Fitting Empirical Distribution to Theoretical Ones with Scipy

Introduction

You have a large dataset of integer values and aim to calculate p-values, the probability of encountering higher values. To determine these probabilities, you seek a theoretical distribution that approximates your data distribution. This article explores how to achieve this using Python's Scipy package.

Fitting Distributions

Scipy's scipy.stats module provides an extensive collection of continuous and discrete probability distributions. Each distribution has its own parameters that characterize its shape and behavior. The goal is to find the distribution that best fits your empirical data based on a goodness-of-fit test.

Goodness-of-Fit Tests

Goodness-of-fit tests measure the discrepancy between an empirical distribution and a theoretical distribution. Common tests include the Kolmogorov-Smirnov test and the chi-square test. Scipy offers functions to perform these tests, allowing you to evaluate the fitness of candidate distributions.

Sum of Squared Error (SSE)

One approach is to utilize the Sum of Squared Error (SSE) as a goodness-of-fit measure. SSE calculates the squared difference between the empirical and theoretical probability density functions. The distribution with the minimal SSE is considered the best fit.

Python Implementation

The following Python code demonstrates how to fit your data to theoretical distributions using SSE:

<br>import pandas as pd<br>import numpy as np<br>import scipy.stats as st<br>import matplotlib.pyplot as plt</p>
<p>data = pd.read_csv('data.csv') # Replace with your data file</p>
<h1>Histogram of the data</h1>
<p>plt.hist(data, bins=50)<br>plt.show()</p>
<h1>Candidate distributions</h1>
<p>dist_names = ['norm', 'expon', 'gamma', 'beta']</p>
<h1>Fit each distribution and calculate SSE</h1>
<p>best_distribution = None<br>min_sse = np.inf<br>for dist in dist_names:</p>
<div class="code" style="position:relative; padding:0px; margin:0px;"><pre class="brush:php;toolbar:false">dist = getattr(st, dist)
params = dist.fit(data)

# Calculate SSE
sse = np.mean((dist.pdf(data, *params) - np.histogram(data, bins=50, density=True)[0]) ** 2)

# Update the best distribution if necessary
if sse < min_sse:
    min_sse = sse
    best_distribution = dist, params

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Print the distribution parameters for the best fit

This code provides the name of the best-fitting distribution along with its estimated parameters. You can use these parameters to calculate p-values and evaluate the distribution's goodness of fit.

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