% Generated by roxygen2: do not edit by hand
% Please edit documentation in R/multivariance-functions.R
\docType{package}
\name{multivariance-package}
\alias{multivariance-package}
\title{multivariance: Measuring Multivariate Dependence Using Distance Multivariance}
\description{
The multivariance package provides basic functions to calculate distance multivariance and related quantities. To test independence use \code{\link{multivariance.test}}, it provides an interface (via its arguments) to all the tests based on distance (m-/total-)multivariance. The package offers also several other functions related to distance multivariance, e.g. a detection and visualization of dependence structures \code{\link{dependence.structure}}. See below for details on the full content of the package.
}
\details{
Distance multivariance is a multivariate dependence measure, which can be used to detect dependencies between an arbitrary number of random vectors each of which can have a distinct dimension. The necessary functions are implemented in this package, and examples are given. For the theoretic background we refer to the papers [1,2,3,4]. Paper [3] includes a summary of the first two. It is the recommended starting point for users with an applied interest. Paper [4] is concerned with new (faster) p-value estimates for the independence tests.

The (current) code is tested and speed improved in comparison to the former releases. Certainly there is still room for improvement and development. Questions, comments and remarks are welcome: \email{bjoern.boettcher@tu-dresden.de}

For infos on the latest changes and/or updates to the package use \code{news(package="multivariance")}.

To cite this package use the standard citation for R packages, i.e., the output of \code{citation("multivariance")}.
}
\section{Multivariance}{


 \code{\link{multivariance}} computes the distance multivariance

 \code{\link{total.multivariance}} computes the total distance multivariance

 \code{\link{m.multivariance}} computes the m-multivariance (introduced in [3])

 It might be convenient to compute these simultaneously using \code{\link{multivariances.all}}.
}

\section{Functions to use and interpret multivariance}{


 \code{\link{rejection.level}} computes a (conservative) rejection level for a given significance level. This can be used for a conservative interpretation of distance multivariance. The counterpart is \code{\link{multivariance.pvalue}}, which computes a conservative p-value for a given distance multivariance. Both methods are distribution-free.

 \code{\link{resample.rejection.level}} and \code{\link{resample.pvalue}} are the distribution dependent versions of the above. They are approximately sharp, but computational more expensive. Any resampling is done by \code{\link{resample.multivariance}}.

 Using the methods developed in [4] approximate p-value estimates are provided by \code{\link{pearson.pvalue}}. This method is much faster than the resampling method.

 \code{\link{multivariance.test}} and \code{\link{independence.test}} provide the corresponding tests of independence. The former provides output as common for tests in R, the latter is a simple significance test returning for a given significance level just \code{TRUE} or \code{FALSE}

\code{\link{cdm}} and \code{\link{cdms}} compute the (doubly) centered distance matrix and matrices, respectively. These can be used to speed up repeated computations of distance multivariance.

In [4] various methods to estimate the moments of the test statistic under H0 were developed, these are (implicitly) implemented in this package only for the moments used in \code{\link{pearson.pvalue}}. Further and explicit functions can be added upon request. Please feel free to contact the author.

For planing of large projects or studies it might be convenient to estimate the computation time of multivariance via \code{\link{multivariance.timing}}.
}

\section{Dependence structures}{


 \code{\link{dependence.structure}} performs the dependence structure detection algorithm as described in [3].

 \code{\link{find.cluster}} is the basic building block of \code{\link{dependence.structure}}. It is recommended to use \code{\link{dependence.structure}}.
}

\section{Examples}{


\code{\link{coins}} and \code{\link{tetrahedron}} generate samples of pairwise independent random variables, with dependence of higher order.

\code{\link{dep_struct_iterated_13_100}}, \code{\link{dep_struct_ring_15_100}}, \code{\link{dep_struct_several_26_100}} and \code{\link{dep_struct_star_9_100}} are example data sets for the dependence structure detection. These might also serve as benchmark examples.
}

\references{
[1] B. Böttcher, M. Keller-Ressel, R.L. Schilling, Detecting independence of random vectors: generalized distance covariance and Gaussian covariance. Modern Stochastics: Theory and Applications 2018, Vol. 5, No. 3, 353-383. \url{https://www.vmsta.org/journal/VMSTA/article/127/info}

[2] B. Böttcher, M. Keller-Ressel, R.L. Schilling, Distance multivariance: New dependence measures for random vectors. Accepted for publication in Annals of Statistics. \url{https://arxiv.org/abs/1711.07775}

[3] B. Böttcher, Dependence and Dependence Structures: Estimation and Visualization Using Distance Multivariance. Preprint. \url{https://arxiv.org/abs/1712.06532}

[4] G. Berschneider, B. Böttcher, On complex Gaussian random fields, Gaussian quadratic forms and sample distance multivariance. Preprint. \url{https://arxiv.org/abs/1808.07280}
}