Package 'DRAYL'

Title: Computation of Rayleigh Densities of Arbitrary Dimension
Description: We offer an implementation of the series representation put forth in "A series representation for multidimensional Rayleigh distributions" by Wiegand and Nadarajah <DOI: 10.1002/dac.3510>. Furthermore we have implemented an integration approach proposed by Beaulieu et al. for 3 and 4-dimensional Rayleigh densities (Beaulieu, Zhang, "New simplest exact forms for the 3D and 4D multivariate Rayleigh PDFs with applications to antenna array geometrics", <DOI: 10.1109/TCOMM.2017.2709307>).
Authors: Martin Wiegand
Maintainer: Martin Wiegand <[email protected]>
License: GPL-2
Version: 1.0
Built: 2024-12-07 06:27:23 UTC
Source: CRAN

Help Index

Computation of Alpha coefficient matrix

Description

The alpha matrix is a necessary intermediate step in the series expansion approach. It lists the different parameter combinations necessary for the series expansion.

Usage

alphamatrix(n)

Arguments

n

Distribution dimension.

Value

Returns a n-1 dimensional matrix that contains the permutations of all indeces.

Examples

alphamatrix(3)

Auxilliary function computing factors.

Description

Auxilliary function, that evaluates coefficents for elements of the indices matrix.

Usage

btcol(col)

Arguments

col

Variables t,a and j to be combined

Value

Coefficients need to be computed for the entire permutation matrix of indices, this is the columnwise evaluation based on t,a and j.

Auxilliary function computing intermediate products.

Description

Auxilliary function. Based on the results of the btcol the row wise results are computed.

Usage

btprod(t,a,Jstar)

Arguments

t

Index number.
a

The respective Alpha matrix value.
Jstar

Matrix of the j-star indeces of the series expansion.

Value

Returns the row-wise multiplication of the coefficients based on the indeces j.

Three Dimensional Rayleigh Density by Integration

Description

A three dimensional Rayleigh density by integration.

Usage

drayl_int3D(r,omega,sigma,cor,method)

Arguments

r

Evaluation point.
omega

Omega construct necessary for the Integration method.
sigma

Variances of the signals.
cor

Correlation structure.
method

Integration methods, either "Kronrod","Clenshaw","Simpson","Romberg","TOMS614" or "mixed".

Value

Evaluates the 3D Rayleigh density at the point r, for the values omega,sigma and cor as specified by Bealieu's method.

Examples

# Matrix
K3 = matrix(0,nrow = 6,ncol = 6)
sigma3 = sqrt(c(0.5,1,1.5))
diag(K3) = c(0.5,0.5,1,1,1.5,1.5)

# rho_12 rho_13 rho_23
rho3<-c(0.9,0.8,0.7)

K3[1,3]=K3[3,1]=K3[2,4]=K3[4,2]=sigma3[1]*sigma3[2]*rho3[1]
K3[1,5]=K3[5,1]=K3[2,6]=K3[6,2]=sigma3[1]*sigma3[3]*rho3[2]
K3[3,5]=K3[5,3]=K3[4,6]=K3[6,4]=sigma3[2]*sigma3[3]*rho3[3]

cor3 = rho3

mat<-diag(3)
mat[1,2]=mat[2,1]=cor3[1]
mat[1,3]=mat[3,1]=cor3[2]
mat[2,3]=mat[3,2]=cor3[3]

omega3=mat

drayl_int3D(c(1,1,1),omega = omega3,sigma = sigma3,cor = cor3, method = "Romberg")

Four Dimensional Rayleigh Density by Integration

Description

A four dimensional Rayleigh density by integration.

Usage

drayl_int4D(r,omega,sigma,cor,method)

Arguments

r

Evaluation point.
omega

Omega construct necessary for the Integration method.
sigma

Variances of the signals.
cor

Correlation structure.
method

Integration methods, either "Romberg","Cubature" or "Quadrature".

Value

Evaluates the 4D Rayleigh density at the point r, for the values omega,sigma and cor as specified by Bealieu's method.

Examples

library("RConics")

K4 = matrix(0,nrow = 8,ncol = 8)
sigma4 = sqrt(c(0.5,1,1.5,1))
rho4<-c(0.7,0.75,0.8,0.7,0.75,0.7)

K4[1,1]=K4[2,2]=sigma4[1]^2
K4[3,3]=K4[4,4]=sigma4[2]^2
K4[5,5]=K4[6,6]=sigma4[3]^2
K4[7,7]=K4[8,8]=sigma4[4]^2

