?Mathematical formulae have been encoded as MathML and are displayed in this HTML version using MathJax in order to improve their display. Uncheck the box to turn MathJax off. This feature requires Javascript. Click on a formula to zoom. ？数学公式已编码为 MathML，并使用 MathJax 在此 HTML 版本中显示，以改进其显示。取消选中该框可关闭 MathJax。此功能需要 JavaScript。单击公式进行缩放。ABSTRACT 抽象的
Analyses of game sports or of performances shown in them require appropriate models. Many game sports can be modelled as complex, dynamic systems. This study investigated how recurrence plots (RPs) – a method to analyse complex systems – and the analyses of RPs can be applied to the game sports golf and soccer. A golfer is treated as a complex system with many unknown components. The scalar variable Shots Saved can describe his/her behaviour. Phase space reconstruction is needed to unfold hidden facets of a golfer’s behaviour. Results indicate that golfers’ performances do not approach a stable state, but seem rather unpredictable. A soccer match can also be treated as a complex system. Some components – the players – and their respective behaviour represented by movement trajectories are known and can be used to describe the system’s behaviour. We propose that no embedding is needed for the RP construction in this case. Results indicate that the more goal shots there are in a game, the more unstructured it is. Furthermore, if several golfers or soccer matches are investigated, we recommend using the same RP parameters to achieve comparable results.
对竞技运动或其中表现的分析需要适当的模型。许多竞技运动可以建模为复杂的动态系统。本研究调查了递归图 (RP)（一种分析复杂系统的方法）以及 RP 分析如何应用于高尔夫和足球运动。高尔夫球手被视为一个具有许多未知组件的复杂系统。标量变量“保存的镜头”可以描述他/她的行为。需要进行相空间重建来揭示高尔夫球手行为的隐藏方面。结果表明，高尔夫球手的表现并没有达到稳定状态，而是显得相当不可预测。一场足球比赛也可以被视为一个复杂的系统。一些组件（玩家）以及他们各自由运动轨迹表示的行为是已知的，并且可以用来描述系统的行为。我们建议在这种情况下 RP 构建不需要嵌入。结果表明，一场比赛中的射门次数越多，比赛就越无组织性。此外，如果调查多个高尔夫球手或足球比赛，我们建议使用相同的 RP 参数来获得可比较的结果。
Introduction 介绍
Analysing performances shown during sports competitions has become a rapidly growing field in the more recent past. For that, appropriate methods are required to analyse performances in different sports. The performance structure differs from sport to sport. For example, performance in endurance sports is mostly based on performance prerequisites such as athletic ability and technical and tactical skill. An athlete’s 100 m sprint performance might be predicted within narrow margins knowing the actual level of reaction time, acceleration, maximum speed and speed endurance from prior, separate testing. In contrast, the performance in many game sports emerges from a dynamic process which is characterized to a great extent by the interaction between the athletes, although their athletic abilities and technical and tactical skills also influence performance.
近年来，分析体育比赛中的表现已成为一个快速发展的领域。为此，需要采用适当的方法来分析不同运动的表现。不同运动项目的表现结构有所不同。例如，耐力运动的成绩主要取决于运动能力和技战术技能等成绩先决条件。通过之前单独的测试了解反应时间、加速度、最大速度和速度耐力的实际水平，可以在很小的范围内预测运动员的 100 米冲刺成绩。相比之下，许多竞技运动的表现是一个动态过程，其特征在很大程度上是运动员之间的相互作用，尽管他们的运动能力和技战术技能也会影响表现。
In the past, different approaches have been used to assess game sport behaviour. One approach focuses on investigating game-specific actions using performance indicators such as ratios of completed passes, or other descriptive methods such as the analysis of pass sequences [Citation1]. Subsequently, these parameters are studied with respect to their relation to match-winning actions such as scoring goals, using inferential statistics, e.g. [Citation2–Citation4]. However, such performance indicators only describe isolated aspects of the respective game sport [Citation5,Citation6].
过去，人们使用不同的方法来评估游戏运动行为。一种方法侧重于使用性能指标（例如完成传球的比率）或其他描述性方法（例如传球序列分析）来调查特定于游戏的动作[引文1 ]。随后，使用推论统计来研究这些参数与获胜行为（例如进球）的关系，例如[引文2 -引文4 ]。然而，此类绩效指标仅描述了相应比赛运动的孤立方面[引文5 、引文6 ]。
Another approach is to model athletes and game sports as complex, dynamic systems. Human behaviour is based on the interaction of many components providing a large number of degrees of freedom. In a simple case, for example, a finger of a person has a limited number of degrees of freedom. Although one might be able to count the degrees of freedom involved, the task plays an important role which the person performs the finger belongs to. For example, the person could play a PC game by clicking the mouse with the finger in question or he/she could play a sophisticated piano concerto. However, it is obvious ‘that the amount of information that is transmitted by […] [the] biomechanical degree[s] of freedom in the complexity of motion is vastly richer than any counting of mechanical degrees of freedom would suggest’ [Citation7, p. 41]. Observable human behaviour emerges from the synergy of a lot of components, such as mechanical or neuronal components [Citation7]. A specific type of behaviour such as a golf shot can reach a stable pattern through learning (if exercised often enough) and can be reproduced within a small range of variability. The relationships between the interacting components are non-linear. A small change in the hip angle might influence the outcome of a golf shot marginally, whereas a slightly rotated arm usually has a large effect on the angle of the club head, and thus the golf ball curves to the right. This complex systems approach has been applied at many levels of the analysis of human movement and performance, see e.g. [Citation7–Citation9], and is appropriate for modelling individual athletes and their performance in game sports [Citation7].
另一种方法是将运动员和竞技运动建模为复杂的动态系统。人类行为基于许多提供大量自由度的组件的相互作用。在简单的情况下，例如，人的手指具有有限数量的自由度。尽管人们可能能够计算所涉及的自由度，但该任务在执行者手指所属的任务中发挥着重要作用。例如，该人可以通过用相关手指单击鼠标来玩电脑游戏，或者他/她可以演奏复杂的钢琴协奏曲。然而，很明显“运动复杂性中的生物力学自由度所传递的信息量比任何机械自由度的计算所暗示的要丰富得多”[引文7 ，p。 41]。可观察到的人类行为是由许多组件（例如机械或神经元组件）的协同作用产生的[引文7 ]。特定类型的行为（例如高尔夫击球）可以通过学习（如果锻炼得足够频繁）达到稳定的模式，并且可以在小范围的变化范围内重现。相互作用的组件之间的关系是非线性的。臀部角度的微小变化可能会稍微影响高尔夫球的击球结果，而轻微旋转的手臂通常会对球杆头的角度产生很大的影响，因此高尔夫球会向右弯曲。这种复杂的系统方法已应用于人类运动和表现分析的许多层面，参见例如 [引文7 –引文9 ]，适合对个体运动员及其在竞技运动中的表现进行建模 [引文7 ]。
Game sports such as soccer can also be modelled as complex, dynamic systems [Citation10,Citation11]. In a soccer match, there are usually 22 players. The players are coupled with each other in different ways. There are intra-couplings as well as inter-couplings [Citation11]. Intra-couplings describe the couplings within a soccer team. For example, each player of a team is initially coupled with his/her immediate neighbours on the pitch. However, there are many more intra-couplings amongst the players of a team and they keep changing depending on the game’s context. Inter-couplings describe relations between players of different teams such as the coupling between a defender and a striker or between two midfielders of different teams. Due to alternating ball possession, a soccer match shows ‘a to-and-fro behaviour more reminiscent of a tidal ebb-and-flow’ [Citation11, p.773]. Depending on this behaviour, intra- as well as inter-couplings are generated or broken regarding the current game context, which is determined by which team is in possession of the ball and the corresponding positioning of all players on the pitch. For instance, team A gains ball possession and starts a counterattack. Then the team B players try to achieve a well-organized defensive line-up. During this, different players from team B (e.g. a striker and a defender) can be direct neighbours and are coupled differently than during a well-organized defensive team line-up.
足球等游戏运动也可以建模为复杂的动态系统[引文10 、引文11 ]。在一场足球比赛中，通常有 22 名球员。玩家以不同的方式相互配合。存在内部耦合和内部耦合[引文11 ]。内部耦合描述了足球队内部的耦合。例如，球队的每个球员最初都与他/她在球场上的近邻配对。然而，团队成员之间存在更多的内部耦合，并且它们会根据游戏环境而不断变化。相互耦合描述了不同球队的球员之间的关系，例如后卫和前锋之间或不同球队的两名中场球员之间的耦合。由于交替控球，足球比赛表现出“一种更让人想起潮汐涨落的来回行为”[引文11 ，第 773 页]。根据这种行为，根据当前比赛环境生成或破坏内部以及相互之间的耦合，这取决于哪支球队拥有球权以及球场上所有球员的相应位置。例如，A队获得控球权并开始反击。然后B队球员努力实现组织严密的防守阵容。在此期间，B 队的不同球员（例如前锋和后卫）可以是直接邻居，并且与组织良好的防守队阵容中的耦合方式不同。
During ball possession, a team tries to keep possession and score a goal, and the other team simultaneously tries to (re)gain ball possession and prevent the team in possession from scoring. Thus, once the defence of team B and the offence of team A have achieved well-organized line-ups, the teams neutralize each other and stable player patterns occur. The couplings are non-linear, since a player’s error can have either nearly no effect or a very large effect if he/she opens up space and, as a consequence, the team in possession of the ball gets a scoring opportunity. Furthermore, the state of a match at time ti+1 emerges from the players’ behaviour at time ti. Additionally, the complex dynamic system soccer match is a nested system since some of its components – the players – are complex dynamic systems themselves.
