Kernel smoothing

A kernel smoother is a statistical technique for estimating a real valued function ${\displaystyle f(X)\,\,\left(X\in \mathbb {R} ^{p}\right)}$ by using its noisy observations, when no parametric model for this function is known. The estimated function is smooth, and the level of smoothness is set by a single parameter.

This technique is most appropriate for low-dimensional (p < 3) data visualization purposes. Actually, the kernel smoother represents the set of irregular data points as a smooth line or surface.

Let ${\displaystyle K_{h_{\lambda }}(X_{0},X)}$ be a kernel defined by

where:

Popular kernels used for smoothing include

Let ${\displaystyle {\hat {Y}}(X):\mathbb {R} ^{p}\to \mathbb {R} }$ be a continuous function of X. For each ${\displaystyle X_{0}\in \mathbb {R} ^{p}}$, the Nadaraya-Watson kernel-weighted average (smooth Y(X) estimation) is defined by

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