Table of Contents

Class PolynomialRegressionOptions<T>

Namespace
AiDotNet.Models.Options
Assembly
AiDotNet.dll

Configuration options for Polynomial Regression, an extension of linear regression that models the relationship between variables using polynomial functions to capture non-linear relationships in data.

public class PolynomialRegressionOptions<T> : RegressionOptions<T>

Type Parameters

T
Inheritance
PolynomialRegressionOptions<T>
Inherited Members

Remarks

Polynomial Regression transforms the original features by adding polynomial terms (squared, cubed, etc.) to the regression equation. This allows the model to fit curved or more complex relationships that cannot be adequately represented by a straight line. While more flexible than linear regression, polynomial models with higher degrees can lead to overfitting if not properly regularized or validated. Polynomial regression is widely used in various fields including economics, social sciences, biology, and engineering when relationships between variables are suspected to be non-linear. The implementation typically involves creating new features by raising the original features to various powers, then applying standard linear regression techniques.

For Beginners: Polynomial Regression is like an upgraded version of regular linear regression that can handle curved relationships.

Imagine you're trying to model this relationship:

  • Regular (linear) regression can only draw straight lines
  • But many real-world relationships follow curves, not straight lines
  • Examples: plant growth over time, diminishing returns on investment, learning curves

What polynomial regression does:

  • It adds "power terms" to your equation (squared, cubed, etc.)
  • This lets your model create curves instead of just straight lines
  • A degree 2 polynomial can make parabolas (U-shapes)
  • A degree 3 polynomial can make S-curves
  • Higher degrees can create more complex shapes

Think of it like drawing tools:

  • Linear regression gives you only a ruler (straight lines)
  • Polynomial regression gives you flexible curve tools
  • The degree setting controls how flexible those curves can be

This class lets you configure how curved or complex your model can be by setting the polynomial degree.

Properties

Degree

Gets or sets the degree of the polynomial used in the regression model.

public int Degree { get; set; }

Property Value

int

The polynomial degree, defaulting to 2.

Remarks

This parameter determines the highest power to which the independent variables will be raised in the polynomial equation. A value of 1 corresponds to linear regression, 2 introduces quadratic terms, 3 adds cubic terms, and so on. The choice of degree significantly impacts model complexity and flexibility. Higher degree polynomials can capture more complex non-linear relationships but require more training data to avoid overfitting. The optimal degree depends on the underlying data generation process and is often determined through cross-validation or by examining validation metrics. Note that computational complexity increases with the degree, as does the risk of numerical instability with very high degrees.

For Beginners: This setting controls how "bendy" your regression line can be.

The default value of 2 means:

  • Your model can create quadratic curves (parabolas or U-shapes)
  • It can model relationships that go up, then down (or vice versa)
  • It's more flexible than a straight line but still fairly constrained

Think of polynomial degree like this:

  • Degree 1: Straight line (can only go up or down consistently)
  • Degree 2: Parabola (can curve once, making U or upside-down U shapes)
  • Degree 3: Cubic curve (can have S-shapes with two changes in direction)
  • Degree 4+: Increasingly wiggly lines with more changes in direction

You might want a higher degree (like 3 or 4):

  • When your data clearly shows multiple changes in direction
  • For complex physical or financial processes with known non-linear behavior
  • When you have plenty of data points to support a more complex model

You might want a lower degree (just 1):

  • When the relationship appears linear in scatter plots
  • When you have limited data and want to avoid overfitting
  • When simpler models are preferred for interpretability or domain knowledge

Warning: Be careful with high degree values!

  • Too high a degree can cause "overfitting" where your model learns the noise in your data
  • This creates wild fluctuations that look good on training data but fail on new data
  • It's like memorizing the training examples instead of learning the underlying pattern
  • As a rule of thumb, be very cautious about using degrees above 5
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