symmoments-package: Symbolically compute and numerically evaluate multivariate...

In symmoments: Symbolic Central and Noncentral Moments of the Multivariate Normal Distribution

Symbolically compute and numerically evaluate multivariate central moments

Description

Symbolically computes and numerically evaluates multivariate normal moments E[X_1^{k_1} \cdots X_n^{k_n}], where (X_1,\ldots,X_n) \sim N(\mu, \Sigma), in terms of mu and S elements.

Details

Produces Latex code for the moment.

Computes numerical moments at specified means and covariance matrices.

Also converts between moment L-matrices, phylo objects, and matching objects.

A representation of a central moment of the multivariate normal distribution, given by a positive integer vector c(k1,k2,...,kn), is obtained from the function callmultmoments(). This function initializes variables and calls the function multmoments() which determines a representation of a multivariate moment using a recursive algorithm. The representation is given class 'moment'.

The print() method prints the representation of a multivariate moment.

The toLatex() method uses the output of callmultmoments() to determine the LaTeX code for the moment sorted lexicographically.

The generic evaluate() method uses the output of callmultmoments() to determine the value of the moment for a specified covariance matrix.

The simulate() method is used to approximate a (possibly non-central) moment using Monte Carlo integration.

The toLatex_noncentral() function computes the Latex representations of a non-central moment.

The evaluate_noncentral() function computes the value of a non-central moment.

The evaluate_expected.polynomial() function evaluates the expected value of a multivariate polynomial defined by a list, multipol object, or mpoly object.

The convert.multipol() function converts between multipol objects and multivariate polynomials defined by lists.

The convert.mpoly() function converts between mpoly objects and multivariate polynomials defined by lists.

The tounsorted() function converts a sorted moment (e.g. m123) to an unsorted moment (e.g. m312).

The make.all.moments() function computes all moments up to a specified size and places them in the symmoments environment.

The integrate.polynomial() function integrates a multivariate polynomial against the normal distribution using ordinary integration.

The functions toMoment(), toNewick(), and toMatching() convert among moment L-matrices, Newick trees, and ape matching objects.

Note

The mvtnorm package must be loaded for the simulate method. The cubature package must be loaded for the integrate.polynomial function. The combinat package must be loaded for the toMoment function.

Author(s)

Kem Phillips [e-mail]

Maintainer: Florian Schuberth [e-mail]

References

\insertRef

Phillips2010symmoments

See Also

Useful links:

• https://github.com/FloSchuberth/symmoments

• Report bugs at https://github.com/FloSchuberth/symmoments/issues

Examples

# Compute the moment for the 4-dimensional moment c(1,2,3,4):
callmultmoments(c(1,2,3,4))

# Print the representation of the 4-dimensional moment c(1,2,3,4):
print(callmultmoments(c(1,2,3,4)))

# Compute the LaTeX representation of the central moment c(1,2,3,4):
toLatex(callmultmoments(c(1,2,3,4)))

# evaluate the moment c(1,2,3,4) at the following variance-covariance matrix
#  4 2 1 1
#  2 3 1 1
#  1 1 2 1
evaluate(callmultmoments(c(1,2,3,4)), c(4,2,1,1,3,1,1,2,1,2))

# Using 10000 samples, estimate the central moment for c(2,4) (not run)
# at the covariance matrix:
#  2 1
#  1 4
# and mean (0,0)
## Not run: 
library(mvtnorm)
simulate(callmultmoments(c(2,4)), 10000, NULL, c(0,0), c(2,1,1,4))

## End(Not run)

## Not run: 
# Compute Latex representation of a non-central moment
as.matrix(toLatex_noncentral(c(1,3)))

# Create all 2-dimensional moment objects with exponents up to 3
symmoments <- new.env()
make.all.moments(c(3,3))

# Evaluate a non-central moment (requires moments of order up to c(1,3)
# to exist in environment symmoments)
evaluate_noncentral(c(1,3), c(1,2), c(1,0,1))

## End(Not run)

# Create an mpoly object
library(mpoly)
library(multipol)
t0 <- mpoly(list(c(coef=3, x1=2), c(coef=2, x1=1, x2=3),
                 c(coef=-4, z=2), c(coef=1, x1=1, x2=2, z=1)))

# Convert an mpoly object to a moment object
t1 <<- convert.mpoly(t0)

# Convert a moment object to a multipol object
t2 <<- convert.multipol(t1)

# Convert from multipol back to mpoly through moment
mpoly(convert.mpoly(convert.multipol(t2)))

## Not run: 
# Evaluate the expected value of a multivariate polynomial
# (required moments must exist in environment symmoments)
evaluate_expected.polynomial(t0, c(1,2,3), c(1,0,0,1,0,1))

## End(Not run)

# Create a Newick representation of a tree
exam.Newick   <- "(((a,b),c),d);"

# Convert to phylo format
library(ape)
exam.phylo    <- read.tree(text = exam.Newick)

# Convert to matching format
exam.matching <- as.matching(exam.phylo)

# Convert to L-matrix format
exam.L.matrix <- toMoment(exam.matching)

symmoments documentation built on May 27, 2026, 9:06 a.m.