a straight line is too rigid for a wiggly world, and a global degree-9 polynomial is a hostage negotiation. 𐃏 locally weighted regression (LWR — and its robust cousin LOWESS) takes the diplomatic route: fit the simplest possible model, but fit it freshly at every query point, paying attention only to the training points nearby.
motivation
- linear regression commits to one \(\theta\) for the whole input space. if the true \(f\) bends, the residue of that commitment is bias everywhere.
- the fix need not be a fancier global family. any smooth function is locally linear — taylor says so — so a linear fit weighted toward a neighbourhood of \(x_0\) can track an arbitrary smooth \(f\).
- the price: there is no longer a “trained model”. LWR is memory-based and non-parametric — like k-nearest neighbours, it keeps the entire training set and does all of its work at prediction time. training is \(O(1)\); every query costs a fresh weighted least-squares solve. 𐃏
the estimator
weighted least squares at a query point
fix a query \(x_0\). assign each training point a weight \(w_i(x_0) \ge 0\) that decays with distance from \(x_0\), then solve the weighted least-squares problem