version 1.0

DESCRIPTION

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+Package: NORMT3
+Title: Evaluates complex erf, erfc and density of sum of Gaussian and Student's t
+Version: 1.0
+Date: 2005-01-07
+Author: Guy Nason <g.p.nason@bristol.ac.uk>
+Maintainer: Guy Nason <g.p.nason@bristol.ac.uk>
+Description: Evaluates the probability density function of the sum of the
+	Gaussian and Student's t density on 3 degrees of freedom.
+	Evaluates the p.d.f. of the sphered Student's t density function. 
+	Also evaluates the erf and erfc functions on complex-valued arguments.
+License: GPL version 2 or newer
+Packaged: Mon Jan 10 09:49:09 2005; magpn

R/dnormt3.R

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+"dnormt3" <-
+function(x, mean=0, sd=1){
+
+if (sd==0)	{
+	f <- rep(0, length(x))
+	f[x==0] <- Inf
+	}
+else	{
+	f <- sqrt(2)*normt3ip(mu=(x-mean)*sqrt(2)/sd, sigma=1)/sd
+	sv <- abs(x - mean)*sqrt(2)/sd >= 37
+	f[sv] <- 0
+	}
+f
+}

R/dst.R

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+"dst" <-
+function(x,nu=3){
+gamma( (nu+1)/2)/(gamma(nu/2)*sqrt(pi)*sqrt(nu-2)*( 1+ (x^2)/(nu-2))^((nu+1)/2))
+}

R/erf.R

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+"erf" <-
+function(z) 1 - erfc(z)

R/erfc.R

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+"erfc" <-
+function(z){
+
+ans <- .C("IPerfcvec",
+		x=as.double(Re(z)),
+		y=as.double(Im(z)),
+		ansx=as.double(Re(z)),
+		ansy=as.double(Im(z)),
+		n = as.integer(length(z)),
+		error = as.integer(0),
+		PACKAGE="NORMT3")
+if (ans$error != 0)
+	stop(paste("Error code from TOMS 680 was ", ans$error))
+
+return(complex(real=ans$ansx, im=ans$ansy))
+
+}

R/firstlib.R

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+.First.lib <- function(lib,pkg)
+{
+   library.dynam("NORMT3",pkg,lib)
+   cat("NORMT3 loaded\n")
+}

R/ic1.R

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+"ic1" <-
+function (p,d) 
+{
+cc <- complex(re=0, im=p/2)
+
+(sqrt(pi)/2)*exp(-(p^2)/4)*(1-Re(erf(cc) - erf(cc-d)))
+
+}

R/is1.R

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+"is1" <-
+function (p,d) 
+{
+
+cc <- complex(re=0, im=p/2)
+
+(sqrt(pi)/2)*exp(-(p^2)/4)*Im(erf(cc-d))
+}

R/normt3ip.R

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+"normt3ip" <-
+function (mu, sigma) 
+{
+a <- 1/(sigma^2)
+b <- sqrt(2)/sigma
+d <- a/b
+p <- mu*b
+
+T1 <- ((1-a)*cos(mu*a) + p*b*sin(mu*a)/2)* ic1(p,d)
+
+T2 <- ((1-a)*sin(mu*a) - p*b*cos(mu*a)/2)*is1(p,d)
+
+T3 <- b*exp( - d^2)/2
+
+total <- b*exp(a/2)*(T1+T2+T3)/pi
+total
+}

README

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+1. Put any C/C++/Fortran code in 'src'
+2. If you have compiled code, add a .First.lib() function in 'R'
+   to load the shared library
+3. Edit the help file skeletons in 'man'
+4. Run R CMD build to create the index files
+5. Run R CMD check to check the package
+6. Run R CMD build to make the package file
+
+
+Read "Writing R Extensions" for more information.

