HermiteN2X {MultiStatM}

Inverse Hermite Polynomial

Description

Compute the inverse of univariate or multivariate Hermite polynomials.

Usage

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.

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)