Combined Fit of Two Distributions

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What will you learn?

In this tutorial, you will master the art of combining fits of two distributions in Python using advanced statistical methods. By the end, you’ll be equipped to analyze complex datasets with mixed patterns and draw valuable insights from them.

Introduction to the Problem and Solution

When faced with data that appears to stem from a blend of two distinct distributions, leveraging techniques like Maximum Likelihood Estimation (MLE) and curve fitting algorithms becomes crucial. By accurately fitting each distribution and merging them effectively, you can unravel underlying trends within your data, empowering you to make well-informed decisions based on these revelations.

To tackle this challenge effectively, we will delve into generating synthetic data by amalgamating samples from two normal distributions, fitting individual distributions through MLE parameters, combining these fits based on specified weights or proportions, and visually representing the results for better comprehension.

Code

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

# Generate sample data by combining two normal distributions with weights 0.3 and 0.7 respectively
np.random.seed(0)
data = np.concatenate([np.random.normal(loc=5, scale=1, size=300),
                       np.random.normal(loc=10, scale=2, size=700)])

# Fit individual distributions using MLE
params1 = norm.fit(data[:300])
params2 = norm.fit(data[300:])

# Plotting individual distributions for visualization
x = np.linspace(0, 20, 1000)
plt.hist(data, bins=30, density=True)

plt.plot(x,norm.pdf(x,*params1), label='Distribution 1')
plt.plot(x,norm.pdf(x,*params2), label='Distribution 2')

plt.legend()
plt.show()

# Combine the fits of both distributions based on their weights or proportions 
combined_fit = lambda x,alpha: alpha*norm.pdf(x,*params1) + (1-alpha)*norm.pdf(x,*params2)

# Plotting the combined fit function along with original data histogram for comparison
plt.hist(data,bins=30,density=True)
plt.plot(x,norm.pdf(x,*params1),label='Distribution 1')
plt.plot(x,norm.pdf(x,*params2),label='Distribution 2')
plt.plot(x,[combined_fit(val,alpha=0.5) for val in x],label='Combined Fit')

plt.legend()
plt.show()

# Copyright PHD

(Code snippet courtesy of PythonHelpDesk.com)

Explanation

Here’s a breakdown of what happens in the code: – Data Generation: Synthetic data is created by blending samples from two normal distributions. – Fitting Distributions: Each distribution is fitted individually using MLE parameters. – Combining Distributions: A custom function is defined to merge both fitted distributions based on specified weights. – Visualization: Plots are generated to visualize individual fits alongside the combined fit compared with original data.

    How do I determine which type of distribution my data follows?

    You can utilize visualizations like histograms or probability plots and perform hypothesis tests such as Kolmogorov-Smirnov test or QQ plot analysis.

    Can I combine more than two distributions using similar methods?

    Certainly! You can extend this approach by fitting multiple distributions and creating a weighted combination accordingly.

    Is it essential for my sample sizes from different populations be equal?

    While not mandatory, having somewhat balanced sample sizes often leads to better estimation results when combining multiple fits.

    Should I always use MLE for fitting individual components?

    MLE is commonly preferred due to its desirable statistical properties; however other methods like Method of Moments (MoM) can also be employed depending on the scenario.

    How do I interpret the weights assigned while combining fits?

    The weights signify the relative importance given to each component distribution in forming the overall combined fit – typically summing up to one.

    Can I automate parameter estimation process instead of manually inputting values?

    Absolutely! You can employ optimization techniques like scipy.optimize.minimize() paired with suitable cost functions tailored towards your specific requirements.

    What if my data exhibits skewness or heavy tails requiring non-normal models?

    Consider utilizing skewed or heavy-tailed distribution models such as skew-normal or t-distribution within a similar framework discussed above.

    Conclusion

    By mastering how to combine fits of multiple probability distributions in Python, you gain a holistic understanding of intricate datasets showcasing mixed patterns. This knowledge equips you with valuable insights necessary for effective decision-making regarding modeling strategies and parameter selections.

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