Computes a two parameter logistic distance function.
logistic.like(
a,
dist,
covars = NULL,
w.lo = units::set_units(0, "m"),
w.hi = max(dist),
series = "cosine",
expansions = 0,
scale = TRUE,
pointSurvey = FALSE
)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,
the log likelihood of all 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
essential 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.
A vector of likelihood parameter values. Length and meaning
depend on whether covariates and expansions are present as follows:
If no covariates and no expansions: a = [a, b] (see Details)
If no covariates and k expansions: a = [a, b, e1, ..., ek]
If p covariates and no expansions: a = [a, b, b1, ..., bp]
If p covariates and k expansions: a = [a, b, b1, ..., bp, e1, ..., ek]
A numeric vector containing observed distances with measurement units.
Data frame containing values of covariates at
each observation in dist.
Scalar value of the lowest observable distance, with measurement
units.
This is the left truncation sighting distance. Values less than
w.lo are allowed in dist, but are ignored and
their likelihood value is set to
NA in the output.
Scalar value of the largest observable distance, with measurement
units.
This is the right truncation sighting distance.
Values greater than w.hi are allowed in dist,
but are ignored and their likelihood value 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 into a density function, i.e., so that 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
values are 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.
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, the likelihood contribution
for the \(i^{th}\) distance is,
$$f(x|a,b,c_1,c_2,\dots,c_k) = f(x|a,b)(1 +
\sum_{j=1}^{k} c_j h_{ij}(x)).$$
The 'logistic' likelihood contains two parameters. Parameter \(a\) determines the scale and is labeled 'threshold' in Rdistance. Parameter \(b\) determines sharpness (slope) of the likelihood's decrease at \(a\) and is labeled 'knee' in Rdistance. This function is sometimes called the heavy side function (e.g., engineering). The technical form of the function is, $$f(x|a,b) = 1 - \frac{1}{1 + \exp(-b(x-a))} = \frac{\exp( -b(x-a) )}{1 + exp( -b(x-a) )},$$
Parameter \(a\) is the location (distance) of
0.5. That is, the inverse likelihood of 0.5
is \(a\) before scaling
(i.e., logistic.like( c(a,b), a, scale=FALSE) equals
0.5).
Parameter \(b\) is slope of function
at \(a\).
Prior to scaling,
slope of the likelihood at \(a\) is \(-b/4\).
If \(b\)
is large, the "knee" is sharp and the likelihood can look
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 \(a\).
See plots in Examples.
dfuncEstim,
halfnorm.like,
hazrate.like,
negexp.like,
Gamma.like
x <- units::set_units(seq(0, 100, length=100), "m")
# Plots showing effects of changes in Threshold
plot(x, logistic.like(c(20, 20), x), type="l", col="red")
lines(x, logistic.like(c(40, 20), x), type="l", col="blue")
# Plots showing effects of changes in Knee
plot(x, logistic.like(c(50, 100), x), type="l", col="red")
lines(x, logistic.like(c(50, 1), x), type="l", col="blue")
lines(x, logistic.like(c(50, .1), x), type="l", col="orange")
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