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2024年美赛35篇特等奖O奖论文-C-2409961.pdf
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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
2409961
Momentum Prediction and Core Feature Extraction
Based on GRU with Integrated Gradients
Summary
Currently, coaches, players, and managers are all aware that "momentum" plays a huge role
in the progress of tennis matches. By analyzing the changes in momentum during the game, we
can evaluate the performance of players during specific time periods and make speculations about
the game results. By understanding the factors that influence the momentum of the game, we can
assess the fluctuations in the game process and construct a player training system. Therefore, it is
crucial to recognize the role of "momentum" in the game and use it to help players cope with events
that affect the progress of the game.
Firstly, we established an Instantaneous Score Rate (ISR) model to capture match points. After
using the cubic spline interpolation method to process discrete data, the model can smoothly estimate
the Instantaneous score rate at any time and accurately capture key moments in the competition.
Based on the results of this model, we can quickly identify the performance of players at specific
time points. In addition, the results of the model clearly demonstrate the first mover advantage of
the serving party in the game, which is consistent with the facts. Then, we establish a model to test
the role of momentum in the competition. The results indicate that there is not independent between
fluctuations in the game and the success of players. In addition to the first mover advantage, there
will be a strong "momentum" in the game that affects the Instantaneous Score Rate.
Next, in order to predict fluctuations in the competition, we use Gated Recurrent Neural Network
Model (GRU) with Integrated Gradients for feature disentanglement engineering. Based on the
GRU model, we can fit the "momentum" function under existing conditions to accurately predict the
change in competition momentum. We manually divided the original indicators into six dimensions.
Using the Integrated Gradient algorithm, the impact of six dimensions on the output is indicated
from the GRU model output. Based on the model results, we have identified 12 important indicators
that have an significant impact on schedule fluctuations. On this basis, we established a competency
evaluation system based on six dimensions, and provided targeted suggestions to players based on
the analysis of real player abilities.
In addition, we tested the predictive ability of the model in real-time data of different genders,
game systems, and ball games. The test results have shown that our model can accurately predict
other competition processes and has high generalization ability.
Finally, in order to help coaches cultivate players to better cope with events that affect the tennis
schedule, we provide targeted recommendations. We hope these suggestions can help players
achieve victory in the new game.
Keywords: Momentum, GRU, Integrated Gradient, Disentanglement.
Team # 2409961 Page 1 of 24
Contents
1 Introduction 2
2 Assumptions and Notations 4
3 Model Preparation 4
3.1 Data Collection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3.2 Data Cleaning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
4 Instantaneous Scoring Rate Model 5
4.1 Cubic spline interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
4.2 Instantaneous Scoring Rate for the match . . . . . . . . . . . . . . . . . . . . . . 7
5 Test of Independence Between Momentum and Victory 8
5.1 Assumption: Momentum and victory are independent . . . . . . . . . . . . . . . . 8
5.2 Verification: Momentum has an impact on victory . . . . . . . . . . . . . . . . . . 9
6 Disentangled feature Engineering via GRU with integrated gradients 9
6.1 Important Data Extraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
6.2 Match Suggestion Based on Capability Evaluation . . . . . . . . . . . . . . . . . . 15
7 Generalization Testing 16
7.1 Basic Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
7.2 Error Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
7.3 Generalization Ability Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
8 Conclusion 19
9 Strengths and Possible Improvements 19
9.1 Strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
9.2 Possible Improvements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
Appendices 23
Team # 2409961 Page 2 of 24
1 Introduction
Background
In the men’s final of the 2023 Wimbledon Open, Novak Djokovic took on rising Spanish star
Carlos Alcaraz. Djokovic seemed destined for an easy win, winning the first set by an absolute
margin of 5 points; while the second set was tense, with Alcaraz winning 7-6 in a tie-break; and
the third set was the opposite of the first, with Alcaraz defeating Djokovic by an absolute margin
of 5 points... After a number of changes in the trend of the tournament, Grand Slam Djokovic, one
of the greatest players in history, lost at Wimbledon for the first time since 2013. [1]
In tennis, "momentum" means the power that a player acquires in a match through movement or
a series of events. When momentum can be used to measure a player’s performance at a particular
point in a match, the players themselves, coaches, and managers have an advantage in determining
how the situation on the court is changing, for example, which player is likely to be the winner of
the next point [2]. As a result, coaches would like to better understand if momentum can be used
to determine player performance and the extent to which it can be used at a given point in time. If
possible, coaches would like to gain a better understanding of the factors that influence fluctuations
in Momentum, so that they can better judge the future course of the game and use it as a reference
in training to develop more successful players.
