Poisson Elo
If we can frame the Elo rating system as logistic regression, why not define other rating systems based on other generalized linear models (GLMs)? We do just that and derive a rating system equivalent to Poisson regression here.
Motivation
Poisson regression models count data as the response variable. This can be useful for sports such as football, where we can predict the number of goals a team will score.
The Poisson model is even used to model binary outcomes in cases where we are interested in relative risk ratios rather than the odds ratio. The latter is easily extracted from the logistic regression model but is non-intuitive, while the former is hard to extract from the logistic regression model but straightforward to extract from the Poisson model.
Poisson Regression
Suppose we have a count response \(y|\mathbf{x}\sim\text{Poisson}(\lambda)\), where \(\mathbf{x}\) is the set of input variables/regressors and \(\lambda\) is the expectation of response in a given interval. If we model this outcome with model parameters \(\mathbf{w}\), the likelihood function is given by $$ \begin{equation} \mathcal{L}(\mathbf{w})= \frac{\lambda(\mathbf{w;x})^y e^{-\lambda(\mathbf{w;x})}}{y!}. \end{equation} $$ As in Elo as Logistic Regression, we are ignoring the product over samples for the moment.
The Poisson model assumes the count response is given by $$ \begin{equation} \lambda(\mathbf{w};\mathbf{x})= e^{\mathbf{w}\cdot x}. \end{equation} $$ We have not included an intercept although this can easily be added using an additional regressor (see the example).
Suppose each observation corresponds to the number of points/goals scored in a given interval between two of \(n\) entities. We can define two ratings for each entity - one for their attacking ability and one for their defensive ability. Let the index \(i\) run over the attacking abilities and the index \(\underbar{i}\) run over the defensive abilities.
We can write our regressors as
where \(\delta_{ij}\) is the Kronecker delta. The table below shows how the data would look like for three observations:
- team \(1\) scores two goals against team \(2\) on 01/01/24,
- team \(2\) scores one goal against team \(1\) on 01/01/24,
- team \(n\) score no goals against team \(1\) on 02/01/24.
| Date | y | Team 1 (A) | Team 2 (A) | Team 3 (A) | ... | Team n (A) | Team 1 (D) | Team 2 (D) | Team 3 (D) | ... | Team n (D) |
|---|---|---|---|---|---|---|---|---|---|---|---|
| 01/01/24 | 2 | 1 | 0 | 0 | ... | 0 | 0 | -1 | 0 | ... | 0 |
| 01/01/24 | 1 | 0 | 1 | 0 | ... | 1 | -1 | 0 | 0 | ... | 0 |
| 02/01/24 | 0 | 0 | 0 | 0 | ... | 1 | -1 | 0 | 0 | ... | 0 |
| ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... |
where (A) indicates attacking ability and (D) defensive ability.
With the regressors described above, we can rewrite (2) as $$ \begin{equation} \lambda(\mathbf{w};\mathbf{x})= \exp(w_i - w_{\underbar{j}}), \end{equation} $$ where \(y_{ij}\) represents the goals scored by team \(i\) against \(j\). Recalling that \(\lambda\) is the expected count, we can then write $$ \begin{equation} \mathbb{E}\left[y_{ij}|w_1,\cdots w_n\right]= \exp(w_i - w_{\underbar{j}}). \end{equation} $$
Stochastic Gradient Descent
It remains to find the update method for the rating system, which is just (mini-batch) stochastic gradient descent (SGD).
We can perform gradient descent for Poisson regression by minimising the negative log-likelihood. The gradient of the negative log likelihood for Poisson regression is given by $$ \begin{align} \nabla_{\mathbf{w}}\mathcal l &= -\sum_a \left[ y_a \frac{\nabla_{\mathbf{w}}\lambda(\mathbf{w};\mathbf{x}_a)}{\lambda(\mathbf{w};\mathbf{x}_a)} - \lambda(\mathbf{w};\mathbf{x}_a) \nabla_a \lambda(\mathbf{w};\mathbf{x}_a) \right],\\ &=-\sum_a\left[ y_a \mathbf{x}_a - (1 - y_a)p(y|\mathbf{w};\mathbf{x}_a)) \right]\mathbf{x}_a,\\ &=\sum_a\left[\lambda(\mathbf{w};\mathbf{x}) - y_a\right]\mathbf{x}_a, \end{align} $$ where \(\lambda(\mathbf{w};\mathbf{x})\) is the expected count (2), \(l\) is the negative log-likelihood and \(a\) runs over the set of observations/games.
The update method for the model parameters is then given by $$ \begin{equation} \mathbf{w}^t =\mathbf{w}^{t-1} + \alpha \sum_a \left(y_a - \lambda(\mathbf{w};\mathbf{x})\right). \end{equation} $$
If we have the regressors described in the Poisson Regression section above and restrict ourselves to stochastic gradient descent, this becomes $$ \begin{equation} \mathbf{w}^t =\mathbf{w}^{t-1} + \alpha \left(y_{ij} - \exp(w_i - w_{\underbar{j}})\right) \end{equation} $$ or $$ \begin{equation} \mathbf{r}^t =\mathbf{r}^{t-1} + k \left(y_{ij} - 10^{(r_i - r_{\underbar{j}}) / 2\beta}\right). \end{equation} $$ where \(r_i:=2\beta w_i / \ln 10\) and \(k:=2\alpha\beta / \ln10\). We perform this change of variables to align with Elo conventions. Now (5) is given by $$ \begin{equation} \mathbb{E}\left[y_{ij}|r_1,\cdots r_n\right]= 10^{(r_i - r_{\underbar{j}}) / 2\beta}. \end{equation} $$
With Additional Regressors
Following Additional Regressors, equation (11) become
with (10) modified accordingly.
Example
We provide a more detailed example,
including details of how data can be pre-processed to pass to the rating system in
examples/football.ipynb.
from elo_grad import PoissonEloEstimator, Regressor
# Input DataFrame with sorted index of Unix timestamps
# and columns entity_1_attacking | entity_2_defensive | score | home | intercept
# where score is the number of points/goals scored
# by entity 1 against entity 2 and home is a Boolean flag indicating home advantage.
# intercept is a column with all 1s which represents the mean number of goals
# when entities are evenly matched and there is no home advantage.
intercept_col = "intercept"
home_col = "home"
df = ...
estimator = PoissonEloEstimator(
k_factor=20,
default_init_rating=1200,
entity_cols=("entity_1_attacking", "entity_2_defensive"),
score_col="result",
# Set the initial rating for home advantage to 0
init_ratings={intercept_col: (None, 0), home_col: (None, 0)},
# Set k-factor/step-size to 1 for the both the mean and home advantage regressor
additional_regressors=[Regressor(name=intercept_col, k_factor=1), Regressor(name=home_col, k_factor=1)],
)
# Get expected scores
expected_scores = estimator.predict(df)
# Get final ratings (of form (Unix timestamp, rating)) for home advantage
ratings = estimator.model.ratings[home_col]
Other Approaches
The Elo rating system is the basis for 538's NFL predictions. They use the traditional, logistic regression-based Elo rating system but also incorporate margin of victory. They do this by including a multiplier in the update method. Given the natural equivalence of the Elo rating system with logistic regression, we believe this is not a very natural way to include margin of victory and that using a Poisson-based Elo should be preferred.
References
A Poisson regression-based Elo rating system was implemented here and provided the inspiration for this. Note that the update methods used there are not the same.