Latin hypercube sampling: Difference between revisions

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{{Short description|Statistical sampling technique}}

'''Latin hypercube sampling''' ('''LHS''') is a [[statistics|statistical]] method for generating a near-random sample of parameter values from a [[multidimensional distribution]]. The [[Sampling (statistics)|sampling method]] is often used to construct [[computer experiment]]s or for [[Monte- Carlo integration]].<ref name = "C3M"/>

The LHS was described by Michael McKay of Los Alamos National Laboratory in 1979.<ref name = "C3M">{{cite journal

| last = McKay

| first = M.D. |author2=Beckman, R.J. |author3=Conover, W.J.

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| issn = 0040-1706

| doi = 10.2307/1268522

| format = [[JSTOR]] Abstract

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| osti = 5236110

| jstor = 1268522

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</ref> An independently equivalent technique was independently proposed by [[:lv:Vilnis Egl%C4%81jsEglājs|Vilnis Eglājs]] in 1977.<ref>{{cite journal|last=Eglajs|first=V.|author2=Audze P. |title=New approach to the design of multifactor experiments|journal=Problems of Dynamics and Strengths|year=1977|series=35|pages=104–107|publisher=Zinatne Publishing House|___location=Riga|language=Russian}}</ref> It was further elaborated by [[Ronald L. Iman]] and coauthors in 1981.<ref>{{cite journal |last=Iman |first=R.L. |author2=Helton, J.C. |author3=[[James Edward Campbell|Campbell, J.E.]] |title=An approach to sensitivity analysis of computer models, Part 1. Introduction, input variable selection and preliminary variable assessment |journal=Journal of Quality Technology |volume=13 |issue=3 |pages=174–183 |year=1981 |doi=10.1080/00224065.1981.11978748 }}</ref> Detailed computer codes and manuals were later published.<ref>{{cite book |last=Iman |first=R.L. |author2=Davenport, J.M. |author3=Zeigler, D.K. |title=Latin hypercube sampling (program user's guide) |year=1980 |osti=5571631}}</ref>

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.<ref name = "C3M"/>

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.

The maximum number of combinations for a Latin Hypercube of <math>M</math> divisions and <math>N</math> variables (i.e., dimensions) can be computed with the following formula:

<math>\left(\prod_{n=0}^{M-1} (M-n)\right)^{N-1} = (M!)^{N-1}</math>

For example, a Latin hypercube of <math>M = 4</math> divisions with <math>N = 2</math> variables (i.e., a square) will have 24 possible combinations. A Latin hypercube of <math>M = 4</math> divisions with <math>N = 3</math> variables (i.e., a cube) will have 576 possible combinations.

'''Orthogonal sampling''' adds the requirement that the entire sample space must be sampled evenly. Although more efficient, orthogonal sampling strategy is more difficult to implement since all random samples must be generated simultaneously.

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

In two dimensions the difference between random sampling, Latin Hypercubehypercube 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 Hypercubehypercube 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 '''Orthogonalorthogonal sampling''', the sample space is dividedpartitioned into equally probable subspaces. All sample points are then chosen simultaneously making sure that the total ensembleset of sample points is a Latin Hypercubehypercube sample and that each subspace is sampled with the same density.

Thus, orthogonal sampling ensures that the ensembleset of random numbers is a very good representative of the real variability, LHS ensures that the ensembleset of random numbers is representative of the real variability whereas traditional random sampling (sometimes called brute force) is just ana ensembleset of random numbers without any guarantees.

==References==

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[[Category:Latin squares]]

[[Category:Design of experiments]]

[[Category:1979 introductions]]