optimMSSD: Optimization of sample configurations for spatial...

In samuel-rosa/spsann: Optimization of Spatial Samples via Simulated Annealing

Optimization of sample configurations for spatial interpolation (I)

Description

Optimize a sample configuration for spatial interpolation with a 'known' auto- or cross-correlation model, e.g. simple and ordinary (co)kriging. The criterion used is the mean squared shortest distance (MSSD) between sample locations and prediction locations.

Usage

optimMSSD(
  points,
  candi,
  eval.grid,
  schedule,
  plotit = FALSE,
  track = FALSE,
  boundary,
  progress = "txt",
  verbose = FALSE
)

objMSSD(points, candi, eval.grid)

Arguments

points	Integer value, integer vector, data frame (or matrix), or list. The number of sampling points (sample size) or the starting sample configuration. Four options are available: Integer value. The required number of sampling points (sample size). The sample configuration used to start the optimization will consist of grid cell centres of candi selected using simple random sampling, i.e. base::sample() with x = 1:nrow(candi) and size = points. Integer vector. A set of row indexes between one (1) and nrow(candi). These row indexes identify the grid cell centres of candi that will form the starting sample configuration for the optimization. The length of the integer vector, length(points), is the sample size. Data frame (or matrix). The Cartesian x- and y-coordinates (in this order) of the starting sample configuration. List. An object with two named sub-arguments: fixed An integer vector or data frame (or matrix) specifying an existing sample configuration (see options above). This sample configuration is kept as-is (fixed) during the optimization and is used only to compute the objective function values. free An integer value, integer vector, data frame or matrix (see options above) specifying the (number of) sampling points to add to the existing sample configuration. These new sampling points are free to be moved around (jittered) during the optimization. Most users will want to set an integer value simply specifying the required sample size. Using an integer vector or data frame (or matrix) will generally be helpful to users willing to evaluate starting sample configurations, test strategies to speed up the optimization, and fine-tune or thin an existing sample configuration. Users interested in augmenting a possibly existing real-world sample configuration or fine-tuning only a subset of the existing sampling points will want to use a list.
candi	Data frame (or matrix). The Cartesian x- and y-coordinates (in this order) of the cell centres of a spatially exhaustive, rectangular grid covering the entire spatial sampling domain. The spatial sampling domain can be contiguous or composed of disjoint areas and contain holes and islands. candi provides the set of (finite) candidate locations inside the spatial sampling domain for a point jittered during the optimization. Usually, candi will match the geometry of the spatial grid containing the prediction locations, e.g. newdata in gstat::krige(), object in raster::predict(), and locations in geoR::krige.conv().
eval.grid	(Experimental) Data frame or matrix with the objective function evaluation locations. Like candi, eval.grid must have two columns in the following order: ⁠[, "x"]⁠, the projected x-coordinates, and ⁠[, "y"]⁠, the projected y-coordinates.
schedule	List with named sub-arguments setting the control parameters of the annealing schedule. See scheduleSPSANN().
plotit	(Optional) Logical for plotting the evolution of the optimization. Plot updates occur at each ten (10) spatial jitters. Defaults to plotit = FALSE. The plot includes two panels: The first panel depicts the changes in the objective function value (y-axis) with the annealing schedule (x-axis). The objective function values should be high and variable at the beginning of the optimization (panel's top left). As the optimization proceeds, the objective function values should gradually transition to a monotone decreasing behaviour till they become virtually constant. The objective function values constancy suggests the end of the optimization (panel's bottom right). The second panel shows the starting (grey circles) and current spatial sample configuration (black dots). Black crosses indicate the fixed (existing) sampling points when a spatial sample configuration is augmented. The plot shows the starting sample configuration to assess the effects on the optimized spatial sample configuration: the latter generally should be independent of the first. The second panel also shows the maximum possible spatial jitter applied to a sampling point in the Cartesian x- (x-axis) and y-coordinates (y-axis).
track	(Optional) Logical value. Should the evolution of the energy state be recorded and returned along with the result? If track = FALSE (the default), only the starting and ending energy states return along with the results.
boundary	(Optional) An object of class SpatialPolygons (see sp::SpatialPolygons()) with the outer and inner limits of the spatial sampling domain (see candi). These SpatialPolygons help depict the spatial distribution of the (starting and current) sample configuration inside the spatial sampling domain. The outer limits of candi serve as a rough boundary when plotit = TRUE, but the SpatialPolygons are missing.
progress	(Optional) Type of progress bar that should be used, with options "txt", for a text progress bar in the R console, "tk", to put up a Tk progress bar widget, and NULL to omit the progress bar. A Tk progress bar widget is useful when using parallel processors. Defaults to progress = "txt".
verbose	(Optional) Logical for printing messages about the progress of the optimization. Defaults to verbose = FALSE.

