction. International Journal of Sport Psychology, 34, 63–79.
克拉克，RD，III。 （2004）。职业高尔夫球手的不稳定：事实还是虚构。国际运动心理学杂志，34，63-79。 - Clark, R. D., III. (2005a). Examination of hole-to-hole streakiness on the PGA Tour. Perceptual and Motor Skills, 100, 806–814. doi:10.2466/pms.100.3.806-814
克拉克，RD，III。 （2005a）。检查 PGA 巡回赛上各洞之间的不稳定性。知觉和运动技能， 100，806–814 。 doi:10.2466/pms.100.3.806-814 - Clark, R. D., III. (2005b). An examination of the “hot hand” in professional golfers. Perceptual and Motor Skills, 101, 935–942. doi:10.2466/pms.101.3.935-942
克拉克，RD，III。 （2005b）。对职业高尔夫球手“热手”的检查。知觉和运动技能， 101，935–942 。 doi:10.2466/pms.101.3.935-942 - Dutt-Mazumder, A., Button, C., Robins, A., & Bartlett, R. (2011). Neural network modelling and dynamical system theory: Are they relevant to study the governing dynamics of association football players? Sports Medicine, 41(12), 1003–1017. doi:10.2165/11593950-000000000-00000
Dutt-Mazumder, A.、Button, C.、Robins, A. 和 Bartlett, R. (2011)。神经网络建模和动力系统理论：它们与研究足球运动员的治理动态相关吗？运动医学， 41 （ 12 ），1003-1017。号码：10.2165/11593950-000000000-00000 - Fahrmeir, L., Kneib, T., & Lang, S. (2009). Regression. Berlin: Springer.
Fahrmeir, L.、Kneib, T. 和 Lang, S. (2009)。回归。柏林：施普林格。 - Glazier, P. (in press). Towards a grand unified theory of sports performance. Human Movement Science. doi:10.1016/j.humov.2015.08.001
Glazier, P.（出版中）。走向运动表现的大统一理论。人体运动科学。 doi:10.1016/j.humov.2015.08.001 - Hamilton, J. D. (1994). Time series analysis. Princeton: Princeton University Press.
汉密尔顿，JD (1994)。时间序列分析。普林斯顿：普林斯顿大学出版社。 - Hughes, M., James, N., Hughes, M. T., Murray, S., Burt, E., & Heath, L. (2015). Momentum and ‘hot hands’. In M. Hughes & I. Franks (Eds.), Essentials of performance analysis in sport (2nd ed., pp. 270–290). London: Routledge.
休斯，M.、詹姆斯，N.、休斯，MT、默里，S.、伯特，E. 和希思，L. (2015)。势头和“热手”。载于 M. Hughes 和 I. Franks（主编），运动表现分析要点（第二版，第 270-290 页）。伦敦：劳特利奇。 - James, N. (2009). Performance analysis of golf: Reflections on the past and a vision of the future. International Journal of Performance Analysis in Sport, 9, 188–209.
詹姆斯，N.（2009）。高尔夫表现分析：对过去的反思和对未来的展望。国际体育表现分析杂志，9，188-209。 - Kennel, M. B., Brown, R., & Abarbanel, H. D. I. (1992). Determining embedding dimension for phase-space reconstruction using a geometrical construction. Physical Review A, 45, 3403–3411. doi:10.1103/PhysRevA.45.3403
Kennel, MB、Brown, R. 和 Abarbanel, HDI (1992)。使用几何构造确定相空间重建的嵌入维数。物理评论 A ，45，3403–3411 。 doi:10.1103/PhysRevA.45.3403 - Koebbe, M., & Mayer-Kress, G. (1992). Use of recurrence plots in the analysis of time-series data. In M. Casdagli & S. Eubank (Eds.), Proceedings of SFI studies in the science of complexity, vol. XXI, Redwood City (pp. 361–378). Reading: Addison-Wesley.
Koebbe, M. 和 Mayer-Kress, G. (1992)。在时间序列数据分析中使用递归图。 M. Casdagli 和 S. Eubank（主编）， SFI 复杂性科学研究论文集，卷。 XXI，雷德伍德城（第 361-378 页）。阅读：艾迪生-韦斯利。 - Lames, M., & McGarry, T. (2007). On the search for reliable performance indicators in game sports. International Journal of Performance Analysis in Sport, 7(1), 62–79.
