Kernel smoother

A kernel smoother is a statistical technique for estimating a real valued function $f(X)\,\,\left( X\in \mathbb{R}^{p} \right)$ 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.

[edit] Definitions

Let $K_{h_\lambda}(X_0 ,X)$ be a kernel defined by

$K_{h_\lambda}(X_0 ,X) = D\left( \frac{\left\| X-X_0 \right\|}{h_\lambda (X_0)} \right)$

where:

$X,X_0 \in \mathbb{R}^p$
$\left\| \cdot \right\|$ is the Euclidean norm
$h λ (X 0)$ 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 X₀.

Popular kernels used for smoothing include

Let $\hat{Y}(X):\mathbb{R}^p \to \mathbb{R}$ be a continuous function of X. For each $X_0 \in \mathbb{R}^p$ , the Nadaraya-Watson kernel-weighted average (smooth Y(X) estimation) is defined by

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

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.

[edit] Nearest neighbor smoother

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, $h_m (X_0)=\left\| X_0 - X_{[m]} \right\|$ , where $X [m]$ is the mth closest to X₀ neighbor, and

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

Example:

In this example, X is one-dimensional. For each X₀, the $\hat{Y}(X_0)$ is an average value of 16 closest to X₀ points (denoted by red). The result is not smooth enough.

[edit] Kernel average smoother

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 $λ$ to X₀ (the closer to X₀ points get higher weights).

Formally, $h λ (X 0) = λ = constant,$ 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.

[edit] Local linear regression

Main article: Local regression

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

For one dimension (p = 1):

$\begin{align} & \min_{\alpha (X_0),\beta (X_0)} \sum\limits_{i=1}^N {K_{h_{\lambda }}(X_0,X_i)\left( Y(X_i)-\alpha (X_0)-\beta (X_{0})X_i \right)^2} \\ & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\Downarrow \\ & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\hat{Y}(X_{0})=\alpha (X_{0})+\beta (X_{0})X_{0} \\ \end{align}$

The closed form solution is given by:

$\hat{Y}(X_0)=\left( 1,X_0 \right)\left( B^{T}W(X_0)B \right)^{-1}B^{T}W(X_0)y$

where:

$y=\left( Y(X_1),\dots,Y(X_N) \right)^T$
$W(X_0)= \operatorname{diag} \left( K_{h_{\lambda }}(X_0,X_i) \right)_{N\times N}$
$B^{T}=\left( \begin{matrix} 1 & 1 & \dots & 1 \ X_{1} & X_{2} & \dots & X_{N} \\end{matrix} \right)$

Example:

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

[edit] Local polynomial regression

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

For p=1, one should minimize:

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

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

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

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

[edit] See also

[edit] References

Li, Q. and J.S. Racine. Nonparametric Econometrics: Theory and Practice. Princeton University Press, 2007, ISBN 0691121613.
T. Hastie, R. Tibshirani and J. Friedman, The Elements of Statistical Learning, Chapter 6, Springer, 2001. ISBN 0387952845 (companion book site).