在控球期间，一支球队试图保持控球权并进球，而另一支球队同时试图（重新）获得控球权并阻止控球球队得分。这样，一旦B队的防守和A队的进攻形成了良好的阵型，两队就会相互抵消，形成稳定的球员格局。这种耦合是非线性的，因为如果球员打开空间，他/她的失误可能几乎没有影响，或者产生非常大的影响，因此，控球球队获得得分机会。此外，时间t i +1时的比赛状态是从玩家在时间t i时的行为中得出的。此外，复杂动态系统足球比赛是一个嵌套系统，因为它的一些组件（球员）本身就是复杂的动态系统。
Physical sciences provide a variety of concepts and tools which are suitable for understanding and analysing the behaviour of complex systems. For example, the dynamical systems theory, widely used in game sport analyses, provides theoretical concepts such as self-organization [Citation12], affordances [Citation13] or constraints [Citation14] which are used to analyse a system’s observable behaviour qualitatively [Citation5]. On the other hand, there is a variety of methods to quantify, and subsequently statistically analyse, a complex system’s behaviour. Amongst others, such methods are the relative phase, measures of complexity and synchrony – such as the approximate entropy – or recurrence analysis. Relative phase, e.g. [Citation15–Citation20], and complexity measures, e.g. [Citation21,Citation22], have been applied to game sports frequently. For example, relative phase analyses have identified soccer [Citation17,Citation18] and basketball [Citation19,Citation20] as in-phase games outlining a very close inter-coupling between teams regarding longitudinal (goal to goal or basket to basket) movement direction. Furthermore, it was possible to identify some phases as scoring opportunities, when the teams were less strongly coupled. However, for technical reasons relative phase only allows the coupling to be analysed taking one movement direction into consideration. The application of recurrence plots (RPs), a tool for visualizing recurring patterns of complex, dynamical system behaviour, and recurrence quantification analyses (RQAs) [Citation23] – which allow the structures in RPs to be quantified – to game sports is relatively new. So far, the authors of this study are only aware of one study which has applied RQA to game sports, its application to tennis. The players’ relative positioning during baseline rallies was investigated, and the results showed that RQA helped to identify rally breaks leading to points [Citation24].
物理科学提供了各种适合理解和分析复杂系统行为的概念和工具。例如，广泛用于游戏运动分析的动力系统理论提供了诸如自组织[引文12 ]、可供性[引文13 ]或约束[引文14 ]等理论概念，用于定性分析系统的可观察行为[引文5 ]。另一方面，有多种方法可以量化复杂系统的行为，并随后进行统计分析。其中，此类方法包括相对相位、复杂性和同步性的度量（例如近似熵）或递归分析。相对相位，例如[引文15 -引文20 ]，和复杂性测量，例如[引文21 ，引文22 ]，已频繁应用于竞技运动。例如，相对阶段分析已将足球 [引文17 、引文18 ] 和篮球 [引文19 、引文20 ] 确定为同阶段游戏，概述了球队之间在纵向（球门到球门或篮筐到篮筐）方面非常紧密的相互耦合。运动方向。 此外，当团队耦合程度较低时，可以将某些阶段确定为得分机会。然而，由于技术原因，相对相位仅允许在考虑一个运动方向的情况下分析耦合。递归图 (RP) 是一种可视化复杂动态系统行为的重复模式的工具，并且递归量化分析 (RQA) [引文23 ]（允许对 RP 中的结构进行量化）在体育运动中的应用相对较新。到目前为止，这项研究的作者只知道一项将 RQA 应用到竞技运动中的研究，即它在网球中的应用。对基线集会期间球员的相对位置进行了调查，结果表明 RQA 有助于识别导致得分的集会休息时间 [引文24 ]。
A similar method to RP and RQA is the T-patterns approach. It allows the analysis of time series data sets regarding recurring structures [Citation25]. This method has been used to detect and to describe recurring sequences of behavioural events. In contrast to the RP approach, where the underlying data are interval-scaled, the T-patterns are based on categorical event data which describe the beginning and the end of certain behaviour performed by an agent. The T-patterns approach has already been applied to investigate some game sports, including soccer [Citation26]. The event categories in this soccer research were pass, tackle, header or run, and the field of play was separated into discrete zones. A correlation showed a close relation (r = .81) between the number of patterns found for a team and an assessment of the team’s performance by experts. The more structured the manner in which a team played, the better it was.
与 RP 和 RQA 类似的方法是 T 模式方法。它允许分析有关重复结构的时间序列数据集 [引文25 ]。该方法已用于检测和描述行为事件的重复序列。与基础数据按区间缩放的 RP 方法相反，T 模式基于描述代理执行的某些行为的开始和结束的分类事件数据。 T 模式方法已应用于研究一些体育运动，包括足球 [引文26 ]。这项足球研究中的赛事类别为传球、铲球、头球或跑动，并且比赛场地被划分为离散的区域。相关性显示团队发现的模式数量与专家对团队绩效的评估之间存在密切关系 ( r = .81)。球队的比赛方式越结构化，效果就越好。
Considering the findings of research [Citation26], and due to the similarity of the methods, RP and RQA seem to be promising approaches for shedding light on game sport behaviour. The RP analyses are based on the positional information of players in this research and thus focus on the dynamic of the game rather than on specific events.
考虑到研究结果 [引文26 ]，并且由于方法的相似性，RP 和 RQA 似乎是阐明游戏运动行为的有希望的方法。 RP 分析基于本研究中玩家的位置信息，因此关注游戏的动态而不是特定事件。
In golf, the developments in the more recent past have led to measurements which allow the performance of a shot to be quantified in terms of its outcome [Citation27,Citation28]. The existence of such measurements allows more detailed analyses regarding the performance structure, such as ‘Are there phases when golfers’ style of play is more stable or unstable?’ or ‘How predictable are performances?’. Furthermore, amongst experts, fans and also athletes there is the belief that ‘success breeds success and failure breeds failure’ [Citation29, p. 525]. This phenomenon, called streakiness or hot-hand, has been investigated already, but only based on round or hole scores due to the data available. However, only weak support was reported for this belief [Citation30–Citation32]. Such open golf questions could be approached on a shot-by-shot level using RP and RQA.
在高尔夫领域，最近的发展导致了一些测量方法，可以根据其结果来量化击球的性能[引文27 、引文28 ]。此类测量的存在允许对表现结构进行更详细的分析，例如“高尔夫球手的打球风格是否存在更稳定或不稳定的阶段？”或“表演的可预测性如何？”。此外，专家、球迷和运动员都相信“成功孕育成功，失败孕育失败”[引文29 ，第 14 页]。 525]。这种称为条纹或热手的现象已经被研究过，但由于现有数据仅基于圆或孔分数。然而，据报道，这一信念的支持率很弱[引文30 -引文32 ]。此类开放性高尔夫问题可以使用 RP 和 RQA 在逐个击球的水平上进行处理。
This paper illustrates the application of the RP method to golf and soccer. In both sports, progress in information technologies has led to the collection of a huge amount of data during competitions. During golf competitions, data on each shot of each golfer is collected, at least during tournaments of the American professional golf tour, the US PGA TOUR. During soccer matches, positional data is collected for every player on the field of play. The existence of such data allows for the progress of the complex, dynamic system golfer as well as the progress of the complex, dynamic system soccer match to be described during a competition. The emphasis of this paper is on the technical aspects of the method’s application to both sports – the construction of RPs – rather than the application of the method from a sport scientific viewpoint, for instance a detailed analysis of performances using RP and RQA. However, for both sports a sport scientific application will be illustrated shortly.
本文阐述了 RP 方法在高尔夫和足球中的应用。在这两项运动中，信息技术的进步导致比赛期间收集大量数据。在高尔夫比赛期间，至少在美国职业高尔夫巡回赛、美国 PGA 巡回赛的比赛期间，每个高尔夫球手的每次击球数据都会被收集。在足球比赛期间，会收集赛场上每位球员的位置数据。这种数据的存在允许在比赛期间描述复杂的动态系统高尔夫球手的进展以及复杂的动态系统足球比赛的进展。本文的重点是该方法在两项运动中的应用的技术方面——RP的构建——而不是从运动科学角度应用该方法，例如使用RP和RQA对表现进行详细分析。然而，对于这两项运动，运动科学应用将很快得到说明。
The paper is outlined as follows: first, there is a short overview on RP and RQA. Then, the technical details of their application to golf are presented, followed by a short sport scientific application; and finally, the technical details of their application to soccer are presented followed by a short sport scientific application.
