GAS ranking models

Introduction

GAS ranking models can be used for pairwise comparisons or competitive activities. The \(GASRank\) model of Taylor, R. (2016) models a point difference between two competitors. The point difference is assumed to follow a particular distribution. For example, suppose that the point difference \(\mu_{t}\) is normally distributed, then we can model the point difference as the location parameter \(\mu\):

\[\mu_{t} = \delta + \alpha_{t,i} - \alpha_{t,j}\]

Where \(\delta_{t}\) is a ‘home advantage’ latent variable, \(i\) and \(j\) refer to home and away competitors, and \(\alpha\) contains the team power rankings. The power rankings are modelled as random walk processes between each match:

\[\alpha_{k,i} = \alpha_{k-1,i} + \eta{U}_{k-1,i}\]

\[\alpha_{k,j} = \alpha_{k-1,j} - \eta{U}_{k-1,j}\]

Where \(k\) is the game index, \(\eta\) is a learning rate or scaling parameter to be estimated.

The model can be extended to a two component model where each competitor has two aspects to their ‘team’. For example, we might model an NFL team along with the Quarterback power rankings in the same game:

\[\mu_{t} = \delta + \alpha_{t,i} - \alpha_{t,j} + \gamma_{t,i} - \gamma_{t,j}\]

Here \(\gamma\) represents the power ranking of the second component. The secondary component power rankings are modelled as random walk processes between each match:

\[\gamma_{k,i} = \gamma_{k-1,i} + \eta{U_2}_{k-1,i}\]

\[\gamma_{k,j} = \gamma_{k-1,j} - \eta{U_2}_{k-1,j}\]

Developer Note

• This model type has yet to be cythonized, so performance can be slow.

Example

We will model the point difference in NFL games with a simple model. Here is the data:

import pyflux as pf
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt

data = pd.read_csv("nfl_data_new.csv")
data["PointsDiff"] = data["HomeScore"] - data["AwayScore"]
data.columns
Index(['Unnamed: 0', 'AFirstDowns', 'AFumbles0', 'AFumbles1', 'AIntReturns0',
    'AIntReturns1', 'AKickoffReturns0', 'AKickoffReturns1', 'ANetPassYards',
    'APassesAttempted', 'APassesCompleted', 'APassesIntercepted',
    'APenalties0', 'APenalties1', 'APossession', 'APuntReturns0',
    'APuntReturns1', 'APunts0', 'APunts1', 'AQB', 'ARushes0', 'ARushes1',
    'ASacked0', 'ASacked1', 'AwayScore', 'AwayTeam', 'Date', 'HFirstDowns',
    'HFumbles0', 'HFumbles1', 'HIntReturns0', 'HIntReturns1',
    'HKickoffReturns0', 'HKickoffReturns1', 'HNetPassYards',
    'HPassesAttempted', 'HPassesCompleted', 'HPassesIntercepted',
    'HPenalties0', 'HPenalties1', 'HPossession', 'HPuntReturns0',
    'HPuntReturns1', 'HPunts0', 'HPunts1', 'HQB', 'HRushes0', 'HRushes1',
    'HSacked0', 'HSacked1', 'HomeScore', 'HomeTeam', 'Postseason',
    'PointsDiff'],
   dtype='object')

We can plot the point difference to get an idea of potentially suitable distributions:

data = pd.read_csv("nfl_data_new.csv")
data["PointsDiff"] = data["HomeScore"] - data["AwayScore"]
plt.figure(figsize=(15,7))
plt.ylabel("Frequency")
plt.xlabel("Points Difference")
plt.hist(data["PointsDiff"],bins=20);

We will use a pf.Normal() families, although we could try a family with heavier tails also. We setup the \(GASRank\) model, referring to the appropriate columns in our DataFrame:

model = pf.GASRank(data=data,team_1="HomeTeam", team_2="AwayTeam",
                   score_diff="PointsDiff", family=pf.Normal())

Next we estimate the latent variables. For this example we will use a maximum likelihood point mass estimate \(z^{MLE}\):

x = model.fit()
x.summary()

NormalGAS Rank
======================================== ==================================================
Dependent Variable: PointsDiff           Method: MLE
Start Date: 0                            Log Likelihood: -10825.1703
End Date: 2667                           AIC: 21656.3406
Number of observations: 2668             BIC: 21674.0079
===========================================================================================
Latent Variable           Estimate   Std Error  z        P>|z|    95% C.I.
========================= ========== ========== ======== ======== =========================
Constant                  2.2405     0.2547     8.795    0.0      (1.7412 | 2.7398)
Ability Scale             0.0637     0.0058     10.9582  0.0      (0.0523 | 0.0751)
Normal Scale              13.9918
===========================================================================================

Once we have fit the model we can plot the power rankings of the teams in our DataFrame over their competitive history using plot_abilities():

model.plot_abilities(["Denver Broncos", "Green Bay Packers", "New England Patriots",
                      "Carolina Panthers"],figsize=(15,8))

model.plot_abilities(["San Francisco 49ers", "Oakland Raiders", "San Diego Chargers"],
                       figsize=(15,8))

We can predict the point difference between two competitors in the future using predict():

model.predict("Denver Broncos","Carolina Panthers",neutral=True)
array(-4.886816685966575)

Our DataFrame also has information on quarterbacks. Let’s extend our model with a second component by including quarterbacks in the model:

model.add_second_component("HQB","AQB")
x = model.fit()
x.summary()

