hermite.expansion function

Calculation of Hermite expansion for detection function likelihoods

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

hermite.expansion(x, 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 ($d <= w$), 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.

Returns

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.

Details

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

• First term :

• Second term :

• Third term :

• Fourth term :

The maximum number of expansion terms computed is 4.

Examples

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

See Also

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