本文概述如下：首先，对 RP 和 RQA 进行简短概述。然后，介绍了它们在高尔夫中的应用的技术细节，随后是简短的运动科学应用；最后，介绍了它们在足球中应用的技术细节，然后是简短的运动科学应用。
Recurrence plots 重现图
The RP concept RP概念
An RP is a tool for visualizing similarities within time series of complex, dynamic systems. Recurring patterns of a system can be identified in a symmetric N×N graphic, where N is the length of the time series. Generally, RPs can be divided into three categories according to the pattern of the recurrence points regarding the whole plot (large-scale texture) [Citation33]: homogeneous plots (stationary process), periodical plots (oscillating process) and drift plots (transmuting process). Furthermore, the local patterns of an RP (small-scale textures) provide detailed information on the complex system’s behaviour during a certain time period. The most important small-scale textures are [Citation23]:
RP 是一种可视化复杂动态系统时间序列内相似性的工具。系统的重复模式可以在对称的N × N图形中识别，其中N是时间序列的长度。一般来说，RP可以根据整个图（大尺度纹理）的重现点模式分为三类：同质图（稳态过程）、周期图（振荡过程）和漂移图（嬗变过程）过程）。此外，RP（小尺度纹理）的局部模式提供了复杂系统在特定时间段内行为的详细信息。最重要的小尺度纹理是 [引文23 ]：
Isolated recurrence points (recurrence points without neighbours) indicate rare states or high fluctuation.
孤立的复发点（没有邻居的复发点）表示罕见状态或高波动。Diagonal lines describe that the investigated system’s trajectories run parallel and close to each other for a certain time.
对角线描述了所研究系统的轨迹在一定时间内彼此平行且接近。Vertical (or horizontal) lines illustrate that the system remains in a state or very slowly moves in the neighbourhood of a state for a certain time.
垂直（或水平）线表示系统在一定时间内保持在某个状态或在某个状态附近非常缓慢地移动。Clusters of recurrence points mean that vertical and horizontal lines occur at the same time, forming blocks.
重复点簇意味着垂直线和水平线同时出现，形成块。White space or bands refer to states which are rare or only occur once.
空白或条带是指罕见或仅出现一次的状态。
Depending on the characteristics of the analysed complex system, the types and the amount of occurring small-scale textures usually vary. For example, deterministic processes cause longer diagonal lines in their RPs than chaotic processes, whose RPs are dominated by isolated recurrence points [Citation23].
根据所分析的复杂系统的特征，出现的小尺度纹理的类型和数量通常会有所不同。例如，确定性过程在其 RP 中导致比混沌过程更长的对角线，混沌过程的 RP 由孤立的重复点主导 [引文23 ]。
RP construction RP施工
The construction of RPs involves several steps which will be introduced in this subsection. The details of the RP construction applied to golf and to soccer will be discussed in the following sections.
RP 的构建涉及本小节将介绍的几个步骤。应用于高尔夫和足球的 RP 构造的细节将在以下部分中讨论。
The investigated complex system’s behaviour is often measured as scalar, discrete time series a1,…,aN, since not all relevant components of a system are known or can be measured. According to Takens [Citation34], the original time series needs to be expanded to a higher dimensional phase space in order to unfold the real system’s dynamics. The so-called phase space reconstruction is often realized using a time delay method [Citation23]. This method requires two parameters to be chosen properly, the embedding dimension m and the time delay tau, [Citation23,Citation35,Citation36]. These parameters are frequently determined using a false nearest neighbour algorithm for the embedding dimension and an autocorrelation or mutual information function for the time delay. False nearest neighbours are phase space points that are close to each other in dimensions smaller than the original dimension of the investigated system in which these points are not close to each other. The method used in this paper [Citation37] considers the amount of false nearest neighbours as a function of the embedding dimension. An appropriate embedding dimension is found when the number of false nearest neighbours approaches zero [Citation37]. The autocorrelation function assesses dependences between the measurements at different times. An appropriate tau is found when the function approaches zero [Citation38]. Mutual information involves a function expressing ‘the average of the information about a value after a delay tau, which can be yielded from the knowledge of the current value’ [Citation38, p. 6]. According to the research [Citation38], an appropriate tau is where the mutual information function has its smallest local minimum.
所研究的复杂系统的行为通常被测量为标量、离散时间序列a 1 , … , a N ，因为并非系统的所有相关组件都是已知或可以测量的。根据 Takens [引文34 ]，原始时间序列需要扩展到更高维的相空间，才能展现真实系统的动力学。所谓的相空间重构通常使用时间延迟方法来实现[引文23 ]。该方法需要适当选择两个参数，即嵌入维度 m和时间延迟 tau ，
[引文23 、引文35 、引文36 ]。这些参数经常使用嵌入维度的伪最近邻算法和时间延迟的自相关或互信息函数来确定。假最近邻是指尺寸小于所研究系统的原始尺寸且彼此不接近的相空间点。本文[引文37 ]中使用的方法将错误最近邻的数量视为嵌入维度的函数。 当错误最近邻的数量接近零时，就会找到合适的嵌入维度[引用37 ]。自相关函数评估不同时间测量值之间的相关性。当函数接近零时，就会找到合适的tau [引文38 ]。互信息涉及一个函数，该函数表示“关于延迟tau后的值的信息的平均值，该平均值可以从当前值的知识中得出”[引文38 ，第 17 页]。 6]。根据研究[引文38 ]，适当的tau是互信息函数具有最小局部最小值的位置。
Based on m and tau, the measured time series is embedded into a higher dimensional phase space where the progress of the system is represented by a series of vectors bi = (ai, ai+1*tau,…, ai+(m−1)*tau)T, i = 1,…,N – (m – 1)*tau. However, there are discussions on whether an embedding is necessary for the RP construction [Citation35,Citation39]. In such cases the RP construction is conducted with tau = 1 and m = 1.
基于m和tau ，测量的时间序列被嵌入到更高维的相空间中，其中系统的进展由一系列向量b i = ( a i , a i +1*tau,… , a i + ( m −1)*tau ) T ， i = 1,…,N – (m – 1)*tau 。然而，对于 RP 构建是否需要嵌入，存在讨论[引文35 、引文39 ]。在这种情况下，RP 构建是在tau = 1 和m = 1 的情况下进行的。
Independent of the decision to embed or not to embed, the similarity of states needs to be determined. It is usually assessed using a distance measure, such as the absolute value norm (finding the lowest number of neighbours), the Euclidean norm (finding an intermediate number of neighbours) or the maximum norm (finding the highest number of neighbours). According to [Citation23], all these norms are appropriate and can be chosen with respect to the computational speed; for instance, the maximum norm is the fastest. States are treated as similar and are also called recurring states, if the distance between them is smaller than a predefined threshold ε [Citation23]. Determining an appropriate ε is crucial since it is important that not every state is similar to all others and, on the other hand, too small a value results in a nearly empty RP. In these cases the RP would not provide meaningful information on the recurring structures anymore. There are several approaches to choosing ε which are accepted for delivering the appropriate thresholds, see [Citation23] for an overview, e.g.:
与嵌入或不嵌入的决定无关，需要确定状态的相似性。通常使用距离度量来评估，例如绝对值范数（找到最少数量的邻居）、欧几里得范数（找到中间数量的邻居）或最大范数（找到最多数量的邻居）。根据[引文23 ]，所有这些规范都是合适的，并且可以根据计算速度来选择；例如，最大范数是最快的。如果状态之间的距离小于预定义的阈值ε ，则状态被视为相似，也称为重复状态 [引用23 ]。确定适当的ε至关重要，因为并非每个状态都与所有其他状态相似，这一点很重要，另一方面，值太小会导致 RP 几乎为空。在这些情况下，RP 将不再提供有关重复结构的有意义的信息。有几种选择ε 的方法被接受用于提供适当的阈值，请参阅 [引文23 ] 了解概述，例如：
A frequently used approach to finding a proper ε is to make the threshold a few per cent of the maximum diameter of the phase space trajectory of the investigated system [Citation40], but not greater than 10% [Citation41].
寻找合适ε 的常用方法是将阈值设置为所研究系统相空间轨迹最大直径的百分之几 [引文40 ]，但不大于 10% [引文41 ]。Marwan [Citation42] states that it is important to achieve a high number of long diagonals while the variability of the lengths stays small to get as much information as possible from an RP. Therefore, the value of ε should be selected as the one where the ratio (number of recurrence points forming diagonal lines)/(variability of the length of the diagonal lines) reaches its maximum. The measurement of a complex system’s process is composed by the real values and noise. If the amount of noise is known, ε should be larger than five times the standard deviation of the observational noise.
Marwan [引文42 ] 指出，重要的是获得大量的长对角线，同时保持长度的可变性较小，以便从 RP 中获取尽可能多的信息。因此， ε的值应选择为（形成对角线的重复点的数量）/（对角线长度的可变性）之比达到最大的值。复杂系统过程的测量由实值和噪声组成。如果噪声量已知，则 ε应大于观测噪声标准差的五倍。
Mathematically, an RP is represented by a matrix with binary entries
在数学上，RP 由具有二进制条目的矩阵表示
where Θ is the Heaviside function and ||.|| is a norm, the chosen distance measure for assessing similarity between two states. The entries of RP are plotted colour coded, black usually indicates that two states are similar and white the opposite.
