Package 'MultiStatM' reference manual

Title:	Multivariate Statistical Methods
Description:	Algorithms to build set partitions and commutator matrices and their use in the construction of multivariate d-Hermite polynomials; estimation and derivation of theoretical vector moments and vector cumulants of multivariate distributions; conversion formulae for multivariate moments and cumulants. Applications to estimation and derivation of multivariate measures of skewness and kurtosis; estimation and derivation of asymptotic covariances for d-variate Hermite polynomials, multivariate moments and cumulants and measures of skewness and kurtosis. The formulae implemented are discussed in Terdik (2021, ISBN:9783030813925), "Multivariate Statistical Methods".
Authors:	Gyorgy Terdik [aut], Emanuele Taufer [aut, cre]
Maintainer:	Emanuele Taufer <[email protected]>
License:	GPL-3
Version:	2.0.0
Built:	2025-02-22 02:54:18 UTC
Source:	https://github.com/cran/MultiStatM

Commutator Index

Description

This function calculates the commutator index based on the specified type. The available types are "Kmn", "Kperm", "Mixing", and "Moment". Depending on the selected type, the corresponding specific function is called.

Usage

CommutatorIndx(Type, ...)
CommutatorIndx(Type, ...)

Arguments

`Type`	a string specifying the type of commutator index to be calculated. Must be one of "Kmn", "Kperm", "Mixing", or "Moment".
`...`	additional arguments passed to the specific commutator function.

Details

The function 'CommutatorIndx' acts as a wrapper to call specific commutator functions based on the input 'Type'.

Type "Kmn":

Parameters:

m - Row-dimension.
n - Col-dimension.

Return:

A vector of indexes to provide the commutation, transforming vec A to vec of the transposed A.

Type "Kperm":

Parameters:

perm - Vector indicating the permutation of the order in the Kronecker product.
dims - Vector indicating the dimensions of the vectors.

Return:

An index vector to produce the permutation of the Kronecker products of vectors of any length.

Type "Mixing":

Parameters:

x - A vector of dimension prod(d1)*prod(d2).
d1 - Dimension of the first group of vectors.
d2 - Dimension of the second group of vectors.

Return:

A vector Kx representing the product of the moment commutator and the vector x.

Type "Moment":

Parameters:

x - A vector of length d^n where n is the length of el_rm.
el_rm - Type of a partition.
d - Dimensionality of the underlying multivariate distribution.

Return:

A vector Kx representing the product of the moment commutator and the vector x.

Value

A vector representing the commutator index.

Examples

# Kmn example
A <- 1:6
CommutatorIndx(Type = "Kmn", m = 3, n = 2)

# Kperm example
a1 <- c(1, 2)
a2 <- c(2, 3, 4)
a3 <- c(1, 3)
p1 <- a1 %x% a2 %x% a3
CommutatorIndx(Type = "Kperm", perm = c(3, 1, 2), dims = c(2, 3, 2))

# Mixing example
d1 <- c(2, 3, 2)
d2 <- c(3, 2, 2)
x <- 1:(prod(d1) * prod(d2))
CommutatorIndx(Type = "Mixing", x = x, d1 = d1, d2 = d2)

# Moment example
n <- 4
r <- 2
m <- 1
d <- 2
PTA <- PartitionTypeAll(n)
el_r <- PTA$eL_r[[r]][m, ]
x <- 1:d^n
CommutatorIndx(Type = "Moment", x = x, el_rm = el_r, d = d)

# Kmn example
A <- 1:6
CommutatorIndx(Type = "Kmn", m = 3, n = 2)

# Kperm example
a1 <- c(1, 2)
a2 <- c(2, 3, 4)
a3 <- c(1, 3)
p1 <- a1 %x% a2 %x% a3
CommutatorIndx(Type = "Kperm", perm = c(3, 1, 2), dims = c(2, 3, 2))

# Mixing example
d1 <- c(2, 3, 2)
d2 <- c(3, 2, 2)
x <- 1:(prod(d1) * prod(d2))
CommutatorIndx(Type = "Mixing", x = x, d1 = d1, d2 = d2)

# Moment example
n <- 4
r <- 2
m <- 1
d <- 2
PTA <- PartitionTypeAll(n)
el_r <- PTA$eL_r[[r]][m, ]
x <- 1:d^n
CommutatorIndx(Type = "Moment", x = x, el_rm = el_r, d = d)

Commutator Matrix

Description

This function generates various types of commutator matrices.

Usage

CommutatorMatr(Type, ...)
CommutatorMatr(Type, ...)

Arguments

`Type`	A string specifying the type of commutator matrix. Choices are "Kmn", "Kperm", "Mixing", or "Moment".
`...`	Additional arguments specific to the type of commutator matrix (see Details).

Details

The function CommutatorMatr supports the following types of commutator matrices:

Kmn

Description: Transforms vec(A) to vec(A^T), where A^T is the transpose of matrix A. An option for sparse matrix is provided. By default, a non-sparse matrix is produced. Using sparse matrices increases computation times but requires far less memory. Arguments:

m (integer): Number of rows of the first matrix.
n (integer): Number of columns of the first matrix.
useSparse (logical, optional): If TRUE, returns a sparse matrix. Default is FALSE.

Kperm

Description: Generates a commutation matrix for a specified permutation of matrix dimensions. An option for sparse matrix is provided. By default, a non-sparse matrix is produced. Using sparse matrices increases computation times but requires far less memory. Arguments:

perm (integer vector): The permutation vector.
dims (integer vector): The dimensions of the matrices involved.
useSparse (logical, optional): If TRUE, returns a sparse matrix. Default is FALSE.

Mixing

Description: Generates the Mixing commutation matrix used in linear algebra transformations involving tensor products. An option for sparse matrix is provided. By default, a non-sparse matrix is produced. Using sparse matrices increases computation times but requires far less memory. Arguments:

d1 (integer vector): Dimensions of the first set.
d2 (integer vector): Dimensions of the second set.
useSparse (logical, optional): If TRUE, returns a sparse matrix. Default is FALSE.

Moment

Description: Generates the Moment commutation matrix based on partitioning of moments. An option for sparse matrix is provided. By default, a non-sparse matrix is produced. Using sparse matrices increases computation times but requires far less memory. Arguments:

el_rm (integer vector): Elements of the partition.
d (integer): Dimension of the partition.
useSparse (logical, optional): If TRUE, returns a sparse matrix. Default is FALSE.

Value

Depending on the type:

Kmn: A commutation matrix of dimension $mn \times mn$ . If useSparse=TRUE, an object of class "dgCMatrix" is produced.
Kperm: A square permutation matrix of size prod(dims). If useSparse=TRUE, an object of class "dgCMatrix" is produced.
Mixing: A square matrix of dimension prod(d1) * prod(d2). If useSparse=TRUE, an object of class "dgCMatrix" is produced.
Moment: A commutator matrix for moment formulae.

