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:
\[ H_{\nu}(x) = e^{\frac{x^2}{2}}(-1)^\nu \frac{d^\nu}{dx^\nu}e^{-\frac{x^2}{2}} \]
The function hermite
generates recursively all the Hermite polynomials of degree \(\nu'\) where \(|\nu'| \leq|\nu|\). The output is a
list of Hermite polynomials of degree \(\nu'\), where each polynomial is
described as a list containing the character
representing the polynomial, the order of the polynomial, and a
data.frame containing the variables, coefficients and
degrees of each term in the polynomial.
In the univariate case, for \(\nu=2\):
hermite(order = 2)
#> $`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 2In the multivariate case, where for simplicity \(\Sigma_{ij}=\delta_{ij}\), \(x=(x_1,x_2)\), and \(|\nu|=2\):
hermite(order = 2, sigma = diag(2), var = c("x1", "x2"))
#> $`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) * x2^1"
#>
#> $`0,1`$order
#> [1] 1
#>
#> $`0,1`$terms
#> var coef degree
#> 0,0 1 0 0
#> 0,1 x2^1 1 1
#> 1,0 x1^1 0 1
#>
#>
#> $`1,0`
#> $`1,0`$f
#> [1] "(1) * x1^1"
#>
#> $`1,0`$order
#> [1] 1
#>
#> $`1,0`$terms
#> var coef degree
#> 0,0 1 0 0
#> 0,1 x2^1 0 1
#> 1,0 x1^1 1 1
#>
#>
#> $`0,2`
#> $`0,2`$f
#> [1] "(-1) * 1 + (1) * x2^2"
#>
#> $`0,2`$order
#> [1] 2
#>
#> $`0,2`$terms
#> var coef degree
#> 0,0 1 -1 0
#> 0,1 x2^1 0 1
#> 1,0 x1^1 0 1
#> 0,2 x2^2 1 2
#> 2,0 x1^2 0 2
#> 1,1 x1^1*x2^1 0 2
#>
#>
#> $`2,0`
#> $`2,0`$f
#> [1] "(-1) * 1 + (1) * x1^2"
#>
#> $`2,0`$order
#> [1] 2
#>
#> $`2,0`$terms
#> var coef degree
#> 0,0 1 -1 0
#> 0,1 x2^1 0 1
#> 1,0 x1^1 0 1
#> 0,2 x2^2 0 2
#> 2,0 x1^2 1 2
#> 1,1 x1^1*x2^1 0 2
#>
#>
#> $`1,1`
#> $`1,1`$f
#> [1] "(1) * x1^1*x2^1"
#>
#> $`1,1`$order
#> [1] 2
#>
#> $`1,1`$terms
#> var coef degree
#> 0,0 1 0 0
#> 0,1 x2^1 0 1
#> 1,0 x1^1 0 1
#> 0,2 x2^2 0 2
#> 2,0 x1^2 0 2
#> 1,1 x1^1*x2^1 1 2If transform is not NULL, the variables
var \(x\) are replaced
with transform \(f(x)\) to
compute the polynomials \(H_{ν}(f(x),\Sigma)\). For example:
\[ f(x_1,x_2)= \begin{bmatrix} x_1+x_2,x_1-x_2 \end{bmatrix} \]
hermite(order = 2, sigma = diag(2), var = c("x1", "x2"), transform = c('x1+x2','x1-x2'))
#> $`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) * x2^1 + (1) * x1^1"
#>
#> $`0,1`$order
#> [1] 1
#>
#> $`0,1`$terms
#> var coef degree
#> 0,0 1 0 0
#> 0,1 x2^1 -1 1
#> 1,0 x1^1 1 1
#>
#>
#> $`1,0`
#> $`1,0`$f
#> [1] "(1) * x2^1 + (1) * x1^1"
#>
#> $`1,0`$order
#> [1] 1
#>
#> $`1,0`$terms
#> var coef degree
#> 0,0 1 0 0
#> 0,1 x2^1 1 1
#> 1,0 x1^1 1 1
#>
#>
#> $`0,2`
#> $`0,2`$f
#> [1] "(-1) * 1 + (1) * x2^2 + (1) * x1^2 + (-2) * x1^1*x2^1"
#>
#> $`0,2`$order
#> [1] 2
#>
#> $`0,2`$terms
#> var coef degree
#> 0,0 1 -1 0
#> 0,1 x2^1 0 1
#> 1,0 x1^1 0 1
#> 0,2 x2^2 1 2
#> 2,0 x1^2 1 2
#> 1,1 x1^1*x2^1 -2 2
#>
#>
#> $`2,0`
#> $`2,0`$f
#> [1] "(-1) * 1 + (1) * x2^2 + (1) * x1^2 + (2) * x1^1*x2^1"
#>
#> $`2,0`$order
#> [1] 2
#>
#> $`2,0`$terms
#> var coef degree
#> 0,0 1 -1 0
#> 0,1 x2^1 0 1
#> 1,0 x1^1 0 1
#> 0,2 x2^2 1 2
#> 2,0 x1^2 1 2
#> 1,1 x1^1*x2^1 2 2
#>
#>
#> $`1,1`
#> $`1,1`$f
#> [1] "(-1) * x2^2 + (1) * x1^2"
#>
#> $`1,1`$order
#> [1] 2
#>
#> $`1,1`$terms
#> var coef degree
#> 0,0 1 0 0
#> 0,1 x2^1 0 1
#> 1,0 x1^1 0 1
#> 0,2 x2^2 -1 2
#> 2,0 x1^2 1 2
#> 1,1 x1^1*x2^1 0 2Guidotti 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
A BibTeX entry for LaTeX users is
@Article{calculus,
title = {{calculus}: High-Dimensional Numerical and Symbolic Calculus in {R}},
author = {Emanuele Guidotti},
journal = {Journal of Statistical Software},
year = {2022},
volume = {104},
number = {5},
pages = {1--37},
doi = {10.18637/jss.v104.i05},
}