Kernel smoother

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

${\displaystyle K_{h_{\lambda }}(X_{0},X)=D\left({\frac {\left\|X-X_{0}\right\|}{h_{\lambda }(X_{0})}}\right)}$

where:

• ${\displaystyle X,X_{0}\in \mathbb {R} ^{p}}$
• ${\displaystyle \left\|\cdot \right\|}$ is the Euclidean norm
• ${\displaystyle h_{\lambda }(X_{0})}$ 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 ${\displaystyle Y(X):\mathbb {R} ^{p}\to \mathbb {R} }$ 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

${\displaystyle {\hat {Y}}(X_{0})={\frac {\sum \limits _{i=1}^{N}{K_{h_{\lambda }}(X_{0},X_{i})Y(X_{i})}}{\sum \limits _{i=1}^{N}{K_{h_{\lambda }}(X_{0},X_{i})}}}}$

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.

The Gaussian kernel is one of the most widely used kernels, and is expressed with the equation below.

${\displaystyle K(x^{*},x_{i})=\exp \left(-{\frac {(x^{*}-x_{i})^{2}}{2b^{2}}}\right)}$

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, ${\displaystyle h_{m}(X_{0})=\left\|X_{0}-X_{[m]}\right\|}$, where ${\displaystyle X_{[m]}}$ is the mth closest to X₀ neighbor, and

${\displaystyle D(t)={\begin{cases}1/m&{\text{if }}|t|\leq 1\\0&{\text{otherwise}}\end{cases}}}$

Example:

In this example, X is one-dimensional. For each X₀, the ${\displaystyle {\hat {Y}}(X_{0})}$ 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 ${\displaystyle \lambda }$ to X₀ (the closer to X₀ points get higher weights).

Formally, ${\displaystyle h_{\lambda }(X_{0})=\lambda ={\text{constant}},}$ 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. 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 X₀) in the window, when the X₀ is close enough to the boundary.

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 ${\displaystyle {\hat {Y}}(X_{0})}$ is provided by the value of this line at X₀ point. By repeating this procedure for each X₀, one can get the estimation function ${\displaystyle {\hat {Y}}(X)}$. Like in previous section, the window width is constant ${\displaystyle h_{\lambda }(X_{0})=\lambda ={\text{constant}}.}$ Formally, the local linear regression is computed by solving a weighted least square problem.

For one dimension (p = 1):

${\displaystyle {\begin{aligned}&\min _{\alpha (X_{0}),\beta (X_{0})}\sum \limits _{i=1}^{N}{K_{h_{\lambda }}(X_{0},X_{i})\left(Y(X_{i})-\alpha (X_{0})-\beta (X_{0})X_{i}\right)^{2}}\\&\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\Downarrow \\&\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\hat {Y}}(X_{0})=\alpha (X_{0})+\beta (X_{0})X_{0}\\\end{aligned}}}$

The closed form solution is given by:

${\displaystyle {\hat {Y}}(X_{0})=\left(1,X_{0}\right)\left(B^{T}W(X_{0})B\right)^{-1}B^{T}W(X_{0})y}$

where:

• ${\displaystyle y=\left(Y(X_{1}),\dots ,Y(X_{N})\right)^{T}}$
• ${\displaystyle W(X_{0})=\operatorname {diag} \left(K_{h_{\lambda }}(X_{0},X_{i})\right)_{N\times N}}$
• ${\displaystyle B^{T}=\left({\begin{matrix}1&1&\dots &1\\X_{1}&X_{2}&\dots &X_{N}\\\end{matrix}}\right)}$

Example:

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

Local linear regression can be applied to any-dimensional space, though the question of what is a local neighborhood becomes more complicated. It is common to use k nearest training points to a test point to fit the local linear regression. This can lead to high variance of the fitted function. To bound the variance, the set of training points should contain the test point in their convex hull (see Gupta et al. reference).

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

For p=1, one should minimize:

${\displaystyle {\underset {\alpha (X_{0}),\beta _{j}(X_{0}),j=1,...,d}{\mathop {\min } }}\,\sum \limits _{i=1}^{N}{K_{h_{\lambda }}(X_{0},X_{i})\left(Y(X_{i})-\alpha (X_{0})-\sum \limits _{j=1}^{d}{\beta _{j}(X_{0})X_{i}^{j}}\right)^{2}}}$

with ${\displaystyle {\hat {Y}}(X_{0})=\alpha (X_{0})+\sum \limits _{j=1}^{d}{\beta _{j}(X_{0})X_{0}^{j}}}$

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

${\displaystyle {\begin{aligned}&{\hat {\beta }}(X_{0})={\underset {\beta (X_{0})}{\mathop {\arg \min } }}\,\sum \limits _{i=1}^{N}{K_{h_{\lambda }}(X_{0},X_{i})\left(Y(X_{i})-b(X_{i})^{T}\beta (X_{0})\right)}^{2}\\&b(X)=\left({\begin{matrix}1,&X_{1},&X_{2},...&X_{1}^{2},&X_{2}^{2},...&X_{1}X_{2}\,\,\,...\\\end{matrix}}\right)\\&{\hat {Y}}(X_{0})=b(X_{0})^{T}{\hat {\beta }}(X_{0})\\\end{aligned}}}$

• Savitzky–Golay filter
• Kernel methods
• Kernel density estimation
• Local regression
• Kernel regression

• Li, Q. and J.S. Racine. Nonparametric Econometrics: Theory and Practice. Princeton University Press, 2007, ISBN 0-691-12161-3.
• T. Hastie, R. Tibshirani and J. Friedman, The Elements of Statistical Learning, Chapter 6, Springer, 2001. ISBN 0-387-95284-5 (companion book site).
• M. Gupta, E. Garcia and E. Chin, "Adaptive Local Linear Regression with Application to Printer Color Management," IEEE Trans. Image Processing 2008.