# 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

1 |

### Arguments

`method` |
Defines the spatial sampling design to be used. The values can be |

`statistic` |
Describes what aspect of design you want calculated. The choices are |

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

`radius` |
Patch radius. Not needed if |

`pdetect` |
Probability detection. Not needed if |

`ssize` |
Sample size. Not needed if |

### 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"). Th`

is 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

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 | ```
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))
``` |