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大学生,数学建模,美国大学生数学建模竞赛,MCM/ICM,2024年美赛特等奖O奖论文
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Problem Chosen
C
2024
MCM/ICM
Summary Sheet
Team Control Number
2409404
Riding the Momentum
Summary
"If you have momentum, you want to try to keep that momentum going, keep that confidence
rolling for you. You know, that’s why the streak is there," said Djokovis. Undoubtedly, momen-
tum is the key to success for Djokovic. Does momentum effectively function? To effectively
tackle this inquiry and offer valuable guidance to coaches and players, we present the fol-
lowing thoughts.
For task one, it is necessary to capture the flow of play and identify crucial moments
while also quantifying the player’s performance. We establish the Momentum Quantiza-
tion Model Based on Sliding Windows (MQ-SW) that assesses the performance level of
each player at given moments, taking into account the influence of past mistakes and other
factors. Additionally, we incorporate the player’s serving advantage into the calculation.
Furthermore, we discern the crucial aspects of the match by considering both the tennis
regulations and the CUSUM algorithm. Figure 4 depicts the visual representation of the
match progression between Carlos Alcaraz and Novak Djokovic. The model demonstrates
proficiency in quantifying a player’s momentum and precisely identifying crucial moments.
For task two, we need to examine the impact of "momentum" on the game and deter-
mine if the fluctuations in the game condition and successive player scores are random.
Thus, we begin by examining the volatility of the game’s condition using the Point Vic-
tor factor and the MQ-SW Model. Subsequently, we employ the Runs Test to assess the
state of the game and the successive scores of the participants. Upon computation, the
p − value < 0.01, indicating that the variability in the game scenario. Next, we discretize
the momentum of both players at each instance and employ the Chi-square Test to com-
pare the aforementioned indices. The experimental p − valu e < 0.01, indicating that the
"momentum" element is highly likely to be the reason, as demonstrated in Table 5.
For task three, we need to predict the volatility of the circumstances in a tennis match.
The dependent variable is the discretized "momentum", whereas the independent variables
consist of other factors associated with players. We established the Prediction for Match
Trend Model Based on PCA&XGBoost (PMT-PX). The model uses the principal compo-
nent analysis to examine the variation in scatter on a two-dimensional coordinate graph
and subsequently utilizes XGBoost to train on the match data. From this, we have identi-
fied the most significant influencing factors, namely "Point Victor" and "Break PT" at the
current moment. The corresponding scores are displayed in Table 6.
For task four, it requires an examination of the model’s effect and its capacity to general-
ize. We assess the model using several metrics, such as the confusion matrix, classification
report, and ROC curve. The area under the curve (AUC) is 0.75, indicating a discernible ef-
fect. We utilize the model to evaluate the U.S. Open women’s final. The results indicate that
the model exhibits a commendable level of generalization and is capable of efficiently and
accurately identifying the target. Simultaneously, we conduct an analysis of the model’s
drawbacks and provide the elements that should be taken into account in the future. Fur-
thermore, the model sensitivity analysis is conducted by altering the dimensions of the
sliding window along with the additional advantage parameter of the player’s serve. The
outcomes indicate that the model exhibits a high level of robustness.
We finally write a memorandum, including our model and strategy, for the coaches and
players. We hope our memorandum can be valuable to them and actually help them.
Keywords: Momentum, MQ-SW, Runs Test, Chi-square Test, PMT-PX
Team # 2409404 Page 2 of 25
Contents
1 Introduction 3
1.1 Probelm Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Restatement of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.4 Our work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Preparation of the Models 5
2.1 Assumptions and Explanations . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.3 Data Preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.3.1 Data Collection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.3.2 Data Pre-processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3 Establish the Momentum Quantization Model Based on Sliding Windows 6
3.1 Quantization of Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3.2 Identification of Key Points in the Game . . . . . . . . . . . . . . . . . . . . . . 7
3.3 Competition Flow Visualization . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
4 Momentum’s Impact on the Game Situation 10
4.1 The Runs Test —Stochastic Analysis . . . . . . . . . . . . . . . . . . . . . . . . 10
4.2 The Chi-square Test —Correlation Analysis . . . . . . . . . . . . . . . . . . . . 11
4.3 Reach a Verdict . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
5 Establish the Prediction for Match Trend Model Based on PCA&XGBoost 12
5.1 Principal Component Analysis —Determine the Model . . . . . . . . . . . . . 12
5.2 XGBoost —Match Trend Prediction . . . . . . . . . . . . . . . . . . . . . . . . . 13
5.3 Factor Importance Extraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
5.4 Tactical Suggestions for the Game . . . . . . . . . . . . . . . . . . . . . . . . . . 15
6 Model Effect Analysis 16
6.1 Model Testing on a New Race . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
6.2 Generalized Ability Analysis —U.S. Open Women’s Singles . . . . . . . . . . 18
6.3 Future Consideration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
7 Sensitivity Analysis 21
8 Model Evaluation 22
8.1 Strengths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
8.2 Weaknesses and Improvements . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
9 Conclusion 22
References 23
Memorandum 24
Team # 2409404 Page 3 of 25
1 Introduction
1.1 Probelm Background
In the men’s singles final of the 2023 Wimbledon Open, Spanish rising star Carlos Al-
caraz ended Novak Djokovic’s legendary player’s unbeaten run at Wimbledon since 2013.
So What factors influenced the outcome of this seemingly unbalanced match?What factors
contribute to the occurrence of analogous scenarios for individuals or players who appear to
have a dominant position in the situation or game? Indeed, these conclusions are frequently
ascribed to the phenomenon of "momentum".
