simple.expansion: Calculate simple polynomial expansion for detection function likelihoods
Description
Computes simple polynomial expansion terms used in the likelihood of a distance analysis. More generally, will compute
polynomial expansions of any numeric vector.
Usage
simple.expansion(x, expansions)
Value
A matrix of size length(x) X expansions. The columns of this matrix are
the Simple polynomial expansions of x. Column 1 is the first expansion term of x,
column 2 is the second x, and so on up to 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 (\(d\leq w\)),
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
The polynomials computed here are:
First term: $$h_1(x)=x^4,$$
Second term: $$h_2(x)=x^6,$$
Third term: $$h_3(x)=x^8,$$
Fourth term: $$h_4(x)=x^{10},$$
The maximum number of expansion terms computed is 4.