Why High-Order Polynomials for Regression are Discouraged: Overfitting, Runges Phenomenon, Numerical Instability, and Better Alternatives

The use of high-order polynomials for regression is often discouraged due to several key reasons. This article will explore these reasons in detail and discuss why alternative methods such as splines or piecewise polynomial regression are often preferred.

Overfitting

High-order polynomials can fit the training data very closely, capturing noise rather than the underlying trend. This phenomenon, known as overfitting, results in a situation where the model performs well on the training data but poorly on unseen data. The model becomes overly complex, leading to poor generalization.

Runges Phenomenon

When using high-order polynomials for interpolation, especially on evenly spaced data points, oscillations can occur at the edges of the interval, a phenomenon known as Runges phenomenon. This can result in poor approximation and large errors, even for points that are close to the data points. This is particularly problematic in real-world applications where smoothness and accuracy are crucial.

Numerical Instability

High-order polynomials can lead to numerical instability, especially when evaluating the polynomial at points far from the data range. Small changes in input can lead to large changes in output, making the model sensitive to noise in the data. This instability can also lead to unreliable and unpredictable model predictions.

Interpretability

Higher-degree polynomials can be difficult to interpret. Simple linear or low-degree polynomial models are more straightforward, allowing for easier understanding of the relationship between variables. In many practical applications, interpretability is a crucial factor, making simpler models preferable.

Increased Complexity

As the degree of the polynomial increases, the model becomes more complex, requiring more data to estimate the parameters reliably. Higher complexity can also lead to longer computation times and increased risk of errors in estimation. This added complexity often doesn't justify the benefits of a more accurate model.

Alternative Methods

There are often better alternatives such as splines or piecewise polynomial regression which provide flexibility and can fit data well without the drawbacks of high-degree polynomial fitting. Splines, for instance, are piecewise polynomials that can be used to model complex relationships while maintaining smoothness and reducing the risk of overfitting.

Conclusion

While high-order polynomials can be useful in certain contexts, their disadvantages often outweigh the benefits, especially in practical applications. It is generally advisable to use simpler models or alternative approaches that balance flexibility and robustness. By choosing more appropriate methods, such as splines, we can ensure more accurate, robust, and interpretable models.