Circular Probability
Functions to answer the question "what radius is expected to include proportion p of points from a circular bivariate distribution corresponding to a given detection function", and the reverse. These functions may be used to relate the scale parameter(s) of a detection function (e.g., σ) to home-range area (specifically, the area within an activity contour for the corresponding simple home-range model) (see Note).
WARNING: the default behaviour of these functions changed in version
2.6.0. Integration is now performed on the cumulative hazard (exposure)
scale for all functions unless hazard = FALSE. Results will
differ.
circular.r (p = 0.95, detectfn = 0, sigma = 1, detectpar = NULL, hazard = TRUE, upper = Inf, ...) circular.p (r = 1, detectfn = 0, sigma = 1, detectpar = NULL, hazard = TRUE, upper = Inf, ...)
p |
vector of probability levels for which radius is required |
r |
vector of radii for which probability level is required |
detectfn |
integer code or character string for shape of detection function 0 = halfnormal, 2 = exponential etc. – see detectfn for other codes |
sigma |
spatial scale parameter of detection function |
detectpar |
named list of detection function parameters |
hazard |
logical; if TRUE the transformation -log(1-g(d)) is applied before integration |
upper |
numeric upper limit of integration |
... |
other arguments passed to |
circular.r is the quantile function of the specified circular
bivariate distribution (analogous to qnorm, for example). The
quantity calculated by circular.r is sometimes called 'circular
error probable' (see Note).
For detection functions with two parameters (intercept and scale) it is
enough to provide sigma. Otherwise, detectpar should be a
named list including parameter values for the requested detection
function (g0 may be omitted, and order does not matter).
Detection functions in secr are expressed in terms of the decline
in probability of detection with distance g(d), and both
circular.r and circular.p integrate this function by
default. Rather than integrating g(d) itself, it may be more
appropriate to integrate g(d) transformed to a hazard i.e. 1
- log(-g(d)). This is selected with hazard = TRUE.
Integration may also fail with the message “roundoff error is detected in the extrapolation table”.
Setting upper to a large number less than infinity sometimes corrects this.
Vector of values for the required radii or probabilities.
The term ‘circular error probable’ has a military origin. It is
commonly used for GPS accuracy with the default probability level set to
0.5 (i.e. half of locations are further than CEP from the true
location). A circular bivariate normal distriubution is commonly assumed
for the circular error probable; this is equivalent to setting
detectfn = "halfnormal".
Closed-form expressions are used for the normal and uniform cases; in the circular bivariate normal case, the relationship is r = (-2*log(1-p))^0.5 * sigma. Otherwise, the probability is computed numerically by integrating the radial distribution. Numerical integration is not foolproof, so check suspicious or extreme values.
When circular.r is used with the default sigma = 1, the result
may be interpreted as the factor by which sigma needs to be inflated to
include the desired proportion of activity (e.g., 2.45 sigma for 95%
of points from a circular bivariate normal distribution fitted on the hazard
scale (detectfn = 14) OR 2.24 sigma on the probability scale (detectfn = 0)).
Calhoun, J. B. and Casby, J. U. (1958) Calculation of home range and density of small mammals. Public Health Monograph No. 55. United States Government Printing Office.
Johnson, R. A. and Wichern, D. W. (1982) Applied multivariate statistical analysis. Prentice-Hall, Englewood Cliffs, New Jersey, USA.
## Calhoun and Casby (1958) p 3.
## give p = 0.3940, 0.8645, 0.9888
circular.p(1:3, hazard = FALSE)
## halfnormal, hazard-rate and exponential
circular.r ()
circular.r (detectfn = "HR", detectpar = list(sigma = 1, z = 4))
circular.r (detectfn = "EX")
circular.r (detectfn = "HHN")
circular.r (detectfn = "HHR", detectpar = list(sigma = 1, z = 4))
circular.r (detectfn = "HEX")
plot(seq(0, 5, 0.01), circular.p(r = seq(0, 5, 0.01)),
type = "l", xlab = "Radius (multiples of sigma)", ylab = "Probability")
lines(seq(0, 5, 0.01), circular.p(r = seq(0, 5, 0.01), detectfn = 2),
type = "l", col = "red")
lines(seq(0, 5, 0.01), circular.p(r = seq(0, 5, 0.01), detectfn = 1,
detectpar = list(sigma = 1,z = 4)), type = "l", col = "blue")
abline (h = 0.95, lty = 2)
legend (2.8, 0.3, legend = c("halfnormal","hazard-rate, z = 4", "exponential"),
col = c("black","blue","red"), lty = rep(1,3))
## in this example, a more interesting comparison would use
## sigma = 0.58 for the exponential curve.Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.