ML Theory: Kernel Regression

Motivation

For observed pairs (x_i, y_i), i = 1, \hdots, n, the relationship between x and y can be defined generally as

    \[ y_i = m(x_i) + \varepsilon_i \]

where f(x_i) = E[y_i | x = x_i] and \varepsilon_i \sim^{iid} (0, \sigma^2). If we are unsure about the form of m(\cdot), our objective may be to estimate m(\cdot) without making too many assumptions about its shape. In other words, we aim to “let the data speak for itself”.

Simulated scatterplot of y = f(x) + \epsilon. Here, x \sim Uniform(0, 10) and \epsilon \sim N(0, 1). The true function f(x) = sin(x) is displayed in green.

Non-parametric approaches require only that m(\cdot) be smooth and continuous. These assumptions are far less restrictive than those of alternative parametric approaches, thereby increasing the number of potential fits and providing additional flexibility. This makes non-parametric models particularly appealing when prior knowledge about m(\cdot)‘s functional form is limited.

Estimating the Regression Function

If multiple values of y were observed at each x, f(x) could be estimated by averaging the value of the response at each x. However, since x is often continuous, it can take on a wide range of values making this quite rare. Instead, a neighbourhood of x is considered.

Result of averaging y_i at each x_i. The fit is extremely rough due to gaps in x and low y frequency at each x.

Define the neighbourhood around x as x \pm \lambda for some bandwidth \lambda > 0. Then, a simple non-parametric estimate of m(x) can be constructed as average of the y_i‘s corresponding to the x_i within this neighbourhood. That is,

(1)   \begin{equation*} \hat{f}_{\lambda}(x) = \frac{\sum_{n} \mathbb{I}(|x - x_i| \leq \lambda)~ y_i}{\sum_{n} \mathbb{I}(|x - x_i| \leq \lambda)} = \frac{\sum_n K\left( \frac{x - x_i}{\lambda} \right) y_i}{\sum_n K\left( \frac{x - x_i}{\lambda} \right) } \end{equation*}

where

    \[ K(u) = \begin{cases} \frac{1}{2} & |u| \leq 1 \\ 0  & \text{o.w.} \end{cases} \]

is the uniform kernel. This estimator, referred to as the Nadaraya-Watson estimator, can be generalized to any kernel function K(u) (see my previous blog bost). It is, however, convention to use kernel functions of degree \nu = 2 (e.g. the Gaussian and Epanechnikov kernels).

The red line is the result of estimating f(x) with a Gaussian kernel and arbitrarily selected bandwidth of \lambda = 1.25. The green line represents the true function sin(x).

Kernel and Bandwidth Selection

The implementation of a kernel estimator requires two choices:

  • the kernel, K(u), and
  • the smoothing parameter, or bandwidth, \lambda.

Kernels are often selected based on their smoothness and compactness. We prefer a compact kernel as it ensures that only data local to the point of interest is considered. The optimal choice, under some standard assumptions, is the Epanechnikov kernel. This kernel has the advantages of some smoothness, compactness, and rapid computation.

The choice of bandwidth \lambda is critical to the performance of the estimator and far more important than the choice of kernel. If the smoothing parameter is too small, the estimator will be too rough; but if it is too large, we risk smoothing out important features of the function. In other words, choosing \lambda involves a significant bias-variance trade-off.


\lambda \uparrow ~~~\Rightarrow smooth curve, low variance, high bias
\lambda \downarrow ~~~\Rightarrow rough curve, high variance, low bias

The simplest way of selecting \lambda is to plot \hat{m}_\lambda(x) for a range of different \lambda and pick the one that looks best. The eye can always visualize additional smoothing, but it is not so easy to imagine what a less smooth fit might look like. For this reason, it is recommended that you choose the least smooth fit that does not show any implausible fluctuations.

Kernel regression fits for various values of \lambda.

Cross-Validation Methods

Selecting the amount of smoothing using subjective methods requires time and effort. Automatic selection of \lambda can be done via cross-validation. The cross-validation criterion is

    \[ CV(\lambda) = \frac{1}{n} \sum_{n} \left( y_j - \hat{m}_{\lambda}^{(-j)} x_j \right)^2 \]

where (-j) indicates that point j is left out of the fit. The basic idea is to leave out observation j and estimate m(\cdot) based on the other n-1 observations. \lambda is chosen to minimize this criterion.

True cross-validation is computationally expensive, so an approximation known as generalized cross-validation (GCV) is often used. GCV approximates CV and involves only one non-parametric fit for each \lambda value (compared to CV which requires n fits at each \lambda).

In order to approximate CV, it is important to note that kernel smooths are linear. That is,

    \[ \hat{Y} = \hat{m}_{\lambda}(x) = S_{\lambda} y \]

where S_{\lambda} is an n \times n smoothing matrix. S_{\lambda} is analogous to the hat matrix H in parametric linear models.

    \[ H = X(X^{T}X)^{-1}X^{T} \]

It can be shown that

    \[ CV(\lambda) = \frac{1}{n} \sum_{n} \left[ \frac{y_i - \hat{m}_{\lambda}(x_i)}{1 - s_{ii}(\lambda)} \right]^2 \]

where s_{ii}(\lambda) is the i^{th} diagonal element of S_{\lambda} (hence s_{ii} is analogous to h_{ii}, the leverage of the i^{th} observation). Using the smoothing matrix,

    \[ GCV(\lambda) = \frac{1}{n} \sum_{i=1}^{n} \left[ \frac{y_i - \hat{m}_{\lambda}(x_i)}{1 - \frac{\text{tr}(S_{\lambda})}{n}}\right] \]

where \text{tr}(S_{\lambda}) is the trace of S_{\lambda}. In this sense, GCV is analogous to the influence matrix.

Automatic methods such as CV often work well but sometimes produce estimates that are clearly at odds with the amount of smoothing that contextual knowledge would suggest. It is therefore extremely important to exercise caution when using them and it is recommended that they be used as a starting point.

Published by

Emma Smith

Emma Smith is a young statistician who's on a mission to convince the masses statistics is as awesome as she *knows* it is! When she's not working on expanding her knowledge of machine learning and mathematical statistics, she's busy petting cats and unsuccessfully convincing her boyfriend to let her adopt them, hiking, concocting indie and folk rock playlists, and kicking butt in roller derby.

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