1\name{U_product} 2\alias{U_product} 3\title{ Inner product in the Hilbert space of U-centered 4distance matrices} 5\description{ 6 Stand-alone function to compute the inner product in the 7 Hilbert space of U-centered distance matrices, as in the definition of 8 partial distance covariance. 9} 10\usage{ 11U_product(U, V) 12} 13\arguments{ 14 \item{U}{ U-centered distance matrix} 15 \item{V}{ U-centered distance matrix} 16} 17\details{ 18 Note that \code{pdcor}, etc. functions include the centering and 19 projection operations, so that these stand alone versions are not 20 needed except in case one wants to check the internal computations. 21 22 Exported from U_product.cpp. 23} 24\value{ 25\code{U_product} returns the inner product, a scalar. 26} 27\references{ 28 Szekely, G.J. and Rizzo, M.L. (2014), 29 Partial Distance Correlation with Methods for Dissimilarities, 30 \emph{Annals of Statistics}, Vol. 42, No. 6, pp. 2382-2412. 31 \cr \url{https://projecteuclid.org/euclid.aos/1413810731} 32} 33\author{ Maria L. Rizzo \email{mrizzo @ bgsu.edu} and 34Gabor J. Szekely 35} 36\examples{ 37 x <- iris[1:10, 1:4] 38 y <- iris[11:20, 1:4] 39 M1 <- as.matrix(dist(x)) 40 M2 <- as.matrix(dist(y)) 41 U <- U_center(M1) 42 V <- U_center(M2) 43 44 U_product(U, V) 45 dcovU_stats(M1, M2) 46 } 47\keyword{ multivariate } 48\concept{ multivariate } 49\concept{ distance correlation } 50\concept{ distance covariance } 51\concept{ energy statistics } 52