Latin hypercube sampling: Difference between revisions

Revision as of 00:24, 1 October 2019

Revision as of 20:59, 7 March 2020

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In the context of statistical sampling, a square grid containing sample positions is a [[Latin square]] if (and only if) there is only one sample in each row and each column. A '''Latin [[hypercube]]''' is the generalisation of this concept to an arbitrary number of dimensions, whereby each sample is the only one in each axis-aligned [[hyperplane]] containing it.

When sampling a function of <math>N</math> variables, the range of each variable is divided into <math>M</math> equally probable intervals. <math>M</math> sample points are then placed to satisfy the Latin hypercube requirements; note that this forces the number of divisions, <math>M</math>, to be equal for each variable. Also note that thisThis sampling scheme does not require more samples for more dimensions (variables); this independence is one of the main advantages of this sampling scheme. Another advantage is that random samples can be taken one at a time, remembering which samples were taken so far.

[[Image:LHSsampling.png|100px|right]]

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In two dimensions the difference between random sampling, Latin Hypercube sampling, and orthogonal sampling can be explained as follows:

#In '''random sampling''' new sample points are generated without taking into account the previously generated sample points. One does not necessarily need to know beforehand how many sample points are needed.

#In '''Latin Hypercube sampling''' one must first decide how many sample points to use and for each sample point remember in which row and column the sample point was taken. Note that suchSuch configuration is similar to having N [[Rook_(chess)|rooks]] on a chess board without threatening each other.

#In '''Orthogonal sampling''', the sample space is divided into equally probable subspaces. All sample points are then chosen simultaneously making sure that the total set of sample points is a Latin Hypercube sample and that each subspace is sampled with the same density.