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When working with machine learning models, you've probably encountered linear regression as one of the simplest and most fundamental algorithms. In one analysis of real-world software engineering datasets, which often involve time, size, or defect counts, at least 50% contained non-linear relationships.
But what happens when the data you’re trying to model doesn’t quite fit a straight line? That’s where polynomial regression comes in.
This blog explores when and why you should use polynomial regression in machine learning, helping you decide whether it's the right tool for your task.
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What is Polynomial Regression?
At its core, polynomial regression is an extension of linear regression. While linear regression attempts to fit a straight line (a linear relationship) to the data, polynomial regression allows for a curved line, enabling the model to capture more complex relationships.
Instead of fitting a model of the form:
y = β₀ + β₁x
You’re fitting a model that uses a polynomial regression formula:
y = β₀ + β₁x + β₂x² + β₃x³ + ... + βₙxⁿ
Here, the degree of the polynomial (n) determines the flexibility of your curve. A higher degree can capture more intricate patterns in the data, but as you'll see later, this comes with its own trade-offs.
When Should You Use Polynomial Regression?
Here are some polynomial regression examples that show how it becomes a compelling choice:
1. The Relationship Between Variables is Non-Linear
If you plot your data and notice that the relationship between independent and dependent variables forms a curve (e.g. parabolic, exponential-like, etc.), linear regression simply won’t capture the trend accurately. In these cases, polynomial regression helps model the underlying structure.
For example, if you're predicting the growth of a plant based on sunlight exposure, a small amount of sunlight might help, more sunlight might help even more, but too much can damage the plant.
2. You're Looking to Improve Model Performance
Sometimes a linear model just doesn’t give you satisfactory results in terms of metrics like Mean Squared Error (MSE) or R² score. By adding polynomial terms, you can capture more variance in the data and potentially improve performance.
However, you should use cross-validation to ensure that you're not just overfitting to your training set.
3. Residual Plots Suggest Non-Linearity
One good practice in regression analysis is to examine residual plots, which represent the differences between the predicted and actual values. If the residuals show a pattern (e.g. a curve), it indicates that your model isn’t capturing all the underlying trends, and adding polynomial terms might help.
4. You're Working with a Small Dataset
Because polynomial regression can model complex relationships without needing vast amounts of data, it may be a good choice when your dataset is small. You can achieve reasonable results without needing deep learning models or more sophisticated techniques.
That said, you still need to be cautious. A high-degree polynomial on a small dataset can overfit quite easily.
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Why Use Polynomial Regression?
You now know when to use linear polynomial regression. But why should you consider it over other techniques like decision trees or neural networks?
1. Simplicity and Interpretability
Polynomial regression remains a parametric model, meaning it has a clear, analytical form. If you value interpretability, polynomial regression offers a middle ground between linear models and more opaque algorithms, such as neural networks.
You can clearly see the impact of each term (e.g. x², x³) on the output, and this can be useful for reporting and diagnostics.
2. Low Computational Cost
Compared to more complex models, polynomial regression is computationally inexpensive. It doesn’t require huge processing power, making it ideal for quick prototypes, experiments, or edge deployments where resources are limited.
3. Smoothness
Polynomial regression creates smooth, continuous curves. This can be particularly useful when modelling phenomena where abrupt jumps are unrealistic or undesirable, for example, predicting physical behaviours like velocity, pressure, or temperature.
4. Foundation for Feature Engineering
Understanding polynomial regression helps you grasp how to engineer polynomial features for use in other models manually. For example, even when using algorithms like Support Vector Machines (SVMs) or Gradient Boosting, creating polynomial features can improve model performance.
Things to Watch Out For
Despite its advantages, polynomial regression isn’t without pitfalls. It’s easy to misuse the model if you’re not careful.
1. Overfitting
The biggest risk with a polynomial regression equation is overfitting. As you increase the degree of the polynomial, the model becomes more flexible, but at the cost of generalisation. You may find that a high-degree polynomial fits your training data perfectly but performs poorly on unseen data.
To avoid this, always validate your model using cross-validation techniques and keep an eye on both training and validation errors. A gap between the two often indicates overfitting.
2. Extrapolation Issues
Polynomial regression can behave unpredictably outside the range of your data. High-degree polynomials often “explode” at the edges, resulting in bizarre predictions. So if your model needs to extrapolate beyond the observed data, polynomial regression might not be reliable.
3. Multicollinearity
When you add polynomial terms (like x², x³, etc.), they are mathematically correlated with each other. This multicollinearity can make the model unstable and the coefficients hard to interpret. To mitigate this, you can use techniques like regularisation (Ridge or Lasso regression) or orthogonal polynomials.
4. Choice of Degree
Choosing the right degree is both an art and a science. Too low, and you underfit; too high, and you overfit. Try multiple degrees, plot the learning curves, and use grid search with cross-validation to find the sweet spot.
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Conclusion
Polynomial regression is a powerful tool in your machine learning toolkit, especially when you're dealing with non-linear relationships in small to moderately sized datasets. Polynomial regression model provides a good balance between complexity and interpretability, allowing you to capture curved trends that linear regression would miss.
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