Examples

# Example for Kmn
CommutatorMatr("Kmn", m = 3, n = 2)

# Example for Kperm
dims <- c(2, 3, 2)
perm <- c(1, 3, 2)
CommutatorMatr("Kperm", perm = perm, dims = dims)

# Example for Mixing
d1 <- c(2, 3, 2)
d2 <- c(3, 2, 2)
CommutatorMatr("Mixing", d1 = d1, d2 = d2)

# Example for Moment
n <- 4
r <- 2
m <- 1
d <- 2
PTA <- PartitionTypeAll(n)
el_r <- PTA$eL_r[[r]][m,]
CommutatorMatr("Moment", el_r = el_r, d = d)
# Example for Kmn
CommutatorMatr("Kmn", m = 3, n = 2)

# Example for Kperm
dims <- c(2, 3, 2)
perm <- c(1, 3, 2)
CommutatorMatr("Kperm", perm = perm, dims = dims)

# Example for Mixing
d1 <- c(2, 3, 2)
d2 <- c(3, 2, 2)
CommutatorMatr("Mixing", d1 = d1, d2 = d2)

# Example for Moment
n <- 4
r <- 2
m <- 1
d <- 2
PTA <- PartitionTypeAll(n)
el_r <- PTA$eL_r[[r]][m,]
CommutatorMatr("Moment", el_r = el_r, d = d)

Convert cumulants to moments (univariate and multivariate)

Description

Obtains a vector of moments from a vector of cumulants for either univariate or multivariate data.

Usage

Cum2Mom(cumulants, Type = c("Univariate", "Multivariate"))
Cum2Mom(cumulants, Type = c("Univariate", "Multivariate"))

Arguments

`cumulants`	Either a vector of univariate cumulants or a list of vectors of multivariate cumulants.
`Type`	A character string specifying the type of cumulants provided. Use "Univariate" for univariate cumulants and "Multivariate" for multivariate cumulants.

Value

The vector of moments if Type is "Univariate" or the list of vectors of moments if Type is "Multivariate".

References

Gy. Terdik, Multivariate statistical methods - going beyond the linear, Springer 2021. Section 3.4.

Examples

# Univariate example
cum_x <- c(1, 2, 3, 4)
Cum2Mom(cum_x, Type = "Univariate")

# Multivariate example
cum <- list(c(0,0), c(1,0,0,1), c(rep(0,8)), c(rep(0,16)), c(rep(0,32)))
Cum2Mom(cum, Type = "Multivariate")

# Univariate example
cum_x <- c(1, 2, 3, 4)
Cum2Mom(cum_x, Type = "Univariate")

# Multivariate example
cum <- list(c(0,0), c(1,0,0,1), c(rep(0,8)), c(rep(0,16)), c(rep(0,32)))
Cum2Mom(cum, Type = "Multivariate")

Distinct values selection vector

Description

Eliminates the duplicated/q-plicated elements in a T-vector of multivariate moments and cumulants. Produces the same results as EliminMatr. Note EliminIndx does not provide the same results as unique()

Usage

EliminIndx(d, q)
EliminIndx(d, q)

Arguments

`d`	dimension of a vector x
`q`	power of the Kronecker product

Value

A vector of indexes of the distinct elements in the T-vector

References

Gy. Terdik, Multivariate statistical methods - going beyond the linear, Springer 2021. Section 1.3.2 Multi-Indexing, Elimination, and Duplication, p.21,(1.32)

Examples

x<-c(1,0,3)
y<-kronecker(x,kronecker(x,x))
y[EliminIndx(3,3)]
## Not the same results as
unique(y)

x<-c(1,0,3)
y<-kronecker(x,kronecker(x,x))
y[EliminIndx(3,3)]
## Not the same results as
unique(y)

Elimination Matrix

Description

Eliminates the duplicated/q-plicated elements in a T-vector of multivariate moments and cumulants.

Usage

EliminMatr(d, q, useSparse = FALSE)
EliminMatr(d, q, useSparse = FALSE)

Arguments

`d`	dimension of a vector x
`q`	power of the Kronecker product
`useSparse`	TRUE or FALSE.

Value

Elimination matrix of order $\eta_{d,q} \times d^q= {d+q-1 \choose q}$ . If useSparse=TRUE an object of the class "dgCMatrix" is produced.

References

Gy. Terdik, Multivariate statistical methods - going beyond the linear, Springer 2021. Section 1.3.2 Multi-Indexing, Elimination, and Duplication, p.21,(1.32)

Examples

x<-c(1,2,3)
y<-kronecker(kronecker(x,x),x)
## Distinct elements of y
z<-as.matrix(EliminMatr(3,3))%*%y
## Restore eliminated elements in z
as.vector(QplicMatr(3,3)%*%z)

x<-c(1,2,3)
y<-kronecker(kronecker(x,x),x)
## Distinct elements of y
z<-as.matrix(EliminMatr(3,3))%*%y
## Restore eliminated elements in z
as.vector(QplicMatr(3,3)%*%z)

EVSK multivariate Skew Normal

Description

Computes the theoretical values of the mean vector, covariance, skewness vector, total skenwness, kurtosis vector and total kurtosis for the multivariate Skew Normal distribution

Usage

EVSKSkewNorm(omega, alpha)
EVSKSkewNorm(omega, alpha)

Arguments

`omega`	A $d \times d$ correlation matrix
`alpha`	shape parameter d-vector

Value

A list of theoretical values for the mean vector, covariance, skewness vector, total skenwness, kurtosis vector and total kurtosis

References

Gy.Terdik, Multivariate statistical methods - Going beyond the linear, Springer 2021 (5.5) p.247

S. R. Jammalamadaka, E. Taufer, Gy. Terdik. On multivariate skewness and kurtosis. Sankhya A, 83(2), 607-644.

Examples

alpha<-c(10,5,0)
omega<-diag(3)
EVSKSkewNorm(omega,alpha)
alpha<-c(10,5,0)
omega<-diag(3)
EVSKSkewNorm(omega,alpha)

EVSK of the Uniform distribution on the sphere or its modulus

Description

Cumulants (up to the 4th order), skewness, and kurtosis of the d-variate Uniform distribution on the sphere or the modulus of the d-variate Uniform distribution on the sphere.

Usage

EVSKUniS(d, nCum = TRUE, Type = c("Standard", "Modulus"))
EVSKUniS(d, nCum = TRUE, Type = c("Standard", "Modulus"))

Arguments

`d`	dimensions
`nCum`	if it is FALSE then moments (up to the 4th order) are calculated.
`Type`	specify the type of distribution: "Standard" for the Uniform distribution on the sphere, or "Modulus" for the modulus of the Uniform distribution on the sphere.

Value

A list of computed moments and cumulants.

