Latin hypercube sampling is a statistical method for generating a
sample of plausible collections of parameter values from a multidimensional
distribution. The sampling method is often used to construct computer
experiments. The LHS was described by McKay in 1979.
An independently equivalent technique has been proposed by Eglājs in 1977. It
was further elaborated by Ronald L. Iman and coauthors in 1981. Detailed computer
codes and manuals were later published. In the context of statistical sampling,
a square grid containing sample positions is a Latin square 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 variables, the range of each variable is divided
into equally probable intervals. sample points are then placed to satisfy
the Latin hypercube requirements; note that this forces the number of
divisions, , to be equal for each variable. Also note that this sampling
scheme does not require more samples for more dimensions; 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 divisions and variables can be computed with the
following formula: For example, a Latin hypercube of
divisions with variables will have 24 possible combinations. A Latin hypercube
of divisions with variables 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. 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.
In Orthogonal sampling, the sample space is divided into equally probable
subspaces. All sample points are then chosen simultaneously making sure that
the total ensemble of sample points is a Latin Hypercube sample and that each
subspace is sampled with the same density.
Thus, orthogonal sampling ensures that the ensemble of random numbers is a very
good representative of the real variability, LHS ensures that the
ensemble of random numbers is representative of the real variability
whereas traditional random sampling is just an ensemble of random numbers
without any guarantees. References
^ McKay, M.D.; Beckman, R.J.; Conover, W.J.. "A Comparison of Three Methods for
Selecting Values of Input Variables in the Analysis of Output from a Computer
Code". Technometrics 21: 239–245. doi:10.2307/1268522. ISSN 0040-1706.
JSTOR 1268522. OSTI 5236110. ^ Eglajs, V.; Audze P.. "New approach to
the design of multifactor experiments". Problems of Dynamics and Strengths. 35:
104–107. ^ Iman, R.L.; Helton, J.C.; and
Campbell, J.E.. "An approach to sensitivity analysis of computer models,
Part 1. Introduction, input variable selection and preliminary variable
assessment". Journal of Quality Technology 13: 174–183.
^ Iman, R.L.; Davenport, J.M.; Zeigler, D.K.. Latin hypercube sampling. OSTI
5571631. Further reading
Tang, B.. "Orthogonal Array-Based Latin Hypercubes". Journal of the American
Statistical Association 88: 1392–1397. doi:10.2307/2291282. JSTOR 2291282.
Owen, A.B.. "Orthogonal arrays for computer experiments, integration and
visualization". Statistica Sinica 2: 439–452.
Ye, K.Q.. "Orthogonal column Latin hypercubes and their application in
computer experiments". Journal of the American Statistical Association 93:
1430–1439. doi:10.2307/2670057. JSTOR 2670057.