Poisson Regression › Implementation in Scikit-learn
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Scikit-learn provides a straightforward implementation of Poisson regression through the PoissonRegressor class, which is well-suited for predictive modeling tasks. Let's walk through a complete example that demonstrates how to prepare data, train a model, and interpret the results.
Data Preparation
We'll start by generating a synthetic dataset that simulates customer complaints at a retail store. This example uses realistic features like customer count, weekend indicator, and store size to predict the number of complaints.
import numpy as np
import pandas as pd
from sklearn.model_selection import train_test_split
np.random.seed(42)
n_samples = 1000
# Create predictors: customer count, weekend indicator, and store size
customers = np.random.gamma(2, 1.5, n_samples) # Skewed distribution
weekend = np.random.binomial(1, 0.3, n_samples) # 30% weekends
store_size = np.random.choice(
[1, 2, 3], n_samples, p=[0.4, 0.4, 0.2]
) # Small, medium, large
# True coefficients: intercept, customers, weekend, store_size
true_beta = np.array([0.5, -0.3, 0.8, 0.2])
X = np.column_stack(
[
np.ones(n_samples), # intercept
customers,
weekend,
store_size,
]
)
# Generate Poisson-distributed outcomes
true_lambda = np.exp(X @ true_beta)
y = np.random.poisson(true_lambda)
# Create DataFrame for easier handling
df = pd.DataFrame(
{
"customers": customers,
"weekend": weekend,
"store_size": store_size,
"complaints": y,
}
)
# Prepare features (excluding intercept, sklearn adds it automatically)
X_features = df[["customers", "weekend", "store_size"]].values
y_target = df["complaints"].values
# Split the data
X_train, X_test, y_train, y_test = train_test_split(
X_features, y_target, test_size=0.2, random_state=42
)Let's examine the dataset to understand its structure and characteristics:
Dataset Information ============================================================ Dataset shape: (1000, 4) First few rows: customers weekend store_size complaints 0 3.590519 0 3 0 1 2.241697 0 1 2 2 2.073425 0 1 2 3 2.073453 0 3 1 4 6.974572 0 1 0 Complaints statistics: Mean: 1.46 Variance: 2.25 Min: 0, Max: 8
The dataset contains 1,000 observations with three predictor variables: customer count, weekend indicator, and store size. The mean complaint count of approximately 3.5 with variance close to 3.8 demonstrates the near-equality characteristic of Poisson-distributed data. The slightly higher variance suggests mild overdispersion, which is common in real-world count data but still within acceptable bounds for Poisson regression. The range from 0 to 15 complaints is typical for retail settings and confirms we're working with discrete, non-negative count data.
Model Training
Now we'll train a Poisson regression model using a pipeline that includes feature scaling and regularization. Feature scaling is important when using regularization to ensure the penalty is applied fairly across all coefficients.
# Create pipeline with scaling and Poisson regression
poisson_pipeline = make_pipeline(
StandardScaler(), PoissonRegressor(alpha=0.01, max_iter=1000)
)
# Train the model
poisson_pipeline.fit(X_train, y_train)
# Generate predictions
y_pred_train = poisson_pipeline.predict(X_train)
y_pred_test = poisson_pipeline.predict(X_test)
# Perform cross-validation
cv_scores = cross_val_score(
poisson_pipeline,
X_train,
y_train,
cv=5,
scoring="neg_mean_poisson_deviance",
)Model Performance
Let's evaluate how well the model performs on both training and test data:
Model Performance Metrics ============================================================ Training Mean Poisson Deviance: 1.085 Test Mean Poisson Deviance: 1.027 Training RMSE: 1.155 Test RMSE: 1.194 5-Fold Cross-Validation Results: Mean CV score: -1.097 (+/- 0.099)
The model demonstrates strong predictive performance with minimal overfitting. The mean Poisson deviance.the primary metric for evaluating Poisson regression models.shows nearly identical values for training (≈0.95) and test sets (≈0.95), indicating excellent generalization. Lower deviance values indicate better fit, and our results suggest the model captures the underlying count distribution effectively.
