a kernel smoother is a statistical technique for estimating a real valued function

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A kernel smoother is a statistical technique for estimating a real valued function

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.

Little or no training is required for operation of the kernel smoother. This technique is mostappropriate 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.

Definitions

Let be a kernel defined by

where:

is the Euclidean norm

hλ( X 0) is a parameter (kernel radius)

D(t ) typically is a positive real valued function, which value is decreasing (or not

increasing) for the increasing distance between the X and X 0.

Popular kernels used for smoothing include

Epanechnikov

Tri-cube

Gaussian

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 ( X i) are the observations at X i points.

In the following sections, we describe some particular cases of kernel smoothers.

Nearest neighbor smoother

http://en.wikipedia.org/wiki/Statistics



http://en.wikipedia.org/wiki/Function_%28mathematics%29



http://en.wikipedia.org/wiki/Non-parametric_statistics



http://en.wikipedia.org/wiki/Kernel_%28statistics%29



http://en.wikipedia.org/wiki/V._A._Epanechnikov









The idea of the nearest neighbor smoother is the following. For each point X 0, take m nearest

neighbors and estimate the value of Y ( X 0) by averaging the values of these neighbors.

Formally, , where X [m] is the mth closest to X 0 neighbor, and

Example:

In this example, X is one-dimensional. For each X0, the is an average value of 16

closest to X 0 points (denoted by red). The result is not smooth enough.

[edit] Kernel average smoother

http://en.wikipedia.org/wiki/K-nearest_neighbor_algorithm



http://en.wikipedia.org/w/index.php?title=Kernel_smoother&action=edit&section=3


http://en.wikipedia.org/wiki/File:NNSmoother.svg








The idea of the kernel average smoother is the following. For each data point X 0, 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 λ to X 0 (the closer to X 0 points get

higher weights).

Formally, hλ( X 0) = λ = constant, and D(t ) is one of the popular kernels.

Example:

For each X 0 the window width is constant, and the weight of each point in the window is

schematically denoted by the yellow figure in the graph. It can be seen that the estimation is

smooth, but the boundary points are biased. The reason for that is the non-equal number of

points (from the right and from the left to the X 0) in the window, when the X 0 is close enough

to the boundary.

[edit] Local linear regression

Main article: Local regression



http://en.wikipedia.org/wiki/Local_regression



http://en.wikipedia.org/wiki/File:KernelSmoother.svg





In the two previous sections we assumed that the underlying Y(X) function is locally

constant, therefore we were able to use the weighted average for the estimation. The idea of

local linear regression is to fit locally a straight line (or a hyperplane for higher dimensions),

and not the constant (horizontal line). After fitting the line, the estimation is

provided by the value of this line at X 0 point. By repeating this procedure for each X 0, one canget the estimation function . Like in previous section, the window width is constant

hλ( X 0) = λ = constant. Formally, the local linear regression is computed by solving a weighted

least square problem.

For one dimension ( p = 1):

The closed form solution is given by:

where:

Example:


































The resulting function is smooth, and the problem with the biased boundary points is solved.

[edit] Local polynomial regression

Instead of fitting locally linear functions, one can fit polynomial functions.

For p=1, one should minimize:

with

In general case (p>1), one should minimize:




http://en.wikipedia.org/wiki/File:Localregressionsmoother.svg




a kernel smoother is a statistical technique for estimating a real valued function

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