其中 θ 是 Heaviside 函数，||.||是一个范数，为评估两个状态之间的相似性而选择的距离度量。 RP的条目以颜色编码绘制，黑色通常表示两个状态相似，白色表示相反。
Recurrence quantification analysis
复发量化分析
A method called RQA was developed to quantify structures in RPs [Citation23,Citation43]. Amongst other things, an RQA involves analyses of diagonal lines, vertical lines and the recurrence point density. The respective RQA parameters used in this paper are the rate of recurrence points in a plot (RR), the proportion of recurrence points forming diagonal (DET) and vertical lines (LAM), the average length of the diagonal (L) and the vertical lines (TT) and the variability in the length of the diagonal lines (ENTR).
开发了一种称为 RQA 的方法来量化 RP 中的结构 [引文23 、引文43 ]。其中，RQA 涉及对角线、垂直线和重复点密度的分析。本文使用的 RQA 参数分别是图中重复点的比率 ( RR )、重复点形成对角线 ( DET ) 和垂直线 ( LAM ) 的比例、对角线 ( L ) 和垂直线的平均长度 (L)。线 ( TT ) 和对角线长度的变化 ( ENTR )。
For the RQA parameters which are based on diagonal and vertical lines, a minimum line length needs to be determined prior to it. The main reason for selecting the minimum line length with caution is in order to restrict the influence of tangential motion which ‘also [includes] points into the neighbourhood which are simple consecutive points on the trajectory’ [Citation23, p. 247]. Furthermore, if the minimum is set too large the detected amount of diagonal lines in the RQA decreases and, consequently, the reliability of DET and LAM declines. Typically, the minimum line length is set two [Citation38].
对于基于对角线和垂直线的RQA参数，需要在其之前确定最小线长度。谨慎选择最小线长度的主要原因是为了限制切向运动的影响，切向运动“还[包括]指向邻域，这些点是轨迹上的简单连续点”[引文23 ，第 17 页]。 247]。此外，如果最小值设置得太大，RQA 中检测到的对角线数量会减少，从而导致DET和LAM的可靠性下降。通常，最小线路长度设置为 2 [引用38 ]。
Applications to golf and soccer
在高尔夫和足球中的应用
This section thoroughly addresses the RP applications to golf and to soccer. In particular, the choice of parameters for the RP construction is described in detail. For the RP and RQA computations, the RP MATLAB toolbox provided by the authors of [Citation23] was used.
本节彻底讨论 RP 在高尔夫和足球中的应用。特别是，详细描述了 RP 构造参数的选择。对于 RP 和 RQA 计算，使用了 [引文23 ] 作者提供的 RP MATLAB 工具箱。
RPs in golf
高尔夫RP
First, Shots Saved, a golf-specific performance indicator which describes the quality of individual golf shots, needs to be introduced briefly [Citation28]. Basically, in this application the investigated dynamical system is the golfer and the system’s behaviour is described by Shots Saved values [Citation28]. Since a golfer usually plays several shots at each golf hole and on a professional level a round is comprised of 18 holes, a round generates a sequence of shots, each of which can be assessed using Shots Saved values. Such a Shots Saved sequence can be considered as a discrete time series describing the respective golfer’s behaviour on the golf course on a shot-by-shot basis. Therefore, the RP approach based on a sequence of Shots Saved values is technically appropriate for studying the characteristics of golfers’ performances during competitions in principle.
首先，需要简单介绍一下“Shots Saved”（保存的击球数），这是一项高尔夫特定的表现指标，用于描述单个高尔夫击球的质量[引文28 ]。基本上，在此应用中，研究的动力系统是高尔夫球手，系统的行为由击球保存值描述[引文28 ]。由于高尔夫球手通常在每个高尔夫球洞打几次球，而在专业水平上，一轮由 18 个洞组成，因此一轮会生成一系列击球，每个击球都可以使用“已保存击球”值进行评估。这样的击球保存序列可以被认为是离散时间序列，其逐杆描述高尔夫球场上的相应高尔夫球手的行为。因此，基于一系列已保存击球值的RP方法原则上在技术上适合研究高尔夫球手在比赛中的表现特征。
In golf, each hole should be played with a certain number of shots, called par. Depending on the distance from the tee to the hole, there are par three, par four and par five holes. Research based on hole-by-hole analyses reveals that golfers on the PGA TOUR play par on about 60% of the holes [Citation44]. Further research reported that in 60% of cases a par is followed by another par, whereas better and worse holes follow a par 20% of the time [Citation32]. Thus, there are often stable phases when players play a few pars in a row. From a performance analysis perspective, this motivated us to conduct a pilot study of the performance of PGA TOUR golfers on a stroke-by-stroke basis using RP aimed at the question of whether there are such stable phases at a shot-by-shot level too. The recurrence analysis application described below is based on data from a PGA TOUR tournament, THE PLAYERS Championship 2011. It involves data from those 74 players who played all four rounds of the tournament (after two rounds there is a cut which only allows the better ranked half of the players to continue playing).
在高尔夫运动中，每个洞都应进行一定数量的击球，称为标准杆。根据从发球台到球洞的距离，有三杆洞、四杆洞和五杆洞。基于逐洞分析的研究表明，PGA 巡回赛上的高尔夫球手在大约 60% 的洞中打出标准杆成绩 [引文44 ]。进一步的研究表明，在 60% 的情况下，一个标准杆之后会紧跟着另一个标准杆，而更好和更差的球洞则有 20% 的情况会跟随一个标准杆 [引文32 ]。因此，当球员连续打出几个标准杆时，通常会出现稳定的阶段。从表现分析的角度来看，这促使我们使用 RP 对美巡赛高尔夫球手的逐杆表现进行试点研究，旨在解决逐杆水平上是否存在这种稳定阶段的问题也。下面描述的重复分析应用程序基于 PGA 巡回赛锦标赛（2011 年球员锦标赛）的数据。它涉及参加锦标赛所有四轮比赛的 74 名球员的数据（两轮之后有一个晋级，只允许排名较好的球员参赛）一半的玩家继续玩）。
The ISOPAR method ISOPAR 方法
The ISOPAR method is an approach developed to assess the quality of golf shots [Citation28]. It relies on concepts of the dynamical systems theory, in the sense that the golfer as a human being can be considered as a dynamic system and that his/her behaviour on the golf course is constrained by influences from different sources [Citation28]. For instance, such influences include the weather (e.g. wind, sun or rain) or the hole type (e.g. distance to the hole, surrounding area such as trees, or hardness and speed of the green). The golfers’ play can be influenced directly or indirectly by such factors. The golfers are directly influenced, for example, by a tree which blocks the straight line between the ball’s location and the hole. A factor indirectly influencing the golfers’ play would be a water hazard from which the golfers aim away for strategic reasons. Although this is not the shortest way to the hole, it limits the possibility of hitting the ball into the water, for which a golfer is penalized by an extra shot. Moreover, the golfers themselves are a source by which their play is constrained. A golfer needs to perceive the external factors, process this information to make a decision for the next shot and finally perform the intended shot as well as possible.
ISOPAR 方法是一种为评估高尔夫球击球质量而开发的方法 [引文28 ]。它依赖于动态系统理论的概念，从某种意义上说，高尔夫球手作为一个人可以被视为一个动态系统，并且他/她在高尔夫球场上的行为受到来自不同来源的影响的约束[引文28 ]。例如，此类影响包括天气（例如风、阳光或雨）或球洞类型（例如到球洞的距离、周围区域（例如树木）或果岭的硬度和速度）。高尔夫球手的比赛会直接或间接地受到这些因素的影响。例如，高尔夫球手会直接受到遮挡球位置和球洞之间直线的树木的影响。间接影响高尔夫球手比赛的一个因素是水障碍区，高尔夫球手出于战略原因而将目标瞄准远离水障碍区。虽然这不是到达球洞的最短路径，但它限制了将球击入水中的可能性，因此高尔夫球手将受到额外击球的处罚。此外，高尔夫球手本身也是他们比赛受到限制的一个来源。高尔夫球手需要感知外部因素，处理这些信息以做出下一次击球的决定，并最终尽可能地击出预期的击球。
The ISOPAR method is based on ball locations and the respective number of remaining shots until the ball is holed. The number of remaining shots from a location represents the difficulty of this because the more difficult the location was, the more shots were needed. The difficulty is the result of the synergy of the abovementioned influencing factors. The ISOPAR method provides an algorithm for calculating the difficulty of holing the ball from any location at a golf hole. The resulting difficulty values are a weighted average of the remaining number of shots and are called ISOPAR values. Since the influencing factors are different from hole to hole, the difficulty is calculated for each hole separately. Furthermore, since a golf tournament is usually played over 4 days (one round a day) and each hole is played once a day, the ISOPAR value calculation is performed for each hole in each round to account for changing factors such as the weather (assuming that the weather during one given day is more stable than across 4 days) or the pin location, which is different in each round at each hole.
