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.
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 |
\( \lambda(d) = \lambda_0
\lbrace 1 - G (d; k, \theta)\rbrace \); \(g(d) = 1-\exp(-\lambda(d))\)
|
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.