Probability of circular patch detection

Description

The function can calculate the probability of detection of a circular patch of specified radius for a specified density of points; the density needed to achieve a specified probability of detection; or the radius of the patch that will be detected with specified probability and sampling density.This is done for random, square lattice, and triangular lattice spatial sampling designs.

Usage

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detect(method, statistic, area=NA, radius=NA, pdetect=NA, ssize=NA)

Arguments

method

Defines the spatial sampling design to be used. The values can be "R" (random), "S" (square lattice) or "T" (triangular lattice). See Barry and Nicholson (1993) for details and formulae for the probabilities of detection for the square lattice and triangular lattice designs. For the random design, prob(detect)=1 - (1 - a/A)^N, where a is the patch area and A is the survey area. This gives similar answers to the formula in Barry and Nicholson, but is exact for fixed sample size.

statistic

Describes what aspect of design you want calculated. The choices are "P" (probability detection); "N" (sample size) or "R" (patch radius).

area

The survey area (same units as distance and radius).

radius

Patch radius. Not needed if statistic="R".

pdetect

Probability detection. Not needed if statistic="P".

ssize

Sample size. Not needed if statistic="N".

Details

The basic idea is that you wish to conduct a survey in an area area to detect some object (patch) of interest. This could be a cockle patch, an area of reef or an archaeological deposit. This function asssumes that the object is circular with radius radius. You have three choices of sampling deign to use: spatial, square lattice and triangular lattice. In terms of patch detection, for a given sample size, the triangular design gives the highest probability - because its points are equi-distant apart.

The simplest application of this function is to assess the patch detection probability for a particular design. This is obtained using the statistic="P" option. However, the problem can be turned around and this function used to calculate the sample size needed to obatain a specific patch detection probability (statistic="N") or the radius of the patch that would be detected with some desired probability (statistic="R"). This last scenario might be useful if there was some particular size of patch that you wanted to be sure (say, 90 percent) of detecting.

Value

prob

Probability of patch detection

ssize

Sample size

rad

Patch radius

sep

Separation distance (for square and triangular lattice designs)

Author(s)

Jon Barry: Jon.Barry@cefas.co.uk

References

Barry J and Nicholson M D (1993) Measuring the probabilities of patch detection for four spatial sampling schemes. Journal of Applied Statistics, 20, 353-362.

Examples

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detect(method='R', statistic='P', area=100, radius=2, ssize=15)$prob
detect(method='R', statistic='N', area=100, radius=2, pdetect=0.95)$ssize
detect(method='R', statistic='R', area=100, pdetect=0.95, ssize=15)$rad

detect(method='S', statistic='P', area=100, radius=1.4, ssize=15)
detect(method='S', statistic='N', area=100, radius=1.4, pdetect=0.6)

# Plot patch detection as a function of radius
square = rep(0,200); rand = square; triang = rand
radius = seq(0.01, 2, 0.01)

for (j in 1:200) {
rand[j] = detect(method='R', statistic='P', area=100, radius=radius[j], ssize=15)$p
square[j] = detect(method='S', statistic='P', area=100, radius=radius[j], ssize=15)$p
triang[j] = detect(method='T', statistic='P', area=100, radius=radius[j], ssize=15)$p
}

plot(radius, rand, ylim=c(0,1), xlab='Patch radius', ylab='Probability detection', type='l')
lines(radius, square, col=2, lty=2)
lines(radius, triang, col=3, lty=3)
legend('topleft', legend=c('Random', 'Square', 'Triangular'), col=c(1,2,3), lty=c(1,2,3))

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