\name{pre.scale, pre.sphere}\alias{pre.sphere}\alias{pre.scale}\title{Pre-sphering and pre-scaling}\description{Pre-sphered or pre-scaled version of data.}\usage{pre.sphere(x, mean.centred=FALSE)pre.scale(x, mean.centred=FALSE)}\arguments{  \item{x}{matrix of data values}  \item{mean.centred}{if TRUE then centre the data values to have zero mean}}\value{Pre-sphered or pre-scaled version of data. These  pre-transformations are required for implementing the plug-in  \code{\link{Hpi}} selectors and the smoothed cross validation  \code{\link{Hscv}} selectors. }\details{ For pre-scaling, the data values are pre-multiplied by  \eqn{\mathbf{S}^{-1/2}}{S^(-1/2)} and for pre-scaling, by  \eqn{(\mathbf{S}_D)^{-1/2}}{S_D^(-1/2)} where  \eqn{\mathbf{S}}{S} is the sample variance and \eqn{\mathbf{S}_D}{S_D}  is \eqn{\mathrm{diag} \, (S_1^2, S_2^2, \dots, S_d^2)}{diag (S_1^2,    S_2^2, ..., S_d^2)} where  \eqn{S_i^2}{S_i^2} is the i-th marginal sample variance.  If \eqn{\mathbf{H}^*}{H*} is the bandwidth matrix for the  pre-transformed data and \eqn{\mathbf{H}}{H} is the bandwidth matrix for the  original data, then  \eqn{\mathbf{H}=\mathbf{S}^{1/2} \mathbf{H}^* \mathbf{S}^{1/2}}{S^(1/2) H* S^(1/2)} or \eqn{\mathbf{H} = \mathbf{S}_D^{1/2} \mathbf{H}^*\mathbf{S}_D^{1/2}}{S_D^(1/2) H* S_D^(1/2)} as appropriate.  }%%\references{ Wand, M.P. \& Jones, M.C. (1994) Multivariate plugin bandwidth%%    selection. \emph{Computational Statistics}, \bold{9}, 97-116.%%  %%  Duong, T. \& Hazelton, M.L. (2003) Plug-in bandwidth matrices for%%    bivariate kernel density estimation. \emph{Journal of Nonparametric%%  Statistics}, \bold{15}, 17-30.%%}\examples{data(unicef)unicef <- as.matrix(unicef)unicef.sp <- pre.sphere(unicef)unicef.sc <- pre.scale(unicef, mean.centred=TRUE)var(unicef.sp)var(unicef.sc)}\keyword{ algebra }