{ "cells": [ { "cell_type": "markdown", "id": "81215b75-f5f8-4c17-9cd4-08b1b0ed4234", "metadata": {}, "source": [ "# Bivariate Poisson Model\n", "\n", "The Bivariate Poisson Model is an extension of the standard Poisson model that accounts for the correlation between the number of goals scored by each team in a football match. \n", "\n", "Unlike independent Poisson models, which assume team goal distributions are unrelated, the Bivariate Poisson approach introduces a dependency structure that better captures real-world interactions, such as defensive and offensive interplay between teams. \n", "\n", "This results in more accurate probability estimates for match outcomes, correct score predictions, and betting markets like Asian handicaps and total goals. \n", "\n", "The model is particularly useful for improving forecasting accuracy in competitive matches where team performances are not entirely independent." ] }, { "cell_type": "code", "execution_count": 1, "id": "1f931497-c1f9-4cb4-969a-058676e42a24", "metadata": { "tags": [] }, "outputs": [], "source": [ "import penaltyblog as pb" ] }, { "cell_type": "markdown", "id": "4a1b5c76-8f47-4f59-8351-d5add2f69309", "metadata": {}, "source": [ "## Get data from football-data.co.uk" ] }, { "cell_type": "code", "execution_count": 2, "id": "949b129d-e4e5-4975-8318-dd601d918e90", "metadata": { "tags": [] }, "outputs": [ { "data": { "text/html": [ "
\n", "\n", "\n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", " \n", "
datedatetimeseasoncompetitiondivtimeteam_hometeam_awayfthgftag...b365_cahhb365_cahapcahhpcahamax_cahhmax_cahaavg_cahhavg_cahagoals_homegoals_away
id
1565308800---liverpool---norwich2019-08-092019-08-09 20:00:002019-2020ENG Premier LeagueE020:00LiverpoolNorwich41...1.911.991.941.981.992.071.901.9941
1565395200---bournemouth---sheffield_united2019-08-102019-08-10 15:00:002019-2020ENG Premier LeagueE015:00BournemouthSheffield United11...1.951.951.981.952.001.961.961.9211
1565395200---burnley---southampton2019-08-102019-08-10 15:00:002019-2020ENG Premier LeagueE015:00BurnleySouthampton30...1.872.031.892.031.902.071.862.0230
1565395200---crystal_palace---everton2019-08-102019-08-10 15:00:002019-2020ENG Premier LeagueE015:00Crystal PalaceEverton00...1.822.081.971.962.032.081.961.9300
1565395200---tottenham---aston_villa2019-08-102019-08-10 17:30:002019-2020ENG Premier LeagueE017:30TottenhamAston Villa31...2.101.702.181.772.211.872.081.8031
\n", "

5 rows × 111 columns

\n", "
" ], "text/plain": [ " date datetime \\\n", "id \n", "1565308800---liverpool---norwich 2019-08-09 2019-08-09 20:00:00 \n", "1565395200---bournemouth---sheffield_united 2019-08-10 2019-08-10 15:00:00 \n", "1565395200---burnley---southampton 2019-08-10 2019-08-10 15:00:00 \n", "1565395200---crystal_palace---everton 2019-08-10 2019-08-10 15:00:00 \n", "1565395200---tottenham---aston_villa 2019-08-10 2019-08-10 17:30:00 \n", "\n", " season competition \\\n", "id \n", "1565308800---liverpool---norwich 2019-2020 ENG Premier League \n", "1565395200---bournemouth---sheffield_united 2019-2020 ENG Premier League \n", "1565395200---burnley---southampton 2019-2020 ENG