Guide to Polynomial Regression in Machine Learning: Implementation and Benefits

Linear regression is a powerful tool for modeling relationships between variables, but it falls short when the data shows non-linear patterns. Polynomial regression, using methods like the Least Squares method, introduces higher-degree terms to capture curves in data. 

This makes it ideal for cases such as predicting housing prices, stock market trends, and sales data, where relationships between variables are nonlinear.

This blog will cover how polynomial regression works, its advantages over linear regression, and key implementation tips for real-world applications.

What Is Polynomial Regression in Machine Learning? Key Benefits and Applications

Polynomial regression in machine learning is used to model relationships between variables when those relationships are not linear. By adding higher-degree terms (such as x^2 and x^), it can capture curvatures in data that linear regression cannot. 

This makes it valuable for cases where data shows acceleration, deceleration, or other forms of non-linear trends. 

Looking to build your machine learning expertise with polynomial regression? Discover upGrad’s leading programs, designed to help you master advanced regression techniques and handle real-world ML challenges: 

• Executive Diploma in Machine Learning and AI
• Master’s Degree in Artificial Intelligence and Data Science
• Master of Science in Machine Learning & AI

Now let’s explore some of the top benefits of using polynomial regression: 

Also Read: 18 Types of Regression in Machine Learning You Should Know [Explained With Examples]

Applications of Polynomial Regression:

• Housing Price Prediction: Polynomial regression can model the non-linear effects of house size, age, and location on pricing.
• Stock Market Forecasting: Used to capture trends in stock prices that exhibit acceleration or deceleration.
• Sales Forecasting: Polynomial regression helps predict sales when growth rates are non-linear and change over time.
• Medical Data Analysis: In studies where the effect of a drug increases or decreases over time, polynomial regression captures these non-linear trends.
• Agricultural Yield Prediction: Modeling crop yields as a function of time, temperature, and rainfall, where relationships are non-linear.

Also Read: 25 Powerful Machine Learning Applications Driving Innovation in 2025

We will now examine how polynomial regression is applied to datasets for a better understanding of the data. 

How Does Polynomial Regression Work

Polynomial regression extends linear regression by adding higher-degree terms to model non-linear relationships between variables.

It uses a polynomial equation to fit the data, capturing patterns that linear regression cannot represent. The degree of the polynomial determines the complexity of the model.

Formula:
The general formula for polynomial regression is:

where,

• y is the dependent variable (eg, salary)
• x is the independent variable (eg, experience)
• β0, β1, β2,....,βn are the coefficients (weights) to be optimized
• n is the degree of the polynomial

Dataset:

Consider the following dataset for experience and salary:

We aim to predict the salary based on the years of experience using polynomial regression with degree 2.

Example:
For a degree-2 polynomial regression, the formula becomes:

We will now calculate the coefficient β0, β1, and β2 using the least squares method.

Steps for Calculation:

Before implementing polynomial regression, it's essential to transform the data by adding higher-degree terms. This transformation allows the model to capture the non-linear relationships that linear regression cannot.

1. Transform the Data:

For polynomial regression, we first need to create new features by adding the square of the explanatory variable.

2. Solve for Coefficients using Least Squares:

In polynomial regression, the coefficients β0, β1, and β2 are found using the following system of equations, which minimizes the sum of squared error:

where,

• X is the matrix of feature (Experience and Expereience^2)
• Y is the vector of salary value
• X^T is the transpose of matrix X

Step-by-Step Calculation (simplified for illustration):

We can solve for the coefficients using a matrix inversion or numerical methods in Python(using libraries like NumPy or scikit-learn). 

Once the coefficients are calculated, we can substitute them into the polynomial formula to predict salaries for any given experience.

Prediction:

Using the obtained coefficients (say β0=30,000, β1=5,000, β2=2,000), the salary prediction for an individual with 6 years of experience would be:

Salary = 30,000 + 5,000 x 6 + 2,000x6x6 = 1,32,000

Also Read: Difference Between Linear and Logistic Regression: A Comprehensive Guide for Beginners in 2025

As you've seen, polynomial regression is a powerful tool for modeling nonlinear relationships. Now let's move on to how you can implement polynomial regression in Python to solve real-world problems.