When Type is "Standard":

`EU1`	Mean vector
`varU`	Covariance matrix
`Skew.U`	Skewness vector (always zero)
`Skew.tot`	Total skewness (always zero)
`Kurt.U`	Kurtosis vector
`Kurt.tot`	Total kurtosis

When Type is "Modulus":

`EU1`	Mean vector
`varU`	Covariance matrix
`EU.k`	List of moments up to 4th order
`cumU.k`	List of cumulants up to 4th order
`skew.U`	Skewness vector
`kurt.U`	Kurtosis vector

References

Gy. Terdik, Multivariate statistical methods - Going beyond the linear, Springer 2021 Proposition 5.3 p.297

S. R. Jammalamadaka, E. Taufer, Gy. Terdik. On multivariate skewness and kurtosis. Sankhya A, 83(2), 607-644.

Gy. Terdik, Multivariate statistical methods - Going beyond the linear, Springer 2021, Lemma 5.12 p.298

Examples

# Example for Standard type
EVSKUniS(d=3, Type="Standard")

# Example for Modulus type
EVSKUniS(d=3, Type="Modulus")
# Example for Standard type
EVSKUniS(d=3, Type="Standard")

# Example for Modulus type
EVSKUniS(d=3, Type="Modulus")

Coefficients of Hermite polynomials

Description

Provides the coefficients of Hermite polynomials, either univariate or multivariate.

Usage

HermiteCoeff(Type, N, d = NULL)
HermiteCoeff(Type, N, d = NULL)

Arguments

`Type`	A character string specifying the type of Hermite polynomial. Must be either "Univariate" or "Multivariate".
`N`	The order of polynomial. Required for both types.
`d`	The dimension of the d-variate X. Required only for multivariate type.

Value

For 'Type = "Univariate"', returns a vector of coefficients of $x^N$ , $x^{N-2}$ , etc. For 'Type = "Multivariate"', returns a list of matrices of coefficients for the d-variate polynomials from 1 to N.

References

Gy. Terdik, Multivariate statistical methods - going beyond the linear, Springer 2021. Sections 4.4 (4.24) and 4.6.2, p. 223, Remark 4.8

Examples

# Univariate example
H_uni <- HermiteCoeff(Type = "Univariate", N = 5)

# Multivariate example
N <- 5; d <- 3
H_multi <- HermiteCoeff(Type = "Multivariate", N = N, d = d)
X <- c(1:3)
X3 <- kronecker(X, kronecker(X, X))
X5 <- kronecker(X3, kronecker(X, X))
Idv <- as.vector(diag(d)) # vector of variance matrix
# value of H5 at X is
vH5 <- H_multi[[1]] %*% X5 + H_multi[[2]] %*% kronecker(Idv, X3) +
  H_multi[[3]] %*% kronecker(kronecker(Idv, Idv), X)
# Univariate example
H_uni <- HermiteCoeff(Type = "Univariate", N = 5)

# Multivariate example
N <- 5; d <- 3
H_multi <- HermiteCoeff(Type = "Multivariate", N = N, d = d)
X <- c(1:3)
X3 <- kronecker(X, kronecker(X, X))
X5 <- kronecker(X3, kronecker(X, X))
Idv <- as.vector(diag(d)) # vector of variance matrix
# value of H5 at X is
vH5 <- H_multi[[1]] %*% X5 + H_multi[[2]] %*% kronecker(Idv, X3) +
  H_multi[[3]] %*% kronecker(kronecker(Idv, Idv), X)

Covariance matrix for multivariate T-Hermite polynomials

Description

Computation of the covariance matrix between d-variate T-Hermite polynomials $H_N(X_1)$ and $H_N(X_2)$ .

Usage

HermiteCov12(SigX12, N)
HermiteCov12(SigX12, N)

Arguments

`SigX12`	Covariance matrix of the Gaussian vectors X1 and X2 respectively of dimensions d1 and d2
`N`	Common degree of the multivariate Hermite polynomials

Value

Covariance matrix of $H_N(X_1)$ and $H_N(X_2)$

References

Gy.Terdik, Multivariate statistical methods - going beyond the linear, Springer 2021. (4.59), (4.66),

Examples

Covmat<-matrix(c(1,0.8,0.8,1),2,2)
Cov_X1_X2 <- HermiteCov12(Covmat,3)

Covmat<-matrix(c(1,0.8,0.8,1),2,2)
Cov_X1_X2 <- HermiteCov12(Covmat,3)

Hermite Polynomials (Univariate and Multivariate)

Description

Computes either univariate or multivariate Hermite polynomials up to a specified order.

Usage

HermiteN(x, N, Type, sigma2 = 1, Sig2 = diag(length(x)))
HermiteN(x, N, Type, sigma2 = 1, Sig2 = diag(length(x)))

Arguments

`x`	A scalar (for univariate) or a vector (for multivariate) at which to evaluate the Hermite polynomials.
`N`	The maximum order of the polynomials.
`Type`	A character string specifying the type of Hermite polynomials to compute. Can be either "Univariate" or "Multivariate".
`sigma2`	The variance for univariate Hermite polynomials. Default is 1. (Only used if Type is "Univariate").
`Sig2`	The covariance matrix for multivariate Hermite polynomials. Default is the unit matrix diag(length(x)). (Only used if Type is "Multivariate").

Details

Depending on the value of the 'Type' parameter, this function computes either the univariate or the multivariate Hermite polynomials.

Value

Depending on the type, the function returns:

Univariate: A vector of univariate Hermite polynomials with degrees from 1 to N evaluated at x.
Multivariate: A list of multivariate polynomials of order from 1 to N evaluated at vector x.

References

Gy.Terdik, Multivariate statistical methods - going beyond the linear, Springer 2021. Section 4.1 (for univariate), Section 4.6.2, (4.73), p.223 (for multivariate).

Examples

# Univariate example
HermiteN(x = 1, N = 3, Type = "Univariate")

# Multivariate example
HermiteN(x = c(1, 3), N = 3, Type = "Multivariate", Sig2 = diag(2))

# Univariate example
HermiteN(x = 1, N = 3, Type = "Univariate")

# Multivariate example
HermiteN(x = c(1, 3), N = 3, Type = "Multivariate", Sig2 = diag(2))

Inverse Hermite Polynomial

Description

Compute the inverse of univariate or multivariate Hermite polynomials.

Usage

HermiteN2X(Type, H_N, N, Sig2 = NULL)
HermiteN2X(Type, H_N, N, Sig2 = NULL)

Arguments

`Type`	A string specifying the type of Hermite polynomial inversion. Must be either "Univariate" or "Multivariate".
`H_N`	Input Hermite polynomials. For univariate, it is a vector. For multivariate, it is a list.
`N`	The highest polynomial order.
`Sig2`	The variance matrix of x for multivariate, or variance for univariate. Defaults to identity matrix for multivariate and 1 for univariate.

