一、题目
（一） 赛题原文
2024 MCM Problem C: Momentum in Tennis
In the 2023 Wimbledon Gentlemen's final, 20-year-old Spanish rising star Carlos Alcaraz defeated 36-year-old Novak Djokovic. The loss was Djokovic's first at Wimbledon since 2013 and ended a remarkable run for one of the all-time great players in Grand Slams.
The match itself was a remarkable battle.[1] Djokovic seemed destined to win easily as he dominated the first set 6 -- 1 (winning 6 of 7 games). The second set, however, was tense and finally won by Alcarez in a tie-breaker 7 -- 6. The third set was the reverse of the first, Alcaraz winning handily 6 -- 1. The young Spaniard seemed in total control as the fourth set started, but somehow the match again changed course with Djokovic taking complete control to win the set 6 -- 3. The fifth and final set started with Djokovic carrying the edge from the fourth set, but again a change of direction occurred and Alcaraz gained control and the victory 6 -- 4. The data for this match is in the provided data set, "match_id" of "2023-wimbledon-1701". You can see all the points for the first set when Djokovic had the edge using the "set_no" column equal to 1. The incredible swings, sometimes for many points or even games, that occurred in the player who seemed to have the advantage are often attributed to "momentum."
One dictionary definition of momentum is "strength or force gained by motion or by a series of events."[2] In sports, a team or player may feel they have the momentum, or "strength/force" during a match/game, but it is difficult to measure such a phenomenon. Further, it is not readily apparent how various events during the match act to create or change momentum if it exists.
Data is provided for every point from all Wimbledon 2023 men's matches after the first 2 rounds. You may choose to include additional player information or other data at your discretion, but you must completely document the sources. Use the data to:
- Develop a model that captures the flow of play as points occur and apply it to one or more of the matches. Your model should identify which player is performing better at a given time in the match, as well as how much better they are performing. Provide a visualization based on your model to depict the match flow. Note: in tennis, the player serving has a much higher probability of winning the point/game. You may wish to factor this into your model in some way.
- A tennis coach is skeptical that "momentum" plays any role in the match. Instead, he postulates that swings in play and runs of success by one player are random. Use your model/metric to assess this claim.
- Coaches would love to know if there are indicators that can help determine when the flow of play is about to change from favoring one player to the other.
a) Using the data provided for at least one match, develop a model that predicts these swings in the match. What factors seem most related (if any)?
b) Given the differential in past match "momentum" swings how do you advise a player going into a new match against a different player? - Test the model you developed on one or more of the other matches. How well do you predict the swings in the match? If the model performs poorly at times, can you identify any factors that might need to be included in future models? How generalizable is your model to other matches (such as Women's matches), tournaments, court surfaces, and other sports such as table tennis.
- Produce a report of no more than 25 pages with your findings and include a one- to two-page memo summarizing your results with advice for coaches on the role of "momentum", and how to prepare players to respond to events that impact the flow of play during a tennis match.
