det_hrz {highriskzone}

Determination of the high-risk zone.

Description

det_hrz determines the high-risk zone through the method of fixed radius (type = "dist" and criterion = "direct"), the quantile-based method (type = "dist" and criterion = "area"/"indirect") and the intensity-based method (type = "intens").

Usage

det_hrz(
  ppdata,
  type,
  criterion,
  cutoff,
  distancemap = NULL,
  intens = NULL,
  nxprob = 0.1,
  covmatrix = NULL
)

Arguments

ppdata

Observed spatial point process of class ppp.

type

Method to use, can be one of "dist" (method of fixed radius or quantile-based method), or "intens" (intensity-based method)

criterion

criterion to limit the high-risk zone, can be one of "area" (giving size of hrz), "indirect" (giving quantile/alpha depending on type), or "direct" (giving radius/threshold c depending on type)

cutoff

Value of criterion (area, radius, quantile, alpha or threshold). Depending on criterion and type: If criterion = "direct" and type = "intens", cutoff is the maximum intensity of unexploded bombs outside the risk zone. If type = "dist" instead, cutoff is the radius of the circle around each exploded bomb. "If criterion = "indirect", cutoff is the quantile for the quantile-based method and the failure probability alpha for the intensity-base method. If criterion = "area", cutoff is the area the high-risk zone should have.

distancemap

(optional) distance map: distance of every pixel to the nearest observation of the point pattern; only needed for type="dist". If not given, it will be computed by distmap.

intens

(optional) estimated intensity of the observed process (object of class "im"), only needed for type="intens". If not given, it will be estimated using density.ppp.

nxprob

Probability of having unobserved events. Default value is 0.1.

covmatrix

(optional) Covariance matrix of the kernel of a normal distribution, only needed for type="intens" if no intensity is given. If not given, it will be estimated using Hscv.

Details

There are different methods implemented to determine a high-risk zone.

Method of fixed radius

In this method, the high-risk zone is determined by drawing a circle around each observed event with a fixed radius. This method will be used when type = "dist" and criterion = "direct". cutoff then is the radius.

Quantile-based method

This method is a development of the above. Here the radius is not fixed. It uses the distance of every observed event to the nearest other event, which is calculated by the nearest-neighbour distance. The radius is assessed by the p-quantile of the empirical distribution function of the nearest-neighbour distance. This method will be used when type = "dist" and criterion = "indirect" or "area". If criterion = "indirect", then cutoff is the quantile that should be used. If criterion = "area" then cutoff is the area that the high-risk zone has to have at the end and from that the quantile/the radii are determined. When the calculation is done via the area, it can not really be classified to the quantile-based method. It is rather a third "distance-based" method.

Intensity-based method

The first step of this method is to estimate the intensity of the observed events. Based on the estimated intensity and the specified probability of unobserved bombs nxprob it is possible to estimate the intensity of unobserved/unexploded bombs. The high-risk zone is then the area in which the estimated intensity of unexploded bombs exceeds a certain value. This value is called threshold c. The method will be used when type = "intens". There are three different ways to construct a high-risk zone:

1. Fixing the threshold c: criterion = "direct"

2. Fixing the area of the high-risk zone: criterion = "area"

3. Fixing the failure probability alpha, which is the probability of having unobserved events outside the high-risk zone: criterion = "indirect" Here, the point process is assumed to be an inhomogeneous Poisson process.

For further information see Mahling et al. (2013) (References).

If there are restriction areas in the observation window, use det_hrz_restr instead. For estimation of intensity based highrikszones with a bigger observation area than area of interest (evaluation area) use det_hrz_eval_ar.

Value

An object of class "highriskzone", which is a list of

typehrz, criterion, cutoff, nxprob

see arguments

zone

Determined high-risk zone: Object of class "owin" based on a binary mask. See owin.

threshold

determined threshold. If type = "dist" and criterion = "direct" it is the specified radius. If criterion = "indirect" or "area" the determined radius used to construct a risk zone fulfilling the specified criterion and cutoff. If type = "dist" it is the specified or calculated threshold c, the maximum intensitiy of unexploded bombs outside the risk zone.

calccutoff

determined cutoff-value. For type="dist" and criterion="area", this is the quantile of the nearest-neighbour distance. For type="intens" and criterion="area" or "direct", it is the failure probability alpha. For all other criterions it is NA.

covmatrix

If not given (and type="intens"), it is estimated. See Hscv.

References

Monia Mahling, Michael Hoehle & Helmut Kuechenhoff (2013), Determining high-risk zones for unexploded World War II bombs by using point process methodology. Journal of the Royal Statistical Society, Series C 62(2), 181-199.

Monia Mahling (2013), Determining high-risk zones by using spatial point process methodology. Ph.D. thesis, Cuvillier Verlag Goettingen, available online: http://edoc.ub.uni-muenchen.de/15886/

See Also

distmap, eval.im, owin, eval_method, det_hrz_restr

Examples

 data(craterA)
## change npixel to 1000 to obtain nicer plots
spatstat.geom::spatstat.options(npixel=100)
## type: dist
hrzd1 <- det_hrz(craterA, type = "dist", criterion = "area", cutoff = 1000000, nxprob = 0.1)
hrzd2 <- det_hrz(craterA, type = "dist", criterion = "indirect", cutoff = 0.9, nxprob = 0.1)
hrzd3 <- det_hrz(craterA, type = "dist", criterion = "direct", cutoff = 100, nxprob = 0.1)

op <- par(mfrow = c(2, 2))
plot(craterA)
plot(hrzd1, zonecol = 2, win = craterA$window, plotwindow = TRUE)
plot(hrzd2, zonecol = 3,  win = craterA$window, plotwindow = TRUE)
plot(hrzd3, zonecol = 4,  win = craterA$window, plotwindow = TRUE)
par(op)

## Not run: 
# or first calculate the distancemap and use it:
distm <- distmap(craterA)
hrzd <- det_hrz(craterA, type = "dist", criterion = "direct", cutoff = 100,
                distancemap = distm, nxprob = 0.1)

## End(Not run)                
## type: intens 
# reduce number of observations for faster computation
thin.craterA <- craterA[1:10]
hrzi1 <- det_hrz(thin.craterA, type = "intens", criterion = "area", cutoff = 100000, nxprob = 0.1)
plot(hrzi1)
plot(thin.craterA, add = TRUE)
plot(thin.craterA$window, add = TRUE)
## Not run: 
hrzi2 <- det_hrz(craterA, type = "intens", criterion = "indirect", cutoff = 0.1, nxprob = 0.1)
hrzi3 <- det_hrz(craterA, type = "intens", criterion = "direct", cutoff = 0.0001, nxprob = 0.1)
plot(hrzi2)
plot(hrzi3)

## End(Not run)
                 
## More detailed examples on http://highriskzone.r-forge.r-project.org/