HermiteCoeff {MultiStatM}

Coefficients of Hermite polynomials

Description

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

Usage

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

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)