R Functions to Symbolically Compute the Central and Non-central Moments of the Multivariate Normal Distribution

The central moments of the multivariate normal distribution are functions of its n×n variance-covariance matrix Σ. These moments can be expressed symbolically as linear combinations of products of powers of the elements of Σ. A formula for these moments derived by differentiating the characteristic function is developed. The formula requires searching integer matrices for matrices whose n successive row and column sums equal the n exponents of the moment. This formula is implemented in R, with R functions to display moments in LTEX and to evaluate moments at specified variance-covariance matrices are included. This vignette also describes how the symmoments package has been augmented to calculate symbolic representations of non-central multivariate moments, and to calculate the expected value of such moments as well as the expected value of multivariate polynomials. In addition, it is shown that first non-central multivariate moments are in a one-to-one relationship to phylogenetic trees. Functions making this correspondence, as well as a function for the correspondence with matchings are provided.