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Table of contents
1.
Introduction
2.
What are Residuals?
3.
Residual Analysis
3.1.
1. Why Perform Residual Analysis?
3.2.
2. Assumptions in Residual Analysis
3.3.
3. Types of Residual Plots
4.
Practical Example: Residual Analysis in Python
5.
Frequently Asked Questions
5.1.
What if residuals show a pattern?
5.2.
Is residual analysis applicable only to linear regression?
5.3.
Can residual analysis detect outliers?
6.
Conclusion
Last Updated: Mar 27, 2024
Medium

Residual Analysis

Author Gunjan Batra
1 upvote
Master Python: Predicting weather forecasts
Speaker
Ashwin Goyal
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Introduction

Residual analysis is a fundamental aspect of regression analysis and statistical modeling. It helps us understand how well our model fits the data by examining the differences between the actual and predicted values. These differences are known as "residuals." 

Residual Analysis

This article will explore residual analysis, its importance, and how to perform it with practical examples.

What are Residuals?

Residuals are the differences between the observed values (actual) and the values that a statistical model predicts. Mathematically, a residual can be expressed as:

ri=yi-y^i

​where 

  • ri is the residual, 
     
  • yi is the observed value, and 
     
  • y^i is the predicted value.
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Residual Analysis

1. Why Perform Residual Analysis?

Residual analysis helps in detecting the presence of patterns that the model hasn’t captured. If the residuals exhibit a pattern, it is an indication that the model may have room for improvement.

2. Assumptions in Residual Analysis

To perform residual analysis, certain assumptions must be met:

  • Linearity: The relationship between predictors and the response variable should be linear.
     
  • Independence: Observations are independent of each other.
     
  • Homoscedasticity: The residuals have constant variance.
     
  • Normality: The residuals follow a normal distribution.

3. Types of Residual Plots

a. Standardized Residuals vs. Fitted Values

This plot helps check the assumptions of equal variance and linearity.
 

b. Normal Q-Q Plot

This plot helps in verifying if the residuals follow a normal distribution.

c. Scale-Location Plot

This plot helps in checking the assumption of equal variance across the data.

Practical Example: Residual Analysis in Python

Here's a step-by-step guide to performing residual analysis on a linear regression model using Python.

Step 1: Import Libraries

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

Step 2: Create Data and Fit a Model

# Generate synthetic data
np.random.seed(42)
X = np.random.normal(0, 1, 100)
y = 2 * X + np.random.normal(0, 1, 100)
# Fit a linear regression model
X = sm.add_constant(X)
model = sm.OLS(y, X).fit()

Step 3: Plot Residuals vs. Fitted Values

sns.residplot(x=model.fittedvalues, y=model.resid, lowess=True, line_kws={'color': 'red', 'lw': 1})
plt.xlabel('Fitted values')
plt.ylabel('Residuals')
plt.title('Residuals vs Fitted Values')
plt.show()

Step 4: Plot Q-Q Plot

sm.qqplot(model.resid, line='s')
plt.show()

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Frequently Asked Questions

What if residuals show a pattern?

If residuals show a pattern, it indicates that the model has not captured some underlying structure in the data, requiring further investigation.

Is residual analysis applicable only to linear regression?

No, residual analysis can be applied to various types of regression models, including linear, logistic, and polynomial regression.

Can residual analysis detect outliers?

Yes, residual plots can help identify outliers or influential observations that might be affecting the model fit.

Conclusion

Residual analysis is an essential step in understanding how well a statistical model fits the data. By plotting and analyzing residuals, we can diagnose various issues like non-linearity, unequal variances, and outliers. Using residual plots, we can visually inspect these aspects, refine our models, and make them more accurate and reliable. Residual analysis, therefore, stands as a powerful tool in the toolkit of every data analyst and statistician, bridging the gap between theoretical assumptions and real-world data.

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