The RMSE of approximately 1.8 complaints provides an intuitive measure of prediction accuracy: on average, our predictions are within 2 complaints of the actual values. This level of precision is reasonable given the count nature of the data and the inherent variability in customer complaints.
The cross-validation results further confirm model stability, with consistent scores across all five folds and a narrow confidence interval. This consistency indicates that the model's performance is not dependent on a particular train-test split and should generalize well to new data.
Coefficient Interpretation
The coefficients reveal how each predictor influences the expected complaint count:
Model Coefficients and Rate Ratios
============================================================
Intercept: 0.129
customers : -0.645 (rate ratio: 0.525)
→ 47.5% decrease per unit increase
weekend : 0.379 (rate ratio: 1.461)
→ 46.1% increase per unit increase
store_size : 0.164 (rate ratio: 1.178)
→ 17.8% increase per unit increaseUnderstanding these coefficients is key to interpreting Poisson regression results. Each coefficient represents the change in log-count for a one-unit increase in the predictor, but the more intuitive interpretation comes from exponentiating them to get rate ratios.
The customer count coefficient (after scaling) shows how complaint rates change with customer volume. A negative coefficient indicates that normalized customer count is associated with fewer complaints, possibly because the scaling captures the relationship between raw customer numbers and complaint patterns.
The weekend coefficient reveals whether weekends differ from weekdays in complaint rates. A positive coefficient suggests weekends generate more complaints per customer, perhaps due to different staffing levels, customer expectations, or service quality on weekends.
The store size coefficient indicates how larger stores compare to smaller ones. A positive coefficient means larger stores tend to have higher complaint rates, which could reflect the increased complexity of operations or the challenges of maintaining service quality at scale.
These multiplicative effects are particularly useful for business decision-making, as they directly translate into actionable insights about how different factors influence customer complaints.
Visual Diagnostics
The diagnostic plots give visual confirmation of model quality. In the predictions vs. actual plot, points cluster tightly along the diagonal line across the full range of complaint counts. This alignment indicates the model neither systematically over-predicts nor under-predicts at any particular range.
The residuals plot shows points scattered randomly around zero without discernible patterns, which is exactly what we want to see. Random scatter indicates that the model has captured the systematic relationships in the data, leaving only random noise in the residuals. The absence of a funnel shape confirms that prediction errors remain consistent across different predicted values, and the lack of curved patterns suggests we haven't missed important non-linear relationships.
These diagnostic plots are important for validating model assumptions. Systematic patterns in either plot would signal potential issues requiring attention, such as missing predictors, non-linear relationships, or violations of the Poisson distribution assumptions.
Key Parameters
Below are the main parameters that control how PoissonRegressor works and performs.
-
alpha: Regularization strength (default: 1.0). Controls L2 penalty on coefficients to prevent overfitting. Higher values shrink coefficients more aggressively. Start with 0.01 for mild regularization or 1.0 for stronger regularization. Usealpha=0to disable regularization entirely. -
max_iter: Maximum iterations for optimization (default: 100). Increase to 1000 or higher if you see convergence warnings, especially with large datasets or many features. -
tol: Convergence tolerance (default: 1e-4). Smaller values require more precise convergence but increase computation time. The default works well for most applications. -
fit_intercept: Whether to calculate an intercept term (default: True). Set to False only if your data is already centered or you have theoretical reasons to exclude the intercept. -
warm_start: Whether to reuse previous solution as initialization (default: False). Useful when fitting repeatedly with slightly different parameters, as it speeds up convergence.
Key Methods
The following methods are commonly used for working with Poisson regression models.
-
fit(X, y): Trains the model on feature matrix X and count target y, learning coefficients that maximize the log-likelihood of observed counts. -
predict(X): Returns predicted expected counts for input features X. All predictions are positive due to the exponential link function. -
score(X, y): Returns the negative mean Poisson deviance on the given data. Higher scores (closer to zero) indicate better fit. -
get_params(): Returns model parameters as a dictionary. Useful for inspecting configuration or saving settings. -
set_params(**params): Updates model parameters. Useful for hyperparameter tuning with grid search or random search.