Details

This function computes the powers of x when Hermite polynomials are given. Depending on the type specified, it handles either univariate or multivariate Hermite polynomials.

Value

A list of x powers: $x$ , $x^{\otimes 2}$ , ... , $x^{\otimes N}$ for multivariate, or a vector of x powers: $x^n$ , $n=1:N$ for univariate.

References

Gy.Terdik, Multivariate statistical methods - going beyond the linear, Springer 2021. Section 4.6.2, (4.72), p.223 and Section 4.4, (4.23), p.198.

Examples

# Univariate example
H_N_x <- c(1, 2, 3, 4)
x_powers <- HermiteN2X(Type = "Univariate", H_N = H_N_x, N = 4, Sig2 = 1)

# Multivariate example
x <- c(1, 3)
Sig2 <- diag(length(x))
N <- 4
H_N_X <- HermiteN(x, N, Type="Multivariate")
x_ad_n <- HermiteN2X(Type = "Multivariate", H_N = H_N_X, N = N, Sig2 = Sig2)

# Univariate example
H_N_x <- c(1, 2, 3, 4)
x_powers <- HermiteN2X(Type = "Univariate", H_N = H_N_x, N = 4, Sig2 = 1)

# Multivariate example
x <- c(1, 3)
Sig2 <- diag(length(x))
N <- 4
H_N_X <- HermiteN(x, N, Type="Multivariate")
x_ad_n <- HermiteN2X(Type = "Multivariate", H_N = H_N_X, N = N, Sig2 = Sig2)

Convert moments to cumulants (univariate and multivariate)

Description

Obtains a vector of cumulants from a vector of moments for either univariate or multivariate data.

Usage

Mom2Cum(moments, Type = c("Univariate", "Multivariate"))
Mom2Cum(moments, Type = c("Univariate", "Multivariate"))

Arguments

`moments`	Either a vector of univariate moments or a list of vectors of multivariate moments.
`Type`	A character string specifying the type of moments provided. Use "Univariate" for univariate moments and "Multivariate" for multivariate moments.

Value

The vector of cumulants if Type is "Univariate" or the list of vectors of cumulants if Type is "Multivariate".

References

Gy. Terdik, Multivariate statistical methods - going beyond the linear, Springer 2021. Section 3.4.

Examples

# Univariate example
mu_x <- c(1, 2, 3, 4)
Mom2Cum(mu_x, Type = "Univariate")

# Multivariate example
mu <- list(c(0,0), c(1,0,0,1), c(0,0,0,0,0,0,0,0), c(3,0,0,1,0,1,1,0,0,1,1,0,1,0,0,3), c(rep(0,32)))
Mom2Cum(mu, Type = "Multivariate")

# Univariate example
mu_x <- c(1, 2, 3, 4)
Mom2Cum(mu_x, Type = "Univariate")

# Multivariate example
mu <- list(c(0,0), c(1,0,0,1), c(0,0,0,0,0,0,0,0), c(3,0,0,1,0,1,1,0,0,1,1,0,1,0,0,3), c(rep(0,32)))
Mom2Cum(mu, Type = "Multivariate")

Moments and cumulants CFUSN

Description

Provides the theoretical cumulants of the multivariate Canonical Fundamental Skew Normal distribution

Usage

MomCumCFUSN(r, d, p, Delta, nMu = FALSE)
MomCumCFUSN(r, d, p, Delta, nMu = FALSE)

Arguments

`r`	The highest cumulant order
`d`	The multivariate dimension and number of rows of the skewness matrix Delta
`p`	The number of cols of the skewness matrix Delta
`Delta`	The skewness matrix
`nMu`	If set to TRUE, the list of the first r d-variate moments is provided

Value

The list of theoretical cumulants in vector form

References

Gy.Terdik, Multivariate statistical methods - Going beyond the linear, Springer 2021, Lemma 5.3 p.251

Examples

r <- 4; d <- 2; p <- 3
Lamd <-  matrix(sample(1:50-25, d*p), nrow=d)
ieg<- eigen(diag(p)+t(Lamd)%*%Lamd)
V <- ieg$vectors
Delta <-Lamd %*% V %*% diag(1/sqrt(ieg$values)) %*% t(V)
MomCum <- MomCumCFUSN(r,d,p,Delta)
r <- 4; d <- 2; p <- 3
Lamd <-  matrix(sample(1:50-25, d*p), nrow=d)
ieg<- eigen(diag(p)+t(Lamd)%*%Lamd)
V <- ieg$vectors
Delta <-Lamd %*% V %*% diag(1/sqrt(ieg$values)) %*% t(V)
MomCum <- MomCumCFUSN(r,d,p,Delta)

Moments and cumulants d-variate Skew Normal

Description

Computes the theoretical values of moments and cumulants up to the r-th order. Warning: if nMu = TRUE it can be very slow

Usage

MomCumSkewNorm(r = 4, omega, alpha, nMu = FALSE)
MomCumSkewNorm(r = 4, omega, alpha, nMu = FALSE)

Arguments

`r`	the highest moment and cumulant order
`omega`	A $d \times d$ correlation matrix
`alpha`	shape parameter d-vector
`nMu`	if it is TRUE then moments are calculated as well

Value

A list of theoretical moments and cumulants

References

Gy.Terdik, Multivariate statistical methods - Going beyond the linear, Springer 2021 (5.5) p.247, Lemma 5.1 p. 246

S. R. Jammalamadaka, E. Taufer, Gy. Terdik. On multivariate skewness and kurtosis. Sankhya A, 83(2), 607-644.

Examples

alpha<-c(10,5,0)
omega<-diag(3)
MomCumSkewNorm(r=4,omega,alpha)
alpha<-c(10,5,0)
omega<-diag(3)
MomCumSkewNorm(r=4,omega,alpha)

Moments and cumulants Uniform Distribution on the Sphere

Description

By default, only moments are provided

Usage

MomCumUniS(r, d, nCum = FALSE)
MomCumUniS(r, d, nCum = FALSE)

Arguments

`r`	highest order of moments and cumulants
`d`	dimension
`nCum`	if it is TRUE then cumulants are calculated

Value

The list of moments and cumulants in vector form

References

Gy.Terdik, Multivariate statistical methods - Going beyond the linear, Springer 2021 Proposition 5.3 p.297

Examples

# The first four moments for d=3
MomCumUniS(4,3,nCum=0)
# The first four moments and cumulants for d=3
MomCumUniS(4,3,nCum=4)
# The first four moments for d=3
MomCumUniS(4,3,nCum=0)
# The first four moments and cumulants for d=3
MomCumUniS(4,3,nCum=4)

Moments and Cumulants of the Central Folded Normal Distribution

Description

Provides the theoretical moments and cumulants of the Central Folded Normal distribution. Depending on the choice of 'Type', either the univariate or d-variate distribution is used.