Details

Generating mechanism

There are multiple mechanism to generate a new sample configuration out of an existing one. The main step consists of randomly perturbing the coordinates of a single sample, a process known as ‘jittering’. These mechanisms can be classified based on how the set of candidate locations for the samples is defined. For example, one could use an infinite set of candidate locations, that is, any location in the spatial domain can be selected as a new sample location after a sample is jittered. All that is needed is a polygon indicating the boundary of the spatial domain. This method is more computationally demanding because every time an existing sample is jittered, it is necessary to check if the new sample location falls in spatial domain.

Another approach consists of using a finite set of candidate locations for the samples. A finite set of candidate locations is created by discretising the spatial domain, that is, creating a fine (regular) grid of points that serve as candidate locations for the jittered sample. This is a less computationally demanding jittering method because, by definition, the new sample location will always fall in the spatial domain.

Using a finite set of candidate locations has two important inconveniences. First, not all locations in the spatial domain can be selected as the new location for a jittered sample. Second, when a sample is jittered, it may be that the new location already is occupied by another sample. If this happens, another location has to be iteratively sought for, say, as many times as the size of the sample configuration. In general, the larger the size of the sample configuration, the more likely it is that the new location already is occupied by another sample. If a solution is not found in a reasonable time, the the sample selected to be jittered is kept in its original location. Such a procedure clearly is suboptimal.

spsann uses a more elegant method which is based on using a finite set of candidate locations coupled with a form of two-stage random sampling as implemented in spcosa::spsample(). Because the candidate locations are placed on a finite regular grid, they can be taken as the centre nodes of a finite set of grid cells (or pixels of a raster image). In the first stage, one of the “grid cells” is selected with replacement, i.e. independently of already being occupied by another sample. The new location for the sample chosen to be jittered is selected within that “grid cell” by simple random sampling. This method guarantees that virtually any location in the spatial domain can be selected. It also discards the need to check if the new location already is occupied by another sample, speeding up the computations when compared to the first two approaches.

Annealing schedule

The search graph corresponds to the set of effective candidate locations for a sample location selected to be jittered. The size of the search graph, i.e. area within which a sample location can be moved around, is related to the concept of temperature. A larger search graph is equivalent to higher temperatures, which potentially result in more movement – or ‘agitation’ – of the set of sample locations.

The current version of the spsann-package uses a linear cooling schedule which depends upon the number of jitters to control the size of the search graph. The equations are

x_max = x_max0 - (chains_i / chains) * (x_max0 - x_min) + x_cellsize + x_min0

and

y_max = y_max0 - (chains_i / chains) * (y_max0 - y_min) + y_cellsize + y_min0,

where $x_max0$ and $y_max0$ are the maximum allowed shifts in the x- and y-coordinates in the first chain, $x_min$ and $y_min$ are the minimum required shifts in the x- and y-coordinates, $x_max$ and $y_max$ are the maximum allowed shifts in the x- and y-coordinates during the next chain, $chains$ and $chain_i$ are the total and current chains, and $x_cellsize$ and $y_cellsize$ are the grid spacing in the x- and y-coordinates. Because $x_cellsize$ and $y_cellsize$ can be equal to zero when a finite set of candidate locations is used, $x_min0$ and $y_min0$ are the maximum nearest neighbour distance in the x- and y-coordinates between candidate locations.

