Computes univariate and multivariate Hermite polynomials.
hermite(order, sigma = 1, var = "x", transform = NULL)the order of the Hermite polynomial.
the covariance matrix of the Gaussian kernel.
character vector giving the variables of the polynomial.
character vector representing a change of variables. See details.
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.
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 \(H_{\nu}(f(x),\Sigma)\)
Guidotti E (2022). "calculus: High-Dimensional Numerical and Symbolic Calculus in R." Journal of Statistical Software, 104(5), 1-37. doi:10.18637/jss.v104.i05
Other polynomials:
taylor()
### univariate Hermite polynomials up to order 3
hermite(3)
#> $`0`
#> $`0`$f
#> [1] "(1) * 1"
#>
#> $`0`$order
#> [1] 0
#>
#> $`0`$terms
#> var coef degree
#> 0 1 1 0
#>
#>
#> $`1`
#> $`1`$f
#> [1] "(1) * x^1"
#>
#> $`1`$order
#> [1] 1
#>
#> $`1`$terms
#> var coef degree
#> 0 1 0 0
#> 1 x^1 1 1
#>
#>
#> $`2`
#> $`2`$f
#> [1] "(-1) * 1 + (1) * x^2"
#>
#> $`2`$order
#> [1] 2
#>
#> $`2`$terms
#> var coef degree
#> 0 1 -1 0
#> 1 x^1 0 1
#> 2 x^2 1 2
#>
#>
#> $`3`
#> $`3`$f
#> [1] "(-3) * x^1 + (1) * x^3"
#>
#> $`3`$order
#> [1] 3
#>
#> $`3`$terms
#> var coef degree
#> 0 1 0 0
#> 1 x^1 -3 1
#> 2 x^2 0 2
#> 3 x^3 1 3
#>
#>
### multivariate Hermite polynomials up to order 2
hermite(order = 2,
sigma = matrix(c(1,0,0,1), nrow = 2),
var = c('z1', 'z2'))
#> $`0,0`
#> $`0,0`$f
#> [1] "(1) * 1"
#>
#> $`0,0`$order
#> [1] 0
#>
#> $`0,0`$terms
#> var coef degree
#> 0,0 1 1 0
#>
#>
#> $`0,1`
#> $`0,1`$f
#> [1] "(1) * z2^1"
#>
#> $`0,1`$order
#> [1] 1
#>
#> $`0,1`$terms
#> var coef degree
#> 0,0 1 0 0
#> 0,1 z2^1 1 1
#> 1,0 z1^1 0 1
#>
#>
#> $`1,0`
#> $`1,0`$f
#> [1] "(1) * z1^1"
#>
#> $`1,0`$order
#> [1] 1
#>
#> $`1,0`$terms
#> var coef degree
#> 0,0 1 0 0
#> 0,1 z2^1 0 1
#> 1,0 z1^1 1 1
#>
#>
#> $`0,2`
#> $`0,2`$f
#> [1] "(-1) * 1 + (1) * z2^2"
#>
#> $`0,2`$order
#> [1] 2
#>
#> $`0,2`$terms
#> var coef degree
#> 0,0 1 -1 0
#> 0,1 z2^1 0 1
#> 1,0 z1^1 0 1
#> 0,2 z2^2 1 2
#> 2,0 z1^2 0 2
#> 1,1 z1^1*z2^1 0 2
#>
#>
#> $`2,0`
#> $`2,0`$f
#> [1] "(-1) * 1 + (1) * z1^2"
#>
#> $`2,0`$order
#> [1] 2
#>
#> $`2,0`$terms
#> var coef degree
#> 0,0 1 -1 0
#> 0,1 z2^1 0 1
#> 1,0 z1^1 0 1
#> 0,2 z2^2 0 2
#> 2,0 z1^2 1 2
#> 1,1 z1^1*z2^1 0 2
#>
#>
#> $`1,1`
#> $`1,1`$f
#> [1] "(1) * z1^1*z2^1"
#>
#> $`1,1`$order
#> [1] 2
#>
#> $`1,1`$terms
#> var coef degree
#> 0,0 1 0 0
#> 0,1 z2^1 0 1
#> 1,0 z1^1 0 1
#> 0,2 z2^2 0 2
#> 2,0 z1^2 0 2
#> 1,1 z1^1*z2^1 1 2
#>
#>
### 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','z1-z2'))
#> $`0,0`
#> $`0,0`$f
#> [1] "(1) * 1"
#>
#> $`0,0`$order
#> [1] 0
#>
#> $`0,0`$terms
#> var coef degree
#> 0,0 1 1 0
#>
#>
#> $`0,1`
#> $`0,1`$f
#> [1] "(-1) * z2^1 + (1) * z1^1"
#>
#> $`0,1`$order
#> [1] 1
#>
#> $`0,1`$terms
#> var coef degree
#> 0,0 1 0 0
#> 0,1 z2^1 -1 1
#> 1,0 z1^1 1 1
#>
#>
#> $`1,0`
#> $`1,0`$f
#> [1] "(1) * z2^1 + (1) * z1^1"
#>
#> $`1,0`$order
#> [1] 1
#>
#> $`1,0`$terms
#> var coef degree
#> 0,0 1 0 0
#> 0,1 z2^1 1 1
#> 1,0 z1^1 1 1
#>
#>
#> $`0,2`
#> $`0,2`$f
#> [1] "(-1) * 1 + (1) * z2^2 + (1) * z1^2 + (-2) * z1^1*z2^1"
#>
#> $`0,2`$order
#> [1] 2
#>
#> $`0,2`$terms
#> var coef degree
#> 0,0 1 -1 0
#> 0,1 z2^1 0 1
#> 1,0 z1^1 0 1
#> 0,2 z2^2 1 2
#> 2,0 z1^2 1 2
#> 1,1 z1^1*z2^1 -2 2
#>
#>
#> $`2,0`
#> $`2,0`$f
#> [1] "(-1) * 1 + (1) * z2^2 + (1) * z1^2 + (2) * z1^1*z2^1"
#>
#> $`2,0`$order
#> [1] 2
#>
#> $`2,0`$terms
#> var coef degree
#> 0,0 1 -1 0
#> 0,1 z2^1 0 1
#> 1,0 z1^1 0 1
#> 0,2 z2^2 1 2
#> 2,0 z1^2 1 2
#> 1,1 z1^1*z2^1 2 2
#>
#>
#> $`1,1`
#> $`1,1`$f
#> [1] "(-1) * z2^2 + (1) * z1^2"
#>
#> $`1,1`$order
#> [1] 2
#>
#> $`1,1`$terms
#> var coef degree
#> 0,0 1 0 0
#> 0,1 z2^1 0 1
#> 1,0 z1^1 0 1
#> 0,2 z2^2 -1 2
#> 2,0 z1^2 1 2
#> 1,1 z1^1*z2^1 0 2
#>
#>