Monte Carlo Simulation of English Premier League — using Bivariate Poisson + Elo

Using a Bivariate Poisson (shared component) to model football matches and Elo as a covariate, a Jupyter notebook (Final_Project.ipynb) then performs Monte Carlo simulations to generate full-season distributions (points, positions, title/top-4/relegation probabilities). Includes lightweight backtesting.

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Table of Contents

1. 🧭 Overview
2. 🗂️ Project Structure
3. ⚙️ Installation
4. 📊 Data
5. 🚀 Quickstart
6. 💡 What It Does
7. 📈 Outputs & Visualizations
8. 🖼️ Project Poster
9. 🔧 Configuration
10. 🛠️ Troubleshooting
11. 🗺️ Roadmap / Future Work
12. 📚 References & Data
    • 📝 Citing & Background
    • 🗃️ Data Sources
13. 🤝 Contributing
14. 📄 License
15. ✉️ Contact

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🧭 Overview

• Scoring model: To capture goal dependence, using a bivariate Poisson with a common latent component $$\lambda_3$$.
• Team effects: attack (α), defence (β), and team-specific home advantage (η_h). We apply ridge regularization and sum-to-zero constraints to α and β; η_h is currently unpenalized.
• Elo with decay: pre-match HomeElo / AwayElo; Elo ratio scales goal rates via exponent γ.
• Monte Carlo: To simulate distributions over points, positions, and important outcomes, model entire seasons.
• Backtests: Utilizing historical fixtures and points data, compare the simulated mean points to the actual to obtain MAE.

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🗂️ Project Structure

project-acm40690-monte-carlo-simulation-of-epl/
├── data/
│   ├── E0_19_20.csv
│   ├── E0_20_21.csv
│   ├── E0_21_22.csv
│   ├── epl_22_23_fixtures.csv
│   ├── epl_23_24_fixtures.csv
│   ├── epl_24_25_fixtures.csv
│   └── epl_25_26_fixtures.csv
├── points/
│   ├── epl_2022_23_points.csv
│   └── epl_2023_24_points.csv
├── images/
│   ├── heatmap.png
│   ├── points_boxplot_output.png
│   └── relegation_output.png
├── Final_Project.ipynb     # main notebook
├── README.md
└── LICENSE

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⚙️ Installation

Use a clean environment (venv or conda).

# clone
git clone https://github.com/your-org/project-acm40690-monte-carlo-simulation-of-epl.git
cd project-acm40690-monte-carlo-simulation-of-epl

# create & activate venv
python -m venv .venv
# Windows:
.venv\Scripts\activate
# macOS/Linux:
# source .venv/bin/activate

python -m pip install --upgrade pip
pip install -r requirements.txt

Example Requirements.txt:

numpy>=1.24
scipy>=1.10
matplotlib>=3.7
pandas>=2.0
seaborn>=0.13
jupyter

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📊 Data

We used football-data style match CSVs with:

• Date (DD/MM/YYYY or parseable), HomeTeam, AwayTeam, FTHG (home goals), FTAG (away goals)

Example:

Date,HomeTeam,AwayTeam,FTHG,FTAG
10/08/2019,Liverpool,Norwich,4,1
10/08/2019,West Ham,Man City,0,5

Backtest “actual points” files:

• Columns: Team,Points (the notebook renames Points -> ActualPts internally).

Team,Points
Manchester City,91
Arsenal,89
...

Preprocessing & EDA:

• Performed an Exploratory Data Analysis (EDA) step to:

  • Check for missing values in critical columns (Date, HomeTeam, AwayTeam, FTHG, FTAG)

  • Remove rows with missing or invalid values

  • Strip extra spaces from team names

  • Ensure goals are integers and non-negative Backtest/forecast fixture files:

• Columns: HomeTeam,AwayTeam (no dates required for simulation).

Name consistency: Verify that team names appear same in all training, fixture, and point files. Results will suffer from simple mismatches ("Man City" vs. "Manchester City"). To lessen these discrepancies, we made a few consistent manual adjustments to our CSV file.

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🚀 Quickstart

1. Open Final_Project.ipynb in Notebook/Jupyter Lab or VS Code.
2. Execute all cells in order:
   • Load & clean data
   • Compute Elo (with decay)
   • Fit Bivariate Poisson model
   • Backtest using fixtures & actual points (prints MAE)
   • Simulate future seasons
   • View plots inline

The notebook saves or loads from the repository root; run it from the project folder.

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💡 What It Does

• Data engineering

  • Parse dates, cast goals to int, sort chronologically.
  • Perform an EDA check for missing values and remove any invalid rows.
  • Strip extra spaces from team names.
  • Collect all unique teams from training + fixture files to ensure complete parameter vectors.

• Elo (with decay)

  • Baseline 1500 per team; update per match with
    • K-factor grid (e.g., 20/30/40), decay factor (e.g., 0.995).
  • Expected home result $E_h = \frac{1}{1+10^{(R_a - R_h)/400}}$; update both teams with a decayed step.