Mean squared shortest distance

This objective function is based on the knowledge that the simple and ordinary (co)kriging prediction error variance only depends upon the separation distance between sample locations: the larger the distance, the larger the prediction error variance. As such, the better the spread of the sample locations in the spatial domain, the smaller the overall simple/ordinary (co)kriging prediction error variance. This is the purpose of using a regular grid of sample locations.

However, a regular grid usually is suboptimal, especially if the spatial domain is irregularly shaped. Thus the need for optimization, that is based on measuring the goodness of the spread of sample locations in the spatial domain. To measure this spread we can compute the distance from every sample location to each of the prediction locations placed on a fine grid covering the entire spatial domain. Next, for every prediction location we find the closest sample location and record its distance. The mean of these squared distances over all prediction location will measure the spread of the sample locations.

During the optimization, we try to reduce this measure – the mean squared shortest distance – between sample and prediction locations. (This is also know as spatial coverage sampling, see the R-package spcosa.)

Value

optimMSSD returns an object of class OptimizedSampleConfiguration: the optimized sample configuration with details about the optimization.

objMSSD returns a numeric value: the energy state of the sample configuration – the objective function value in square map units, generally m^2 or km^2.

Note

Distance between two points

spsann always computes the distance between two locations (points) as the Euclidean distance between them. This computation requires the optimization to operate in the two-dimensional Euclidean space, i.e. the coordinates of the sample, candidate and evaluation locations must be Cartesian coordinates, generally in metres or kilometres. spsann has no mechanism to check if the coordinates are Cartesian: you are the sole responsible for making sure that this requirement is attained.

Sample configuration for spatial interpolation

A sample configuration optimized for spatial interpolation such as simple and ordinary (co)kriging is not necessarily appropriate for estimating the parameters of the spatial autocorrelation model, i.e. the parameters of the variogram model. See optimPPL for more information on the optimization of sample configurations for variogram identification and estimation.

Author(s)

Alessandro Samuel-Rosa [e-mail]

References

Brus, D. J.; de Gruijter, J. J.; van Groenigen, J.-W. Designing spatial coverage samples using the k-means clustering algorithm. In: P. Lagacherie,A. M.; Voltz, M. (Eds.) Digital soil mapping – an introductory perspective. Elsevier, v. 31, p. 183-192, 2006.

de Gruijter, J. J.; Brus, D.; Bierkens, M.; Knotters, M. Sampling for natural resource monitoring. Berlin: Springer, p. 332, 2006.

Walvoort, D. J. J.; Brus, D. J.; de Gruijter, J. J. An R package for spatial coverage sampling and random sampling from compact geographical strata by k-means. Computers and Geosciences. v. 36, p. 1261-1267, 2010.

See Also

[distanceFromPoints](https://CRAN.R-project.org/package=raster), [stratify](https://CRAN.R-project.org/package=spcosa).

Examples

#####################################################################
# NOTE: The settings below are unlikely to meet your needs.         #
#####################################################################
data(meuse.grid, package = 'sp')
candi <- meuse.grid[, 1:2]
schedule <- scheduleSPSANN(
  chains = 1, initial.temperature = 5000000,
  x.max = 1540, y.max = 2060, x.min = 0, y.min = 0, cellsize = 40)
set.seed(2001)
res <- optimMSSD(points = 10, candi = candi, schedule = schedule)
data.frame(
  expected = 247204.8,
  objSPSANN = objSPSANN(res),
  objMSSD = objMSSD(candi = candi, points = res)
)

samuel-rosa/spsann documentation built on Nov. 6, 2023, 12:48 p.m.