Locally Weighted Linear Regression (LWR)

Locally Weighted Linear Regression (LWR) is a non-parametric regression technique designed to address the limitations of traditional linear regression, especially when the data relationship is not well modeled by a simple global linear function.

Basic Idea: Instead of fitting a single global linear model, LWR fits a linear model locally around the query point x. It places more weight on training examples close to x, and less weight on examples farther away.
Weighting Scheme: Each training example (x(i), y(i)) is assigned a weight w(i) based on its distance from the query point x, typically using a kernel function like the Gaussian:

w(i)=exp(−2τ2∥x(i)−x∥2),

where τ is a bandwidth parameter controlling how quickly the weight decreases with distance.

Fitting and Prediction: To predict y at x, LWR:

Solves a weighted least squares problem minimizing:

∑iw(i)(y(i)−θTx(i))2,

where each data point's contribution is scaled by its weight.

Uses the fitted parameters θ to output the prediction:

y^=θTx.

Non-Parametric Nature: Unlike standard linear regression that produces a single set of parameters θ, LWR adapts parameters locally for each query point. It requires retaining the entire training set for prediction, making it a non-parametric method.

Advantages:

Can handle complex, non-linear relationships without explicitly defining a global model.
Makes prediction sensitive to the local structure of data.
Reduces the dependency on carefully selecting features.

Considerations:

Choosing the bandwidth τ is critical; too small leads to high variance (overfitting), too large leads to high bias (underfitting).
Computationally expensive for large datasets since it fits a model for each query point.

This method smooths between fitting the data globally and simply using nearest neighbor predictions, providing a flexible approach to regression when data relationships vary locally

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Mashtishk Vigyan Anusandhan

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