uniform.like: Uniform likelihood function for distance analyses

Description Usage Arguments Details Value Author(s) See Also Examples

Description

This function computes likelihood contributions for sighting distances, scaled appropriately, for use as a distance likelihood.

Usage

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uniform.like(a, dist, covars = NULL, w.lo = 0, w.hi = max(dist),
  series = "cosine", expansions = 0, scale = TRUE,
  pointSurvey = FALSE)

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 uniform likelihood is not technically uniform. This function is continuous at its upper limit (a true uniform is discontinuous at its upper limit) which allows better estimation of the upper limit. The function has two parameters (the upper limit or 'threshold' and the 'knee') and can look similar to a uniform or a negative exponential.

The uniform likelihood used here is actually the heavy side or logistic function of the form,

f(x|a,b) = 1 - 1 / (1 + exp(-b*(x-a))) = exp(-b*(x-a)) / (1 + exp(-b*(x-a))),

where a and b are the parameters to be estimated.

Parameter a, the "threshold", is the location of the approximate upper limit of a uniform distribution's support. The inverse likelihood of 0.5 is a before scaling (i.e., uniform.like(c(a,b),a,scale=FALSE) equals 0.5).

Parameter b, the "knee", is the sharpness of the bend at a and estimates the degree to which observations decline at the outer limit of sightability. Note that, prior to scaling for g.x.scl, the slope of the likelihood at a is -b/4. After scaling for g.x.scl, the inverse of g.x.scl/2 is close to a/f(0). If b is large, the "knee" is sharp and the likelihood looks uniform with support from w.lo to a/f(0). If b is small, the "knee" is shallow and the density of observations declines in an elongated "S" shape pivoting at a/f(0). As b grows large and assuming f(0) = 1, the effective strip width approaches a from above.

See Examples for plots using large and small values of b.

Expansion Terms: If expansions = k (k > 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), ..., 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,...,c_k) = f(x|a,b)(1 + c(1) h_i1(x) + c(2) h_i2(x) + ... + c(k) h_ik(x)).

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.

Author(s)

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

See Also

dfuncEstim, halfnorm.like, hazrate.like, negexp.like, Gamma.like

Examples

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x <- seq(0, 100, length=100)

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

# Plots showing effects of changes in Knee
plot(x, uniform.like(c(50, 100), x), type="l", col="red")
plot(x, uniform.like(c(50, 1), x), type="l", col="blue")

         

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