Multispectral and RGB images are simulated with multispectral/RGB simulator using the software “POV-Ray 3.7” (https. Were randomly drawn using a “Latin Hypercube” sampling scheme. The parameters were assumed to follow a uniform distribution within their range of variation (Abichou. Struik PC (2005). This is sampling utility implementing Latin hypercube sampling from multivariate normal, uniform & empirical distribution. Correlation among variables can be sprecified.
Latin Hypercube Sampling LHC
JOSE LOPEZ-COLLADO
JULY, 2015
After searching fow a while, I finally found a paper describing the LHC sampling (Swiler and Wyss 2004). LHC is a re-scaling function in the domain of a random uniform variate so to have a better dispersion of the input numbers used to generate the pdf deviates. The paper of Swiler & Wyss presents a detailed example of the algorithm so anybody can check the results and the algorithm itself (pages 2-9 in the paper).
In essence, the sample size ss serves to divide the sampling space into ss categories and then the U values are re-scaled to the new limits:
(* HERE IS THE MATHEMATICA CODE WITH COMMENTS *)
SetDirectory[NotebookDirectory[]];
wdist = WeibullDistribution[1.5, 3];
dname = wdist;
(* ss is the sample size*)
ss= 1500;
(*scaleu is the LHS re-scaling, very simple indeed! *)
scaleu[u_, i_, ss_] := u (1/ss) + ((i - 1)/ss);
(* set function as listable, capable of handling lists*)
SetAttributes[scaleu, Listable];
(*get a list of uniform random numbers*)
un = RandomVariate[UniformDistribution[{0, 1}], ss];
(*Get a sequence of integers, 1,2,3,... ,ss*)
strata = Range[ss];
(*Re-distribute the random numbers using LHC*)
usc = scaleu[un, strata, ss];
(*Obtain a random number from the list above, in this example is wdist, a Weibull Distribution with shape and scale parameters of 1.5 and 3 respectively*)
(*Obtain a list of random numbers using LHC, check that we use the inverse of the cumulative density function CDF to translate from the re-scaled U values to their pdf values *)
pvL = Map[InverseCDF[dname, #] &, usc];
(* get some statistics, mean and standard deviation *)
mL = Mean[pvL]
2.70744
sdL = StandardDeviation[pvL]
1.83532
(*The next call is is the conventional random sampling, NOT the LHC, it uses the built-in Mathematica function RandomVariate and the distribution name and sample size as arguments *)
pvR = RandomVariate[dname, ss];
(* get the same statistics: mean and standard deviation*)
mR = Mean[pvR]
2.62812
sdR = StandardDeviation[pvR]
1.7902
(* Draw the distributions, Latin Hypercube on the left, regular sampling on the right *)
GraphicsRow[{Histogram[pvL], Histogram[pvR]}]
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