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