ISOPAR 方法基于球的位置以及球进洞前的相应剩余击球次数。某个地点剩余的拍摄次数代表了该场景的难度，因为地点越困难，需要拍摄的镜头就越多。困难是上述影响因素综合作用的结果。 ISOPAR 方法提供了一种计算从高尔夫球洞任意位置击球难度的算法。所得难度值是剩余射击次数的加权平均值，称为 ISOPAR 值。由于每个洞的影响因素不同，所以难度是针对每个洞单独计算的。此外，由于高尔夫锦标赛通常持续 4 天（每天一轮），并且每个洞每天打一次，因此每轮中的每个洞都会执行 ISOPAR 值计算，以考虑天气等变化因素（假设某一天的天气比 4 天的天气更稳定）或旗杆位置，每个洞的每一轮比赛中的旗杆位置都不同。
Based on the ISOPAR values, this method allows the quality of each individual shot to be assessed through the performance indicator Shots Saved [Citation28]. For each ball location, an ISOPAR value can be determined. Considering the ball location before and after a shot allows a golfer’s performance to be assessed as shot quality (SQ) = ISOPAR valuebefore – ISOPAR valueafter [Citation28]. The Shots Saved value for a shot is defined as SQ – SQave, where SQave describes the average SQ of all golfers playing at the respective hole [Citation28]. Thus, a Shots Saved value describes how much better or worse a golfer’s shot was compared to an average performance at a hole. A Shots Saved value of a shot is the ‘measurable’ outcome of the behaviour of the dynamic system golfer at this shot.
基于 ISOPAR 值，该方法允许通过性能指标“已保存的镜头”来评估每个镜头的质量 [引文28 ]。对于每个球位置，可以确定 ISOPAR 值。考虑击球前后的球位置，可以根据击球质量 (SQ) = 击球前的 ISOPAR 值 – 击球后的ISOPAR 值来评估高尔夫球手的表现 [引文28 ]。每次击球的“保存的击球数”值定义为 SQ – SQ ave ，其中 SQ ave描述了在相应洞打球的所有高尔夫球手的平均 SQ [引用28 ]。因此，“节省的击球”值描述了高尔夫球手的击球与球洞的平均表现相比好多少或差了多少。击球 击球的保存值是动态系统高尔夫球手在该击球时的行为的“可测量”结果。
As described above, the quality of a golfer’s shot sequence in a tournament can be described by his/her Shots Saved series x1,…,xN, where N describes the number of shots the golfer took during the tournament. This Shots Saved series was used as the time series describing the golfer’s behaviour on which the RP is based.
如上所述，高尔夫球手在锦标赛中的击球序列的质量可以通过他/她的击球保存系列x 1 , …,x N来描述，其中N描述高尔夫球手在锦标赛期间击球的次数。此击球保存序列用作描述高尔夫球手行为的时间序列，RP 以此为基础。
RP construction RP施工
For the application to golf data, the use of phase space reconstruction was chosen to construct the RPs. The complex system golfer comprises a lot of unknown components and the interactions between them, and additionally the measurement describing the golfer’s performance is scalar. According to Takens [Citation34], embedding phase space reconstruction is recommended. It was conducted using the time delay method. The respective parameters were determined using the mutual information algorithm and the false nearest neighbour algorithm both implemented in the used MATLAB toolbox [Citation23]. The mutual information plots were visually inspected for all golfers in the data set (e.g. see
对于高尔夫数据的应用，选择使用相空间重建来构建 RP。高尔夫球手的复杂系统包含许多未知组件以及它们之间的相互作用，此外，描述高尔夫球手表现的测量是标量的。根据 Takens [引文34 ]，建议使用嵌入相空间重建。它是使用时间延迟方法进行的。各自的参数是使用互信息算法和虚假最近邻算法确定的，这两种算法都在所用的 MATLAB 工具箱中实现 [引用23 ]。对数据集中所有高尔夫球手的互信息图进行了目视检查（例如，参见). The mutual information functions had their smallest local minima at time delay tau = 2. Subsequently, the embedding dimensions were determined based on the false nearest neighbours approach using tau = 2 (e.g. see
）。互信息函数在时间延迟tau = 2 时具有最小局部最小值。随后，使用tau = 2 基于虚假最近邻方法确定嵌入维度（例如，参见). The embedding dimension, where either no false nearest neighbours were left or the number of false nearest neighbours approached zero, was m = 2 for all golfers.
）。对于所有高尔夫球手来说，嵌入维度为m = 2，其中要么没有留下任何虚假最近邻居，要么虚假最近邻居的数量接近零。
Figure 1. Mutual information function of one of the golfers at THE PLAYERS Championship 2011.
图 1. 2011 年球员锦标赛上一名高尔夫球手的互信息函数。
Figure 2. False nearest neighbour function of one of the golfers at THE PLAYERS Championship 2011 using tau = 2.
图 2. 2011 年球员锦标赛上一名高尔夫球手的虚假最近邻函数，使用tau = 2。
Using the phase space reconstruction based on m and tau, a recurrence threshold ε had to be determined. A frequently used approach was used to determine ε as a few per cent of the maximum diameter of the phase space trajectory of the investigated system, but not greater than 10%. Since for all golfers in the data set the reconstructed phase space had the same dimension, ε was decided to be the same for all golfers. Otherwise, the similarity of states would be assessed differently and the RPs would no longer be comparable. For instance, two states can be 0.05 Shots Saved apart from each other for one golfer and two states can be 0.2 Shots Saved apart from each other for another golfer, but in both cases the states are considered similar because of individual recurrence thresholds. Such unwanted effects affect the analysis of the RPs. Therefore, the following approach was used to determine ε, which is valid for all golfers:
使用基于m和tau 的相空间重建，必须确定复发阈值ε 。一种常用的方法用于将ε确定为所研究系统相空间轨迹最大直径的百分之几，但不大于 10%。由于对于数据集中的所有高尔夫球手来说，重建的相空间具有相同的维度，因此ε对于所有高尔夫球手来说都是相同的。否则，将对状态的相似性进行不同的评估，并且 RP 将不再具有可比性。例如，对于一名高尔夫球手来说，两个状态可以彼此相距 0.05 Shots Saved，对于另一高尔夫球手来说，两个状态可以彼此相距 0.2 Shots Saved，但是在这两种情况下，由于各个重复阈值，状态被认为是相似的。这种不良影响会影响 RP 的分析。因此，使用以下方法来确定ε ，这对所有高尔夫球手都有效：
For each golfer the trajectory bi, i = 1,…,N – (m – 1)*tau, in the reconstructed phase space was determined.
对于每个高尔夫球手，在重建的相空间中确定轨迹b i , i = 1, …,N – (m – 1 )*tau 。Extreme outliers were removed from each trajectory to ensure that ε meets the maximum criterion of 10%. To identify extreme outliers, the approach suggested in [Citation45] was used. There extreme outliers eo are defined as
从每个轨迹中删除极端异常值，以确保ε满足 10% 的最大标准。为了识别极端异常值，使用了[引文45 ]中建议的方法。极端异常值eo定义为
where IQ is the interquartile distance and Qi denotes the ith quartile of the distribution.
其中IQ是四分位数距离， Q i表示分布的第 i个四分位数。
For each golfer the recurrence threshold εi was determined as 10% of the diameter of the leftover trajectory.
对于每个高尔夫球手，复发阈值ε i被确定为剩余轨迹直径的 10%。Finally, the recurrence threshold was computed as
.
最后，复发阈值计算为
The resulting ε was 0.14. In this application, the Euclidean norm was chosen as the distance measure for assessing the similarity of states in the reconstructed phase space as well as for the calculation of ε.
所得的ε为0.14。在此应用中，选择欧几里得范数作为距离度量，用于评估重构相空间中状态的相似性以及计算ε 。
Using the parameters determined above, the resulting average RR based on all golfers’ RPs equals 6.9% (SD = 1.0%). Thus, the resulting RPs are sparse enough to allow an appropriate RQA according to Webber and Zbilut [Citation46]. Therefore, the parameters for the RP construction were chosen sufficiently.
使用上面确定的参数，基于所有高尔夫球手的 RP 得出的平均RR等于 6.9% (SD = 1.0%)。因此，根据 Webber 和 Zbilut [引文46 ]，生成的 RP 足够稀疏以允许适当的 RQA。因此，RP 结构的参数选择充分。
Sport scientific application
运动科学应用
shows the RP of a golfer, K. J. Choi who won the tournament the data was taken from. The RP is characterized by many isolated recurrence points, a few clusters and short vertical lines as well as a few short diagonal lines. Therefore, Choi’s play can be interpreted as very fluctuating, but alternating with some short stable phases (vertical lines) from a sport scientific viewpoint using the suggestions of [Citation23]. More-or-less white bands or sparsely populated bands can be identified in the RP as well, e.g. before shot 150. Those bands indicate that the quality of performance changed abruptly [Citation23]; in other words, Choi played a few consecutive shots which he rarely played in this tournament, e.g. alternating extremely good and extremely bad shots. However, the RP itself does not provide information on how well or how bad Choi played at any given time.
