What is kernel smoothing?

A way to fit a smooth curve through noisy points: each prediction is a weighted average of nearby data, with closer points weighted more; the bandwidth sets how far "nearby" reaches.

kernel() — Kernel smoothing

Predicts each point as a weighted average of nearby points.

kernel(bandwidth, x, y)

Key idea

Predict at a point by averaging nearby values, giving closer points more weight. A kernel is the weighting shape that fades with distance. (e.g. a bell-shaped kernel fades with distance)
The bandwidth h sets how wide the neighbourhood is — the only real knob.
Small h = jittery (chases noise); large h = very smooth (washes out detail). (e.g. bias-variance trade-off again)
It is non-parametric: no fixed formula is assumed for the curve.

Definition

$\hat{f} (x_{0}) = \frac{\sum _{i} K _{h} ( x _{0} , x _{i} ) y _{i}}{\sum _{i} K _{h} ( x _{0} , x _{i} )}$

$K_{h}$: a kernel weight that fades with distance
$h$: the bandwidth (neighbourhood width)
$x_{0}$: the point you predict at

In plain words

To guess the value here, peek at the neighbours and average them — but let the closest ones have the loudest say.

Where you'd use it

Smoothing noisy sensor or financial data into a readable trend.
Estimating a relationship when you have no idea of its formula.
The idea behind LOESS scatterplot smoothers and density estimation.

Frequently asked questions

What is kernel smoothing?: A way to fit a smooth curve through noisy points: each prediction is a weighted average of nearby data, with closer points weighted more; the bandwidth sets how far "nearby" reaches.