# Exercise - Bundesliga Game Prediction

## Table of Contents

## Introduction

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`

or `assert_almost_equal`

. These statements raise exceptions, as long as the calculated result is not yet correct.

## Requirements

### Knowledge

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

### Python Modules

```
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
```

## Exercises

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)]`

`relevant_games`

```
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[1]
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.

#### Poisson distribution

Probability for outcome$ k \in \{0, 1, 2, \dots\} $

$ P_\lambda (k) = \frac{\lambda^k}{k!}\, \mathrm{e}^{-\lambda} $

with parameter$ \lambda>0 $ -$ \lambda $ 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_)
```

#### Probabilistic Model

Each team$ i $ has a offence and defence strength (distribution). (Note that the average goals per game$ \approx 3 \Rightarrow \Delta \mu=1.5 $):

$ offence_i \sim \mathcal N(\mu=1.5, \tau=1) $ $ defence_i \sim \mathcal N(\mu=0, \tau=1) $ $ \mathcal N $ is the Gaussian distribution with parameters

- mean:$ \mu $
- precision:$ \tau=1/\sigma^2 $ (variance:$ \sigma^2 $)

Model: The number of goals that team$ i $ scores against team$ j $ is Poisson distributed with

$ goals_{ij} = Poisson \left(\lambda = (offence_i-defence_j) \right) $

```
import daft
def plot_model():
pgm = daft.PGM([6.3, 4.05], origin=[-1., -1.], aspect=1.)
pgm.add_node(daft.Node("mu_o", r"$\mu_o$", .5, .5, fixed=True))
pgm.add_node(daft.Node("tau_o", r"$\tau_o$", .5, 1.5, fixed=True))
pgm.add_node(daft.Node("o_i", r"o$_i$", 1.5, 1))
pgm.add_node(daft.Node("tau_d", r"$\tau_d$", 2., 3., fixed=True))
pgm.add_node(daft.Node("mu_d", r"$\mu_d$", 3., 3., fixed=True))
pgm.add_node(daft.Node("d_j", r"d$_j$", 2.5, 2.2))
#pgm.add_node(daft.Node("Delta", r"$\Delta_{ij}$", 2.5, 1))
pgm.add_node(daft.Node("g", r"g$_{ij}$", 2.5, 1., observed=True))
# Add in the edges.
pgm.add_edge("mu_o", "o_i")
pgm.add_edge("tau_o", "o_i")
pgm.add_edge("mu_d", "d_j")
pgm.add_edge("tau_d", "d_j")
pgm.add_edge("o_i", "g")
pgm.add_edge("d_j", "g")
#pgm.add_edge("Delta", "g")
# And plates.
pgm.add_plate(daft.Plate([2., 0.2, 1., 2.5], label=r"$j$", shift=0.))
pgm.add_plate(daft.Plate([1., 0.5, 2.2, 1.1], label=r"$i$", shift=0.))
pgm.render()
```

### Graphical representation of the model

`plot_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)
```

`offence`

```
# 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)
```

```
# don't use the first samples - already considered by tune
burn = 0
trace = trace[burn:]
```

`trace.get_values("offence")`

#### Sampling histograms

```
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")
```

## Licenses

### 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:

The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software.

THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.