• Model fit (Bivariate Poisson)

  • For match $(h,a)$:
    • $$\lambda_1 = \exp(\alpha_h - \beta_a + \eta_h)\cdot\left(\frac{\mathrm{Elo}_h}{\mathrm{Elo}_a}\right)^{\gamma}$$
    • Here, η_h denotes a team-specific home-advantage term (one per home team).
    • $$\lambda_2 = \exp(\alpha_a - \beta_h)\cdot\left(\frac{\mathrm{Elo}_h}{\mathrm{Elo}_a}\right)^{-\gamma}$$
    • $$\lambda_3 = \exp(\theta)$$ shared component
  • Penalized log-likelihood with ridge on $\alpha,\beta$, fitted via BFGS with zero-mean constraints on $\alpha$ and $\beta$.

• Hyper-parameter search

  • Small grid for speed: $K \in {20,30,40}$, $\gamma \in {0.04,0.06}$, $\lambda_{\text{ridge}} = 0.02$.
  • Backtests on two seasons; objective = average MAE between simulated mean points and actual points.

• Monte Carlo

  • Sample $k \sim \mathrm{Poisson}(\lambda_3)$, $x \sim \mathrm{Poisson}(\lambda_1)$, $y \sim \mathrm{Poisson}(\lambda_2)$; score $=(x+k,; y+k)$.
  • Simulate each fixture list $N$ times; aggregate to points tables and rank distributions.

• Reproducibility

  • Simulation RNGs seeded: _rng = np.random.default_rng(1) for match sims; a separate default_rng(0) for tie-break jitter in ranking.

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📈 Outputs & Visualizations

• The median simulated points for the current league cohort are shown in the Predicted table (printed).

• Each team's distribution of points is shown in a horizontal boxplot.

• Heatmap of finish-position probability

• The best teams are at the top; positions 1…N left→right; darker = higher likelihood.

• The bars showing the outcome probabilities are P(Title), P(Top-4), and P(Relegation).

• The boxplot output of team points after simulation.

• Finishing probability of teams after simulation.

• The chances of team getting relegated/title/top 4.

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🖼️ Project Poster

📄 View the poster (PDF)

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🔧 Configuration

Edit the first Config cell in Final_Project.ipynb (example below mirrors the notebook variables):

from pathlib import Path

ROOT = Path(__file__).resolve().parent if "__file__" in globals() else Path.cwd().resolve()
DATA_DIR   = ROOT / "data"
POINTS_DIR = ROOT / "points"

training_csvs = [DATA_DIR / "E0_19_20.csv", DATA_DIR / "E0_20_21.csv", DATA_DIR / "E0_21_22.csv"]
backtest_fixtures_csvs   = [DATA_DIR / "epl_22_23_fixtures.csv", DATA_DIR / "epl_23_24_fixtures.csv"]
backtest_actual_pts_csvs = [POINTS_DIR / "epl_2022_23_points.csv", POINTS_DIR / "epl_2023_24_points.csv"]
future_fixtures_csvs     = [DATA_DIR / "epl_24_25_fixtures.csv", DATA_DIR / "epl_25_26_fixtures.csv"]

# Monte Carlo simulations per season
N_SIMS = 500

Model / search defaults (inside the notebook):

• Elo decay: 0.995 (effective step ≈ (1 - decay) * K)
• Search grids (for speed; expand for final tuning):
  • K ∈ {20, 30, 40}
  • γ ∈ {0.04, 0.06}
  • λ_ridge = 0.02

Tips

• Runtime scales roughly linearly with N_SIMS.
• Keep N_SIMS small while iterating; bump for final figures.

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🛠️ Troubleshooting

• Seaborn theming
  Use:

  import seaborn as sns
  sns.set_theme(style="whitegrid", rc={"figure.dpi": 120})

• Pandas observed warning
  For pivot_table, pass observed=False (already set), or switch to .pivot() if categories are fixed.

• Plots not showing
  In some environments, add %matplotlib inline at the top of a notebook.

• Paths / working directory
  The notebook uses project-relative paths via pathlib.Path. Run from the repo root (cd your-repo) or adjust ROOT.

• Team name mismatches
  Make sure that training, fixtures, and points files use identical team names as strings.

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🗺️ Roadmap / Future Work

• Dixon–Coles time decay in the likelihood (down-weight older matches).
• More covariates: injuries, transfers, schedule congestion (rest days).
• Bayesian / hierarchical variants for team effects and ( \lambda_3 ).
• Calibration checks (50/80/95% interval coverage) & reliability plots.
• Expanded hyper-parameter search and cross-league support.

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📚 References & Data

📝 Citing & Background

• Dixon & Coles (1997). Modelling Association Football Scores and Inefficiencies in the Football Betting Market. JRSS C 46(2): 265–280. DOI: 10.2307/2986290.
• Related reading: https://royalsocietypublishing.org/doi/10.1098/rsos.210617

🗃️ Data Sources

• Football-Data.co.uk — England (EPL) match results & odds
• Fixture Download — Premier League results/fixtures 2024–25

Accessed: August 12, 2025. Check each site’s terms before redistribution.

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🤝 Contributing

PRs welcome — loaders, metrics (rank corr, Brier), visual polish, or tuning refactors.

Steps

1. Fork.
2. Create a branch.
3. Commit changes with examples.
4. Open a Pull Request.

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📄 License

Released under the MIT License. See LICENSE.

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✉️ Contact

Authors: Anusha Sarla & Sanmesh Shintre
Emails: anusha.sarla@ucdconnect.ie · sanmesh.shintre@ucdconnect.ie