The Poisson distribution is a valuable tool for modeling count data, representing the probability of a specific number of events within a fixed timeframe or space, assuming a constant average rate and event independence. However, real-world datasets often deviate from these ideal conditions. This article explores fitting Poisson distributions to diverse datasets in Python, addressing challenges like overdispersion.
Table of Contents
- Basic Poisson Distribution Fitting in Python
- Binned Least Squares Method
- Addressing Overdispersion with the Negative Binomial Distribution
- Conclusion
Basic Poisson Distribution Fitting in Python
Let’s begin with the fundamental process of fitting a Poisson distribution using the scipy.stats library. The core function is poisson.fit(), which estimates λ (lambda), representing the average event rate.
import numpy as np
from scipy.stats import poisson
import matplotlib.pyplot as plt
# Sample data: Number of cars passing a point per minute (100 minutes)
data = np.random.poisson(lam=5, size=100)
# Fit the Poisson distribution
lambda_fit, = poisson.fit(data)
# Display the fitted lambda
print(f"Fitted lambda: {lambda_fit}")
# Prepare for plotting
x = np.arange(0, max(data) + 1)
# Plot histogram and fitted distribution
plt.hist(data, bins=range(max(data) + 2), density=True, alpha=0.6, label='Data')
plt.plot(x, poisson.pmf(x, lambda_fit), 'r-', label=f'Fitted Poisson (λ={lambda_fit:.2f})')
plt.xlabel('Number of Cars')
plt.ylabel('Probability')
plt.legend()
plt.title('Poisson Distribution Fit')
plt.show()
This straightforward approach works well when data closely follows a Poisson distribution. However, real-world data often deviates.
Binned Least Squares Method
While poisson.fit() is convenient, the Binned Least Squares method offers a more robust alternative, particularly with limited data or significant deviations from Poisson assumptions. This method involves binning the data and minimizing the squared differences between observed and expected frequencies. Implementation requires iterative optimization (e.g., using scipy.optimize.minimize) and is beyond this concise example’s scope but is readily found in dedicated statistical packages.
Addressing Overdispersion with the Negative Binomial Distribution
Overdispersion arises when the data variance exceeds its mean, violating a key Poisson assumption (variance equals mean). The negative binomial distribution, accommodating overdispersion, provides a superior fit in such cases.
from scipy.stats import nbinom
# Example of overdispersed data
overdispersed_data = np.random.negative_binomial(n=2, p=0.5, size=100)
# Fit the Negative Binomial distribution
n_fit, p_fit = nbinom.fit(overdispersed_data)
# Display fitted parameters
print(f"Fitted n: {n_fit}")
print(f"Fitted p: {p_fit}")
# Prepare for plotting
x = np.arange(0, max(overdispersed_data) + 1)
# Plot histogram and fitted distribution
plt.hist(overdispersed_data, bins=range(max(overdispersed_data) + 2), density=True, alpha=0.6, label='Data')
plt.plot(x, nbinom.pmf(x, n_fit, p_fit), 'r-', label=f'Fitted Negative Binomial (n={n_fit:.2f}, p={p_fit:.2f})')
plt.xlabel('Number of Events')
plt.ylabel('Probability')
plt.legend()
plt.title('Negative Binomial Fit for Overdispersed Data')
plt.show()
This demonstrates how the negative binomial distribution effectively captures the characteristics of overdispersed data, offering a more accurate model than forcing a Poisson fit.
Conclusion
Effective Poisson distribution fitting requires careful data analysis. While poisson.fit() provides a simple starting point, recognizing and addressing overdispersion using the negative binomial distribution is crucial for accurate count data modeling. The choice of distribution hinges on the dataset’s specific characteristics. Visual inspection of the fit using plots ensures the chosen distribution accurately represents the data.