hazrate.like: hazrate.like - Hazard rate likelihood

Description

Computes the hazard rate likelihood of off-transect distances, given parameters. Primarily used as a minimization objective during distance function estimation.

Usage

hazrate.like(
  a,
  dist,
  covars = NULL,
  w.lo = units::set_units(0, "m"),
  w.hi = max(dist),
  series = "cosine",
  expansions = 0,
  scale = TRUE,
  pointSurvey = FALSE
)

Value

A numeric vector the same length and order as dist containing the likelihood contribution for corresponding distances in dist. Assuming L is the returned vector from one of these functions, the full log likelihood of all the data is -sum(log(L), na.rm=T). Note that the returned likelihood value for distances less than

w.lo or greater than w.hi is NA, and thus it is prudent to use na.rm=TRUE in the sum. If scale = TRUE, the integral of the likelihood from w.lo to w.hi is 1.0. If scale = FALSE, the integral of the likelihood is arbitrary.

Arguments

a: A vector of likelihood parameter values. Length and meaning depend on series and expansions. If no expansion terms were called for (i.e., expansions = 0), the distance likelihoods contain one or two canonical parameters (see Details). If one or more expansions are called for, coefficients for the expansion terms follow coefficients for the canonical parameters. If p is the number of canonical parameters, coefficients for the expansion terms are a[(p+1):length(a)].
dist: A numeric vector containing the observed distances.
covars: Data frame containing values of covariates at each observation in dist.
w.lo: Scalar value of the lowest observable distance. This is the left truncation of sighting distances in dist. Same units as dist. Values less than w.lo are allowed in dist, but are ignored and their contribution to the likelihood is set to NA in the output.
w.hi: Scalar value of the largest observable distance. This is the right truncation of sighting distances in dist. Same units as dist. Values greater than w.hi are allowed in dist, but are ignored and their contribution to the likelihood is set to NA in the output.
series: A string specifying the type of expansion to use. Currently, valid values are 'simple', 'hermite', and 'cosine'; but, see dfuncEstim about defining other series.
expansions: A scalar specifying the number of terms in series. Depending on the series, this could be 0 through 5. The default of 0 equates to no expansion terms of any type.
scale: Logical scalar indicating whether or not to scale the likelihood so it integrates to 1. This parameter is used to stop recursion in other functions. If scale equals TRUE, a numerical integration routine (integration.constant) is called, which in turn calls this likelihood function again with scale = FALSE. Thus, this routine knows when its values are being used to compute the likelihood and when its value is being used to compute the constant of integration. All user defined likelihoods must have and use this parameter.
pointSurvey: Boolean. TRUE if dist is point transect data, FALSE if line transect data.

Details

The hazard rate likelihood is $$f(x|\sigma,k) = 1 - \exp(-(x/\sigma)^{-k})$$ where $\sigma$ determines location (i.e., distance at which the function equals 1 - exp(-1) = 0.632), and $k$ determines slope of the function at $\sigma$ (i.e., larger k equals steeper slope at $\sigma$). For distance analysis, the valid range for both $\sigma$ and k is $\geq 0$.

Expansion Terms: If expansions = e (e > 0), the expansion function specified by series is called (see for example cosine.expansion). Assuming $h_{ij}(x)$ is the $j^{th}$ expansion term for the $i^{th}$ distance and that $c_1, c_2, \dots, c_k$ are (estimated) coefficients for the expansion terms, the likelihood contribution for the $i^{th}$ distance is, $$f(x|a,b,c_1,c_2,\dots,c_e) = f(x|a,b)(1 + \sum_{j=1}^{e} c_j h_{ij}(x)).$$

Examples

Run this code

if (FALSE) {
x <- seq(0, 100, length=100)

# Plots showing effects of changes in sigma
plot(x, hazrate.like(c(20, 5), x), type="l", col="red")
plot(x, hazrate.like(c(40, 5), x), type="l", col="blue")

# Plots showing effects of changes in beta
plot(x, hazrate.like(c(50, 20), x), type="l", col="red")
plot(x, hazrate.like(c(50, 2), x), type="l", col="blue")
}