Implementation of Polynomial Regression in Python

Polynomial regression is ideal when the relationship between variables is non-linear. In this case study, we’ll model the relationship between years of experience and salary, where the trend is likely non-linear. 

We will first implement linear regression, then switch to polynomial regression to better capture the non-linear nature of the data. 

By comparing the two models, we can see how polynomial regression enhances the model's ability to predict different types of data.

Data Setup and Visualization

Before applying polynomial regression, we visualize the data to understand the relationship between experience and salary. 

Linear regression is used as a baseline, fitting a straight line to the data, allowing us to evaluate its performance before moving to the more flexible polynomial regression model.

import numpy as np
import matplotlib.pyplot as plt
from sklearn.linear_model import LinearRegression
from sklearn.preprocessing import PolynomialFeatures

# Define the data for experience and salary
experience = np.array([[1], [3], [5], [7], [10]])  # Experience in years
salary_inr = np.array([4000000, 5000000, 6000000, 7000000, 9000000])  # Salary in INR

# Linear Regression Model
linear_regressor = LinearRegression()
linear_regressor.fit(experience, salary_inr)

# Predicting with Linear Regression
linear_pred = linear_regressor.predict(experience)

# Plotting the Linear Regression results
plt.scatter(experience, salary_inr, color='blue')
plt.plot(experience, linear_pred, color='red', label='Linear Regression')
plt.title('Linear Regression: Experience vs Salary (INR)')
plt.xlabel('Years of Experience')
plt.ylabel('Salary (INR)')
plt.legend()
plt.show()

Output: 

Explanation:

The linear regression model fits a straight line to the data. From the plot, we can see how the linear regression model may not capture the increasing rate of salary effectively as experience increases. This suggests that a polynomial regression might fit the data better.

Also Read: Algorithm Complexity and Data Structure: Types of Time Complexity

Introducing Polynomial Regression in Python 

Polynomial regression introduces higher-degree terms to the model to capture non-linear trends. Here, we’ll implement a polynomial regression model to fit the data better and potentially improve predictive analysis of the model.

Code: 

# Polynomial Regression Model (degree 2)
poly = PolynomialFeatures(degree=2)  # Use degree 2 polynomial
experience_poly = poly.fit_transform(experience)

# Fit the polynomial regression model
poly_regressor = LinearRegression()
poly_regressor.fit(experience_poly, salary_inr)

# Predicting with Polynomial Regression
poly_pred = poly_regressor.predict(experience_poly)

# Plotting the Polynomial Regression results
plt.scatter(experience, salary_inr, color='blue')
plt.plot(experience, poly_pred, color='green', label='Polynomial Regression')
plt.title('Polynomial Regression: Experience vs Salary (INR)')
plt.xlabel('Years of Experience')
plt.ylabel('Salary (INR)')
plt.legend()
plt.show()

Output:

Explanation:

In this example, polynomial regression is used to fit a curve to the data. The model includes a squared term for the experience feature, which allows it to capture the curvature in the relationship between experience and salary. 

The polynomial regression curve provides a better fit, as seen in the plot, compared to the linear regression model.

Also Read: Multiple Linear Regression in R: A Complete Guide

Polynomial regression offers significant benefits, but it also presents challenges, such as overfitting and model complexity. Let’s explore these issues and the strategies to address them.

Challenges and Solutions for Using Polynomial Regression

Polynomial regression, while powerful for capturing non-linear relationships, introduces challenges that can affect model accuracy and generalization. Key issues include overfitting,model complexity, and computational challenges. 

Addressing these challenges requires careful handling of model complexity, data transformations, and validation techniques to ensure the model is both accurate and reliable.

1. Overfitting: Risk of Excessive Model Complexity

Overfitting occurs when a polynomial regression model is too complex, capturing not only the underlying pattern in the data but also the noise or random fluctuations. 

This results in a model that performs well on training data but poorly on new, unseen data.

Solutions:

• Limit Polynomial Degree: Keep the polynomial degree low (e.g., degree 2 or 3) to avoid overfitting.
• Cross-Validation: Utilize k-fold cross-validation to assess model performance across various subsets of the data.
• Regularization: Apply Lasso (L1) or Ridge (L2) regularization to penalize large coefficients, reducing model complexity and improving generalization.