Momentum is defined in the dictionary as "strength or force gained by motion or by a
series of events", but in actual sports, a team or player may feel energized during a game,
but it is difficult to accurately measure this phenomenon. Furthermore, assuming that such
momentum exists, it is difficult to determine how the various events of a game generate
or alter momentum. So, is there a computable and estimable frontal model of momentum
available to aid players in achieving improved outcomes?
1.2 Restatement of the Problem
Given the problem’s background information and constrained conditions, we must
complete the following tasks:
• Task 1: Build a model that captures the flow of the match as it occurs and apply it to
one or more matches to visualize the flow of the match. The model needs to be able to
identify the time period in the match when a player performs better and the degree of
excellence within that time period.
• Task 2: A coach is skeptical about the role of momentum, arguing that fluctuations
in match situations and consecutive wins by players occur randomly, use a model to
evaluate this coach’s claim.
• Task 3: Coaches need to be clear on whether there are indicators that can assist in
determining when a game’s trend will shift.
– Using data from at least one match, build a model that predicts the movement
of a match and identify which factors, if any, have the greatest impact on the
movement of a match.
– Based on the difference in "momentum" in previous matches, help a player de-
termine the correct strategy for a new match against a different opponent.
• Task 4: Test the current model in one or more other games and evaluate the model’s
prediction of in-game fluctuations; if the prediction is poor, can factors be identified
for inclusion in future models? How generalizable is the model to other games, tour-
naments, field types, and other sports?
• Task 5: Write a memo summarizing the findings. Explain to a coach the role of "mo-
mentum" and make recommendations to prepare players for events that affect the
course of a tennis match.
Team # 2409404 Page 4 of 25
1.3 Literature Review
The situation of a tennis match is rapidly changing. In the face of the variability of
environment, strategy, and process, momentum, a factor that may turn the match around,
has become a focus of research.
Ben Moss et al. argue that the uncertainty of the momentum effect in tennis, which may
vary from player to player, needs to be considered at an individual level
[1]
. In addition, if a
player breaks his opponent’s serve when facing a game point, he is more likely to hold his
serve in the next round of serve reception.
Zhang Xiaofei suggests that the gain or loss of a key score will have a serious impact
on a player’s momentum, thus affecting the direction of the match. In the face of changes
in the score, some players may increase their momentum, attack courageously, and take the
initiative. While some players may attach too much importance to the change of score, and
whenever they encounter the change of key score, their momentum may be reduced, and
then the game will turn around
[2]
.
Chen Liang suggests that there is a direct "attack-defense relationship" in tennis and
that in order to win a match, one must maximize one’s own performance and inhibit one’s
opponents to the maximum extent possible. Under the joint effect of oneself, opponent,
and competition environment, the momentum of both sides usually will not remain stable
during the competition, and the change of the contrast relationship will form the stage "rise
and fall"
[3]
of the competitive performance.
In conclusion, momentum affects different players in different ways, which is impor-
tant for the psychological support of athletes.
1.4 Our work
For ease of description and visualization, we have drawn a flow chart (Figure 1) to
represent our work.
Prepare
Data collection
Data pre-processing
Task 1
Quantification of the momentum
Identification of key points in the game
Visual analysis of match flows
Set the size of
the moving window
Selection of
player-related
indicator factors
Weighing
of indicators
Game Point
Tunning point of
the momentum (CUSUM)
Task 2
Stochastic analysis of
situational fluctuations and
player continuity scores
Correlation analysis of momentum
The Runs Test
Chi-square Test
Task 3
Prediction of the
swings in the match
PCA & XGBOOST
Importance extraction
of factors
Tactical suggestions for
the game
Task 4
Testing of the Model
Confusion
Matrix
Classification
Report
ROC Curve
Application capacity
Model Enhancement
(Prospects for the future)
Final
Advice for
Coaches and Athletes
Figure 1: Flow chart of our work
Team # 2409404 Page 5 of 25
2 Preparation of the Models
2.1 Assumptions and Explanations
To simplify the issue, we have made the following assumptions, each of which is ap-
propriately reasonable.
• Assumption 1: The entire game is not affected by external factors such as unfair deci-
sions by the referee.
,→ Explanation: An unfair judgement by the referee will mislead our judgement of
the player’s score, thus not accurately calculating the impact of momentum on the
athlete.
• Assumption 2: Players were in good physical condition throughout the game.
,→ Explanation: Physical condition will affect an athlete’s performance in a game,
while existing data makes it difficult to measure a player’s moment-to-moment con-
dition.
• Assumption 3: The athlete has not used illegal drugs such as doping and has not
cheated in competition.
,
→
Explanation:
Doping can substantially increase an athlete’s level of competitive-
ness, deviating from reality and affecting the calculation of momentum. Doping is
also strictly prohibited in competition.
2.2 Notations
The primary notations used in this paper are listed in Table 1.
Table 1: Notations
Symbol Definition
α Extra Advantages of the Player’s Serve
β Sliding Window Size
i Scoring Moment
E(·) Data Average
V(·) Data Variance
* Some variables are not listed here because they have different meanings in different places, so we will
discuss them in detail in each section.
2.3 Data Preparation
2.3.1 Data Collection
To enhance the credibility of our analysis, we have gathered a substantial quantity of
information from the following sources, in addition to the data provided by the contest
questions. Refer to Table 2.
The data from the US Open Women’s Singles Final between C. Gauff and A. Sabalenka
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