5 Best Ways to Plot Gamma Distribution with Alpha and Beta Parameters in Python Using Matplotlib

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πŸ’‘ Problem Formulation: Visualizing the Gamma distribution is essential for statisticians and data scientists working with datasets where timing or the interval until an event occurs is important. Gamma distributions are described by two parameters, alpha (shape) and beta (scale), which dictate their shape and scale. This article demonstrates how Python’s Matplotlib library can be used to plot Gamma distributions with varying alpha and beta parameters. For example, if given alpha=2.0 and beta=1.0, one should be able to create a visualization representing the corresponding Gamma distribution.

Method 1: Using scipy.stats.gamma

This method involves using the gamma class from the scipy.stats module to generate random variables following a Gamma distribution, which are then plotted with Matplotlib. It enables greater control over the distribution’s properties.

Here’s an example:

import numpy as np
import matplotlib.pyplot as plt
from scipy.stats import gamma

alpha, beta = 2.0, 1.0
x = np.linspace(0, 20, 1000)
pdf = gamma.pdf(x, a=alpha, scale=1/beta)

plt.plot(x, pdf, 'r-', lw=2, label=f'Gamma PDF')
plt.title('Gamma Distribution')
plt.legend()
plt.show()

The output is a plot displaying the Gamma distribution’s probability density function (PDF) with the specified alpha and beta values.

This snippet creates an array of x values and computes the corresponding probability density function values for a Gamma distribution with alpha=2.0 and beta=1.0. A plot is then generated using Matplotlib, showcasing the PDF curve with proper labeling and a legend.

Method 2: Directly using matplotlib.pyplot.hist

Matplotlib offers a direct way to plot histograms of samples. This method uses random gamma-distributed samples generated by Numpy’s np.random.gamma function and plots them using the plt.hist method, giving a histogram representation of the Gamma distribution.

Here’s an example:

import numpy as np
import matplotlib.pyplot as plt

alpha, beta = 2.0, 1.0
samples = np.random.gamma(alpha, beta, 1000)

plt.hist(samples, bins=30, density=True, alpha=0.6, color='g', label='Sample Histogram')
plt.title('Sampled Gamma Distribution')
plt.legend()
plt.show()

The output is a histogram approximating the Gamma distribution based on the sampled data.

Using 1000 samples drawn from the Gamma distribution defined by the given alpha and beta parameters, this code creates a histogram that is normalized to form a probability density, thereby giving a visual approximation of the Gamma distribution’s shape.

Method 3: Overlaid Histogram and PDF Plot

Combining methods 1 and 2, this approach overlays the histogram of random samples from the Gamma distribution with the theoretical probability density function (PDF), allowing for comparison of empirical data with the theoretical curve.

Here’s an example:

import numpy as np
import matplotlib.pyplot as plt
from scipy.stats import gamma

# Parameters and sample generation
alpha, beta = 2.0, 1.0
samples = np.random.gamma(alpha, beta, 1000)

# PDF for theoretical curve
x = np.linspace(0, max(samples), 1000)
pdf = gamma.pdf(x, a=alpha, scale=1/beta)

# Plot histogram of samples
plt.hist(samples, bins=30, density=True, alpha=0.4, color='blue', label='Sample Histogram')

# Plot theoretical PDF
plt.plot(x, pdf, 'r-', lw=2, label='Theoretical PDF')

# Final plot adjustments
plt.title('Empirical vs Theoretical Gamma Distribution')
plt.legend()
plt.show()

The output is a plot that overlays the histogram of the sampled data with the theoretical PDF curve.

This snippet combines the empirical histogram from the sample data with the theoretical gamma PDF, facilitating a visual comparison to assess the fit between the sampled data and the statistical model.

Method 4: Plotting Multiple Gamma Distributions

This method shows how to plot multiple Gamma distributions with different alpha and beta parameters on the same axes for comparison. It’s particularly useful for demonstrating the effect of these parameters on the shape of the distribution.

Here’s an example:

import numpy as np
import matplotlib.pyplot as plt
from scipy.stats import gamma

# Define a range of x values
x = np.linspace(0, 20, 1000)

# Plot multiple gamma distributions
params = [(1.0, 1.0), (2.0, 1.0), (3.0, 1.0), (2.0, 2.0)]
for alpha, beta in params:
    plt.plot(x, gamma.pdf(x, a=alpha, scale=1/beta), label=f'Ξ±={alpha}, Ξ²={beta}')

# Plotting details
plt.title('Multiple Gamma Distributions')
plt.xlabel('Value')
plt.ylabel('Probability Density')
plt.legend()
plt.show()

The output includes multiple curves, each representing a different Gamma distribution based on the various alpha and beta parameter pairs.

By iterating over a list of alpha and beta pairs and plotting the corresponding gamma PDF for each, this code effectively illustrates the impact that varying these parameters has on the distribution’s shape. As the plot is generated, each curve is labeled with its parameter values for easy identification.

Bonus One-Liner Method 5: Using seaborn.distplot

Seaborn, a statistical data visualization library built on top of Matplotlib, provides a high-level interface for drawing attractive and informative statistical graphics. The distplot function can take a dataset and fit a Gamma distribution automatically, simplifying the whole process.

Here’s an example:

import seaborn as sns
import numpy as np

# Generate gamma distributed samples
alpha, beta = 2.0, 1.0
samples = np.random.gamma(alpha, beta, 1000)

# Use seaborn to plot a histogram with the Gamma kernel density estimate
sns.distplot(samples, kde_kws={"label": "Gamma KDE"}, hist_kws={"label": "Histogram"})

The output is a histogram overlaid with a kernel density estimate (KDE) fitted to a Gamma distribution.

In this one-liner approach, Seaborn’s distplot function is provided with the generated samples from a Gamma distribution and uses its built-in KDE mechanism to estimate and plot the distribution’s density curve over a histogram of the data, offering a quick and visually pleasing result.

Summary/Discussion

  • Method 1: scipy.stats.gamma. Offers precise control over the PDF plot with the flexibility to specify the distribution’s parameters. However, it requires knowledge of the underlying statistical functions.
  • Method 2: Matplotlib Histogram. Good for visualizing actual data distributions through histogram representation. Less theoretical and more practical but relies on sufficient sample data.
  • Method 3: Overlaid Histogram and PDF Plot. Combines empirical data visualization with theoretical models, allowing for comparison between them. It serves both demonstration and analysis purposes but can be busy if overused.
  • Method 4: Plotting Multiple Distributions. Allows easy comparison between multiple Gamma distributions. It’s quite informative for educational purposes but can become cluttered if too many distributions are plotted.
  • Bonus One-Liner Method 5: Seaborn’s distplot. Provides a simple way to plot a distribution with minimal configuration. Suitable for quick exploratory data analysis but offers less customization than Matplotlib.