2. Model Interpretability

As the degree of the polynomial increases, the model becomes more difficult to interpret. With higher-degree polynomials, the relationship between input variables and the target variable becomes less transparent, making it harder to explain the model’s behavior in simple terms.

Solutions:

• Use Lower-Degree Polynomials: Opt for lower-degree polynomials (degree 2 or 3) to maintain a balance between fit and interpretability.
• Visualization Tools: Utilize tools such as Partial Dependence Plots (PDPs) or feature importance rankings to interpret the influence of each feature on the output.
• Model Simplicity: If high complexity is required, consider using simpler models with higher interpretability, such as decision trees or linear models with feature engineering.

3. Computational Challenges

Polynomial regression models, especially those with high degrees, require more computational power for fitting and prediction. 

As the degree of the polynomial increases, the model becomes more computationally expensive, which can be a limitation for large datasets or real-time predictions.

Solutions:

• Dimensionality Reduction: Apply techniques like Principal Component Analysis (PCA) to reduce the number of features before applying polynomial transformations.
• Efficient Optimization: Use more efficient optimization methods like stochastic gradient descent (SGD) for large datasets.
• Parallelization: Distribute computations across multiple processors or machines to speed up model training and prediction.

Also Read: 15 Dimensionality Reduction in Machine Learning Techniques To Try!

4. Risk of Multicollinearity

Polynomial regression introduces the risk of multicollinearity, where higher-degree terms (e.g., x^2, x^3) become highly correlated with the original features. 

This makes it difficult to estimate the coefficients reliably, leading to inflated standard errors and unstable model estimates.

Solutions:

• Centering Data: Subtract the mean from features before applying polynomial transformations to reduce correlations between features.
• Regularization: Apply Ridge regression, which penalizes large coefficients, reducing multicollinearity and stabilizing the model.
• Principal Component Regression: Use principal component regression (PCR) to reduce dimensionality and mitigate multicollinearity.

5. Handling Sparse or Noisy Data

Polynomial regression models are sensitive to noisy or sparse data. When there are many outliers or the data is sparse, polynomial models can produce erratic predictions, especially with higher-degree polynomials.

Solutions:

• Outlier Detection and Removal: Use techniques like IQR (Interquartile Range) or Z-table to identify and remove outliers before training the model.
• Robust Regression: Use robust regression methods, which down-weight the influence of outliers, ensuring the model is not overly affected by noise.
• Data Augmentation: In sparse datasets, augment data with synthetic samples or use bootstrapping to improve model performance

Also Read: Top Machine Learning Algorithms - Real World Applications & Career Insights [Infographic]

With the right strategies in place, mastering polynomial regression becomes achievable. upGrad offers structured learning and expert mentorship to support your machine learning journey.

How can upGrad Support you in Your ML Journey? 

Polynomial regression models non-linear relationships, like predicting real estate prices, by capturing complex patterns in data. To learn, start with the theory of linear and polynomial regression. 

Practice with datasets like housing prices and sales, and evaluate models using MSE. Use cross-validation to avoid overfitting, and compare polynomial regression with models like decision trees to deepen your understanding.

A common challenge in machine learning is the lack of structured guidance, which makes it difficult to understand complex algorithms and avoid pitfalls such as overfitting. 

upGrad’s courses offer structured learning, from basics to advanced models like polynomial regression, with hands-on projects and expert mentorship to guide your progress effectively.

Some additional courses include: 

• DBA in Emerging Technologies with Concentration in Generative AI
• Master of Science in Data Science
• Gen AI Mastery Certificate for Data Analysis

upGrad offers tailored guidance and offline centers for in-person interaction, ensuring that you can bridge the gap between theory and practical application. Our approach ensures you gain the confidence and skills to excel in machine learning.

Expand your expertise with the best resources available. Browse the programs below to find your ideal fit in Best Machine Learning and AI Courses Online.

Best Machine Learning and AI Courses Online

Discover in-demand Machine Learning skills to expand your expertise. Explore the programs below to find the perfect fit for your goals.

In-demand Machine Learning Skills

Discover popular AI and ML blogs and free courses to deepen your expertise. Explore the programs below to find your perfect fit.

Popular AI and ML Blogs & Free Courses

Reference:
https://medium.com/geekculture/polynomial-regression-can-improve-accuracy-of-a-linear-regression-model-dfaa0d062a61