Usage

MomCumZabs(r, d, Type, nCum = FALSE)
MomCumZabs(r, d, Type, nCum = FALSE)

Arguments

`r`	The highest moment (cumulant) order.
`d`	Integer; the dimension of the distribution. Must be 1 when 'Type' is "Univariate" and greater than 1 when 'Type' is "Multivariate".
`Type`	Character; specifies the type of distribution. Must be either "Univariate" or "Multivariate".
`nCum`	Logical; if TRUE, then cumulants are calculated.

Value

A list containing moments and optionally cumulants.

For "Univariate" type:
- MuZ: The moments of the univariate Central Folded Normal distribution.
- CumZ: The cumulants of the univariate Central Folded Normal distribution.
For "Multivariate" type:
- MuZ: The moments of the d-variate Central Folded Normal distribution.
- CumZ: The cumulants of the d-variate Central Folded Normal distribution.

References

Gy.Terdik, Multivariate statistical methods - Going beyond the linear, Springer 2021, Proposition 5.1 p.242 and formula: p. 301

Examples

# Univariate case: The first three moments
MomCumZabs(3, 1, Type = "Univariate")
# Univariate case: The first three moments and cumulants
MomCumZabs(3, 1, Type = "Univariate",nCum = TRUE)
# d-variate case: The first three moments
MomCumZabs(3, 2, Type = "Multivariate" )
# d-variate case: The first three moments and cumulants
MomCumZabs(3, d=2, Type = "Multivariate",  nCum = TRUE)
# Univariate case: The first three moments
MomCumZabs(3, 1, Type = "Univariate")
# Univariate case: The first three moments and cumulants
MomCumZabs(3, 1, Type = "Univariate",nCum = TRUE)
# d-variate case: The first three moments
MomCumZabs(3, 2, Type = "Multivariate" )
# d-variate case: The first three moments and cumulants
MomCumZabs(3, d=2, Type = "Multivariate",  nCum = TRUE)

Standardize multivariate data

Description

For data formed by d-variate vectors x with sample covariance S and sample mean M, it computes the values $z=S^{-1/2}(x-M)$

Usage

MVStandardize(x)
MVStandardize(x)

Arguments

`x`	a multivariate data matrix, sample size is the number of rows

Value

a matrix of multivariate data with null mean vector and identity sample covariance matrix

Examples

x<-MASS::mvrnorm(1000,c(0,0,1,3),diag(4))
z<-MVStandardize(x)
mu_z<- apply(z,2,mean)
cov_z<- cov(z)

x<-MASS::mvrnorm(1000,c(0,0,1,3),diag(4))
z<-MVStandardize(x)
mu_z<- apply(z,2,mean)
cov_z<- cov(z)

General Partition Function

Description

A unified function to compute different types of partitions. Depending on the partition type specified, it calls the appropriate function: Partition_2Perm, Partition_DiagramsClosedNoLoops, Partition_Indecomposable, or Partition_Pairs.

Usage

Partitions(Type, ...)
Partitions(Type, ...)

Arguments

Type

A character string specifying the type of partion to compute. Choose from "2Perm", "Diagram", "Indecomp", "Pairs".

...

Additional arguments passed to the specific partition function:

For "2Perm", "Diagram" and "Indecomp":

L: A partition matrix.

For "Pairs":

N: An integer specifying the number of elements to be partitioned.

Value

Depending on the commutator type:

2Perm: A vector with the elements 1 to N permuted according to L.
Diagram: The list of partition matrices indecomposable with respect to L, representing diagrams without loops.
Indecomp: A list of partition matrices indecomposable with respect to L and a vector indicating the number of indecomposable partitions by sizes.
Pairs: The list of partition matrices with blocks containing two elements. The list is empty if N is odd.

Examples

# Example for 2Perm
PA <- PartitionTypeAll(4)
Partitions("2Perm", L = PA$Part.class[[3]])

# Example for Diagram
L <- matrix(c(1,1,0,0,0,0,1,1),2,4,byrow=TRUE)
Partitions("Diagram", L = L)

# Example for Indecomp
L <- matrix(c(1,1,0,0,0,0,1,1),2,4,byrow=TRUE)
Partitions("Indecomp", L = L)

# Example for Pairs
Partitions("Pairs", N = 4)
# Example for 2Perm
PA <- PartitionTypeAll(4)
Partitions("2Perm", L = PA$Part.class[[3]])

# Example for Diagram
L <- matrix(c(1,1,0,0,0,0,1,1),2,4,byrow=TRUE)
Partitions("Diagram", L = L)

# Example for Indecomp
L <- matrix(c(1,1,0,0,0,0,1,1),2,4,byrow=TRUE)
Partitions("Indecomp", L = L)

# Example for Pairs
Partitions("Pairs", N = 4)

Partitions, type and number of partitions

Description

Generates all partitions of N numbers and classify them by type

Usage

PartitionTypeAll(N)
PartitionTypeAll(N)

Arguments

`N`	The (integer) number of elements to be partitioned

Value

Part.class The list of all possible partitions given as partition matrices

S_N_r A vector with the number of partitions of size r=1, r=2, etc. (Stirling numbers of second kind )

eL_r A list of partition types with respect to partitions of size r=1, r=2, etc.

S_r_j Vectors of number of partitions with given types grouped by partitions of size r=1, r=2, etc.

References

Gy. Terdik, Multivariate statistical methods - going beyond the linear, Springer 2021. Case 1.4, p.31 and Example 1.18, p.32.

Examples

# See Example 1.18, p. 32, reference below
PTA<-PartitionTypeAll(4)
# Partitions generated
PTA$Part.class
# Partitions of size 2 includes two types
PTA$eL_r[[2]]
# Number of partitions with r=1 blocks, r=2 blocks, etc-
PTA$S_N_r
# Number of different types collected by  partitions of size r=1, r=2, etc.
PTA$S_r_j
# Partitions with size r=2, includes two types (above) each with number
PTA$S_r_j[[2]]

# See Example 1.18, p. 32, reference below
PTA<-PartitionTypeAll(4)
# Partitions generated
PTA$Part.class
# Partitions of size 2 includes two types
PTA$eL_r[[2]]
# Number of partitions with r=1 blocks, r=2 blocks, etc-
PTA$S_N_r
# Number of different types collected by  partitions of size r=1, r=2, etc.
PTA$S_r_j
# Partitions with size r=2, includes two types (above) each with number
PTA$S_r_j[[2]]

Inverse of a Permutation

Description

Inverse of a Permutation

Usage

PermutationInv(permutation0)
PermutationInv(permutation0)

Arguments

permutation0

A permutation of numbers 1:n

Value

A vector containing the inverse permutation of permutation0

References

Gy. Terdik, Multivariate statistical methods - going beyond the linear, Springer 2021, Remark 1.1, p.2

Qplication vector

Description

Restores the duplicated/q-plicated elements which are eliminated by EliminMatr or EliminIndx in a T-product of vectors of dimension d. It produces the same results as QplicMatr.