NormalGAS Rank
======================================== ==================================================
Dependent Variable: PointsDiff           Method: MLE
Start Date: 0                            Log Likelihood: -10799.4544
End Date: 2667                           AIC: 21606.9087
Number of observations: 2668             BIC: 21630.4651
===========================================================================================
Latent Variable           Estimate   Std Error  z        P>|z|    95% C.I.
========================= ========== ========== ======== ======== =========================
Constant                  2.2419     0.2516     8.9118   0.0      (1.7488 | 2.735)
Ability Scale 1           0.0186     0.0062     2.9904   0.0028   (0.0064 | 0.0307)
Ability Scale 2           0.0523     0.0076     6.8492   0.0      (0.0373 | 0.0673)
Normal Scale              13.8576
==========================================================================================================

We can plot the power rankings of the QBs in our DataFrame over their competitive history using plot_abilities():

model.plot_abilities(["Cam Newton", "Peyton Manning"],1,figsize=(15,8))

We can predict the point difference between two competitors in the future using predict():

model.predict("Denver Broncos","Carolina Panthers","Peyton Manning","Cam Newton",neutral=True)
array(-7.33759714587138)

And some more power rankings for fan interest…

model.plot_abilities(["Aaron Rodgers", "Tom Brady", "Russell Wilson"],1,figsize=(15,8))

model.plot_abilities(["Peyton Manning","Michael Vick", "David Carr", "Carson Palmer"
                     ,"Eli Manning","Alex Smith","JaMarcus Russell","Matthew Stafford"
                     ,"Sam Bradford","Cam Newton","Andrew Luck","Jameis Winston"],1,
                     figsize=(15,8))

Class Description

class GASRank(data, team_1, team_2, family, score_diff)

Generalized Autoregressive Score Ranking Models (GASRank).

Parameter	Type	Description
data	pd.dataframe	Containing the competitive data
team_1	string	Column name for home team names
team_2	string	Column name for away team names
family	pf.Family instance	The distribution for the time series, e.g pf.Normal()
score_diff	string	Column name for the point difference

Attributes

latent_variables

A pf.LatentVariables() object containing information on the model latent variables, prior settings. any fitted values, starting values, and other latent variable information. When a model is fitted, this is where the latent variables are updated/stored. Please see the documentation on Latent Variables for information on attributes within this object, as well as methods for accessing the latent variable information.

Methods

add_second_component(team_1, team_2)

Adds a second component to the model

Parameter	Type	Description
team_1	string	Column name for team 1 second component
team_2	string	Column name for team 2 second component

Returns : void - changes model to a second component model

adjust_prior(index, prior)

Adjusts the priors for the model latent variables. The latent variables and their indices can be viewed by printing the latent_variables attribute attached to the model instance.

Parameter	Type	Description
index	int	Index of the latent variable to change
prior	pf.Family instance	Prior distribution, e.g. pf.Normal()

Returns: void - changes the model latent_variables attribute

fit(method, **kwargs)

Estimates latent variables for the model. User chooses an inference option and the method returns a results object, as well as updating the model’s latent_variables attribute.

Parameter	Type	Description
method	str	Inference option: e.g. ‘M-H’ or ‘MLE’

See Bayesian Inference and Classical Inference sections of the documentation for the full list of inference options. Optional parameters can be entered that are relevant to the particular mode of inference chosen.

Returns: pf.Results instance with information for the estimated latent variables

plot_abilities(team_ids)

Plots power rankings of the model components. Optional arguments include figsize, the dimensions of the figure to plot.

Parameter	Type	Description
team_ids	list	Of strings (team names) or indices

For a two component model, arguments are:

Parameter	Type	Description
team_ids	list	Of strings (team names) or indices
component_id	int	0 for component 1, 1 for component 2

Returns : void - shows a matplotlib plot

plot_fit(**kwargs)

Plots the fit of the model against the data. Optional arguments include figsize, the dimensions of the figure to plot.

Returns : void - shows a matplotlib plot

plot_z(indices, figsize)

Returns a plot of the latent variables and their associated uncertainty.

Parameter	Type	Description
indices	int or list	Which latent variable indices to plot
figsize	tuple	Size of the matplotlib figure

Returns : void - shows a matplotlib plot

predict(team_1, team_2, neutral=False)

Returns predicted point differences. For a one component model, arguments are:

Parameter	Type	Description
team_1	string or int	If string, team name, else team index
team_2	string or int	If string, team name, else team index
neutral	boolean	If True, disables home advantage

For a two component model, arguments are:

Parameter	Type	Description
team_1	string or int	If string, team name, else team index
team_2	string or int	If string, team name, else team index
team1b	string or int	If string, team 1, player 2 name
team2b	string or int	If string, team 2, player 2 name
neutral	boolean	If True, disables home advantage

Returns : np.ndarray - point difference predictions

References

Creal, D; Koopman, S.J.; Lucas, A. (2013). Generalized Autoregressive Score Models with Applications. Journal of Applied Econometrics, 28(5), 777–795. doi:10.1002/jae.1279.

Harvey, A.C. (2013). Dynamic Models for Volatility and Heavy Tails: With Applications to Financial and Economic Time Series. Cambridge University Press.

Taylor, R. (2016). A Tour of Time Series Analysis (and a model for predicting NFL games). https://github.com/RJT1990/PyData2016-SanFrancisco