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!date
import numpy as np
import pandas as pd
try:
from StringIO import StringIO
except ImportError:
from io import StringIO
%matplotlib inline
import pymc3 as pm3, theano.tensor as tt
This is a Rugby prediction exercise. So we'll input some data
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data_csv = StringIO("""home_team,away_team,home_score,away_score
Wales,Italy,23,15
France,England,26,24
Ireland,Scotland,28,6
Ireland,Wales,26,3
Scotland,England,0,20
France,Italy,30,10
Wales,France,27,6
Italy,Scotland,20,21
England,Ireland,13,10
Ireland,Italy,46,7
Scotland,France,17,19
England,Wales,29,18
Italy,England,11,52
Wales,Scotland,51,3
France,Ireland,20,22""")
The league is made up by a total of T= 6 teams, playing each other once in a season. We indicate the number of points scored by the home and the away team in the g-th game of the season (15 games) as $y_{g1}$ and $y_{g2}$ respectively.
The vector of observed counts $\mathbb{y} = (y_{g1}, y_{g2})$ is modelled as independent Poisson: $y_{gi}| \theta_{gj} \tilde\;\; Poisson(\theta_{gj})$ where the theta parameters represent the scoring intensity in the g-th game for the team playing at home (j=1) and away (j=2), respectively.
We model these parameters according to a formulation that has been used widely in the statistical literature, assuming a log-linear random effect model: $$log \theta_{g1} = home + att_{h(g)} + def_{a(g)} $$ $$log \theta_{g2} = att_{a(g)} + def_{h(g)}$$ the parameter home represents the advantage for the team hosting the game and we assume that this effect is constant for all the teams and throughout the season.
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df = pd.read_csv(data_csv)
teams = df.home_team.unique()
teams = pd.DataFrame(teams, columns=['team'])
teams['i'] = teams.index
df = pd.merge(df, teams, left_on='home_team', right_on='team', how='left')
df = df.rename(columns = {'i': 'i_home'}).drop('team', 1)
df = pd.merge(df, teams, left_on='away_team', right_on='team', how='left')
df = df.rename(columns = {'i': 'i_away'}).drop('team', 1)
observed_home_goals = df.home_score.values
observed_away_goals = df.away_score.values
home_team = df.i_home.values
away_team = df.i_away.values
num_teams = len(df.i_home.drop_duplicates())
num_games = len(home_team)
g = df.groupby('i_away')
att_starting_points = np.log(g.away_score.mean())
g = df.groupby('i_home')
def_starting_points = -np.log(g.away_score.mean())
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In [5]:
model = pm3.Model()
with pm3.Model() as model:
# global model parameters
home = pm3.Normal('home', 0, .0001)
tau_att = pm3.Gamma('tau_att', .1, .1)
tau_def = pm3.Gamma('tau_def', .1, .1)
intercept = pm3.Normal('intercept', 0, .0001)
# team-specific model parameters
atts_star = pm3.Normal("atts_star",
mu =0,
tau =tau_att,
shape=num_teams)
defs_star = pm3.Normal("defs_star",
mu =0,
tau =tau_def,
shape=num_teams)
atts = pm3.Deterministic('atts', atts_star - tt.mean(atts_star))
defs = pm3.Deterministic('defs', defs_star - tt.mean(defs_star))
home_theta = tt.exp(intercept + home + atts[away_team] + defs[home_team])
away_theta = tt.exp(intercept + atts[away_team] + defs[home_team])
# likelihood of observed data
home_points = pm3.Poisson('home_points', mu=home_theta, observed=observed_home_goals)
away_points = pm3.Poisson('away_points', mu=away_theta, observed=observed_away_goals)
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with model:
start = pm3.find_MAP()
step = pm3.NUTS(state=start)
trace = pm3.sample(2000, step, start=start, progressbar=True)
pm3.traceplot(trace)
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