Polynomial Regression in Machine Learning
Polynomial Regression is an extension of linear regression used to model non-linear relationships between the independent variable(s) and the dependent variable. Unlike linear regression, which fits a straight line, polynomial regression fits a curve to capture the underlying trends in the data.
In simple terms, polynomial regression helps answer questions where the relationship between variables isn’t a straight line, like:
- “How does temperature affect the growth rate of a plant over time?”
- “What is the trajectory of a rocket based on time and speed?”
How Does Polynomial Regression Work?
Mathematical Model:
Polynomial regression is an extension of linear regression, where the relationship between the input features and the output is modelled as a polynomial function. This allows it to capture nonlinear relationships in the data.
The equation for Polynomial Regression is:
\( y = \beta_0 + \beta_1x + \beta_2x^2 + \beta_3x^3 + \dots + \beta_nx^n \)
Here:
y: Predicted value (dependent variable)x: Input feature (independent variable)n: Degree of the polynomial- \( \beta_0, \beta_1, \dots, \beta_n \): Coefficients or weights of the polynomial terms
Unlike simple linear regression, which fits a straight line to the data, polynomial regression can fit curves, allowing it to better model data with more complex patterns.
The degree of the polynomial (\(n\)) determines the flexibility of the model. A higher degree allows for more complex relationships but risks overfitting the data.
Data Transformation:
Before fitting the model, polynomial regression transforms the input feature x into additional polynomial features such as \(x^2, x^3, \dots, x^n\). For example:
Original feature: \( x \)
Transformed features: \( [x, x^2, x^3, \dots, x^n] \)
These transformed features are then used in a standard linear regression model, allowing the linear regression algorithm to find a curve that best fits the data.
Training the Model:
The training process in polynomial regression involves finding the coefficients (\(\beta_0, \beta_1, \beta_2, \dots, \beta_n\)) that minimize the error between the predicted values and the actual values. This is done using a cost function, typically the Mean Squared Error (MSE), defined as:
\( \text{MSE} = \frac{1}{n} \sum_{i=1}^n (y_i – \hat{y}_i)^2 \)
Here:
- \( y_i\): Actual target value for the \(i^{th}\) data point
- \( \hat{y}_i \): Predicted target value for the \(i^{th}\) data point
n: Number of data points
To minimize the MSE, optimization techniques like Gradient Descent are used. The coefficients are updated iteratively as:
\( \beta_j \leftarrow \beta_j – \alpha \frac{\partial}{\partial \beta_j} \text{MSE} \)
Where:
- \( \beta_j \): The \(j^{th}\) coefficient
- \( \alpha \): Learning rate (controls the step size)
- \( \dfrac{\partial}{\partial \beta_j} \text{MSE} \): Gradient of the MSE with respect to \(\beta_j\)
Prediction:
After training, the model predicts target values for new inputs by applying the polynomial equation:
\( \hat{y} = \beta_0 + \beta_1x + \beta_2x^2 + \beta_3x^3 + \dots + \beta_nx^n \)
Here, \(\hat{y}\) is the predicted value for a given input x. By leveraging the learned coefficients, the model can generalize to predict for unseen data.
While polynomial regression is powerful for modelling nonlinear relationships, careful attention must be given to selecting the appropriate degree of the polynomial to avoid overfitting or under-fitting the data.
When to Use Polynomial Regression?
- When the data shows a non-linear relationship between the independent and dependent variables.
- Example: In real-world scenarios like weather patterns or stock price movements, where trends often follow curved paths.
Comparison with Linear Regression
| Feature | Linear Regression | Polynomial Regression |
|---|---|---|
| Relationship Type | Assumes a straight-line relationship | Captures curved relationships |
| Flexibility | Limited to linear trends | Handles non-linear trends effectively |
| Overfitting Risk | Low | High with higher-degree polynomials |
Advantages of Polynomial Regression
- Flexibility: Captures non-linear trends that linear regression cannot.
- Simplicity: Still relatively simple to implement compared to more advanced machine learning models.
- Interpretability: Provides a clear mathematical formula for the relationship between variables.
Limitations of Polynomial Regression
- Overfitting: Using a high-degree polynomial can lead to overfitting, where the model performs well on training data but poorly on new data.
- Complexity: Higher-degree polynomials can make the model harder to interpret and more computationally expensive.
- Extrapolation Issues: Predictions outside the range of the training data can become wildly inaccurate.