Poisson Regression

Poisson regression is a statistical technique used to model and analyze count data, where the outcome variable represents the number of times an event occurs in a fixed interval of time, space, or any other dimension. It is most appropriate when the values of the response variable are non-negative whole numbers (0, 1, 2, ...) and the average rate at which events occur is constant.

Poisson regression is used when you want to predict things that are counts, like "how many" or "how much," and these counts can't be negative.
Number of customers arriving at a store
Number of clicks on a website
Number of defects in a batch

Key Assumptions of Poisson Regression

The response variable is a count such as number of visits, accidents, or purchases.
Counts follow a Poisson distribution.
The mean and variance of the distribution are equal.
Observations are independent of each other.
Events occur at a constant average rate.

Mathematical Formulation of Poisson Regression

In Poisson regression, the output Y is assumed to follow a Poisson distribution:

P(Y = y) = \frac{e^{-\lambda} \lambda^{y}}{y!}

Where:

is the count variable
y is a particular count
lambda is the expected rate of occurrence
e is Euler’s number (approximately 2.718)

Instead of modeling Y directly, we model the log of the expected value:

\log(\lambda) = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \dots + \beta_k x_k

Or, in exponential form: \lambda = e^{\beta_0 + \beta_1 x_1 + \beta_2 x_2 + \cdots + \beta_k x_k}

Where:

\lambda: The expected count
X_i: Independent variables
\beta_i: Coefficients to be learned

When to Use Poisson Regression

Poisson Regression is appropriate when:

The dependent variable is a count (e.g., 0, 1, 2, …)
Counts are not negative.
The counts follow a Poisson distribution (i.e., mean ≈ variance).
The observations are independent.

Implementation of Poisson Regression in Python

Step 1: Import Required Libraries

We start by importing NumPy for data, Statsmodels for building the Poisson regression model, and Matplotlib for plotting.

Python

import numpy as np
import statsmodels.api as sm
import matplotlib.pyplot as plt

Step 2: Create Sample Data

We generate input values (x) and simulate corresponding count data (y) that follow a Poisson distribution, where counts increase with x.

Python

np.random.seed(42)
x = np.linspace(0, 10, 100)
X = sm.add_constant(x)
lambda_ = np.exp(0.5 + 0.3 * x)
y = np.random.poisson(lambda_)

Step 3: Fit the Poisson Regression Model

We use the GLM function with a Poisson family to build and fit the model.

Python

model = sm.GLM(y, X, family=sm.families.Poisson())
results = model.fit()

Step 4: View Model Summary

This gives us details about the model, including the learned coefficients and model performance.

Python

print(results.summary())

Output:

Generalized_linear_model_regression_result — Generalized Linear Model Regression Results

Step 5: Predict and Plot the Results

We use the model to predict counts and then plot the actual data vs. the fitted curve.

Python

y_pred = results.predict(X)
plt.scatter(x, y, color='orange', label='Observed')
plt.plot(x, y_pred, color='red', label='Poisson Fit')
plt.xlabel('x')
plt.ylabel('Count (y)')
plt.title('Poisson Regression')
plt.legend()
plt.show()

Output: