hermite.expansion: Hermite expansion factors

In Rdistance: Density and Abundance from Distance-Sampling Surveys

Hermite expansion factors

Description

Computes Hermite expansion terms for use in distance analysis. The Hermite (and other expansions) allow "wiggle" in estimated distance functions.

Usage

hermite.expansion(x, expansions)

Arguments

x	A numeric matrix of distances at which to evaluate the expansion series. For distance analysis, x should be the proportion of the maximum sighting distance at which a group was sighted, i.e., x = d/w, where d is sighting distance and w is maximum sighting distance.
expansions	A scalar specifying the number of expansion terms to compute. Must be one of the integers 1, 2, 3, 4, or 5.

Details

There are, in general, several expansions that can be called Hermite. Let w = 4x - 2. Rdistance's Hermite expansions are:

• First term:

  h_1(w) = w + 2,

• Second term:

  h_2(w) = w^2 - 4,

• Third term:

  h_3(w) = w^3 - 3w + 2,

• Fourth term:

  h_4(w) = w^4 - 6w^2 + 8,

The maximum number of expansion terms computed is 4.

Value

A 3D array of size nrow(x) X ncol(x) X expansions. The 'pages' (3rd dimension) of this array are the cosine expansions of x. i.e., page 1 is the first expansion term of x, page 2 is the second expansion term of x, etc.

See Also

dfuncEstim , cosine.expansion , sine.expansion , simple.expansion.

Examples

x <- matrix(seq(0, 1, length = 200), ncol = 1)
herm.expn <- hermite.expansion(x, 4)
plot(range(x), range(herm.expn), type="n")
matlines(x, herm.expn[,1,1:4], col=rainbow(4), lty = 1)

Rdistance documentation built on Jan. 10, 2026, 1:07 a.m.