Usage

QplicIndx(d, q)
QplicIndx(d, q)

Arguments

`d`	dimension of the vectors in the T-product
`q`	power of the Kronecker product

Value

A vector (T-vector) with all elements previously eliminated by EliminIndx

References

Gy. Terdik, Multivariate statistical methods - going beyond the linear, Springer 2021, p.21, (1.31)

Examples

x<-c(1,2,3)
y<-kronecker(kronecker(x,x),x)
## Distinct elements of y
z<-y[EliminIndx(3,3)]
## Restore eliminated elements in z
z[QplicIndx(3,3)]

x<-c(1,2,3)
y<-kronecker(kronecker(x,x),x)
## Distinct elements of y
z<-y[EliminIndx(3,3)]
## Restore eliminated elements in z
z[QplicIndx(3,3)]

Qplication Matrix

Description

Restores the duplicated/q-plicated elements which are eliminated by EliminMatr in a T-product of vectors of dimension d.

Usage

QplicMatr(d, q, useSparse = FALSE)
QplicMatr(d, q, useSparse = FALSE)

Arguments

`d`	dimension of a vector x
`q`	power of the Kronecker product
`useSparse`	TRUE or FALSE.

Details

Note: since the algorithm of elimination is not unique, q-plication works together with the function EliminMatr only.

Value

Qplication matrix of order $d^q \times \eta_{d,q}$ , see (1.30), p.15. If useSparse=TRUE an object of the class "dgCMatrix" is produced.

References

Gy. Terdik, Multivariate statistical methods - going beyond the linear, Springer 2021, p.21, (1.31)

Examples

x<-c(1,2,3)
y<-kronecker(kronecker(x,x),x)
## Distinct elements of y
z<-as.matrix(EliminMatr(3,3))%*%y
## Restore eliminated elements in z
as.vector(QplicMatr(3,3)%*%z)

x<-c(1,2,3)
y<-kronecker(kronecker(x,x),x)
## Distinct elements of y
z<-as.matrix(EliminMatr(3,3))%*%y
## Restore eliminated elements in z
as.vector(QplicMatr(3,3)%*%z)

Random multivariate CFUSN

Description

Generate random d-vectors from the multivariate Canonical Fundamental Skew-Normal (CFUSN) distribution

Usage

rCFUSN(n, Delta)
rCFUSN(n, Delta)

Arguments

`n`	The number of variates to be generated
`Delta`	Correlation matrix, the skewness matrix Delta

Value

A random matrix $n \times d$

References

Gy.Terdik, Multivariate statistical methods - Going beyond the linear, Springer 2021 (5.5) p.247

S. R. Jammalamadaka, E. Taufer, Gy. Terdik. On multivariate skewness and kurtosis. Sankhya A, 83(2), 607-644.

Examples

d <- 2; p <- 3
Lamd <-  matrix(sample(1:50-25, d*p), nrow=d)
ieg<- eigen(diag(p)+t(Lamd)%*%Lamd)
V <- ieg$vectors
Delta <-Lamd %*% V %*% diag(1/sqrt(ieg$values)) %*% t(V)
x<-rCFUSN(20,Delta)

d <- 2; p <- 3
Lamd <-  matrix(sample(1:50-25, d*p), nrow=d)
ieg<- eigen(diag(p)+t(Lamd)%*%Lamd)
V <- ieg$vectors
Delta <-Lamd %*% V %*% diag(1/sqrt(ieg$values)) %*% t(V)
x<-rCFUSN(20,Delta)

Random multivariate CFUSSD

Description

Generate random d-vectors from the multivariate Canonical Fundamental Skew-Spherical distribution (CFUSSD) with Gamma generator

Usage

rCFUSSD(n, d, p, a, b, Delta)
rCFUSSD(n, d, p, a, b, Delta)

Arguments

`n`	sample size
`d`	dimension
`p`	dimension of the first term of (5.5)
`a`	shape parameter of the Gamma generator
`b`	scale parameter of the Gamma generator
`Delta`	skewness matrix

Value

A matrix of $n \times d$ random numbers

References

Gy.Terdik, Multivariate statistical methods - Going beyond the linear, Springer 2021, (5.36) p. 266, (see p.247 for Delta)

Examples

n <- 10^3; d <- 2; p <- 3 ; a <- 1; b <- 1
Lamd <-  matrix(sample(1:50-25, d*p), nrow=d)
ieg<- eigen(diag(p)+t(Lamd)%*%Lamd)
V <- ieg$vectors
Delta <-Lamd %*% V %*% diag(1/sqrt(ieg$values)) %*% t(V)
rCFUSSD(20,d,p,1,1,Delta)
n <- 10^3; d <- 2; p <- 3 ; a <- 1; b <- 1
Lamd <-  matrix(sample(1:50-25, d*p), nrow=d)
ieg<- eigen(diag(p)+t(Lamd)%*%Lamd)
V <- ieg$vectors
Delta <-Lamd %*% V %*% diag(1/sqrt(ieg$values)) %*% t(V)
rCFUSSD(20,d,p,1,1,Delta)

Random Multivariate Skew Normal

Description

Generate random d-vectors from the multivariate Skew Normal distribution

Usage

rSkewNorm(n, omega, alpha)
rSkewNorm(n, omega, alpha)

Arguments

`n`	sample size
`omega`	correlation matrix with d dimension
`alpha`	shape parameter vector of dimension d

Value

A random matrix $n \times d$

References

Azzalini, A. with the collaboration of Capitanio, A. (2014). The Skew-Normal and Related Families. Cambridge University Press, IMS Monographs series.

Gy.H.Terdik, Multivariate statistical methods - Going beyond the linear, Springer 2021, Section 5.1.2

Examples

alpha<-c(10,5,0)
omega<-diag(3)
x<-rSkewNorm(20,omega,alpha)

alpha<-c(10,5,0)
omega<-diag(3)
x<-rSkewNorm(20,omega,alpha)

Random Uniform on the sphere

Description

Generate random d-vectors from the Uniform distribution on the sphere

Usage

rUniS(n, d)
rUniS(n, d)

Arguments

`n`	sample size
`d`	dimension

Value

A random matrix $n \times d$

References

Gy.Terdik, Multivariate statistical methods - Going beyond the linear, Springer 2021

S. R. Jammalamadaka, E. Taufer, Gy. Terdik. On multivariate skewness and kurtosis. Sankhya A, 83(2), 607-644.

Estimation of multivariate Mean, Variance, T-Skewness and T-Kurtosis vectors

Description

Provides estimates of mean, variance, skewness and kurtosis vectors for d-variate data

Usage

SampleEVSK(X)
SampleEVSK(X)

Arguments

`X`	d-variate data vector

Value

The list of the estimated mean, variance, skewness and kurtosis vectors

References

Gy.Terdik, Multivariate statistical methods - going beyond the linear, Springer 2021, Sections 6.4.1 and 6.5.1

Examples

x<- MASS::mvrnorm(100,rep(0,3), 3*diag(rep(1,3)))
EVSK<-SampleEVSK(x)
names(EVSK)
EVSK$estSkew
x<- MASS::mvrnorm(100,rep(0,3), 3*diag(rep(1,3)))
EVSK<-SampleEVSK(x)
names(EVSK)
EVSK$estSkew

Gram-Charlier approximation to a multivariate density

Description

Provides the truncated Gram-Charlier approximation to a multivariate density. Approximation can be up to the first k=8 cumulants.

