detectfn  R Documentation 
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 homerange
centre. In secr only simple 2 or 3parameter 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)–(20) 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)–(19) 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.
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 <= \sigma;
g(d) = 0, \mbox{otherwise} 
5 WEX  w exponential  g0, sigma, w  g(d) = g_0, d < w;
g(d) = g_0 \exp \left( \frac{dw}{\sigma} \right), \mbox{otherwise}

6 ANN  annular normal  g0, sigma, w  g(d) = g_0 \exp
\lbrace \frac{(dw)^2} {2\sigma^2} \rbrace 
7 CLN  cumulative lognormal  g0, sigma, z  g(d) = g_0
[ 1  F \lbrace(d\mu)/s \rbrace ]

8 CG  cumulative gamma  g0, sigma, z  g(d) = g_0
\lbrace 1  G (d; k, \theta)\rbrace

9 BSS  binary signal strength  b0, b1  g(d) = 1  F
\lbrace  ( b_0 + b_1 d) \rbrace 
10 SS  signal strength  beta0, beta1, sdS  g(d) =1 
F[\lbrace c  (\beta_0 + \beta_1 d) \rbrace / s] 
11 SSS  signal strength spherical  beta0, beta1, sdS 
g(d) = 1  F [ \lbrace c  (\beta_0 + \beta_1 (d1)  10 \log
_{10} d^2 ) \rbrace / s ] 
14 HHN  hazard halfnormal  lambda0, sigma  \lambda(d) = \lambda_0 \exp
\left(\frac{d^2} {2\sigma^2} \right) ; g(d) = 1\exp(\lambda(d)) 
15 HHR  hazard hazard rate  lambda0, sigma, z  \lambda(d)
= \lambda_0 (1  \exp \{ (^d/_\sigma)^{z} \}) ; g(d) = 1\exp(\lambda(d)) 
16 HEX  hazard exponential  lambda0, sigma  \lambda(d)
= \lambda_0 \exp \{ (^d/_\sigma) \} ; g(d) = 1\exp(\lambda(d)) 
17 HAN  hazard annular normal  lambda0, sigma, w  \lambda(d) = \lambda_0 \exp
\lbrace \frac{(dw)^2} {2\sigma^2} \rbrace ; g(d) = 1\exp(\lambda(d)) 
18 HCG  hazard cumulative gamma  lambda0, sigma, z  \lambda(d) = \lambda_0
\lbrace 1  G (d; k, \theta)\rbrace ; g(d) = 1\exp(\lambda(d))

19 HVP  hazard variable power  lambda0, sigma, z  \lambda(d)
= \lambda_0 \exp \{ (^d/_\sigma)^{z} \} ; g(d) = 1\exp(\lambda(d)) 
20 HPX  hazard pixelar  lambda0, sigma  g(d') = 1exp(\lambda(d')), d' <= \sigma;
g(d') = 0, \mbox{otherwise} 
Functions (1) and (15), the "hazardrate" detection functions described by Hayes and Buckland (1983), are not recommended for SECR because of their long tail, and care is also needed with (2) and (16).
Function (3), the compound halfnormal, follows Efford and Dawson (2009).
Function (4) uniform is defined only for simulation as it poses problems for likelihood maximisation by gradient methods. Uniform probability implies uniform hazard, so there is no separate function ‘HUN’.
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 functions (8) and (18), ‘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). Note that (9) does
not require signalstrength data or c
.
Function (11) includes an additional ‘hardwired’ 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.
Function (19) has been used in some published papers and is included for comparison (see e.g. Ergon and Gardner 2014).
Function (20) assigns positive probability of detection only to points within a square pixel (cell) with side 2 sigma that is centred on the detector. (Typically used with fixed sigma = detector spacing / 2).
The parameters of function (7) are potentially confusing. The fitted
parameters describe a latent threshold variable on the natural scale:
sigma (mean) = \exp(\mu + s^2 / 2)
and z
(standard deviation) = \sqrt{\exp(s^2 + 2
\mu)(\exp(s^2)1)}
. As with other
detection functions, sigma is a spatial scale parameter, although in
this case it corresponds to the mean of the threshold variable; the
standard deviation of the threshold variable (z) determines the shape
(roughly 1/max(slope)) of the detection function.
Efford, M. G. and Dawson, D. K. (2009) Effect of distancerelated 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.
Ergon, T. and Gardner, B. (2014) Separating mortality and emigration: modelling space use, dispersal and survival with robustdesign spatial capture–recapture data. Methods in Ecology and Evolution 5, 1327–1336.
Hayes, R. J. and Buckland, S. T. (1983) Radialdistance models for the linetransect 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.
detectfnplot
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