拉姆斯，M. 和麦加里，T. (2007)。寻找竞技运动中可靠的表现指标。国际体育表现分析杂志，7(1), 62–79。 - Livingston, J. A. (2012). The hot hand and the cold hand in professional golf. Journal of Economic Behavior & Organization, 81(1), 172–184. doi:10.1016/j.jebo.2011.10.001
新泽西州利文斯顿 (2012)。职业高尔夫中的热手和冷手。经济行为与组织杂志， 81 （ 1 ），172-184。 doi:10.1016/j.jebo.2011.10.001 - Marwan, N., Romano, M. C., Thiel, M., & Kurths, J. (2007). Recurrence plots for the analysis of complex systems. Physics Reports, 438, 237–329. doi:10.1016/j.physrep.2006.11.001
Marwan, N.、Romano, MC、Thiel, M. 和 Kurths, J. (2007)。用于分析复杂系统的递归图。物理报告， 438，237–329 。 doi:10.1016/j.physrep.2006.11.001 - McGarry, T., Anderson, D., Wallace, S. A., Hughes, M. D., & Franks, I. M. (2002). Sport competition as a dynamical self-organizing system. Journal of Sports Sciences, 20, 771–781. doi:10.1080/026404102320675620
McGarry, T.、Anderson, D.、Wallace, SA、Hughes, MD 和 Franks, IM (2002)。体育竞赛作为一个动态的自组织系统。体育科学杂志， 20，771–781 。号码：10.1080/026404102320675620 - McGarry, T., Khan, M. A., & Franks, I. A. (1999). On the presence and absence of behavioural traits in sport: An example from championship squash match-play. Journal of Sports Sciences, 17, 297–311. doi:10.1080/026404199366019
McGarry, T.、Khan, MA 和 Franks, IA (1999)。关于运动中行为特征的存在和缺失：壁球冠军赛的一个例子。体育科学杂志， 17，297-311 。号码：10.1080/026404199366019 - Mindlin, G. M., & Gilmore, R. (1992). Topological analysis and synthesis of chaotic time series. Physica D, 58(1–4), 229–242. doi:10.1016/0167-2789(92)90111-Y
总经理明德林 (Mindlin) 和 R 吉尔摩 (Gilmore, R.) (1992)。混沌时间序列的拓扑分析与综合。物理学 D ， 58 （ 1-4 ），229-242。号码：10.1016/0167-2789(92)90111-Y - Palut, Y., & Zanone, P.-G. (2005). A dynamical analysis of tennis: Concepts and data. Journal of Sports Sciences, 23(10), 1021–1032. doi:10.1080/02640410400021682
Palut, Y. 和 Zanone, P.-G. （2005）。网球的动态分析：概念和数据。体育科学杂志， 23 （ 10 ），1021-1032。号码：10.1080/02640410400021682 - Rees, C., & James, N. (2006). A new approach to evaluating streakiness in golf. In H. Dancs, M. D. Hughes, & P. O’Donoghue (Eds.), Performance analysis for sport 7 (pp. 352–360). Cardiff: UWIC CPA Press.
里斯，C. 和詹姆斯，N. (2006)。评估高尔夫条纹的新方法。见 H. Dancs、MD Hughes 和 P. O'Donoghue（编辑），运动表现分析 7 （第 352-360 页）。卡迪夫：UWIC CPA 出版社。 - Robertson, S. J., Burnett, A. F., Newton, R. U., & Knight, P. W. (2012). Development of the Nine-Ball Skills Test to discriminate elite and high-level amateur golfers. Journal of Sports Sciences, 30(5), 431–437. doi:10.1080/02640414.2012.654398
罗伯逊，SJ、伯内特，AF、牛顿，RU 和奈特，PW (2012)。开发九球技能测试以区分精英和高水平业余高尔夫球手。体育科学杂志， 30 （ 5 ），431-437。号码：10.1080/02640414.2012.654398 - Stöckl, M., Lamb, P. F., & Lames, M. (2011). The ISOPAR method—A new approach to performance analysis in golf. Journal of Quantitative Analysis in Sport, 7(1), article 10. doi:10.2202/1559-0410.1289
Stöckl, M.、Lamb, PF 和 Lames, M. (2011)。 ISOPAR 方法——高尔夫表现分析的新方法。体育定量分析杂志， 7 ( 1 )，第 10 条。doi:10.2202/1559-0410.1289 - Stöckl, M., Lamb, P. F., & Lames, M. (2012). A model for visualizing difficulty in golf and subsequent performance rankings on the PGA Tour. International Journal of Golf Science, 1(1), 10–24. doi:10.1123/ijgs.1.1.10
Stöckl, M.、Lamb, PF 和 Lames, M. (2012)。用于可视化高尔夫难度和 PGA 巡回赛后续表现排名的模型。国际高尔夫科学杂志， 1 （ 1 ），10-24。 doi:10.1123/ijgs.1.1.10 - Stöckl, M., Lamb, P. F., & Lames, M. (2013). Analysing individual performance in golf using the ISOPAR Method. In P. O’Donoghue & D. Peters (Eds.), Performance analysis of sport IX (pp. 196–202). London: Routledge.
Stöckl, M.、Lamb, PF 和 Lames, M. (2013)。使用 ISOPAR 方法分析高尔夫中的个人表现。见 P. O'Donoghue 和 D. Peters（编辑）， 《运动表现分析 IX》 （第 196-202 页）。伦敦：劳特利奇。 - Stockton, D., & Rudy, M. (2014). Own your game: How to use your mind to play winning golf. New York: Penguin.
斯托克顿，D.，＆鲁迪，M.（2014）。掌控你的比赛：如何用你的头脑打赢高尔夫。纽约：企鹅。 - Takens, F. (1981). Detecting strange attractors in turbulence. In D. Rand & L.-S. Young (Eds.), Dynamical systems and turbulence, lecture notes in mathematics (Vol. 898, pp. 366–381). Berlin: Springer.
塔肯斯，F.（1981）。检测湍流中的奇怪吸引子。在D.兰德和L.-S。 Young（主编），动力系统和湍流，数学讲义（第 898 卷，第 366-381 页）。柏林：施普林格。 - Webber, C. L., & Zbilut, J. P. (2007). Recurrence quantifications: Feature extractions from recurrence plots. International Journal of Bifurcation and Chaos, 17(10), 3467–3475. doi:10.1142/S0218127407019226
Webber, CL 和 Zbilut, JP (2007)。循环量化：从循环图中提取特征。国际分岔与混沌杂志， 17 （ 10 ）， 3467–3475 。号码：10.1142/S0218127407019226