Description Usage Arguments Details Value Author(s) References Examples
Spatial simulated annealing uses slight perturbations of previous sampling designs and a random search technique to solve spatial optimization problems.
Candidate measurement locations are iteratively moved around and optimized by minimizing the mean universal kriging variance (calculateMukv
). The approach relies on a known, prespecified
model for underlying spatial variation (variogramModel
).
1 2 3 
observations 
object of class 
predGrid 
object of class 
candidates 
candidates is the study area of class

action 
character string indicating which type of action to perform:

nDiff 
number of stations to add or delete 
model 
variogram model:object of class 
nr_iterations 
number of iterations to process before stopping. The default

plotOptim 
logical; if TRUE, creates a plot of the result as optimization progresses; TRUE by default 
formulaString 
formula that defines the dependent variable as a linear model
of independent variables; suppose the dependent variable has name 
coolingFactor 
variable defining how fast the algorithm will cool down. With SSA,
worsening designs are accepted with a decreasing probability (generally set to

covariates 
character string defining whether possible covariates should be found by "over" or "krige", see also details below 
... 
other arguments to be passed on to lower level functions 
This function applies spatial simulated annealing for the special case of
optimization with the MUKV criterion (calculateMukv
).
With covariates, the function takes a universal kriging model into account.
When optimizing a sampling design using SSA and criterion = "mukv"
,
measurement values at new sampling locations are not required in order to
calculate prediction error variance (criterion = "mukv"
).
This attractive property allows one to estimate the kriging prediction error
variance prior to collecting the data (i.e., the dependent variable
is unknown at new candidate locations), and it is this property that is used in
the SSA optimization procedure after (Brus & Heuvelink 2007, Melles et al. 2011).
A stopping criterion countMax
is implemented in lower level functions to
end the optimization procedure after 200 search steps
have occurred without an improvement in the design. If this stopping criterion
is reached before nr_iterations
, SSA will terminate.
method = "ssa"
with criterion = "mukv"
makes it possible to assume
a linear relationship between independent variables in predGrid
and dependent variables at observation locations using universal kriging
(krige
). However, a correct estimate of mean universal
kriging variance requires that the independent
covariate variables be
known at candidate locations. Thus it is necessary to have complete spatial
coverage for all covariate predictors in the linear model. Covariate information
must be available at both new candidate measurement locations and prediction locations.
This is not possible (except for the measurement locations themselves).
Instead, these are estimated from the prediction locations.
There are two possible methods to attain information on covariates at the
candidate locations, and the method can be set using the argument covariates
:
over
and krige
. over
finds the value of
covariates at new locations by overlaying candidate locations on the prediction grid
and taking the value of the nearest neighbour. The second method uses kriging to
estimate covariate values at new locations from predGrid. The first method is
generally faster, the second method is most likely more exact, particularly if
the resolution of predGrid is low in relation to the spatial correlation lengths of the covariates.
Both methods are approximations that may influence the criterion used for
optimization with increasing numbers of points added.
SpatialPointsDataFrame with optimized locations
O. Baume, S.J. Melles, J. Skoien
D. J. Brus, G. B. M. Heuvelink (2007). Optimization of sample patterns for universal kriging of environmental variables, Geoderma, 138: 8695 (2007).
S. J. Melles, G. B. M. Heuvelink, C. J. W. Twenhofel, U. Stohlker (2011). Optimizing the spatial pattern of networks for monitoring radioactive releases, Computers and Geosciences, 37: 280288 (2011).
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39  ## Not run:
# load data:
data(meuse)
coordinates(meuse) = ~x+y
data(meuse.grid)
coordinates(meuse.grid) = ~x+y
gridded(meuse.grid) = TRUE
predGrid = meuse.grid
# estimate variograms (OK/UK):
vfitOK = fit.variogram(variogram(zinc~1, meuse), vgm(1, "Exp", 300, 1))
vfitUK = fit.variogram(variogram(zinc~x+y, meuse), vgm(1, "Exp", 300, 1))
vfitRK = fit.variogram(variogram(zinc~dist+ffreq+soil, meuse), vgm(1, "Exp", 300, 1))
# study area of interest:
bb = bbox(predGrid)
boun = SpatialPoints(data.frame(x=c(bb[1,1],bb[1,2],bb[1,2],bb[1,1],bb[1,1]),
y=c(bb[2,1],bb[2,1],bb[2,2],bb[2,2],bb[2,1])))
Srl = Polygons(list(Polygon(boun)),ID = as.character(1))
candidates = SpatialPolygonsDataFrame(SpatialPolygons(list(Srl)),
data = data.frame(ID=1))
# add 20 more points assuming OK model (SSA method):
optimOK < ssaOptim(meuse, meuse.grid, candidates = candidates, covariates = "over",
nDiff = 20, action = "add", model = vfitOK, nr_iterations = 10000,
formulaString = zinc~1, nmax = 40, countMax = 200)
# add 20 more points assuming UK model (SSA method):
optimUK < ssaOptim(meuse, meuse.grid, candidates = candidates, covariates = "over",
nDiff = 20, action = "add", model = vfitUK, nr_iterations = 10000,
formulaString = zinc~x+y, nmax = 40, countMax = 200)
# add 20 more points with auxiliary variables (SSA method):
optimRK < ssaOptim(meuse, meuse.grid, candidates = candidates, covariates = "over",
nDiff = 20, action = "add", model = vfitRK, nr_iterations = 10000,
formulaString = zinc~dist+ffreq+soil, nmax = 40, countMax = 200)
## End(Not run)

Add the following code to your website.
For more information on customizing the embed code, read Embedding Snippets.