Usage

SampleGC(X, k = 4, cum = NULL)
SampleGC(X, k = 4, cum = NULL)

Arguments

`X`	A matrix of d-variate data
`k`	the order of the approximation, by default set to 4; (k must not be smaller than 3 or greater than 8)
`cum`	if NULL (default) the cumulant vector is estimated from X. If `cum` is provided no estimation of cumulants is performed.

Value

The vector of the Gram-Charlier density evaluated at X

References

Gy.Terdik, Multivariate statistical methods - Going beyond the linear, Springer 2021. Section 4.7.

Examples

# Gram-Charlier density approximation (k=4) of data generated from
# a bivariate skew-gaussian distribution
n<-50
alpha<-c(10,0)
omega<-diag(2)
X<-rSkewNorm(n,omega,alpha)
EC<-SampleEVSK(X)
fy4<-SampleGC(X[1:5,],cum=EC)
# Gram-Charlier density approximation (k=4) of data generated from
# a bivariate skew-gaussian distribution
n<-50
alpha<-c(10,0)
omega<-diag(2)
X<-rSkewNorm(n,omega,alpha)
EC<-SampleEVSK(X)
fy4<-SampleGC(X[1:5,],cum=EC)

Estimate the N-th d-variate Hermite polynomial

Description

The vector x is standardized and the N-th d-variate polynomial is computed

Usage

SampleHermiteN(x, N)
SampleHermiteN(x, N)

Arguments

`x`	a d-variate data vector
`N`	the order of the d-variate Hermite polynomial

Value

The vector of the N-th d-variate polynomial

Examples

x<-MASS::mvrnorm(100,rep(0,3),diag(3))
H3<-SampleHermiteN(x,3)
x<-MASS::mvrnorm(100,rep(0,3),diag(3))
H3<-SampleHermiteN(x,3)

Estimation of Sample Kurtosis (Mardia, MRSz, Total)

Description

Estimates the sample kurtosis index based on the specified method: Mardia, MRSz, or Total.

Usage

SampleKurt(x, Type = c("Mardia", "MRSz", "Total"))
SampleKurt(x, Type = c("Mardia", "MRSz", "Total"))

Arguments

`x`	A matrix of multivariate data.
`Type`	A character string specifying the type of kurtosis index to estimate. Use "Mardia" for Mardia's kurtosis index, "MRSz" for the Mori-Rohatgi-Szekely kurtosis matrix, or "Total" for the total kurtosis index.

Value

A list containing the estimated kurtosis index or matrix and the associated p-value under the Gaussian hypothesis.

`Mardia.Kurtosis`	The kurtosis index when `Type` is "Mardia".
`MRSz.Kurtosis`	The kurtosis matrix when `Type` is "MRSz".
`Total.Kurtosis`	The total kurtosis index when `Type` is "Total".
`p.value`	The p-value under the Gaussian hypothesis for the estimated kurtosis.

References

Gy.Terdik, Multivariate statistical methods - going beyond the linear, Springer 2021. Example 6.1 and 6.9.

Examples

# Mardia's kurtosis example
x <- matrix(rnorm(100*5), ncol=5)
SampleKurt(x, Type = "Mardia")

# MRSz's kurtosis example
SampleKurt(x, Type = "MRSz")

# Total kurtosis example
SampleKurt(x, Type = "Total")

# Mardia's kurtosis example
x <- matrix(rnorm(100*5), ncol=5)
SampleKurt(x, Type = "Mardia")

# MRSz's kurtosis example
SampleKurt(x, Type = "MRSz")

# Total kurtosis example
SampleKurt(x, Type = "Total")

Estimation of multivariate T-Moments and T-Cumulants

Description

Provides estimates of univariate and multivariate moments and cumulants up to order r. By default data are standardized; using only demeaned or raw data is also possible.

Usage

SampleMomCum(X, r, centering = FALSE, scaling = TRUE)
SampleMomCum(X, r, centering = FALSE, scaling = TRUE)

Arguments

`X`	d-vector data
`r`	The highest moment order (r >2)
`centering`	set to T (and scaling = F) if only centering is needed
`scaling`	set to T (and centering=F) if standardization of multivariate data is needed

Value

estMu.r: the list of the multivariate moments up to order r

estCum.r: the list of the multivariate cumulants up to order r

References

Gy.Terdik, Multivariate statistical methods - going beyond the linear, Springer 2021.

Examples

## generate random data from a 3-variate skew normal distribution
alpha<-c(10,5,0)
omega<-diag(3)
x<-rSkewNorm(50,omega,alpha)
## estimate the first three moments and cumulants from raw (uncentered and unstandardized) data
SampleMomCum(x,3,centering=FALSE,scaling=FALSE)
## estimate the first three moments and cumulants from standardized data
SampleMomCum(x,3,centering=FALSE,scaling=TRUE)

## generate random data from a 3-variate skew normal distribution
alpha<-c(10,5,0)
omega<-diag(3)
x<-rSkewNorm(50,omega,alpha)
## estimate the first three moments and cumulants from raw (uncentered and unstandardized) data
SampleMomCum(x,3,centering=FALSE,scaling=FALSE)
## estimate the first three moments and cumulants from standardized data
SampleMomCum(x,3,centering=FALSE,scaling=TRUE)

Estimation of Sample Skewness (Mardia, MRSz)

Description

Estimates the sample skewness index based on the specified method: Mardia or MRSz.

Usage

SampleSkew(x, Type = c("Mardia", "MRSz"))
SampleSkew(x, Type = c("Mardia", "MRSz"))

Arguments

`x`	A matrix of multivariate data.
`Type`	A character string specifying the type of skewness index to estimate. Use "Mardia" for Mardia's skewness index or "MRSz" for the Mori-Rohatgi-Szekely skewness vector and index.

Value

A list containing the estimated skewness index or vector and the associated p-value under the Gaussian hypothesis.

`Mardia.Skewness`	The skewness index when `Type` is "Mardia".
`MRSz.Skewness.Vector`	The skewness vector when `Type` is "MRSz".
`MRSz.Skewness.Index`	The skewness index when `Type` is "MRSz".
`p.value`	The p-value under the Gaussian hypothesis for the estimated skewness.