man/dnormt3.Rd

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+\name{dnormt3}
+\alias{dnormt3}
+\title{Density function of sum of Gaussian and Student's t on 3 df}
+\description{
+	Computes the probability density function of the sum of the
+	Gaussian distribution and the Student's t distribution on
+	3 degrees of freedom. 
+}
+\usage{
+dnormt3(x, mean = 0, sd = 1)
+}
+\arguments{
+  \item{x}{ Where to evaluate the density function}
+  \item{mean}{ The mean of the Gaussian}
+  \item{sd}{ The standard deviation of the Gaussian}
+}
+\details{
+	Computes the probability density function of the sum of the
+	Gaussian distribution and the Student's t distribution on
+	3 degrees of freedom. 
+}
+\value{The appropriate pdf value.
+}
+\references{ Nason, G.P. (2005) On the sum of the Gaussian and Student's t
+        random variables. \emph{Technical Report}, 05:01,
+        Statistics Group, Department of Mathematics, University of Bristol.}
+\author{ Guy Nason }
+
+\seealso{ \code{\link{normt3ip}},\code{\link{dst}},\code{\link{dnorm}}}
+
+\examples{
+dnormt3(0)
+#
+# Should be 0.4501582 = sqrt(2)/pi
+#
+x <- seq(from=-5, to=5, length=100)
+plot(x, dnormt3(x), type="l")	# Density plot
+}
+\keyword{ distribution }% at least one, from doc/KEYWORDS

man/dst.Rd

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+\name{dst}
+\alias{dst}
+\title{ Density function of sphered Student's t distribution}
+\description{
+	Evaluates probability density function of sphered Student's
+	t distribution on nu degrees of freedom. This is just the
+	standard Student's t distribution rescaled to have unit variance.
+}
+\usage{
+dst(x, nu = 3)
+}
+\arguments{
+  \item{x}{ Where to evaluate the density}
+  \item{nu}{ The degrees of freedom }
+}
+\details{Description says it all.
+}
+\value{
+	The appropriate probability density.
+}
+\references{Nason, G.P. (2001) Robust projection indices. \emph{Journal of
+	the Royal Statistical Society}, Series B, \bold{63}, 551--567. }
+\author{ Guy Nason}
+
+\seealso{ \code{\link{dnorm}}, \code{\link{dnormt3}}}
+\examples{
+dst(0)
+#
+# Should be  2/pi = 0.6366198
+#
+x <- seq(from=-5, to=5, length=100)
+plot(x, dst(x), type="l")	# Produces a density plot
+}
+\keyword{ distribution }% at least one, from doc/KEYWORDS

man/erf.Rd

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+\name{erf}
+\alias{erf}
+\title{Error function}
+\description{
+	Computes the error function of a (possibly) complex
+	valued argument. This function is 1-erfc(z).
+}
+\usage{
+erf(z)
+}
+\arguments{
+  \item{z}{Argument of error function}
+}
+\details{
+	Computes the error function of a (possibly) complex
+	valued argument by computing the complementary error function
+	and subtracting the answer from 1.
+}
+\value{
+	The error function of z
+}
+\references{
+Poppe, G.P.M. and Wijers, C.M.J. (1990) More efficient computation of the
+	complex error function. \emph{ACM Transactions on Mathematical
+	Software}, \bold{16}, 38--46.
+}
+\author{Guy P. Nason, Department of Mathematics, University of Bristol}
+
+
+\seealso{ \code{\link{erfc}}}
+\examples{
+erf(0)
+#
+# Should give 0
+#
+erf(1)
+#
+# Should give 0.8427008+0i
+#
+erf(complex(re=1, im=1))
+#
+# Should give 1.316151+0.1904535i
+#
+}
+\keyword{math}

