detectfn: Detection Functions
Description
A detection function relates the probability of detection to the distance
of a detector from a point. The reference point is usually thought of as
an animal's home-range centre. In secr only simple 2- or
3-parameter functions are used. Each type of function is identified by a
numeric code (see below). In most cases the name may also be used (as a
quoted string).
Some functions such as (4) uniform are defined only for simulation as
the pose problems for maximum likelihood.
For function (7), `F' is the standard normal distribution function and
$\mu$ and $s$ are the mean and standard deviation on the
log scale of a latent variable representing a threshold of detection
distance. See Note for the relationship to the fitted parameters sigma
and z.
For function (8), `G' is the cumulative distribution function of the
gamma distribution with shape parameter k ( = z) and scale
parameter $\theta$ ( = sigma/z). See R's
pgamma.
For functions (9), (10) and (11), `F' is the standard normal
distribution function and `c' is an arbitrary signal threshold. The two
parameters of (9) are functions of the parameters of (10) and (11):
$b_0 = (\beta_0 - c) / sdS$ and $b_1 =
\beta_1 / s$ (see Efford et al. 2009).
Function (11) includes an additional `hard-wired' term for sound
attenuation due to spherical spreading. Detection probability at
distances less than 1 m is given by $g(d) = 1 - F \lbrace(c -
\beta_0) / sdS \rbrace$
The hazard-rate detection function was described by Hayes and Buckland
(1983). The compound halfnormal detection function follows Efford and
Dawson (2009). The signal strength and binary signal strength functions
are from Efford et al. (2009).
llll{
Code Name Parameters Function
0 halfnormal g0, sigma $g(d) = g_0 \exp
\left(\frac{-d^2} {2\sigma^2} \right)$
1 hazard rate g0, sigma, z $g(d) = g_0 [1 - \exp{
{-(^d/_\sigma)^{-z}} }]$
2 exponential g0, sigma $g(d) = g_0 \exp {
-(^d/_\sigma) }$
3 compound halfnormal g0, sigma, z $g(d) = g_0 [1
- {1 - \exp \left(\frac{-d^2} {2\sigma^2} \right)} ^ z]$
4 uniform g0, sigma $g(d) = g_0, d References
Efford, M. G. and Dawson, D. K. (2009) Effect of distance-related
heterogeneity on population size estimates from point counts. Auk
126, 100--111.
Efford, M. G., Dawson, D. K. and Borchers, D. L. (2009) Population
density estimated from locations of individuals on a passive detector
array. Ecology 90, 2676--2682.
Hayes, R. J. and Buckland, S. T. (1983) Radial-distance models for the
line-transect method. Biometrics 39, 29--42.