detectfn: Detection Functions
Description
A detection function relates the probability of detection $g$ or the
expected number of detections $\lambda$ for an animal to the
distance of a detector from a point usually thought of as its home-range
centre. In secr only simple 2- or 3-parameter functions are
used. Each type of function is identified by its number or by a 2--3
letter code (version $\ge$ 2.6.0; see below). In most cases the name
may also be used (as a quoted string).
Choice of detection function is usually not critical, and the default
`HN' is usually adequate.
Functions (14)--(18) are parameterised in terms of the expected number
of detections $\lambda$, or cumulative hazard, rather than
probability. `Exposure' (e.g. Royle and Gardner 2011) is another term
for cumulative hazard. This parameterisation is natural for the `count'
detector type or if the function is to be interpreted as a
distribution of activity (home range). When one of the functions
(14)--(18) is used to describe detection probability (i.e., for the binary
detectors `single', `multi',`proximity',`polygonX' or
`transectX'), the expected number of detections is internally
transformed to a binomial probability using $g(d) =
1-\exp(-\lambda(d))$.
The hazard halfnormal (14) is similar to the halfnormal exposure function
used by Royle and Gardner (2011) except they omit the factor of 2 on
$\sigma^2$, which leads to estimates of $\sigma$ that are larger
by a factor of sqrt(2). The hazard exponential (16) is identical to their
exponential function.
llll{
Code Name Parameters Function
0 HN halfnormal g0, sigma $g(d) = g_0 \exp
\left(\frac{-d^2} {2\sigma^2} \right)$
1 HR hazard rate g0, sigma, z $g(d) = g_0 [1 - \exp{
{-(^d/_\sigma)^{-z}} }]$
2 EX exponential g0, sigma $g(d) = g_0 \exp {
-(^d/_\sigma) }$
3 CHN compound halfnormal g0, sigma, z $g(d) = g_0 [1
- {1 - \exp \left(\frac{-d^2} {2\sigma^2} \right)} ^ z]$
4 UN uniform g0, sigma $g(d) = g_0, d <= 1="" 5="" 6="" 7="" 8="" 9="" 10="" 11="" 14="" 15="" 16="" 17="" 18="" \sigma;="" g(d)="0," \mbox{otherwise}$="" wex="" w="" exponential="" g0,="" sigma,="" $g(d)="g_0," d="" <="" w;="" \exp="" \left(="" -\frac{d-w}{\sigma}="" \right),="" ann="" annular="" normal="" \lbrace="" \frac{-(d-w)^2}="" {2\sigma^2}="" \rbrace$="" cln="" cumulative="" lognormal="" z="" [="" -="" f="" \lbrace(d-\mu)="" s="" \rbrace="" ]$="" cg="" gamma="" g="" (d;="" k,="" \theta)\rbrace$="" bss="" binary="" signal="" strength="" b0,="" b1="" (="" b_0="" +="" b_1="" d)="" ss="" beta0,="" beta1,="" sds="" f[\lbrace="" c="" (\beta_0="" \beta_1="" s]$="" sss="" spherical="" (d-1)="" \log="" _{10}="" d^2="" )="" hhn="" hazard="" halfnormal="" lambda0,="" sigma="" $\lambda(d)="\lambda_0" \left(\frac{-d^2}="" \right)$;="" hhr="" rate="" (1="" {="" -(^d="" _\sigma)^{-z}="" })$;="" hex="" _\sigma)="" }$;="" han="" \rbrace$;="" hcg="" \theta)\rbrace$;="" }="" functions="" (1)="" and="" (15),="" the="" "hazard-rate"="" detection="" described="" by="" hayes="" buckland="" (1983),="" are="" not="" recommended="" for="" secr="" because="" of="" their="" long="" tail,="" care="" is="" also="" needed="" with="" (2)="" (16).="" function="" (3),="" compound="" halfnormal,="" follows="" efford="" dawson="" (2009).="" (4)="" uniform="" defined="" only="" simulation="" as="" it="" poses="" problems="" likelihood="" maximisation="" gradient="" methods.="" probability="" implies="" hazard,="" so="" there="" no="" separate="" `hun'.="" (7),="" `f'="" standard="" distribution="" $\mu$="" $s$="" mean="" deviation="" on="" log="" scale="" a="" latent="" variable="" representing="" threshold="" distance.="" see="" note="" relationship="" to="" fitted="" parameters="" z.="" (8)="" (18),="" `g'="" 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). Note that (9) does
not require signal-strength data or $c$.
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 $g(d) = 1 - F \lbrace(c -
\beta_0) / sdS \rbrace$
Functions (12) and (13) are undocumented methods for sound attenuation.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.
Royle, J. A. and Gardner, B. (2011) Hierarchical spatial
capture--recapture models for estimating density from trapping
arrays. In: A.F. O'Connell, J.D. Nichols & K.U. Karanth (eds)
Camera Traps in Animal Ecology: Methods and Analyses. Springer,
Tokyo. Pp. 163--190.