| detectfn {secr} | 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 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)–(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{d-w}{\sigma} \right), \mbox{otherwise}
|
| 6 ANN | annular normal | g0, sigma, w | g(d) = g_0 \exp
\lbrace \frac{-(d-w)^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 (d-1) - 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{-(d-w)^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)) |
Functions (1) and (15), the "hazard-rate" 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 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.
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 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.
Ergon, T. and Gardner, B. (2014) Separating mortality and emigration: modelling space use, dispersal and survival with robust-design spatial capture–recapture data. Methods in Ecology and Evolution 5, 1327–1336.
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.