bellpoly: Evaluate partial Bell polynomials

In kschurter/logdensity: Estimate the (Logarithm) of the Density and its Derivatives

Description

Evaluates the partial Bell polynomials, also known as the incomplete Bell polynomials. The complete Bell polynomial is the sum of the partial Bell polynomials over k=1,…,n. The partial polynomials are evaluted using the recurrence relation

k B_{n,k}(x_1,…,x_{n-k+1}) = ∑_{r=k-1}^{n-1} choose(n,r)x_{n-r}B_{r,k-1}(x_1,…,x_{r-k+2})

with B_{0,0} = 1, B_{0,k} = 0 for k ≥ 1, and B_{n,0} = 0 for n ≥ 1.

Usage

bellpoly(x, n = nrow(x), k = min(1, n):n)

Arguments

x	a matrix or object that can be coerced into one by as.matrix
n	nonnegative integer representing the number of elements
k	vector of integers no greater than n representing the sizes of partition

Value

bellpoly returns a vector or matrix depending on the dimensions of x that contains the value of the partial Bell polynomial(s) evaluated at columns of as.matrix(x).

Examples

n <- 5
k <- 1:n

## Idempotent numbers
bellpoly(k)
choose(n, k) * k^(n - k) # equivalent

## Stirling numbers (unsigned) of the first kind
bellpoly(x = factorial(k-1))

## Stirling numbers of the second kind
bellpoly(x = rep(1, n))

kschurter/logdensity documentation built on Feb. 22, 2022, 1:07 p.m.