Premier League \n", "1565395200---crystal_palace---everton 2019-2020 ENG Premier League \n", "1565395200---tottenham---aston_villa 2019-2020 ENG Premier League \n", "\n", " div time team_home \\\n", "id \n", "1565308800---liverpool---norwich E0 20:00 Liverpool \n", "1565395200---bournemouth---sheffield_united E0 15:00 Bournemouth \n", "1565395200---burnley---southampton E0 15:00 Burnley \n", "1565395200---crystal_palace---everton E0 15:00 Crystal Palace \n", "1565395200---tottenham---aston_villa E0 17:30 Tottenham \n", "\n", " team_away fthg ftag \\\n", "id \n", "1565308800---liverpool---norwich Norwich 4 1 \n", "1565395200---bournemouth---sheffield_united Sheffield United 1 1 \n", "1565395200---burnley---southampton Southampton 3 0 \n", "1565395200---crystal_palace---everton Everton 0 0 \n", "1565395200---tottenham---aston_villa Aston Villa 3 1 \n", "\n", " ... b365_cahh b365_caha pcahh \\\n", "id ... \n", "1565308800---liverpool---norwich ... 1.91 1.99 1.94 \n", "1565395200---bournemouth---sheffield_united ... 1.95 1.95 1.98 \n", "1565395200---burnley---southampton ... 1.87 2.03 1.89 \n", "1565395200---crystal_palace---everton ... 1.82 2.08 1.97 \n", "1565395200---tottenham---aston_villa ... 2.10 1.70 2.18 \n", "\n", " pcaha max_cahh max_caha \\\n", "id \n", "1565308800---liverpool---norwich 1.98 1.99 2.07 \n", "1565395200---bournemouth---sheffield_united 1.95 2.00 1.96 \n", "1565395200---burnley---southampton 2.03 1.90 2.07 \n", "1565395200---crystal_palace---everton 1.96 2.03 2.08 \n", "1565395200---tottenham---aston_villa 1.77 2.21 1.87 \n", "\n", " avg_cahh avg_caha goals_home \\\n", "id \n", "1565308800---liverpool---norwich 1.90 1.99 4 \n", "1565395200---bournemouth---sheffield_united 1.96 1.92 1 \n", "1565395200---burnley---southampton 1.86 2.02 3 \n", "1565395200---crystal_palace---everton 1.96 1.93 0 \n", "1565395200---tottenham---aston_villa 2.08 1.80 3 \n", "\n", " goals_away \n", "id \n", "1565308800---liverpool---norwich 1 \n", "1565395200---bournemouth---sheffield_united 1 \n", "1565395200---burnley---southampton 0 \n", "1565395200---crystal_palace---everton 0 \n", "1565395200---tottenham---aston_villa 1 \n", "\n", "[5 rows x 111 columns]" ] }, "execution_count": 2, "metadata": {}, "output_type": "execute_result" } ], "source": [ "fb = pb.scrapers.FootballData(\"ENG Premier League\", \"2019-2020\")\n", "df = fb.get_fixtures()\n", "\n", "df.head()" ] }, { "cell_type": "markdown", "id": "9257f0fc-5f2b-402f-9209-d005d14880be", "metadata": {}, "source": [ "## Train the Model" ] }, { "cell_type": "code", "execution_count": 3, "id": "7d39d92f-6fa0-4a2a-8a48-22d214e38efc", "metadata": { "tags": [] }, "outputs": [], "source": [ "clf = pb.models.BivariatePoissonGoalModel(\n", " df[\"goals_home\"], df[\"goals_away\"], df[\"team_home\"], df[\"team_away\"]\n", ")\n", "clf.fit()" ] }, { "cell_type": "markdown", "id": "63a12589-0066-431f-8444-92e2944b55a4", "metadata": {}, "source": [ "## The model's parameters" ] }, { "cell_type": "code", "execution_count": 4, "id": "ffe48c5e-3e8c-4a99-be9f-a1b46307c981", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "Module: Penaltyblog\n", "\n", "Model: Bivariate Poisson\n", "\n", "Number of parameters: 42\n", "Log Likelihood: -1059.876\n", "AIC: 2203.752\n", "\n", "Team Attack Defence \n", "------------------------------------------------------------\n", "Arsenal 1.134 -0.987 \n", "Aston Villa 0.835 -0.652 \n", "Bournemouth 0.802 -0.687 \n", "Brighton 0.767 -0.877 \n", "Burnley 0.882 -0.939 \n", "Chelsea 1.356 -0.85 \n", "Crystal Palace 0.535 -0.957 \n", "Everton 0.885 -0.843 \n", "Leicester 1.325 -1.119 \n", "Liverpool 1.55 -1.347 \n", "Man City 1.745 -1.249 \n", "Man United 1.294 -1.274 \n", "Newcastle 0.747 -0.803 \n", "Norwich 0.361 -0.552 \n", "Sheffield United 0.756 -1.212 \n", "Southampton 1.059 -0.751 \n", "Tottenham 1.221 -1.003 \n", "Watford 0.705 -0.699 \n", "West Ham 1.014 -0.724 \n", "Wolves 1.025 -1.187 \n", "------------------------------------------------------------\n", "Home Advantage: 0.237\n", "Correlation: -3.0" ] }, "execution_count": 4, "metadata": {}, "output_type": "execute_result" } ], "source": [ "clf" ] }, { "cell_type": "code", "execution_count": 5, "id": "fc93ec32-d113-4155-a516-abfe58dc8469", "metadata": { "tags": [] }, "outputs": [ { "data": { "text/plain": [ "{'attack_Arsenal': np.float64(1.133560802574037),\n", " 'attack_Aston Villa': np.float64(0.8350898400984498),\n", " 'attack_Bournemouth': np.float64(0.8015404096015637),\n", " 'attack_Brighton': np.float64(0.7674328208660928),\n", " 'attack_Burnley': np.float64(0.8822591220552471),\n", " 'attack_Chelsea': np.float64(1.3562926053643534),\n", " 'attack_Crystal Palace': np.float64(0.5348767383835888),\n", " 'attack_Everton': np.float64(0.8854463751643209),\n", " 'attack_Leicester': np.float64(1.3253210988334698),\n", " 'attack_Liverpool': np.float64(1.5500067231127181),\n", " 'attack_Man City': np.float64(1.7450410219723662),\n", " 'attack_Man United': np.float64(1.2938830262372865),\n", " 'attack_Newcastle': np.float64(0.7470789004929902),\n", " 'attack_Norwich': np.float64(0.36056444644586966),\n", " 'attack_Sheffield United': np.float64(0.7561475016812931),\n", " 'attack_Southampton': np.float64(1.0594558601023647),\n", " 'attack_Tottenham': np.float64(1.2214463479299582),\n", " 'attack_Watford': np.float64(0.7054564312709098),\n", " 'attack_West Ham': np.float64(1.014009505646218),\n", " 'attack_Wolves': np.float64(1.0250904221669037),\n", " 'defense_Arsenal': np.float64(-0.9867989766505458),\n", " 'defense_Aston Villa': np.float64(-0.6517511449587307),\n", " 'defense_Bournemouth': np.float64(-0.6874367252280075),\n", " 'defense_Brighton': np.float64(-0.877458942752178),\n", " 'defense_Burnley': np.float64(-0.9385627948200023),\n", " 'defense_Chelsea': np.float64(-0.8499169474665187),\n", " 'defense_Crystal Palace': np.float64(-0.9570569580061927),\n", " 'defense_Everton': np.float64(-0.8426424728271178),\n", " 'defense_Leicester': np.float64(-1.1187653346121664),\n", " 'defense_Liverpool': np.float64(-1.3469873675965822),\n", " 'defense_Man City': np.float64(-1.2485881916058201),\n", " 'defense_Man United': np.float64(-1.2737712763058457),\n", " 'defense_Newcastle': np.float64(-0.8028015712987764),\n", " 'defense_Norwich': np.float64(-0.5523345592912768),\n", " 