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Concept
Kernel smoother
Summary
A kernel smoother is a statistical technique to estimate a real valued function as the weighted average of neighboring observed data. The weight is defined by the kernel, such that closer points are given higher weights. The estimated function is smooth, and the level of smoothness is set by a single parameter.
Kernel smoothing is a type of weighted moving average.
Let be a kernel defined by
where:
is the Euclidean norm
is a parameter (kernel radius)
D(t) is typically a positive real valued function, whose value is decreasing (or not increasing) for the increasing distance between the X and X0.
Popular kernels used for smoothing include parabolic (Epanechnikov), Tricube, and Gaussian kernels.
Let be a continuous function of X. For each , the Nadaraya-Watson kernel-weighted average (smooth Y(X) estimation) is defined by
where:
N is the number of observed points
Y(Xi) are the observations at Xi points.
In the following sections, we describe some particular cases of kernel smoothers.
The Gaussian kernel is one of the most widely used kernels, and is expressed with the equation below.
Here, b is the length scale for the input space.
The idea of the nearest neighbor smoother is the following. For each point X0, take m nearest neighbors and estimate the value of Y(X0) by averaging the values of these neighbors.
Formally, , where is the mth closest to X0 neighbor, and
Example:
In this example, X is one-dimensional. For each X0, the is an average value of 16 closest to X0 points (denoted by red). The result is not smooth enough.
The idea of the kernel average smoother is the following. For each data point X0, choose a constant distance size λ (kernel radius, or window width for p = 1 dimension), and compute a weighted average for all data points that are closer than to X0 (the closer to X0 points get higher weights).
Formally, and D(t) is one of the popular kernels.
Example:
For each X0 the window width is constant, and the weight of each point in the window is schematically denoted by the yellow figure in the graph.
Official source
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