optimPPL: Optimization of sample configurations for variogram...

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

Optimization of sample configurations for variogram identification and estimation

Description

Optimize a sample configuration for variogram identification and estimation. A criterion is defined so that the optimized sample configuration has a given number of points or point-pairs contributing to each lag-distance class (PPL).

Usage

optimPPL(
  points,
  candi,
  lags = 7,
  lags.type = "exponential",
  lags.base = 2,
  cutoff,
  distri,
  criterion = "distribution",
  pairs = FALSE,
  schedule,
  plotit = FALSE,
  track = FALSE,
  boundary,
  progress = "txt",
  verbose = FALSE
)

objPPL(
  points,
  candi,
  lags = 7,
  lags.type = "exponential",
  lags.base = 2,
  cutoff,
  distri,
  criterion = "distribution",
  pairs = FALSE,
  x.max,
  x.min,
  y.max,
  y.min
)

countPPL(
  points,
  candi,
  lags = 7,
  lags.type = "exponential",
  lags.base = 2,
  cutoff,
  pairs = FALSE,
  x.max,
  x.min,
  y.max,
  y.min
)

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().
lags	Integer value, the number of lag-distance classes. Alternatively, a vector of numeric values with the lower and upper bounds of each lag-distance class, the lowest value being larger than zero (e.g. 0.0001). Defaults to lags = 7.
lags.type	Character value, the type of lag-distance classes, with options "equidistant" and "exponential". Defaults to lags.type = "exponential".
lags.base	Numeric value, base of the exponential expression used to create exponentially spaced lag-distance classes. Used only when lags.type = "exponential". Defaults to lags.base = 2.
cutoff	Numeric value, the maximum distance up to which lag-distance classes are created. Used only when lags is an integer value. If missing, it is set to be equal to the length of the diagonal of the rectangle with sides x.max and y.max as defined in scheduleSPSANN().
distri	Numeric vector, the distribution of points or point-pairs per lag-distance class that should be attained at the end of the optimization. Used only when criterion = "distribution". Defaults to a uniform distribution.
criterion	Character value, the feature used to describe the energy state of the system configuration, with options "minimum" and "distribution". Defaults to objective = "distribution".
pairs	Logical value. Should the sample configuration be optimized regarding the number of point-pairs per lag-distance class? Defaults to pairs = 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.
x.max, x.min, y.max, y.min	Numeric value defining the minimum and maximum quantity of random noise to be added to the projected x- and y-coordinates. The minimum quantity should be equal to, at least, the minimum distance between two neighbouring candidate points. The units are the same as of the projected x- and y-coordinates. If missing, they are estimated from candi.

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.

Lag-distance classes

Two types of lag-distance classes can be created by default. The first are evenly spaced lags (lags.type = "equidistant"). They are created by simply dividing the distance interval from 0.0001 to cutoff by the required number of lags. The minimum value of 0.0001 guarantees that a point does not form a pair with itself. The second type of lags is defined by exponential spacings (lags.type = "exponential"). The spacings are defined by the base b of the exponential expression b^n, where n is the required number of lags. The base is defined using the argument lags.base. See pedometrics::vgmLags() for other details.

Using the default uniform distribution means that the number of point-pairs per lag-distance class (pairs = TRUE) is equal to n \times (n - 1) / (2 \times lag), where n is the total number of points and lag is the number of lags. If pairs = FALSE, then it means that the number of points per lag is equal to the total number of points. This is the same as expecting that each point contributes to every lag. Distributions other than the available options can be easily implemented changing the arguments lags and distri.

There are two optimizing criteria implemented. The first is called using criterion = "distribution" and is used to minimize the sum of the absolute differences between a pre-specified distribution and the observed distribution of points or point-pairs per lag-distance class. The second criterion is called using criterion = "minimum". It corresponds to maximizing the minimum number of points or point-pairs observed over all lag-distance classes.

Value

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

objPPL returns a numeric value: the energy state of the sample configuration – the objective function value.

countPPL returns a data.frame with three columns: a) the lower and b) upper limits of each lag-distance class, and c) the number of points or point-pairs per lag-distance class.

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

Bresler, E.; Green, R. E. Soil parameters and sampling scheme for characterizing soil hydraulic properties of a watershed. Honolulu: University of Hawaii at Manoa, p. 42, 1982.

Pettitt, A. N.; McBratney, A. B. Sampling designs for estimating spatial variance components. Applied Statistics. v. 42, p. 185, 1993.

Russo, D. Design of an optimal sampling network for estimating the variogram. Soil Science Society of America Journal. v. 48, p. 708-716, 1984.

Truong, P. N.; Heuvelink, G. B. M.; Gosling, J. P. Web-based tool for expert elicitation of the variogram. Computers and Geosciences. v. 51, p. 390-399, 2013.

Warrick, A. W.; Myers, D. E. Optimization of sampling locations for variogram calculations. Water Resources Research. v. 23, p. 496-500, 1987.

Examples

#####################################################################
# NOTE: The settings below are unlikely to meet your needs.         #
#####################################################################
if (interactive() & require(sp)) {
  # This example takes more than 5 seconds
  data(meuse.grid, package = "sp")
  schedule <- scheduleSPSANN(
    chains = 1,
    initial.acceptance = c(0.8, 0.99),
    initial.temperature = 9.5,
    x.max = 1540, y.max = 2060, x.min = 0,
    y.min = 0, cellsize = 40)
  set.seed(2001)
  res <- optimPPL(points = 10, candi = meuse.grid[, 1:2],
    schedule = schedule)
  objSPSANN(res)
}

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