This function computes likelihood contributions for sighting distances, scaled appropriately, for use as a distance likelihood.
uniform.like(a, dist, covars = NULL, w.lo = 0, w.hi = max(dist),
series = "cosine", expansions = 0, scale = TRUE,
pointSurvey = FALSE)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)].
A numeric vector containing the observed distances.
Data frame containing values of covariates at
each observation in dist.
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.
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.
A string specifying the type of expansion to
use. Currently, valid values are 'simple', 'hermite', and
'cosine'; but, see dfuncEstim about
defining other series.
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.
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.
Boolean. TRUE if dist is point
transect data, FALSE if line transect data.
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.
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, $$f(x|a,b) = 1 - \frac{1}{1 + \exp(-b(x-a))} = \frac{\exp( -b(x-a) )}{1 + exp( -b(x-a) )},$$ 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
\(c_1, c_2, \dots, c_k\) are (estimated)
coefficients for the expansion terms, the likelihood contribution
for the \(i^{th}\) distance is,
$$f(x|a,b,c_1,c_2,\dots,c_k) = f(x|a,b)(1 +
\sum_{j=1}^{k} c_j h_{ij}(x)).$$
dfuncEstim,
halfnorm.like,
hazrate.like,
negexp.like,
Gamma.like
# NOT RUN {
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")
# }
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