Exercise - Bundesliga Game Prediction
In this exercises we will define a simple model for predicting soccer games for the German "Bundesliga".
Remark: In order to detect errors in your own code, execute the notebook cells containing
assert_almost_equal. These statements raise exceptions, as long as the calculated result is not yet correct.
To complete this exercise notebook, you should possess knowledge about the following topics.
- Basics of Bayesian Inference, see e.g. Introduction into Bayesian Inference with PyMc3
- Knowledge about the Gaussian and Poisson distribution: TODO: Add Link
import numpy as np import pandas as pd import pymc3 as pm import scipy.stats import theano from theano import tensor as T from matplotlib import pyplot as plt from IPython.core.pylabtools import figsize %matplotlib inline
Simple model for the predictions of soccer games: How many goals a team scores.
As data only the results from prior games are used.
Loading and processing of data
# some simple preprocessing of the data url_vereine_csv = "https://github.com/hsro-wif-prg2/hsro-wif-prg2.github.io/raw/master/examples/src/main/resources/bundesliga_Verein.csv" clubs = pd.read_csv(url_vereine_csv, sep=';') # for convinience the club id should start with 0 clubs.V_ID = clubs.V_ID - 1 clubs = clubs.set_index("V_ID")
# just 1. liga club_ids = clubs[clubs.Liga==1].index club_ids
url_spiele_csv = "https://github.com/hsro-wif-prg2/hsro-wif-prg2.github.io/raw/master/examples/src/main/resources/bundesliga_Spiel.csv" games = pd.read_csv(url_spiele_csv, sep=';') #del(games["Unnamed: 8"]) ### not existent anymore? # for convinience the club id should start with 0 games.Heim = games.Heim-1 games.Gast = games.Gast-1
relevant_games = games[games.Heim.isin(club_ids)]
actual_date = "2018-01-01" relevant_games = relevant_games[games.Datum < actual_date] len(relevant_games)
def get_goal_results(gh="Tore_Gast"): result = list() for i in relevant_games.iterrows(): r = i result.append((r.Heim, r.Gast, r[gh])) return result away_goals_ = get_goal_results("Tore_Gast") home_goals_ = get_goal_results("Tore_Heim")
low = 1e-10
Idea: The number of goals a team scores can be modeled with a Poisson distribution.
Probability for outcome
- is also the expectation and variance of the distribution
import scipy.stats k=np.arange(0,10) lambda_= 3.1 plt.figure(figsize=(8,6)) plt.plot(k, scipy.stats.poisson.pmf(k, lambda_), 'bo', ms=6, label='poisson pmf') plt.xlabel("k") plt.ylabel("probability mass") scipy.stats.poisson.pmf(k, lambda_)
Each team has a offence and defence strength (distribution). (Note that the average goals per game ):
is the Gaussian distribution with parameters
- precision: (variance: )
Model: The number of goals that team scores against team is Poisson distributed with
Graphical representation of the model
Implementation with pymc
nb_clubs = len(club_ids) nb_clubs
model = pm.Model() with model: offence = pm.Normal("offence", tau=1., mu=1.5, shape=nb_clubs) defence = pm.Normal("defence", tau=1., mu=0., shape=nb_clubs) home_goals =  away_goals =  hv =  for i,(heim, gast, goals) in enumerate(home_goals_): home_value = offence[heim]-defence[gast] home_value = T.switch(T.lt(home_value, 0.), low, home_value) hv.append(home_value) home_goals.append(goals) hv_ = T.stack(hv) mu_h = pm.Deterministic("home_rate", hv_) pm.Poisson("home_goals", observed=home_goals, mu=mu_h) av =  for i,(heim, gast, goals) in enumerate(away_goals_): away_value = offence[gast]-defence[heim] away_value = T.switch(T.lt(away_value, 0.), low, away_value) av.append(away_value) away_goals.append(goals) av_ = T.stack(av) mu_a = pm.Deterministic("away_rate", av_) pm.Poisson("away_goals", observed=away_goals, mu=mu_a)
# start the sampling procedure #map_estimate = pm.find_MAP(model=model)
Sampling with pymc
# para nb_samples=10000
with model: trace = pm.sample(draws=nb_samples) #20000 5000
# don't use the first samples burn = 1000 trace = trace[burn:]
nb_clubs = club_ids.max() + 1 bins=40 fig, axes = plt.subplots(nrows=nb_clubs, ncols=2, figsize=(10, 50)) for i in club_ids: title = "Offence of " + clubs[clubs.index==i]["Name"][i] axes[i, 0].set_title(title) axes[i, 0].hist(trace.get_values("offence")[:,i], bins=bins, range=(0,4.2)) axes[i, 1].hist(trace.get_values("defence")[:,i], bins=bins, range=(-2.,2.2)) title = "Defence of " + clubs[clubs.index==i]["Name"][i] axes[i, 1].set_title(title) #fig.suptitle("Offence and defence distribution of the clubs.") fig.subplots_adjust(hspace=0.5) fig.tight_layout()
Exercise: Distribution of expected goals
Use the model and the sampling trace to predict how many goals a teams scores agains another team.
What is the expected number of the goals?
Implement the corresponding python (plot) functions, e.g.
# Expectation of number of goals scored by team 0, mean of strength print((np.arange(len(p_goals_1)) * p_goals_1).sum(), d1.mean()) # Expectation of number of goals scored by team 1, mean of strength print((np.arange(len(p_goals_2)) * p_goals_2).sum(), d2.mean())
# probability that team 0 scores 0,1,2, ... goals against team 8 plot_goal_diffs(0, 17)
Exercice: Extension of the model
Extend the model with "home advantage":
At home is a team in general a little bit stronger as away. Modify the model to take this into account.
How strong is the home advantage in your model?
nb_samples = 10000 with model_home_advantage: trace_ha = pm.sample(draws=nb_samples, tune=1000)
# don't use the first samples burn = 1000 trace_ha = trace_ha[burn:]
# This depends on your model! trace_ha.get_values("home_advantage").mean()
plt.hist(trace_ha.get_values("home_advantage"), bins=20) plt.title("Home advantage distribution")
Notebook License (CC-BY-SA 4.0)
The following license applies to the complete notebook, including code cells. It does however not apply to any referenced external media (e.g., images).
Exercise - Bundesliga Game Prediction
by Christian Herta, Klaus Strohmenger
is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.
Based on a work at https://gitlab.com/deep.TEACHING.
Code License (MIT)
The following license only applies to code cells of the notebook.
Copyright 2018 Christian Herta, Klaus Strohmenger
Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the "Software"), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions:
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