K4[1,3]=K4[3,1]=K4[2,4]=K4[4,2]=sigma4[1]*sigma4[2]*rho4[1]
K4[1,5]=K4[5,1]=K4[2,6]=K4[6,2]=sigma4[1]*sigma4[3]*rho4[2]
K4[1,7]=K4[7,1]=K4[2,8]=K4[8,2]=sigma4[1]*sigma4[4]*rho4[3]
K4[3,5]=K4[5,3]=K4[4,6]=K4[6,4]=sigma4[2]*sigma4[3]*rho4[4]
K4[3,7]=K4[7,3]=K4[4,8]=K4[8,4]=sigma4[2]*sigma4[4]*rho4[5]
K4[5,7]=K4[7,5]=K4[8,6]=K4[6,8]=sigma4[3]*sigma4[4]*rho4[6]

sigma4 = c(sqrt(c(K4[1,1],K4[3,3],K4[5,5],K4[7,7])))

cor4 = c(K4[1,3]/(sigma4[1]*sigma4[2]),
         K4[1,5]/(sigma4[1]*sigma4[3]),
         K4[1,7]/(sigma4[1]*sigma4[4]),
         K4[3,5]/(sigma4[2]*sigma4[3]),
         K4[3,7]/(sigma4[2]*sigma4[4]),
         K4[5,7]/(sigma4[3]*sigma4[4]))

omega4=omega4<-matrix(data = c(1,cor4[1],cor4[2],cor4[3],cor4[1],1,cor4[4],
                      cor4[5],cor4[2],cor4[4],1,cor4[6],cor4[3],cor4[5],cor4[6],1),nrow = 4)

drayl_int4D(c(1,1,1,1),omega = omega4,sigma = sigma4,cor = cor4, method = "Cubature")

Three dimensional Rayleigh density by series expansion

Description

Returns a 3D Rayleigh density for arbitrary covariance values. The resulting function can then be evaluated at arbitrary points.

Usage

drayl3D(dK,Ccomp,lim)

Arguments

dK

Determinant of the covariance matrix.
Ccomp

"Compressed" cofactor matrix, leaving out zero value entries.
lim

Number of series terms.

Value

The 3D Rayleigh density for the compressed cofactor matrix Ccomp of the covariance matrix. The function can then be evaluated for 3-dimensional vectors r.

Examples

library("RConics")

# Matrix
K3 = matrix(0,nrow = 6,ncol = 6)
sigma3 = sqrt(c(0.5,1,1.5))
diag(K3) = c(0.5,0.5,1,1,1.5,1.5)

# rho_12 rho_13 rho_23
rho3<-c(0.9,0.8,0.7)

K3[1,3]=K3[3,1]=K3[2,4]=K3[4,2]=sigma3[1]*sigma3[2]*rho3[1]
K3[1,5]=K3[5,1]=K3[2,6]=K3[6,2]=sigma3[1]*sigma3[3]*rho3[2]
K3[3,5]=K3[5,3]=K3[4,6]=K3[6,4]=sigma3[2]*sigma3[3]*rho3[3]


C3=adjoint(K3)
n = nrow(K3)/2
Ccomp3<-C3[seq(1,(2*n-1),2),][,seq(1,(2*n-1),2)]
dK3<-det(K3)


pdf3D<-drayl3D(dK = dK3, Ccomp = Ccomp3, lim = 3)

pdf3D(rep(1,3))

Four dimensional Rayleigh density by series expansion

Description

Returns a 4D Rayleigh density for arbitrary covariance values. The resulting function can then be evaluated at arbitrary points.

Usage

drayl4D(dK,Ccomp,lim)

Arguments

dK

Determinant of the covariance matrix.
Ccomp

"Compressed" cofactor matrix, leaving out zero value entries.
lim

Number of series terms.

Value

The 4D Rayleigh density for the compressed cofactor matrix Ccomp of the covariance matrix. The function can then be evaluated for 4-dimensional vectors r.

Examples

library("RConics")

K4 = matrix(0,nrow = 8,ncol = 8)
sigma4 = sqrt(c(0.5,1,1.5,1))
rho4<-c(0.7,0.75,0.8,0.7,0.75,0.7)

K4[1,1]=K4[2,2]=sigma4[1]^2
K4[3,3]=K4[4,4]=sigma4[2]^2
K4[5,5]=K4[6,6]=sigma4[3]^2
K4[7,7]=K4[8,8]=sigma4[4]^2

K4[1,3]=K4[3,1]=K4[2,4]=K4[4,2]=sigma4[1]*sigma4[2]*rho4[1]
K4[1,5]=K4[5,1]=K4[2,6]=K4[6,2]=sigma4[1]*sigma4[3]*rho4[2]
K4[1,7]=K4[7,1]=K4[2,8]=K4[8,2]=sigma4[1]*sigma4[4]*rho4[3]
K4[3,5]=K4[5,3]=K4[4,6]=K4[6,4]=sigma4[2]*sigma4[3]*rho4[4]
K4[3,7]=K4[7,3]=K4[4,8]=K4[8,4]=sigma4[2]*sigma4[4]*rho4[5]
K4[5,7]=K4[7,5]=K4[8,6]=K4[6,8]=sigma4[3]*sigma4[4]*rho4[6]

C4=adjoint(K4)
n = nrow(K4)/2
Ccomp4<-C4[seq(1,(2*n-1),2),][,seq(1,(2*n-1),2)]
dK4<-det(K4)

pdf4D<-drayl4D(dK = dK4, Ccomp = Ccomp4, lim = 3)
pdf4D(rep(1,4))

Non-zero value determination

Description

Determines the contribution of sum terms, based on the index j, rho and the matrix A.

Usage

zerooneoutput(j,rho,A)

Arguments

j

Vector of j indeces.
rho

Vector of the rho index.
A

Alpha matrix.

Value

Either 0 or 1, computes the integral contribution based on the alphamatrix A.

Examples

A = alphamatrix(3)
zerooneoutput(c(0,0,0),c(-1,-1,-1),A)