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, ...)

Arguments

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.

See Also

Other Commutators: CommutatorMatr()

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)

Commutator Matrix

Description

This function generates various types of commutator matrices.

Usage

CommutatorMatr(Type, ...)

Arguments

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.

See Also

Other Commutators: CommutatorIndx()

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)

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"))

Arguments

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.

See Also

Other Moments and cumulants: EVSKSkewNorm(), EVSKUniS(), Mom2Cum(), MomCumCFUSN(), MomCumSkewNorm(), MomCumUniS(), MomCumZabs()

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")

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)

Arguments

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.

See Also

Other Moments and cumulants: Cum2Mom(), EVSKUniS(), Mom2Cum(), MomCumCFUSN(), MomCumSkewNorm(), MomCumUniS(), MomCumZabs()

Examples

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"))

Arguments

Value

A list of computed moments and cumulants.

When Type is "Standard":

When Type is "Modulus":

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

See Also

Other Moments and cumulants: Cum2Mom(), EVSKSkewNorm(), Mom2Cum(), MomCumCFUSN(), MomCumSkewNorm(), MomCumUniS(), MomCumZabs()

Examples

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

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

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)

Arguments

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)

See Also

Other Matrices and commutators: EliminMatr(), QplicIndx(), QplicMatr(), SymIndx(), SymMatr(), UnivMomCum()

Examples

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)

Arguments

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)

See Also

Other Matrices and commutators: EliminIndx(), QplicIndx(), QplicMatr(), SymIndx(), SymMatr(), UnivMomCum()

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)

Coefficients of Hermite polynomials

Description

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

Usage

HermiteCoeff(Type, N, d = NULL)

Arguments

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

See Also

Other Hermite Polynomials: HermiteCov12(), HermiteN(), HermiteN2X()

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)

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)

Arguments

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),

See Also

Other Hermite Polynomials: HermiteCoeff(), HermiteN(), HermiteN2X()

Examples

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)))

Arguments

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).

See Also

Other Hermite Polynomials: HermiteCoeff(), HermiteCov12(), HermiteN2X()

Examples

# 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)

Arguments

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.

See Also

Other Hermite Polynomials: HermiteCoeff(), HermiteCov12(), HermiteN()

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)

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)

Arguments

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)

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"))

Arguments

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.

See Also

Other Moments and cumulants: Cum2Mom(), EVSKSkewNorm(), EVSKUniS(), MomCumCFUSN(), MomCumSkewNorm(), MomCumUniS(), MomCumZabs()

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")

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)

Arguments

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

See Also

Other Moments and cumulants: Cum2Mom(), EVSKSkewNorm(), EVSKUniS(), Mom2Cum(), MomCumSkewNorm(), MomCumUniS(), MomCumZabs()

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)

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)

Arguments

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.

See Also

Other Moments and cumulants: Cum2Mom(), EVSKSkewNorm(), EVSKUniS(), Mom2Cum(), MomCumCFUSN(), MomCumUniS(), MomCumZabs()

Examples

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)

Arguments

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

See Also

Other Moments and cumulants: Cum2Mom(), EVSKSkewNorm(), EVSKUniS(), Mom2Cum(), MomCumCFUSN(), MomCumSkewNorm(), MomCumZabs()

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)

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)

Arguments

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

See Also

Other Moments and cumulants: Cum2Mom(), EVSKSkewNorm(), EVSKUniS(), Mom2Cum(), MomCumCFUSN(), MomCumSkewNorm(), MomCumUniS()

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)

Partitions, type and number of partitions

Description

Generates all partitions of N numbers and classify them by type

Usage

PartitionTypeAll(N)

Arguments

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.

See Also

Other Partitions: Partitions(), PermutationInv()

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]]

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, ...)

Arguments

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.

See Also

Other Partitions: PartitionTypeAll(), PermutationInv()

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)

Inverse of a Permutation

Description

Inverse of a Permutation

Usage

PermutationInv(permutation0)

Arguments

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

See Also

Other Partitions: PartitionTypeAll(), Partitions()

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)

Arguments

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)

See Also

Other Matrices and commutators: EliminIndx(), EliminMatr(), QplicMatr(), SymIndx(), SymMatr(), UnivMomCum()

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)]

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)

Arguments

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)

See Also

Other Matrices and commutators: EliminIndx(), EliminMatr(), QplicIndx(), SymIndx(), SymMatr(), UnivMomCum()

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)

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)

Arguments

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

See Also

Other Estimation: SampleKurt(), SampleMomCum(), SampleSkew(), VarianceKurt(), VarianceSkew()

Examples

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)

Arguments

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)

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)

Arguments

Value

The vector of the N-th d-variate polynomial

Examples

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"))

Arguments

Value

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

References

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

See Also

Other Estimation: SampleEVSK(), SampleMomCum(), SampleSkew(), VarianceKurt(), VarianceSkew()

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")

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)

Arguments

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.

See Also

Other Estimation: SampleEVSK(), SampleKurt(), SampleSkew(), VarianceKurt(), VarianceSkew()

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)

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"))

Arguments

Value

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

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.

See Also

Other Estimation: SampleEVSK(), SampleKurt(), SampleMomCum(), VarianceKurt(), VarianceSkew()

Examples

# 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)

Arguments

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)

Arguments

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)

See Also

Other Matrices and commutators: EliminIndx(), EliminMatr(), QplicIndx(), QplicMatr(), SymMatr(), UnivMomCum()

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)

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)

Arguments

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)

See Also

Other Matrices and commutators: EliminIndx(), EliminMatr(), QplicIndx(), QplicMatr(), SymIndx(), UnivMomCum()

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)

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)

Arguments

Value

A vector of indexes

See Also

Other Matrices and commutators: EliminIndx(), EliminMatr(), QplicIndx(), QplicMatr(), SymIndx(), SymMatr()

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)]

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)

Arguments

Value

The matrix of theoretical/estimated variance

References

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

See Also

Other Estimation: SampleEVSK(), SampleKurt(), SampleMomCum(), SampleSkew(), VarianceSkew()

Asymptotic Variance of the estimated skewness vector

Description

Asymptotic Variance of the estimated skewness vector

Usage

VarianceSkew(cum)

Arguments

Value

The matrix of theoretical/estimated variance

References

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

See Also

Other Estimation: SampleEVSK(), SampleKurt(), SampleMomCum(), SampleSkew(), VarianceKurt()

Examples

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

Random multivariate CFUSN

Description

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

Usage

rCFUSN(n, Delta)

Arguments

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.

See Also

Other Random generation: rCFUSSD(), rSkewNorm(), rUniS()

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)

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)

Arguments

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)

See Also

Other Random generation: rCFUSN(), rSkewNorm(), rUniS()

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)

Random Multivariate Skew Normal

Description

Generate random d-vectors from the multivariate Skew Normal distribution

Usage

rSkewNorm(n, omega, alpha)

Arguments

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

See Also

Other Random generation: rCFUSN(), rCFUSSD(), rUniS()

Examples

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)

Arguments

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.

See Also

Other Random generation: rCFUSN(), rCFUSSD(), rSkewNorm()