显示了赢得数据的高尔夫球手KJ Choi 的 RP。 RP的特点是许多孤立的复发点、一些簇和短垂直线以及一些短对角线。因此，崔的比赛可以被解释为非常波动，但从运动科学的角度来看，使用[引文23 ]的建议与一些短暂的稳定阶段（垂直线）交替。 RP 中也可以识别出或多或少的白色条带或稀疏的条带，例如在第 150 个镜头之前。这些条带表明表演质量突然发生变化[引文23 ]；换句话说，崔顺实连续打了几杆，这是他在本次比赛中很少打出的，例如交替打出极好球和极差球。然而，RP 本身并没有提供有关崔在任何特定时间的表现有多好或有多差的信息。
Figure 3. Recurrence plot of K. J. Choi’s performance at THE PLAYERS Championship 2011 (m = 2, tau = 2, ε = 0.14); black dots represent recurring states.
图 3. KJ Choi 在 2011 年球员锦标赛上的表现的递归图（ m = 2， tau = 2， ε = 0.14）；黑点代表重复出现的状态。
The absence of longer diagonal lines parallel to the line of identity raises the suspicion that Choi’s behaviour was not deterministic, but rather chaotic. The RQA analysis of Choi’s plot supports this assumption since DET = 16.2% and L = 2.4 (for DET the minimum line length was set two). Such small values indicate that the investigated system, and hence this golfer, behaved rather unpredictably/chaotically [Citation45]. Choi’s LAM = 22.7% and TT = 2.5 reveal that about a fifth of his recurring shots comprised vertical lines which can be interpreted as a stable performance. However, the stable phases only lasted two to three shots on average, except his longest stable phase. The latter lasted seven shots on the holes 16 and 17 of round one, both of which he played par. This stable phase was comprised by shots of 0.06 Shots Saved on average.
由于没有与身份线平行的较长对角线，人们怀疑崔的行为不是确定性的，而是混乱的。 Choi 图的 RQA 分析支持这一假设，因为DET = 16.2% 且L = 2.4（对于DET，最小线长度设置为 2）。如此小的值表明所研究的系统以及该高尔夫球手的行为相当不可预测/混乱[引文45 ]。 Choi 的LAM = 22.7% 和TT = 2.5 显示他的重复投篮中大约有五分之一是垂直线，这可以解释为稳定的表现。然而，除了他最长的稳定阶段外，稳定阶段平均只持续两到三枪。后者在第一轮第16洞和第17洞持续了7杆，两洞均打出标准杆。这个稳定阶段由平均保存 0.06 次射门组成。
With respect to all golfers in the data set, the average DET = 13.5% (SD = 2.6%), the average L = 2.4 (SD = 0.1), the average LAM = 17.5% (SD = 4.6%) and the average TT = 2.2 (SD = 0.1). These values suggest that Choi’s behaviour was rather typical, since all golfers generally behaved rather unpredictably or quite variably during this tournament. In some cases it was only possible to predict the quality of a shot based on the previous shot(s), and there were only a few short phases in which the quality of the shots remained similar. Furthermore, Spearman correlations revealed that DET (p = .979; r = .003), L (p = .907; r = –.014) and LAM (p = .885; r = .017) were not correlated with the tournament ranking. Thus, the fact that the players’ performances were unpredictably fluctuating was independent of their rank, as was the fact that about a fifth of the golfers’ recurring shots composed stable phases. The Spearman correlation of TT (p = .035; r = .246) with the tournament ranks only weakly supported that the worse a player was ranked, the longer his stable phases were. A tournament ranking is determined by the number of shots a player takes, and therefore the better-ranked players play fewer and on average better shots. Thus, one could speculate whether the stable phases of the worse-ranked players were of worse quality. Moreover, the shot sequence at a hole is comprised of different shot types which require different motions. Further RP analyses should aim at this and analyse golfers’ performance regarding the shot types. This might support the identification of weaknesses to focus on in training sessions.
对于数据集中的所有高尔夫球手，平均DET = 13.5% (SD = 2.6%)、平均L = 2.4 (SD = 0.1)、平均LAM = 17.5% (SD = 4.6%) 和平均TT = 2.2（标准差 = 0.1）。这些值表明崔的行为相当典型，因为所有高尔夫球手在这次比赛中通常表现得相当不可预测或变化很大。在某些情况下，只能根据之前的镜头来预测镜头的质量，并且只有几个短暂的阶段镜头的质量保持相似。此外，Spearman 相关性显示DET ( p = .979; r = .003)、 L ( p = .907; r = –.014) 和LAM ( p = .885; r = .017) 与赛事排名。因此，球员的表现不可预测地波动的事实与他们的排名无关，事实上，大约五分之一的高尔夫球手的重复击球构成了稳定阶段。 TT （ p = .035； r = .246）与锦标赛排名的 Spearman 相关性仅弱地支持玩家排名越差，其稳定阶段越长。锦标赛排名由选手的击球次数决定，因此排名较高的选手的击球次数较少，平均击球次数也较多。由此，我们可以推测排名较差的玩家稳定阶段的质量是否较差。此外，一洞的击球序列由需要不同运动的不同击球类型组成。 进一步的 RP 分析应针对这一点并分析高尔夫球手在击球类型方面的表现。这可能有助于识别培训课程中要重点关注的弱点。
Concluding, RP is a tool which allows performance analysis describing the composition of golfers’ performances using the performance indicator Shots Saved. A behaviour including stable phases, which is known from hole-by-hole analysis, cannot be found as distinct on a stroke-by-stroke level. A shot’s performance is less predictable based on the previous shot(s) than a hole score is based on the previous hole score. Further RP analyses should consider the Shots Saved values of the recurring shots to add more context. Then, in particular, RQA might help to shed light on the structure of golfers’ performances, such as analysing the stability and variability of golfers’ play, and might enable the psychological concept of momentum (see [Citation29] for an overview) in golf to be investigated. Hence, there are several sport scientific questions in golf which can be approached using RPs and RQA.
总之，RP 是一种工具，可以使用表现指标“保存的击球数”来进行表现分析，描述高尔夫球手的表现构成。从逐孔分析中得知的包括稳定阶段的行为在逐笔划水平上不能被发现是明显的。与基于前一洞得分的洞得分相比，基于之前的击球的击球表现的可预测性更差。进一步的 RP 分析应考虑重复镜头的镜头保存值，以添加更多上下文。然后，特别是，RQA 可能有助于阐明高尔夫球手表现的结构，例如分析高尔夫球手比赛的稳定性和可变性，并且可能启用动量的心理概念（概述请参见[引文29 ]）高尔夫有待调查。因此，高尔夫中存在一些可以使用 RP 和 RQA 来解决的运动科学问题。
RPs in soccer 足球中的 RP
As outlined in the introduction, a soccer match can be modelled as a complex, dynamic system. In contrast to the golf example from above, here it is possible to use measurements of the behaviour of components the system is comprised of, in fact all the players on the pitch.
正如简介中所述，足球比赛可以建模为一个复杂的动态系统。与上面的高尔夫示例相反，这里可以使用系统组成组件（实际上是球场上所有球员）的行为测量。
Nowadays, companies such as Prozone Sports or ChyronHego provide tracking technologies to collect the positional information of each player during soccer matches. Thus, for the player-related components there is spatio-temporal information describing their behaviour. In soccer there is a line-up based on tactical targets for each team consisting of playing positions such as central defender, central midfielder or forward. Each player is associated with a playing position. However, the players sometimes switch playing positions during matches, which can lead to unwanted artefacts if an RP is calculated based purely on player information. Therefore, the playing positions represent the components better than merely the players themselves.
如今，Prozone Sports 或 ChyronHego 等公司提供跟踪技术来收集足球比赛期间每个球员的位置信息。因此，对于与玩家相关的组件，存在描述其行为的时空信息。在足球比赛中，每支球队都有一个基于战术目标的阵容，包括中后卫、中前卫或前锋等比赛位置。每个球员都与一个比赛位置相关联。然而，球员有时会在比赛期间更换比赛位置，如果纯粹根据球员信息计算 RP，可能会导致不必要的假象。因此，比赛位置比仅仅球员本身更能代表各个组成部分。
Since there is information on each individual player during a match, the system’s behaviour can be described higher dimensional from the beginning rather than using a scalar summary measurement. Combining the positional information of each playing position to a multidimensional variable vi describes the system’s behaviour at time i very precisely. Hence, a series v1,…,vN represents the system’s evolution during a time period of length N.
由于比赛期间存在有关每个球员的信息，因此可以从一开始就以更高的维度描述系统的行为，而不是使用标量汇总测量。将每个演奏位置的位置信息组合到多维变量vi可以非常精确地描述系统在时间i的行为。因此，序列v 1 ,..., v N表示系统在长度为N的时间段内的演化。
In this study, data – to be exact the locations pi = (x,y) of each player – from 12 matches of the 2009/2010 German Bundesliga season were used. The goalkeepers were excluded because of their specific role in a match. Thus, the different states i were represented by matrices vi . In case of a substitution the x,y coordinates of the new player were inserted at the row of the substituted player at the appropriate time i in matrix vi. Furthermore, a change within a team’s tactical line-up was recognized by video inspection and the respective components in vi were switched accordingly. There were no sending-offs in the investigated soccer matches. However, how to deal with such incidents has not been resolved yet. Finally, the original data (10 Hz) were aggregated to 1 Hz for computational reasons, since a system’s state was not expected to change considerably within any given second.
