Accumulation function for parametric Rao's index of quadratic entropy (Q)

RaoAUC computes the accumulation function (integral or area under the curve) of the parametric version of Rao's index of quadratic entropy (Q) on different classes of numeric matrices using a moving window algorithm.

Usage

RaoAUC(
  alphas = 1:5,
  x,
  dist_m = "euclidean",
  window = 9,
  method = "classic",
  rasterAUC = TRUE,
  lambda = 0,
  na.tolerance = 1,
  rescale = FALSE,
  diag = TRUE,
  simplify = 0,
  np = 1,
  cluster.type = "SOCK",
  debugging = FALSE
)

Arguments

alphas: A continuous vector of alphas in the form start:end over which integrated the parametric Rao's index. Default value is 1:5.
x: Input data may be a matrix, a Spatial Grid Data Frame, a SpatRaster, or a list of these objects. In the latter case, if method="classic" only the first element of the list will be considered.
dist_m: Define the type of distance to be calculated between numerical categories. dist_m can be a character string which defines the name of the distance to derive such as "euclidean". The distance names allowed are the same as for proxy::dist. Alternatively, dist_m can be a function which calculates a user-defined distance, (i.e., function(x,y) {return(cos(y-x)-sin(y-x))}) or a matrix of distances. If method="multidimension" then only "euclidean", "manhattan", "canberra", "minkowski" and "mahalanobis" can be used. Default value is "euclidean". If proxy::dist is a matrix then the function will assume that this is the distance matrix, and therefore no distance will be derived.
window: The side of the square moving window, it must be an odd numeric value greater than 1 to ensure that the target pixel is in the centre of the moving window. Default value is 3.
method: Currently, there are two ways to calculate the parametric version of Rao's index. If method="classic", then the normal parametric Rao's index will be calculated on a single matrix. If method="multidimension" (experimental!) a list of matrices must be provided as input. In the latter case, the overall distance matrix will be calculated in a multi- or hyper-dimensional system by using the distance measure defined through the function argument dist_m. Each pairwise distance is then multiplied by the inverse of the squared number of pixels in the considered moving window, and the Rao's Q is finally derived by applying a summation. Default value is "classic".
rasterAUC: Boolean, if TRUE the output will be a SpatRaster object with x as a raster template.
lambda: The value of the lambda of Minkowski's distance. Considered only if dist_m = "minkowski" and method="multidimension". Default value is 0.
na.tolerance: Numeric value \((0.0-1.0)\) which indicates the proportion of NA values that will be tolerated to calculate parametric Rao's index in each moving window over x. If the relative proportion of NA's in a moving window is bigger than na.tolerance, then the value of the window will be set as NA, otherwise Rao's index will be calculated considering the non-NA values. Default values are 1.0 (i.e., no tolerance for NA's). Default value is 1.0.
rescale: Boolean. Considered only if method="multidimension". If TRUE, each element of x is rescaled and centred.
diag: Boolean. If TRUE then the diagonal of the distance matrix is filled with 0's, otherwise with NA's. If diag=TRUE and alpha=0, the output matrix will inexorably be composed of 0's.
simplify: Number of decimal places to be retained to calculate distances in Rao's index. Only if x is floats.
np: The number of processes (cores) which will be spawned. Default value is 2.
cluster.type: The type of cluster which will be created. The options are "MPI" (which calls "makeMPIcluster"), "FORK" (which calls "makeForkCluster"), and "SOCK" (which calls "makeCluster"). Default type is "SOCK".
debugging: A boolean variable set to FALSE by default. If TRUE, additional messages will be printed. For debugging only.

Value

A matrix of dimension dim(x). If rasterAUC=TRUE, then the output is a SpatRaster with x as a template.

Details

The accumulation function for the parametric Rao's Index (\(Q\)) is calculated integrating numerically over a range of alphas. *RaoAUC* is therefore equal to \((\int_{a}^{b} {1\over{N^4}}\cdot{d_{i,j}^{\alpha}})^{1\over{\alpha}} dx\). Where N is the number of pixels in a moving window, and alpha is a weight assigned to distances.

References

Rocchini, D., M. Marcantonio, and C. Ricotta (2017). Measuring Rao’s Q diversity index from remote sensing: An open source solution. Ecological Indicators. 72: 234–238.

Author

Matteo Marcantonio marcantoniomatteo@gmail.com

Examples

# Minimal example; RaoAUC with alphas ranging from 1 to 10
a <- matrix(c(10,10,10,20,20,20,20,30,30), ncol=3, nrow=3)
out <- RaoAUC(alphas=1:10, x=a, window=3, dist_m="euclidean", na.tolerance=1, rasterAUC=TRUE)
#> Warning: Simplify=0. Rounding data to 0 decimal places.
#> Warning: Input data are float numbers. Converting data to integer matrices...
#> 
#> 
#> Processing alpha: 1 Moving Window: 3
#> 
#> 
#> Processing alpha: 1 Moving Window: 3
#> 
#> 
#> Processing alpha: 2 Moving Window: 3
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#> 
#> Processing alpha: 2 Moving Window: 3
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#> 
#> Processing alpha: 3 Moving Window: 3
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#> 
#> Processing alpha: 3 Moving Window: 3
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#> 
#> Processing alpha: 4 Moving Window: 3
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#> 
#> Processing alpha: 4 Moving Window: 3
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#> 
#> Processing alpha: 5 Moving Window: 3
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#> 
#> Processing alpha: 5 Moving Window: 3
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#> 
#> Processing alpha: 6 Moving Window: 3
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#> 
#> Processing alpha: 6 Moving Window: 3
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#> Processing alpha: 7 Moving Window: 3
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#> 
#> Processing alpha: 7 Moving Window: 3
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#> Processing alpha: 8 Moving Window: 3
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#> 
#> Processing alpha: 8 Moving Window: 3
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#> Processing alpha: 9 Moving Window: 3
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#> 
#> Processing alpha: 9 Moving Window: 3
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#> Processing alpha: 10 Moving Window: 3
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#> 
#> Processing alpha: 10 Moving Window: 3
#> 
#> Integrating numerically Rao values over alphas...