hermite {calculus}  R Documentation 
Computes univariate and multivariate Hermite polynomials.
hermite(order, sigma = 1, var = "x", transform = NULL)
order 
the order of the Hermite polynomial. 
sigma 
the covariance 
var 

transform 

Hermite polynomials are obtained by differentiation of the Gaussian kernel:
H_{\nu}(x,\Sigma) = exp \Bigl( \frac{1}{2} x_i \Sigma_{ij} x_j \Bigl) ( \partial_x )^\nu exp \Bigl( \frac{1}{2} x_i \Sigma_{ij} x_j \Bigl)
where \Sigma
is a d
dimensional square matrix and
\nu=(\nu_1 \dots \nu_d)
is the vector representing the order of
differentiation for each variable x = (x_1\dots x_d)
.
In the case where \Sigma=1
and x=x_1
the formula reduces to the
standard univariate Hermite polynomials:
H_{\nu}(x) = e^{\frac{x^2}{2}}(1)^\nu \frac{d^\nu}{dx^\nu}e^{\frac{x^2}{2}}
If transform
is not NULL
, the variables var
x
are replaced with
transform
f(x)
to compute the polynomials H_{\nu}(f(x),\Sigma)
list
of Hermite polynomials with components:
the Hermite polynomial.
the order of the Hermite polynomial.
data.frame
containing the variables, coefficients and degrees of each term in the Hermite polynomial.
Guidotti E (2022). "calculus: HighDimensional Numerical and Symbolic Calculus in R." Journal of Statistical Software, 104(5), 137. doi:10.18637/jss.v104.i05
Other polynomials:
taylor()
### univariate Hermite polynomials up to order 3
hermite(3)
### multivariate Hermite polynomials up to order 2
hermite(order = 2,
sigma = matrix(c(1,0,0,1), nrow = 2),
var = c('z1', 'z2'))
### multivariate Hermite polynomials with transformation of variables
hermite(order = 2,
sigma = matrix(c(1,0,0,1), nrow = 2),
var = c('z1', 'z2'),
transform = c('z1+z2','z1z2'))