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%
% Copyright 2007-2018 by the individuals mentioned in the source code history
%
% Licensed under the Apache License, Version 2.0 (the "License");
% you may not use this file except in compliance with the License.
% You may obtain a copy of the License at
%
% http://www.apache.org/licenses/LICENSE-2.0
%
% Unless required by applicable law or agreed to in writing, software
% distributed under the License is distributed on an "AS IS" BASIS,
% WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
% See the License for the specific language governing permissions and
% limitations under the License.
\name{omxMnor}
\alias{omxMnor}
\title{Multivariate Normal Integration}
\description{
Given a covariance matrix, a means vector, and vectors of lower and upper bounds, returns the multivariate normal integral across the space between bounds.
}
\usage{
omxMnor(covariance, means, lbound, ubound)
}
\arguments{
\item{covariance}{the covariance matrix describing the multivariate normal distribution.}
\item{means}{a row vector containing means of the variables of the underlying distribution.}
\item{lbound}{a row vector containing the lower bounds of the integration in each variable.}
\item{ubound}{a row vector containing the upper bounds of the integration in each variable.}
}
\details{
The order of columns in the \sQuote{means}, \sQuote{lbound}, and \sQuote{ubound} vectors are assumed to be the same as that of the covariance matrix. That is, means[i] is considered to be the mean of the variable whose variance is in covariance[i,i]. That variable will be integrated from lbound[i] to ubound[i] as part of the integration.
The value of ubound[i] or lbound[i] may be set to Inf or -Inf if a boundary at positive or negative infinity is desired.
For all i, ubound[i] must be strictly greater than lbound[i].
The algorithm for multivariate normal integration we use is Alan Genz's FORTRAN implementation of the SADMVN routine described by Genz (1992).
}
\references{
Genz, A. (1992). Numerical Computation of Multivariate Normal Probabilities. \emph{Journal of Computational Graphical Statistics, 1,} 141-149.
}
\examples{
data(myFADataRaw)
covariance <- cov(myFADataRaw[,1:3])
means <- colMeans(myFADataRaw[,1:3])
lbound <- c(-Inf, 0, 1) # Integrate from -Infinity to 0 on first variable
ubound <- c(0, Inf, 2.5) # From 0 to +Infinity on second, and from 1 to 2.5 on third
omxMnor(covariance, means, lbound, ubound)
# 0.0005995
# An alternative specification of the bounds follows
# Integrate from -Infinity to 0 on first variable
v1bound = c(-Inf, 0)
# From 0 to +Infinity on second
v2bound = c(0, Inf)
# and from 1 to 2.5 on third
v3bound = c(1, 2.5)
bounds <- cbind(v1bound, v2bound, v3bound)
lbound <- bounds[1,]
ubound <- bounds[2,]
omxMnor(covariance, means, lbound, ubound)
}
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