Academics have paid attention to the issue of "momentum" in sports events. Some scholars
have looked at the psychology of athletes [3], the point of serve [4], Strategic momentum [5]
and other aspects of momentum in mental tennis. Existing empirical research suggests that in
tennis, players can control the match to benefit themselves from momentum, and that the amount
of countermovement will result in a player having a significantly lower chance of winning the
underdog set than his opponent [6] . Generalized linear mixed-effects models have also been
constructed to demonstrate the legacy effects of score, game, and set [7] . Therefore, by analyzing
a large amount of game flow data, we can explore the relationship between momentum and its
scoring rate and winning rate based on determining the performance of players in different time
periods, and visualize the game flow from the perspective of momentum, with a view to providing
references for coaches, players, the public and the media.
Some scholars have pointed out that the factors that cause changes in momentum are multiple,
including errors and psychological changes during the game [8] . Some empirical scholars have
also focused on the indicator of break points in the data as an important variable in the study
of momentum shifts in matches. [5] On this basis, by examining the mechanisms by which
different factors influence "momentum", we can analyze which factors have the greatest impact on
match volatility and suggest that coaches, players, the public, and the media pay attention to these
indicators.
A preliminary finding of some studies is that, from a psychological point of view, there is little
difference in the ability of male and female tennis players to overcome the influence of momentum
in order to win the match. [9] In examining the influence of momentum, gender, venue, and type of
match may affect the results of the analysis, which leads us to test the generalizability of the model
to changing contexts.
Team # 2409961 Page 3 of 24
Problem Restatement
• Build model 1 to capture match points in the flow of the game and identify the players who
performed better and to what extent at a given point in the game.
- Test Provide a visualization of the flow of the game based on Model 1.
- Judging whether the fluctuation of momentum in the match is independent of the subsequent
scoring situation.
• Build Model 2 to predict fluctuations in game "momentum" and identify better metrics to
help determine game situations.
- Test the accuracy and generalizability in predicting the degree of fluctuation in a match.
- Gives tactical strategies for winning tennis matches against different opponents based on
fluctuations in momentum.
• A memo written by summarizes the results and advises coaches on the following directions:
- Influencing the direction of play through the "momentum" of the game.
- Preparing players for events that affect the tennis schedule.
Our Work
We examined the extent to which "momentum" reflects player performance in tennis, and
based on its impact indicators, we provide tennis coaches with suggestions for focusing on the role
of "momentum" and developing good players based on this concept, as described below:
• Build Model 1 to visualize the game schedule.
• Based on Model 1, evaluate the coach’s stochastic assumption that fluctuations in momentum
during a match are independent of a player’s subsequent score.
• Builds Model 2, which predicts game fluctuations and identifies the relevant indicators that
have the most significant impact on the direction of the game.
• Based on Model 2, provide advice to players in a new round of the tournament on how to
play against different opponents.
• Test the above model, examine its generalizability, and point out the influencing factors that
are lacking in consideration.
• Summarizes the results of the study and makes recommendations to coaches
STEP 1:Establishing the Instantaneous Scoring Rate Model (ISR)
Processing
Discrete Data
Smoothly Estimating
the Scoring Rate
Identifying Players
Who Perform Well at
a Specific Moment.
Using Cubic Spline
Interpolation
STEP 2:Establish Flow-Independent Models
and Non-Independent Models
STEP 3 :Predicting match fluctuations and identifying the most significant correlated indicators
affecting the course of the game.
GRU model
Backpropagation using
Integrated Gradients
algorithm
Data
Dimensionality
Reduction
Analyzing the Magnitude of
Dimensional Influence
Selecting 12
Key Indicators
Establishing a
Competency
Evaluation System
STEP 4 Testing Model Generalizability and Making
Suggestions to the Coach
Evaluate the
Coach’s
Stochastic
Assumption
Flow Potential
Independent
Models
Non-
Independent
Models
CORRECT
FAIL
Mentality
Endurance
Error Rate
Offensive
Skill
Key Point
Independence Test
Searching for
Data on Different
Genders, Other
Ball Games
Conducting
Generalization
Studies on the
Model
Making
Suggestions for the
Coach
Proving the Existence
of a First-Mover
Advantage
Figure 1: Our Work Frame
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