（二）赛题翻译
问题 C：网球运动中的"动量"
在 2023 年温布尔登网球公开赛男子组决赛中，20岁的西班牙新星卡洛斯-阿尔卡拉斯击败了 36岁的诺瓦克-德约科维奇。这是德约科维奇自2013年以来首次在温布尔登输掉比赛，也结束了这位大满贯历史上最伟大球员之一的辉煌战绩。
德约科维奇似乎注定要轻松获胜，因为他在第一盘以 6比1 的比分占据优势（7 局比赛中赢了 6 局）。然而，第二盘比赛却十分紧张，最终阿尔卡拉斯在决胜盘中以 7 - 6 获胜。第三盘与第一盘相反，阿尔卡拉斯以 6-1 的比分轻松获胜。第四盘开始后，年轻的西班牙人似乎完全控制了局面，但不知何故，比赛的走势再次发生了变化，德约科维奇完全控制了局面，以 6 - 3 的比分赢得了这一盘。第五盘也是最后一盘比赛开始后，德约科维奇延续了第四盘的优势，但比赛的走向再次发生了变化，阿尔卡拉斯取得了控制权，并以 6 - 4 的比分赢得了胜利。
本场比赛的数据在提供的数据集中，"match_id "为 "2023- wimbledon-1701"。您可以使用"set_no"列=1 查看第一盘德约科维奇占优时的所有得分情况。 在一场比赛中的大部分的分，甚至是许多回合，这都归功于那些令人难以置信的挥拍，这些发生在某一位运动员身上的优势，成为 "势"。
在字典中，"动量 （势）"的定义是 "通过运动或一系列事件获得的力量或作用力。"在体育运动中，一支球队或一名球员可能会觉得他们在比赛中拥有动量或 "力量/作用力"，但很难衡量这种现象。此外，如果存在"动量 （势）"的话，比赛中那些产生或改变"动量 （势）"的瞬间，也不是一目了然的。
提供 2023 年温布尔登网球公开赛前两轮之后所有男子比赛中每一分的数据。您可以自行决定加入其他球员信息或其他数据，但必须完整记录数据来源。
使用这些数据来：
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建立一个模型，捕捉赛点发生时的比赛流程，并将其应用到一场或多场比赛中。您的模型应能确定哪位球员在比赛中的某个特定时间段表现更好，以及他们的表现好到什么程度。根据您的模型提供可视化的比赛流程描述。注意：在网球比赛中，发球的一方赢得赛点/比赛的概率要高得多。您可能希望以某种方式将这一因素考虑到您的模型中。
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一位网球教练对"动量 （势）"在比赛中的作用持怀疑态度。相反，他认为比赛中的波动和一名球员的成功是随机的。请使用您的模型/度量来评估这一说法。
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教练们很想知道，是否有一些指标可以帮助判断比赛的流程何时会从偏向一名球员变为偏向另一名球员。
a) 利用提供的至少一场比赛的数据，建立一个模型来预测比赛中的这些波动。哪些因素似乎最有关联（如果有的话）？
b) 鉴于过去比赛 "势头 "波动的差异，您如何建议球员在新的比赛中对阵不同的球员？
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在一场或多场其他比赛中测试您开发的模型。您对比赛中的波动预测得如何？如果模型有时表现不佳，您是否能找出未来模型中可能需要包含的任何因素？您的模型对其他比赛（如女子比赛）、锦标赛、球场表面和其他运动（如乒乓球）的通用性如何？
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撰写一份不超过 25 页的报告，介绍您的研究结果，并附上一至两页的备忘录，总结您的研究结果，并就"动量 （势）"的作用以及如何让球员做好准备，应对网球比赛中影响比赛进程的事件，向教练提出建议。
二、赛题分析
这个数学建模赛题围绕2023年温网男子单打决赛展开，其中20岁的西班牙新星卡洛斯·阿尔卡拉斯击败了36岁的诺瓦克·德约科维奇，结束了德约科维奇自2013年以来在温网的连胜纪录。比赛过程中出现了许多令人惊讶的变化，这种变化通常被归因于"势头"现象。赛题要求利用提供的数据集，开发一个模型来捕捉比赛中的战局流向，评估势头在比赛中的作用，并预测比赛中势头的变化。解决这个问题需要建立数学模型来分析比赛中的得分情况和势头变化，可能涉及到时间序列分析、统计模型、以及机器学习方法等。对于每个小问题，需要利用提供的数据集来进行建模和分析，并提供可视化展示和统计指标来支持结论。
针对第一个问题，需要开发一个模型来捕捉比赛中的战局流向，评估每个时刻哪位球员表现更好以及有多好。可能的方法包括建立一个时间序列模型来跟踪比赛中的得分情况，并考虑到发球方的优势。可视化可以通过绘制得分情况的图表来展示比赛流向。
针对第二个问题，涉及评估"势头"在比赛中的作用，可以利用模型来比较实际得分情况与模拟随机得分情况的差异，进而评估势头是否具有统计意义。
针对第三个问题，需要开发一个模型来预测比赛中势头的变化，可能需要考虑到比赛中的各种因素如球员状态、场地情况等。可以使用机器学习方法来挖掘与势头变化相关的因素，并建立预测模型。
针对第四个问题，涉及将模型应用到其他比赛，并评估模型的泛化能力。可能需要考虑到不同比赛、不同球员以及不同球场的情况，并对模型进行调整和改进。