Kernel Smoothing In Matlab: Theory And Practice Of Kernel Smoothing

Theory and Practice of Kernel Smoothing

Ivanka Horová, Jan Koláček, Jiří Zelinka

• Publisher: WSPC
• Year: 2012
• ISBN:
  9789814405508
• Topic: Biological Sciences
• Subtopic: Science General
• Index: Biological Sciences

Methods of kernel estimates represent one of the most effective nonparametric smoothing techniques. These methods are simple to understand and they possess very good statistical properties. This book provides a concise and comprehensive overview of statistical theory and in addition, emphasis is given to the implementation of presented methods in Matlab. All created programs are included in a special toolbox which is an integral part of the book. This toolbox contains many Matlab scripts useful for kernel smoothing of density, cumulative distribution function, regression function, hazard function, indices of quality and bivariate density. Specifically, methods for choosing a choice of the optimal bandwidth and a special procedure for simultaneous choice of the bandwidth, the kernel and its order are implemented. The toolbox is divided into six parts according to the chapters of the book.

All scripts are included in a user interface and it is easy to manipulate with this interface. Each chapter of the book contains a detailed help for the related part of the toolbox too. This book is intended for newcomers to the field of smoothing techniques and would also be appropriate for a wide audience: advanced graduate, PhD students and researchers from both the statistical science and interface disciplines.

Contents:

• Introduction
• Univariate Kernel Density Estimation
• Kernel Estimation of a Distribution Function
• Kernel Estimation and Reliability Assessment
• Kernel Estimation of a Hazard Function
• Kernel Estimation of a Regression Function
• Multivariate Kernel Density Estimation

Readership: Advanced graduate students, researchers in mathematics or statistics.

Chapter 1

Introduction

1.1 Kernels and their properties

Kernel smoothing belongs to a general category of techniques for nonparametric curve estimation including nonparametric regression, nonparametric density estimators and nonparametric hazard functions. These estimations depend on a smoothing parameter called a bandwidth which controls the smoothness of the estimate and on a kernel which plays a role of weight function. As far as the kernel function is concerned, a key parameter is its order which is related both to the number of its vanishing moments and to the number of existing derivatives for the underlying curve to be estimated.

In this chapter, we introduce a definition of the kernel and show some of its useful properties. Various aspects of the choice of the kernel function have been discussed, e.g., in Wand and Jones (1995); Müller (1988). As concerns a bandwidth choice – it is a crucial problem in the kernel smoothing and it will be discussed in the next chapters.

Throughout this book the following definition of a kernel is suitable for our considerations.

Definition 1.1. Let v, k be nonnegative integers, 0 ≤ v < k. Let K be a real valued function satisfying K ∈ S_v, _k,

where

(1.1)

Such a function is called a kernel of order k. The integral conditions are often called moment conditions. We will use the short notation β_k instead of β_k(K) if there cannot be any misunderstanding.

A commonly used kernel function is the Gaussian kernel

But this kernel has an unbounded support and thus it does not belong to the class S_v,k. For our purpose the kernels with bounded support will be more useful. Figure 1.2shows the most popular polynomial kernel – the Epanechnikov kernel (see Epanechnikov (1969)).

Example 1.1. Figures 1.1 – 1.4 present some kernels from the class S_0,2. I_A denotes the indicator function of the set A.

Fig. 1.1 K(x) =

(x), uniform kernel.

Fig. 1.2 K(x) =

(x), Epanechnikov kernel.

Fig. 1.3 K(x) = (1 – |x|)I_[–1,1](x), triangle kernel.

Fig. 1.4 K(x) =

(1—x²)²I_[–1,1](x), quartic kernel.

Example 1.2. Figures 1.5 and 1.6 present some kernels f...

Table of contents

• Cover
• Halftitle
• Title
• Copyright
• Dedication
• Preface
• Contents
• Chapter 1: Introduction
• Chapter 2: Univariate kernel density estimation
• Chapter 3: Kernel estimation of a distribution function
• Chapter 4: Kernel estimation and reliability assessment
• Chapter 5: Kernel estimation of a hazard function
• Chapter 6: Kernel estimation of a regression function
• Chapter 7: Multivariate kernel density estimation
• Notation
• Bibliography
• Index