在这项研究中，使用了 2009/2010 德国德甲赛季 12 场比赛的数据，确切地说是每位球员的位置p i = ( x , y )。守门员因其在比赛中的特殊角色而被排除在外。因此，不同的状态i由矩阵v i表示
。在替换的情况下，新球员的x 、 y坐标在矩阵vi中的适当时间i处插入被替换球员的行。此外，通过视频检查识别出球队战术阵容的变化，并相应地更换了vi中的各个组成部分。在接受调查的足球比赛中没有出现被罚下场的情况。然而，如何处理此类事件尚未得到解决。最后，出于计算原因，原始数据 (10 Hz) 被聚合为 1 Hz，因为系统的状态预计不会在任何给定秒内发生显着变化。
RP construction RP施工
In this application to soccer, it was decided not to use a phase space reconstruction. On the one hand, Iwanski and Bradley [Citation35] criticize that embedding can lead to correlations which are often hard to understand. They argue that unembedded RPs contain the same information as embedded ones [Citation35,Citation39]. On the other hand, in this soccer application 20×2 matrices were used to describe the system’s states instead of one-dimensional measurements as in golf. Whereas the latter might need to be embedded to a higher dimensional phase space to unwrap hidden facets of a system’s behaviour, in this application the system’s behaviour is described relatively precisely by the matrices vi containing information on the measurable components’ behaviour.
在足球应用中，决定不使用相空间重建。一方面，Iwanski 和 Bradley [引文35 ] 批评嵌入可能导致通常难以理解的相关性。他们认为未嵌入的 RP 包含与嵌入的 RP 相同的信息 [引文35 、引文39 ]。另一方面，在此足球应用中，使用 20×2 矩阵来描述系统的状态，而不是像高尔夫中那样使用一维测量。尽管后者可能需要嵌入到更高维的相空间中以解开系统行为的隐藏方面，但在本应用中，系统的行为由包含可测量组件行为信息的矩阵vi相对精确地描述。
The method suggested by Marwan [Citation42] was used to determine the threshold ε. To get as much information as possible from an RP, it is important to achieve a high number of long diagonal lines while the variability of the lengths remains small. Therefore, the value of ε should be selected depending on the ratio DET/ENTR for which the minimum line length was set at two. To find the right threshold, this ratio was plotted as a curve depending on an increasing ε, starting with zero up to a value meaningful for the investigated system. Furthermore, it was decided to use the same ε for all matches to avoid the possibility of similar states being 10 m apart from each other in one RP and similar states being 8 m apart from each other in another RP. It was calculated as the average of the thresholds εi of the investigated matches and equalled 9 m (SD = 0.43 m). However, from a soccer expert’s viewpoint one could question whether a player pattern can be similar to another if they are shifted by about 9 m. With respect to pitch size (usually 68 m wide and 105 m long), 9 m is a quite large threshold itself. On the other hand, this threshold illustrates how dynamic soccer matches are and how difficult it is to find similar player patterns in the sense of nearly equal player patterns.
Marwan [引文42 ] 建议的方法用于确定阈值ε 。为了从 RP 中获取尽可能多的信息，重要的是要获得大量的长对角线，同时长度的变化保持较小。因此，应根据最小线长度设置为 2 的比率DET / ENTR来选择ε的值。为了找到正确的阈值，根据ε 的增加，将该比率绘制为一条曲线，从零开始，直到对所研究的系统有意义的值。此外，决定对所有比赛使用相同的ε ，以避免一个RP中相似状态彼此相距10m而另一RP中相似状态彼此相距8m的可能性。它被计算为所调查匹配的阈值ε i的平均值，等于 9 m (SD = 0.43 m)。然而，从足球专家的角度来看，人们可能会质疑，如果球员模式移动约 9 m，是否可以与另一个球员模式相似。就球场尺寸（通常为 68 m 宽、105 m 长）而言，9 m 本身就是一个相当大的阈值。另一方面，这个阈值说明了足球比赛的动态性以及在几乎相等的球员模式的意义上找到相似的球员模式是多么困难。
As for golf, the Euclidean norm was chosen to identify close states in the phase space. However, the computation of the RP had to be modified because 20×2 matrices were used instead of vectors or scalars. We decided to assess the similarity between two states according to the mean positional change of all players. Thus, Equation (1) turns into
对于Golf ，选择欧几里得范数来识别相空间中的接近状态。然而，由于使用 20×2 矩阵而不是向量或标量， RP的计算必须进行修改。我们决定根据所有球员的平均位置变化来评估两种状态之间的相似性。因此，方程（1）变为
where denotes the nth row of vi.
在哪里
表示vi的第 n行。
Sport scientific application
运动科学应用
Exemplarily for the 12 investigated matches,
以 12 场调查比赛为例， shows the RP of a match which resulted in a 0–0 draw. Basically, three different blocks can be recognized in the RP separated by more-or-less white bands. The first block starts with the kick off and lasts until about minute 35. The second block starts at about minute 42 and ends at about minute 80. Finally, the last few minutes of the match form the third block. The gap between the first two blocks occurred because there was a sequence of longer game interruptions. In contrast to the first block, the match only ran for some seconds between the interruptions. They were free kicks at different locations on the pitch, a corner kick and a goal kick, all executed by the same team. The last 10 min of the match, the crunch time, were characterized by many interruptions accompanied by some yellow cards (four of five in this match) and two substitutions. Between those interruptions there were only short running match phases. Therefore, at the end of the match the pattern of the players differed from the other identified blocks. In particular, the first and the last blocks were different from each other. Inspecting the video of this match, a reason for this could be that the teams fulfilled precise tactical tasks at the beginning of the match whereas they behaved in a less ordered manner at the end of the match, when the home team tried to score in order to win the game by all available means. Another reason for this could be that the players were fatigued at the end of a match and usually ran shorter distances and less quickly during the final minutes [Citation47,Citation48].
显示 0-0 平局比赛的 RP。基本上，可以在 RP 中识别出由或多或少的白色带分隔开的三个不同块。第一个区块从开球开始，一直持续到第 35 分钟左右。第二个区块从第 42 分钟左右开始，到第 80 分钟左右结束。最后，比赛的最后几分钟形成第三个区块。前两个区块之间出现间隙是因为存在一系列较长的游戏中断。与第一个区块相比，比赛在中断之间只进行了几秒钟。分别是球场上不同位置的任意球、角球和球门球，全部由同一支球队执行。比赛的最后 10 分钟，即关键时刻，出现了多次中断，并伴有一些黄牌（本场比赛五张黄牌中的四张）和两次换人。在这些中断之间，只有短暂的比赛阶段。因此，在比赛结束时，球员的模式与其他已识别的区块不同。特别是，第一个和最后一个块彼此不同。从本场比赛的视频来看，造成这种情况的一个原因可能是比赛开始时双方战术任务精准，而比赛结束时主队试图有序得分，表现得比较混乱。用一切可用的手段赢得比赛。造成这种情况的另一个原因可能是球员在比赛结束时感到疲劳，并且通常在最后几分钟跑得更短，速度也更慢[引文47 ，引文48 ]。
Figure 4. Recurrence plot of one of the investigated matches (result 0–0; m = 1, tau = 1, ε = 9); black dots represent recurring states.
图 4.所研究的匹配之一的重现图（结果 0-0； m = 1， tau = 1， ε = 9）；黑点代表重复出现的状态。
Because the analysis above suggests that the structure of RPs might be influenced by game interruptions, a detailed analysis of the recurrence points per second was conducted. For reasons of effort, three matches of the data set were selected and in each match 20 randomly chosen minutes were observed.
由于上述分析表明RP的结构可能会受到游戏中断的影响，因此对每秒的重复点进行了详细分析。出于努力的原因，选择了数据集的三场比赛，并且在每场比赛中观察随机选择的 20 分钟。 shows the number of recurrence points per second and the respective discrete states (build-up game, attacking game, throw-in, corner kick, goal shot, etc.) of the game from above for a snippet of 10 min. Generally, the graph suggests that there were more recurrence points per second during open play phases (build-up game and attacking game) than during set pieces (corner kick, free kick, goal kick and throw-in). A t-test based on the sample of the three matches supports this assumption (open play: mean = 182.4, SD = 167.8; set-play: mean = 106.2, SD = 107.6; p < .001). This means that the teams showed similar patterns during open play phases more often. This could be due to several reasons. First of all, soccer matches are usually interrupted for fewer minutes than they are running [Citation49]. Additionally, there are a lot of different reasons for set pieces, in particular free kicks, corner kicks and throw-ins, which take place at different locations on the field of play and cause different player patterns. Thus, it is obvious that there are fewer states set pieces can be similar to, and that there are fewer recurrence points during set pieces. Another reason why there were more recurrence points during open play could be related to the data set the study is based on. It consists of home games of a team which was one of the best teams in that season and which played quite dominantly. This probably biases the results towards the behaviour of this team. Moreover, the team’s playing style was characterized by long phases of ball possession during which the players were constrained to keep their tactical playing positions as unchanged as possible. Thus, the defending teams’ patterns also remained quite stable during such phases.