'defense_Sheffield United': np.float64(-1.2120340284112339),\n", " 'defense_Southampton': np.float64(-0.7508665627531523),\n", " 'defense_Tottenham': np.float64(-1.003192069120226),\n", " 'defense_Watford': np.float64(-0.6985873243764104),\n", " 'defense_West Ham': np.float64(-0.7242617861788704),\n", " 'defense_Wolves': np.float64(-1.1868096051415054),\n", " 'home_advantage': np.float64(0.23692061130011463),\n", " 'correlation_log': np.float64(-2.999999999999999),\n", " 'lambda3': np.float64(0.049787068367863986)}" ] }, "execution_count": 5, "metadata": {}, "output_type": "execute_result" } ], "source": [ "clf.get_params()" ] }, { "cell_type": "markdown", "id": "43bb1f12-7010-421b-bf93-bb8e1dba2df6", "metadata": {}, "source": [ "## Predict Match Outcomes" ] }, { "cell_type": "code", "execution_count": 6, "id": "3a047b77-707d-46b6-bcf8-57f3356efee3", "metadata": { "tags": [] }, "outputs": [ { "data": { "text/plain": [ "Module: Penaltyblog\n", "\n", "Class: FootballProbabilityGrid\n", "\n", "Home Goal Expectation: [1.82233333]\n", "Away Goal Expectation: [0.72477288]\n", "\n", "Home Win: 0.6357320764164407\n", "Draw: 0.2213346734110996\n", "Away Win: 0.14290631670926676" ] }, "execution_count": 6, "metadata": {}, "output_type": "execute_result" } ], "source": [ "probs = clf.predict(\"Liverpool\", \"Wolves\")\n", "probs" ] }, { "cell_type": "markdown", "id": "2a5274e7-d13e-455b-8e77-a6f51ba6f830", "metadata": {}, "source": [ "### 1x2 Probabilities" ] }, { "cell_type": "code", "execution_count": 7, "id": "cc1d6199-c35e-4ea3-bf82-a89c31a7277d", "metadata": { "tags": [] }, "outputs": [ { "data": { "text/plain": [ "[np.float64(0.6357320764164407),\n", " np.float64(0.2213346734110996),\n", " np.float64(0.14290631670926676)]" ] }, "execution_count": 7, "metadata": {}, "output_type": "execute_result" } ], "source": [ "probs.home_draw_away" ] }, { "cell_type": "code", "execution_count": 8, "id": "eef96983-d83d-4c39-bd49-47cb4a704ab4", "metadata": { "tags": [] }, "outputs": [ { "data": { "text/plain": [ "np.float64(0.6357320764164407)" ] }, "execution_count": 8, "metadata": {}, "output_type": "execute_result" } ], "source": [ "probs.home_win" ] }, { "cell_type": "code", "execution_count": 9, "id": "e08561b2-07ed-47b3-89d7-14c0a05cf854", "metadata": { "tags": [] }, "outputs": [ { "data": { "text/plain": [ "np.float64(0.2213346734110996)" ] }, "execution_count": 9, "metadata": {}, "output_type": "execute_result" } ], "source": [ "probs.draw" ] }, { "cell_type": "code", "execution_count": 10, "id": "594e21a7-9a75-49a3-b3e8-50fa4bd8ac51", "metadata": { "tags": [] }, "outputs": [ { "data": { "text/plain": [ "np.float64(0.14290631670926676)" ] }, "execution_count": 10, "metadata": {}, "output_type": "execute_result" } ], "source": [ "probs.away_win" ] }, { "cell_type": "markdown", "id": "9996be1b-acf8-4305-9bf0-6e4832505d47", "metadata": {}, "source": [ "### Probablity of Total Goals >1.5" ] }, { "cell_type": "code", "execution_count": 11, "id": "8da5ea91-ff28-4c6d-b6bf-0d5ef417da2b", "metadata": { "tags": [] }, "outputs": [ { "data": { "text/plain": [ "np.float64(0.7356970375004634)" ] }, "execution_count": 11, "metadata": {}, "output_type": "execute_result" } ], "source": [ "probs.total_goals(\"over\", 1.5)" ] }, { "cell_type": "markdown", "id": "5a0876d3-9d69-4b63-ae8a-d2b3b8f40aa6", "metadata": {}, "source": [ "### Probability of Asian Handicap 1.5" ] }, { "cell_type": "code", "execution_count": 12, "id": "280e7570-5010-4b39-8104-71ca27e4005a", "metadata": { "tags": [] }, "outputs": [ { "data": { "text/plain": [ "np.float64(0.3756034531458142)" ] }, "execution_count": 12, "metadata": {}, "output_type": "execute_result" } ], "source": [ "probs.asian_handicap(\"home\", 1.5)" ] }, { "cell_type": "markdown", "id": "f1205e38-8afc-45fc-ba5f-59292aad9e21", "metadata": {}, "source": [ "## Probability of both teams scoring" ] }, { "cell_type": "code", "execution_count": 13, "id": "1b63af09-9383-4c5a-ae1c-dadb1a57193a", "metadata": { "tags": [] }, "outputs": [ { "data": { "text/plain": [ "np.float64(0.45978389607673564)" ] }, "execution_count": 13, "metadata": {}, "output_type": "execute_result" } ], "source": [ "probs.both_teams_to_score" ] }, { "cell_type": "markdown", "id": "8d8ace53-efc0-4227-bbdc-8c54e8b1e05c", "metadata": {}, "source": [ "## Train the model with more recent data weighted to be more important" ] }, { "cell_type": "code", "execution_count": 14, "id": "c5fd1d29-cdac-4b70-b4a7-04f64cb87eea", "metadata": { "tags": [] }, "outputs": [], "source": [ "weights = pb.models.dixon_coles_weights(df[\"date\"], 0.001)\n", "\n", "clf = pb.models.BivariatePoissonGoalModel(\n", " df[\"goals_home\"], df[\"goals_away\"], df[\"team_home\"], df[\"team_away\"], weights\n", ")\n", "clf.fit()" ] }, { "cell_type": "code", "execution_count": 15, "id": "954c880f-e861-406c-9ef5-b3b05bfa7f6e", "metadata": { "tags": [] }, "outputs": [ { "data": { "text/plain": [ "Module: Penaltyblog\n", "\n", "Model: Bivariate Poisson\n", "\n", "Number of parameters: 42\n", "Log Likelihood: -881.987\n", "AIC: 1847.973\n", "\n", "Team Attack Defence \n", "------------------------------------------------------------\n", "Arsenal 1.146 -1.008 \n", "Aston Villa 0.813 -0.675 \n", "Bournemouth 0.813 -0.685 \n", "Brighton 0.758 -0.882 \n", "Burnley 0.877 -0.963 \n", "Chelsea 1.363 -0.852 \n", "Crystal Palace 0.526 -0.94 \n", "Everton 0.88 -0.858 \n", "Leicester 1.302 -1.095 \n", "Liverpool 1.548 -1.321 \n", "Man City 1.752 -1.292 \n", "Man United 1.321 -1.3 \n", "Newcastle 0.773 -0.798 \n", "Norwich 0.292 -0.555 \n", "Sheffield United 0.756 -1.197 \n", "Southampton 1.082 -0.784 \n", "Tottenham 1.222 -1.026 \n", "Watford 0.726 -0.701 \n", "West Ham 1.03 -0.739 \n", "Wolves 1.02 -1.215 \n", "------------------------------------------------------------\n", "Home Advantage: 0.245\n", "Correlation: -3.0" ] }, "execution_count": 15, "metadata": {}, "output_type": "execute_result" } ], "source": [ "clf" ] }, { "cell_type": "code", "execution_count": null, "id": "22cb6d4c-bc32-4a75-b803-b96c959184a8", "metadata": {}, "outputs": [], "source": [] } ], "metadata": { "kernelspec": { "display_name": "venv", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.13.1" } }, "nbformat": 4, "nbformat_minor": 5 }