optimFREQ: Optimization of sample configurations for spatial trend...

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

Optimization of sample configurations for spatial trend identification and estimation (II)

Description

Optimize a sample configuration for spatial trend identification and estimation. A criterion is defined so that the sample reproduces the frequency marginal distribution of the covariates (FREQ).

Usage

optimFREQ(
  points,
  candi,
  covars,
  strata.type = "area",
  use.coords = FALSE,
  schedule,
  plotit = FALSE,
  track = FALSE,
  boundary,
  progress = "txt",
  verbose = FALSE
)

objFREQ(points, candi, covars, strata.type = "area", use.coords = FALSE)

optimDIST(
  points,
  candi,
  covars,
  strata.type = "area",
  use.coords = FALSE,
  schedule,
  plotit = FALSE,
  track = FALSE,
  boundary,
  progress = "txt",
  verbose = FALSE
)

objDIST(points, candi, covars, strata.type = "area", use.coords = FALSE)

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().
covars	Data frame or matrix with the spatially exhaustive covariates in the columns.
strata.type	(Optional) Character value setting the type of stratification that should be used to create the marginal sampling strata (or factor levels) for the numerical covariates. Two options are available: "area" (Default) Equal-area marginal sampling strata. "range" Equal-range marginal sampling strata. The first option ("area") is equivalent to drawing the frequency histogram of the numerical covariates with bins of variable width but equal area. The second, however, would result in a frequency histogram with bins of equal width but variable area such as when using graphics::hist() with its default options. Strata of equal area will include virtually the same number of individual covariate grid cells per stratum, while equal-range strata aim for the same number of unique covariate values in each stratum.
use.coords	(Optional) Logical value. Should the projected spatial x- and y-coordinates be used as spatially exhaustive covariates? Defaults to use.coords = FALSE.
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.

Frequency marginal distribution of covariates

Reproducing the frequency marginal distribution of the numeric covariates depends upon the definition of marginal sampling strata. These marginal sampling strata are also used to define the factor levels of all numeric covariates that are passed together with factor covariates. Two types of marginal sampling strata can be used: equal-area and equal-range.

Equal-area marginal sampling strata are defined using the sample quantiles estimated with stats::quantile() using a discontinuous function(type = 3). Using a discontinuous function avoids creating breakpoints that do not occur in the population of existing covariate values.

Depending on the level of discretization of the covariate values, stats::quantile() produces repeated breakpoints. A breakpoint will be repeated if that value has a relatively high frequency in the population of covariate values. The number of repeated breakpoints increases with the number of marginal sampling strata. Repeated breakpoints result in empty marginal sampling strata. To avoid this, only the unique breakpoints are used.

Equal-range marginal sampling strata are defined by breaking the range of covariate values into pieces of equal size. Depending on the level of discretization of the covariate values, this method creates breakpoints that do not occur in the population of existing covariate values. Such breakpoints are replaced with the nearest existing covariate value identified using Euclidean distances.

Like the equal-area method, the equal-range method can produce empty marginal sampling strata. The solution used here is to merge any empty marginal sampling strata with the closest non-empty marginal sampling strata. This is identified using Euclidean distances as well.

The approaches used to define the marginal sampling strata result in each numeric covariate having a different number of marginal sampling strata, some of them with different area/size. Because the goal is to have a sample that reproduces the frequency marginal distribution of the covariate, each marginal sampling strata will have a different number of sample points. The wanted distribution of the number of sample points per marginal strata is estimated empirically as the proportion of points in the population of existing covariate values that fall in each marginal sampling strata.

Value

optimFREQ (optimDIST) returns an object of class OptimizedSampleConfiguration: the optimized sample configuration with details about the optimization.

objFREQ (objDIST) returns a numeric value: the energy state of the sample configuration – the objective function value.

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.

Author(s)

Alessandro Samuel-Rosa [e-mail]

References

Hyndman, R. J.; Fan, Y. Sample quantiles in statistical packages. The American Statistician, v. 50, p. 361-365, 1996.

Everitt, B. S. The Cambridge dictionary of statistics. Cambridge: Cambridge University Press, p. 432, 2006.

See Also

optimACDC()

Examples

#####################################################################
# NOTE: The settings below are unlikely to meet your needs.         #
#####################################################################
if (interactive() & require(sp)) {
  data(meuse.grid, package = "sp")
  schedule <- scheduleSPSANN(
    initial.temperature = 1, chains = 1,
    x.max = 1540, y.max = 2060, x.min = 0,
    y.min = 0, cellsize = 40)
  set.seed(2001)
  res <- optimDIST(points = 10, candi = meuse.grid[, 1:2],
    covars = meuse.grid[, 5], use.coords = TRUE, schedule = schedule)
  objSPSANN(res) -
    objDIST(points = res, candi = meuse.grid[, 1:2],
    covars = meuse.grid[, 5],
    use.coords = TRUE)
}

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