# hermite.expansion: Calculation of Hermite expansion for detection function... In Rdistance: Distance-Sampling Analyses for Density and Abundance Estimation

 hermite.expansion R Documentation

## 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)


### 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.

### 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.

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)


Rdistance documentation built on July 9, 2023, 6:46 p.m.