Hermite polynomials

library(calculus)

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 Σ is a d-dimensional square matrix and ν = (ν₁…ν_d) is the vector representing the order of differentiation for each variable x = (x₁…x_d). In the case where Σ = 1 and x = x₁ 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 ν′ where |ν′| ≤ |ν|. The output is a list of Hermite polynomials of degree ν′, 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 ν = 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      2

In the multivariate case, where for simplicity Σ_ij = δ_ij, x = (x₁, x₂), and |ν| = 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      2

If transform is not NULL, the variables var x are replaced with transform f(x) to compute the polynomials H_ν(f(x), Σ). 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      2

Cite as

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

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},
}

- Cite as