Rdistance (version 3.0.0)

hermite.expansion: Calculation of Hermite expansion for detection function likelihoods

Description

Computes the Hermite expansion terms used in the likelihood of a distance analysis. More generally, will compute a Hermite expansion of any numeric vector.

Usage

hermite.expansion(x, expansions)

Value

A matrix of size length(x) X expansions. The columns of this matrix are the Hermite polynomial expansions of x. Column 1 is the first expansion term of x, column 2 is the second expansion term of x, and so on up to expansions.

Arguments

x

In a distance analysis, x is a numeric vector containing the proportion of a strip transect's half-width at which a group of individuals was sighted. If \(w\) is the strip transect half-width or maximum sighting distance, and \(d\) is the perpendicular off-transect distance to a sighted group (\(d\leq w\)), x is usually \(d/w\). More generally, x is a vector of numeric values.

expansions

A scalar specifying the number of expansion terms to compute. Must be one of the integers 1, 2, 3, or 4.

Details

There are, in general, several expansions that can be called Hermite. The Hermite expansion used here is:

• First term: $$h_1(x)=x^4 - 6x^2 + 3,$$

• Second term: $$h_2(x)=x^6 - 15x^4 + 45x^2 - 15,$$

• Third term: $$h_3(x)=x^8 - 28x^6 + 210x^4 - 420x^2 + 105,$$

• Fourth term: $$h_4(x)=x^10 - 45x^8 + 630x^6 - 3150x^4 + 4725x^2 - 945,$$

The maximum number of expansion terms computed is 4.

See Also

dfuncEstim, cosine.expansion, simple.expansion, and the discussion of user defined likelihoods in dfuncEstim.

Examples

set.seed(83828233)
  x <- rnorm(1000) * 100
  x <- x[0 < x & x < 100]
  herm.expn <- hermite.expansion(x, 3)