Hermite_CoeffMulti {MultiStatM}

Coefficients of multivariate T-Hermite polynomials for standardized variate

Description

Provides the matrix of coefficients of x^{\otimes N}, \kappa_2^{\otimes} x^{\otimes (N-2)}... for the d-variate T-Hermite polynomials up to order N.

Usage

Hermite_CoeffMulti(N, d)

Arguments

N

the maximum order of polynomials

d

the dimension of d-variate X

Value

The 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. Section 4.6.2, p. 223, Remark 4.8,

See Also

Other Hermite: Hermite_Coeff(), Hermite_N_Cov_X1_X2(), Hermite_Nth(), Hermite_Poly_HN_Multi(), Hermite_Poly_HN(), Hermite_Poly_NH_Inv(), Hermite_Poly_NH_Multi_Inv()

Examples

N <- 5; d <- 3
H_N_Xc <- Hermite_CoeffMulti(N,d) # coefficients
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_N_Xc[[1]] %*% X5 + H_N_Xc[[2]] %*%kronecker(Idv,X3) +
  H_N_Xc[[3]] %*%kronecker(kronecker(Idv,Idv),X)