Bandwidth Parameter (τ or h) in LWR and KDE  Python for Integrated Circuits   An Online Book  

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================================================================================= In Locally Weighted Regression (LWR), the goal is to fit the parameter vector θ in such a way that it minimizes the weighted sum of squared errors (also known as the cost function). The specific cost function that LWR aims to minimize is as follows:  [3887a] where,
The common choice of w^{(i)}, shows in Figure 3887a, is,  [3887b] where,
(a) (b)
The bandwidth parameter (often denoted as τ or h) in locally weighted regression (LWR) and kernel density estimation (KDE) does indeed have an effect on the tradeoff between overfitting and underfitting. Understanding this effect requires an understanding of how LWR and KDE work. In LWR and KDE (kernel density estimation), the bandwidth parameter determines the width or spread of the kernel function used to assign weights to data points. A narrower bandwidth assigns higher weights to data points that are very close to the prediction point, making the regression or density estimation highly sensitive to local variations in the data. In contrast, a wider bandwidth assigns more uniform weights to data points within a larger neighborhood, resulting in a smoother and more global estimation. Here's how the bandwidth parameter affects overfitting and underfitting:
The choice of bandwidth is a critical hyperparameter in LWR and KDE, and selecting the right bandwidth value is often done through crossvalidation or other model selection techniques. The goal is to strike a balance between capturing important local information while avoiding the pitfalls of overfitting or underfitting. ============================================


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