显示每秒重复点数以及 10 分钟内比赛的各个离散状态（进攻比赛、进攻比赛、界外球、角球、射门等）。一般来说，该图表明，在开放比赛阶段（进攻比赛和进攻比赛）每秒出现的重复点比定位球阶段（角球、任意球、球门球和界外球）更多。基于三场比赛样本的t检验支持这一假设（开场比赛：平均值 = 182.4，SD = 167.8；固定比赛：平均值 = 106.2，SD = 107.6； p < .001）。这意味着球队在公开比赛阶段更频繁地表现出类似的模式。这可能是由于多种原因造成的。首先，足球比赛中断的时间通常比比赛进行的时间短[引文49 ]。此外，定位球有很多不同的原因，特别是任意球、角球和界外球，它们发生在比赛场地的不同位置，并导致不同的球员模式。因此，很明显，设定片段可以相似的状态较少，并且设定片段期间的重复点也较少。公开赛期间出现更多重复点的另一个原因可能与该研究所基于的数据集有关。它由一支球队的主场比赛组成，该球队是该赛季最好的球队之一，并且发挥着相当大的统治力。这可能会使结果偏向于该团队的行为。此外，球队的比赛风格的特点是长时间控球，球员被迫尽可能保持战术位置不变。 因此，在这个阶段，防守球队的格局也保持相当稳定。
Figure 5. Recurrence points per second from one of the matches of the data set (result 0–0); discrete game phases are annotated.
图 5.数据集某一匹配项的每秒重现点（结果 0-0）；离散的游戏阶段被注释。
RQAs were conducted to analyse the small-scale textures of the investigated RPs. RR was 2.3% on average (SD = 0.6). Thus, there were a lot of different states and only a few of them were similar to each other. Although DET was 96.3% on average (SD = 0.4%) and LAM was 98.4% (SD = 0.2%) on average, the diagonal lines (L = 5.5, SD = 0.3) and vertical lines (TT = 6.4, SD = 0.3) were not very long on average. The RQA parameter DET suggests that the progress of the match was rather predictable. However, considering L the match could only be predicted for a few seconds. Considering TT there were stable states which were repeatedly approached during matches. However, this could be an artefact caused by the recurrence threshold of 9 m together with the match’s velocity. For instance, if nearly all players were moving slowly it would have taken a few seconds until every player had moved further than 9 m. In particular, the matches reached such phases when the dominant team was attacking and trying to find an opening while the defending team was well organized.
进行 RQA 来分析所研究的 RP 的小规模纹理。 RR平均为 2.3% (SD = 0.6)。因此，有很多不同的状态，只有少数状态是彼此相似的。尽管DET平均为 96.3% (SD = 0.4%)， LAM平均为 98.4% (SD = 0.2%)，但对角线 ( L = 5.5，SD = 0.3) 和垂直线 ( TT = 6.4，SD = 0.3) ）平均来说不是很长。 RQA 参数DET表明比赛的进展是相当可预测的。然而，考虑到L，比赛只能预测几秒钟。考虑到TT，比赛中会反复接近稳定状态。然而，这可能是由 9 m 的重复阈值和匹配速度引起的人为因素。例如，如果几乎所有玩家都缓慢移动，则需要几秒钟的时间才能让每个玩家移动超过 9 m。尤其是比赛到了这样的阶段，即优势队进攻并试图寻找空档，而防守队则组织良好。
Finally, the RQA parameters were correlated with a few game statistics from the investigated matches (
最后，RQA 参数与所调查比赛的一些比赛统计数据相关（). Only for a few correlations, significant results were found. The covered distances of the teams were moderately to strongly correlated with TT. This was surprising since the more the players move during 90 min the faster the match could be and the less time the players stay at similar locations. Furthermore, the number of goals in a match seemed to influence the RP structure, the more goals were scored the less structured was the match. Although the number of recurrence points is not correlated with the number of goals, DET and LAM significantly decreased when more goals were scored. Finally, there was a tendency that the more shots on goal there were in a match the fewer recurring states existed. This allowed us to assume that goal shots were probably not results of repeatedly performed, practised passing sequences.
）。仅对于少数相关性，发现了显着的结果。各队的行走距离与TT呈中度至强相关。这是令人惊讶的，因为玩家在 90 分钟内移动的次数越多，比赛的速度就越快，并且玩家在相似位置停留的时间也就越少。此外，一场比赛的进球数似乎会影响 RP 结构，进球数越多，比赛的结构就越松散。尽管复发点数与进球数不相关，但当进球数增多时， DET和LAM显着下降。最后，有一种趋势是，一场比赛中射门次数越多，重复出现的状态就越少。这让我们可以假设射门可能不是反复执行、练习的传球序列的结果。
Table 1. Pearson correlations between the game statistics number of goal shots, number of goals, number of corner kicks, number of fouls and covered distances of all players during a match and the RQA parameters RR, DET, L, LAM and TT of all investigated games.
表 1一场比赛中所有球员的射门次数、进球数、角球数、犯规数和覆盖距离的比赛统计数据与所有球员的 RQA 参数RR、DET、L、LAM和TT之间的 Pearson 相关性调查了游戏。
Concluding, RP and RQA can be useful methods to analyse the dynamic, complex system soccer match. Using a more heterogeneous data set, the resulting RPs could help to shed light on the structure of soccer matches. In particular, investigating the RPs and their structure prior to important game actions such as goal shots or goals is an obvious question that should be approached. Furthermore, RPs should be analysed with respect to their potential regarding identifying and analysing tactical behaviour.
总之，RP 和 RQA 可以成为分析动态、复杂系统足球比赛的有用方法。使用更加异构的数据集，生成的 RP 可以帮助阐明足球比赛的结构。特别是，在重要的比赛动作（例如射门或进球）之前调查 RP 及其结构是一个应该解决的明显问题。此外，应分析 RP 在识别和分析战术行为方面的潜力。
Conclusion 结论
The complex, dynamic systems approach is frequently used in sports to model competitions or athletes. This paper illustrates how RPs – a tool frequently used in physics or earth science – can be applied to, and subsequently used, to analyse complex, dynamic sport systems. The application of the tool depends on the investigated sport. If a system is composed of a lot of unknown and not measureable parts, according to our case study a summary variable could be determined which describes the synergy of the components and is measurable. Based on such a measurement, the phase space reconstruction technique should be used for the RP construction to unfold hidden facets of a system’s dynamic. If several systems of the same type – such as golfers – are investigated, we recommend using the same phase space reconstruction. On the other hand, if there are many known and measureable parts a complex game sport system is composed of, the behaviour of those components should be measured and subsequently used as a multidimensional description of the system’s behaviour. We propose that such a detailed knowledge of the system’s behaviour does not require phase space reconstruction for the RP computation.
复杂的动态系统方法经常在体育运动中用于模拟比赛或运动员。本文阐述了 RP（物理学或地球科学中常用的工具）如何应用于并随后用于分析复杂的动态运动系统。该工具的应用取决于所调查的运动。如果一个系统由许多未知且不可测量的部分组成，根据我们的案例研究，可以确定一个汇总变量，该变量描述了组件的协同作用并且是可测量的。基于这样的测量，相空间重构技术应该用于 RP 构造，以揭示系统动态的隐藏方面。如果研究同一类型的多个系统（例如高尔夫球手），我们建议使用相同的相空间重建。另一方面，如果一个复杂的游戏运动系统由许多已知和可测量的部分组成，则应该测量这些组件的行为，并随后将其用作系统行为的多维描述。我们提出，对系统行为的如此详细的了解不需要为 RP 计算进行相空间重建。
The choice of the recurrence threshold according to which states are assessed as similar is crucial because it determines how much information can be derived from the resulting RPs. Its determination depends on various factors such as whether there are several systems which are going to be compared in the context of sport scientific analyses. It is therefore recommended to use the same threshold for the different game sport systems under investigation. Using the average of the individual system’s thresholds turned out to be an appropriate choice.
根据哪些状态被评估为相似来选择复发阈值至关重要，因为它决定了可以从生成的 RP 中得出多少信息。其确定取决于多种因素，例如是否有多个系统将在运动科学分析的背景下进行比较。因此，建议对正在研究的不同游戏运动系统使用相同的阈值。事实证明，使用单个系统阈值的平均值是一个合适的选择。
The application of RPs to game sports allows a comprehensive view on sports performance, in particular its development over time. In particular, RQAs can help to shed light on the sporting performance of teams and individuals and can assist in answering sport scientific questions.
RP 在竞技运动中的应用可以全面了解运动表现，特别是其随时间的发展。特别是，RQA 可以帮助揭示团队和个人的运动表现，并有助于回答运动科学问题。
Acknowledgements 致谢
The authors would like to thank the PGA Tour for providing access to the ShotLinkTM database.
作者要感谢 PGA 巡回赛提供 ShotLink TM数据库的访问权限。
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。
Additional information 附加信息
Funding 资金
这项研究没有得到任何资助的支持。
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