# cosine.expansion: calculation of cosine expansion for detection function... In Rdistance: Distance-Sampling Analyses for Density and Abundance Estimation

## Description

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

## Usage

 `1` ```cosine.expansion(x, expansions) ```

## Arguments

 `x` In a distance analysis, `x` is a numeric vector of the proportion of a strip transect's half-width at which a group of individuals were 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, 4, or 5.

## Details

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

• First term:

h1(x) = cos(2*Pi*x),

• Second term:

h2(x) = cos(3*Pi*x),

• Third term:

h3(x) = cos(4*Pi*x),

• Fourth term:

h4(x) = cos(5*Pi*x),

• Fifth term:

h5(x) = cos(6*Pi*x),

The maximum number of expansion terms computed is 5.

## Value

A matrix of size `length(x)` X `expansions`. The columns of this matrix are the cosine 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`, `hermite.expansion`, `simple.expansion`, and the discussion of user defined likelihoods in `dfuncEstim`.
 ```1 2 3 4``` ```set.seed(33328) x <- rnorm(1000) * 100 x <- x[ 0 < x & x < 100 ] cos.expn <- cosine.expansion(x, 5) ```