ks: Kernel density estimation for bivariate data, 2009. 178

Kernel density estimation is a popular tool for visualising the distribution of data. See Simonoff (1996), for example, for an overview. When multivariate kernel density estimation is considered it is usually in the constrained context with diagonal bandwidth matrices, e.g. in the R packages sm (Bowman and Azzalini, 2007) and KernSmooth (Wand, 2006). We introduce a new R package ks which implements diagonal and unconstrained data-driven bandwidth matrices for kernel density estimation, which can also be used for multivariate kernel discriminant analysis. The ks package implements selectors for 1- to 6-dimensional data. This vignette contains only a brief introduction to using ks for kernel density estimation for 2-dimensional data. See Duong (2007) for a more detailed account. For a bivariate random sample X1,X2,...,Xn drawn from a density f, the kernel density estimate is defined by f̂(x;H) = n−1 n∑ i=1 KH(x−Xi) where x = (x1, x2) T and Xi = (Xi1, Xi2) T, i = 1, 2,..., n. Here K(x) is the kernel which is a symmetric probability density function, H is the bandwidth matrix which is symmetric and positive-definite, and KH(x) = |H|−1/2K(H−1/2x). The choice of K is not crucial: we take K(x) = (2pi)−1 exp(−12xTx) the standard normal throughout. In contrast, the choice of H is crucial in determining the performance of f ̂. The most common parameterisations of the bandwidth matrix are the diagonal and the general or unconstrained which has no restrictions on H provided that H remains positive definite and symmetric, that is H = h21 0 0 h22 or H = h21 h12 h12 h 2 2 This latter parameterisation allows kernels to have an arbitrary orientation whereas the former only allows kernels which are oriented to the co-ordinate axes. For our target density, we use the ‘dumbbell ’ density, given by the normal mixture