References

Gy.Terdik, Multivariate statistical methods - going beyond the linear, Springer 2021. Example 6.1 and 6.2.

S. R. Jammalamadaka, E. Taufer, Gy. Terdik. On multivariate skewness and kurtosis. Sankhya A, 83(2), 607-644.

Examples

# Mardia's skewness example
x <- matrix(rnorm(100*5), ncol=5)
SampleSkew(x, Type = "Mardia")

# MRSz's skewness example
SampleSkew(x, Type = "MRSz")

# Mardia's skewness example
x <- matrix(rnorm(100*5), ncol=5)
SampleSkew(x, Type = "Mardia")

# MRSz's skewness example
SampleSkew(x, Type = "MRSz")

Estimated Variance of skewness and kurtosis vectors

Description

Provides the estimated covariance matrices of the data-estimated skewness and kurtosis vectors.

Usage

SampleVarianceSkewKurt(X)
SampleVarianceSkewKurt(X)

Arguments

`X`	A matrix of d-variate data

Value

The list of covariance matrices of the skewness and kurtosis vectors

References

Gy.Terdik, Multivariate statistical methods - going beyond the linear, Springer 2021.

Symmetrizing vector

Description

Vector symmetrizing a T-product of vectors of the same dimension d. Produces the same results as SymMatr

Usage

SymIndx(x, d, n)
SymIndx(x, d, n)

Arguments

`x`	the vector to be symmetrized of dimension d^n
`d`	size of the single vectors in the product
`n`	power of the T-product

Value

A vector with the symmetrized version of x of dimension d^n

References

Gy. Terdik, Multivariate statistical methods - going beyond the linear, Springer 2021.Section 1.3.1 Symmetrization, p.14. (1.29)

Examples

a<-c(1,2)
b<-c(2,3)
c<-kronecker(kronecker(a,a),b)
## The symmetrized version of c is
SymIndx(c,2,3)


a<-c(1,2)
b<-c(2,3)
c<-kronecker(kronecker(a,a),b)
## The symmetrized version of c is
SymIndx(c,2,3)

Symmetrizer Matrix

Description

Based on Chacon and Duong (2015) efficient recursive algorithms for functionals based on higher order derivatives. An option for sparse matrix is provided. By using sparse matrices far less memory is required and faster computation times are obtained

Usage

SymMatr(d, n, useSparse = FALSE)
SymMatr(d, n, useSparse = FALSE)

Arguments

`d`	dimension of a vector x
`n`	power of the Kronecker product
`useSparse`	TRUE or FALSE. If TRUE an object of the class "dgCMatrix" is produced.

Value

A Symmetrizer matrix with order ${d^n} \times d^n$ . If useSparse=TRUE an object of the class "dgCMatrix" is produced.

References

Chacon, J. E., and Duong, T. (2015). Efficient recursive algorithms for functionals based on higher order derivatives of the multivariate Gaussian density. Statistics and Computing, 25(5), 959-974.

Gy. Terdik, Multivariate statistical methods - going beyond the linear, Springer 2021.Section 1.3.1 Symmetrization, p.14. (1.29)

Examples

a<-c(1,2)
b<-c(2,3)
c<-kronecker(kronecker(a,a),b)
## The symmetrized version of c is
as.vector(SymMatr(2,3)%*%c)

a<-c(1,2)
b<-c(2,3)
c<-kronecker(kronecker(a,a),b)
## The symmetrized version of c is
as.vector(SymMatr(2,3)%*%c)

Univariate moments and cumulants from T-vectors

Description

A vector of indexes to select the moments and cumulants of the single components of the random vector X for which a T-vector of moments and cumulants is available

Usage

UnivMomCum(d, q)
UnivMomCum(d, q)

Arguments

`d`	dimension of a vector X
`q`	power of the Kronecker product

Value

A vector of indexes

Examples

## For a 3-variate skewness and kurtosis vectors estimated from data, extract
## the skewness and kurtosis estimates for each of the single components of the vector
alpha<-c(10,5,0)
omega<-diag(rep(1,3))
X<-rSkewNorm(200, omega, alpha)
EVSK<-SampleEVSK(X)
## Get the univariate skewness and kurtosis for X1,X2,X3
EVSK$estSkew[UnivMomCum(3,3)]
EVSK$estKurt[UnivMomCum(3,4)]
## For a 3-variate skewness and kurtosis vectors estimated from data, extract
## the skewness and kurtosis estimates for each of the single components of the vector
alpha<-c(10,5,0)
omega<-diag(rep(1,3))
X<-rSkewNorm(200, omega, alpha)
EVSK<-SampleEVSK(X)
## Get the univariate skewness and kurtosis for X1,X2,X3
EVSK$estSkew[UnivMomCum(3,3)]
EVSK$estKurt[UnivMomCum(3,4)]

Asymptotic Variance of the estimated kurtosis vector

Description

Warning: the function requires 8! computations, for d>3, the timing required maybe large.

Usage

VarianceKurt(cum)
VarianceKurt(cum)

Arguments

cum

The theoretical/estimated cumulants up to the 8th order in vector form

Value

The matrix of theoretical/estimated variance

References

Gy.Terdik, Multivariate statistical methods - going beyond the linear, Springer 2021. Ch. 6, formula (6.26)

Asymptotic Variance of the estimated skewness vector

Description

Asymptotic Variance of the estimated skewness vector

Usage

VarianceSkew(cum)
VarianceSkew(cum)

Arguments

cum

The theoretical/estimated cumulants up to order 6 in vector form

Value

The matrix of theoretical/estimated variance

References

Gy.Terdik, Multivariate statistical methods - Going beyond the linear, Springer 2021. Ch.6, formula (6.13)

Examples

alpha<-c(10,5)
omega<-diag(rep(1,2))
MC <- MomCumSkewNorm(r = 6,omega,alpha)
cum <- MC$CumX
VS <- VarianceSkew(cum)
alpha<-c(10,5)
omega<-diag(rep(1,2))
MC <- MomCumSkewNorm(r = 6,omega,alpha)
cum <- MC$CumX
VS <- VarianceSkew(cum)

Package 'MultiStatM'

Help Index

Commutator Index

Description

Usage

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Details

Value

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Examples

Commutator Matrix

Description

Usage

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Details

Value

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Examples

Convert cumulants to moments (univariate and multivariate)

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Value

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Distinct values selection vector

Description

Usage

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Value

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Examples

Elimination Matrix

Description

Usage

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Value

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Examples

EVSK multivariate Skew Normal

Description

Usage

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Value

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Examples

EVSK of the Uniform distribution on the sphere or its modulus

Description

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Value

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Examples

Coefficients of Hermite polynomials

Description

Usage

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Value

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Examples

Covariance matrix for multivariate T-Hermite polynomials

Description

Usage

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Value

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Examples

Hermite Polynomials (Univariate and Multivariate)

Description

Usage

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Details

Value