man/erfc.Rd

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+\name{erfc}
+\alias{erfc}
+\title{Complementary error function}
+\description{
+	Computes the complementary error function of a (possibly) complex
+	valued argument. This function is
+	$2/\sqrt{\pi} \int_{z}^{\infty} \exp^{-t^2} dt$.
+}
+\usage{
+erfc(z)
+}
+\arguments{
+  \item{z}{Argument of complementary error function}
+}
+\details{
+	Computes the complementary error function of a (possibly) complex
+	valued argument. This function is
+	$2/\sqrt{\pi} \int_{z}^{\infty} \exp^{-t^2} dt$. This function
+	actually calls FORTRAN code (algorithm TOMS 680) which computes
+	the Faddeeva's function and then with a slight modification
+	obtains the erfc function of a complex-valued argument.
+}
+\value{
+	The complementary error function of z
+}
+\references{
+Poppe, G.P.M. and Wijers, C.M.J. (1990) More efficient computation of the
+	complex error function. \emph{ACM Transactions on Mathematical
+	Software}, \bold{16}, 38--46.
+}
+\author{Guy P. Nason, Department of Mathematics, University of Bristol}
+
+
+\seealso{\code{\link{erf}}}
+\examples{
+erfc(0)
+#
+# Should give 1
+#
+erfc(1)
+#
+# Should give 0.1572992+0i
+#
+erfc(complex(re=1, im=1))
+#
+# Should give -0.3161513-0.1904535i
+#
+}
+\keyword{math}

man/ic1.Rd

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+\name{ic1}
+\alias{ic1}
+\title{Compute IC1 formula from Nason (2005) }
+\description{
+	Computes $I_{C1}(p,d) = \int_{d}^{\infty} \cos (px) \exp (-x^2) dx$}
+}
+\usage{
+ic1(p, d)
+}
+%- maybe also 'usage' for other objects documented here.
+\arguments{
+  \item{p}{ Argument for IC1 function}
+  \item{d}{ Argument for IC1 function}
+}
+\details{
+	An intermediate function for the computation of the inner product
+	between Gaussian and Student's t distribution.
+}
+\value{
+	Value of the function IC1
+}
+\references{ Nason, G.P. (2005) On the sum of the Gaussian and Student's t
+	random variables. \emph{Technical Report}, 05:01,
+	Statistics Group, Department of Mathematics, University of Bristol.}
+\author{ Guy Nason }
+
+\seealso{ \code{\link{normt3ip}}}
+\examples{
+ic1(1,1)
+#
+# Should give 0.03401986.
+#
+}
+\keyword{ math }

man/is1.Rd

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+\name{is1}
+\alias{is1}
+\title{Compute IS1 formula from Nason (2005) }
+\description{
+	Computes $I_{S1}(p,d) = \int_{d}^{\infty} \cos (px) \exp (-x^2) dx$}
+}
+\usage{
+is1(p, d)
+}
+%- maybe also 'usage' for other objects documented here.
+\arguments{
+  \item{p}{ Argument for IS1 function}
+  \item{d}{ Argument for IS1 function}
+}
+\details{
+	An intermediate function for the computation of the inner product
+	between Gaussian and Student's t distribution.
+}
+\value{
+	Value of the function IS1
+}
+\references{ Nason, G.P. (2005) On the sum of the Gaussian and Student's t
+	random variables. \emph{Technical Report}, 05:01,
+	Statistics Group, Department of Mathematics, University of Bristol.}
+\author{ Guy Nason }
+
+\seealso{ \code{\link{normt3ip}}}
+\examples{
+is1(1,1)
+#
+# Should give 0.1297382.
+#
+}
+\keyword{ math }

man/normt3ip.Rd

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+\name{normt3ip}
+\alias{normt3ip}
+\title{Compute inner product function of Gaussian and t3 distribution}
+\description{This function evaluates the inner product function of
+	the (sphered) Student's t distribution on 3 df and the Gaussian
+	distribution as defined by Theorem~1 of Nason, 2005
+}
+\usage{
+normt3ip(mu, sigma)
+}
+\arguments{
+  \item{mu}{ Mean of the Gaussian}
+  \item{sigma}{ Standard deviation of the Gaussian}
+}
+\details{No extra details  }
+\value{
+	The evaluated function.
+}
+
+\references{ Nason, G.P. (2005) On the sum of the Gaussian and Student's t
+        random variables. \emph{Technical Report}, 05:01,
+        Statistics Group, Department of Mathematics, University of Bristol.}
+\author{ Guy Nason }
+
+
+%\seealso{\code{\link{ic1}},\code{\link{is1}},\code{\link{dnormt3}}}
+\examples{
+#normt3ip(0,1)
+#
+# Answer should be 0.3183099 which is 1/pi
+#
+x <- 3
+}
+\keyword{ math }