detectfn: Detection Functions

In secr: Spatially Explicit Capture-Recapture

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)–(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).

Note

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.

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.

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.

See Also

detectfnplot

secr documentation built on Nov. 4, 2024, 9:06 a.m.