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

## 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

 `1` ```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.

## Details

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

• First term:

h1(x) = x^4 - 6*(x)^2 +3,

• Second term:

h2(x) = (x)^6 - 15*(x)^4 + 45*(x)^2 - 15,

• Third term:

h3(x) = (x)^8 - 28*(x)^6 + 210*(x)^4 - 420*(x)^2 + 105,

• Fourth term:

h4(x) = (x)^10 - 45*(x)^8 + 630*(x)^6 - 3150*(x)^4 + 4725*(x)^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`.

## Author(s)

Trent McDonald, WEST Inc. tmcdonald@west-inc.com Aidan McDonald, WEST Inc. aidan@mcdcentral.org

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

## Examples

 ```1 2 3 4``` ```set.seed(83828233) x <- rnorm(1000) * 100 x <- x[0 < x & x < 100] herm.expn <- hermite.expansion(x, 3) ```

### Example output

```Rdistance (version 1.3.2)
```

Rdistance documentation built on May 2, 2019, 3:49 a.m.