What is a Latin Hypercube?

In statistical sampling, a square grid containing sample positions is a Latin Square if, and only if, each row and column contains only one sample. A Latin hypercube (LH) is the extension of this concept to an arbitrary number of dimensions, where each sample is the only one in each axis-aligned hyperplane that contains it.

For a sampling of M samples in a space of dimension P there is no unique LH. In Simcenter Amesim, it is generated randomly. This is why you are asked to indicate the seed required by the random generator.

An Optimized Latin Hypercube (OLH) is an LH for which samples are uniformly spread through the design space. To obtain an OLH, we start from a LH. Then we switch the sample hyperplanes until we get a satisfactory dispersion of the samples.

Obtaining an OLH is a time consuming process. Fortunately, every sampling of M samples in a space of dimension P can be obtained from the same OLH. It is just a matter of translating each axis to get the right range and then applying distortion at intervals to follow the right distribution. This is why Simcenter Amesim allows you to manage a database of OLH.

Figure 92: Illustration of axis distortion:

Sample obtained with a random generator	Sample obtained with LH	Sample obtained with OLH

Advantages of OLH

LH is a constrained Monte Carlo Sampling which avoids the sample being concentrated on a certain area of the design space. OLH is optimized to cover the whole design space uniformly. Thus there is no part of the design space not taken into account when computing statistics or RSM.

Therefore, compared with a crude sampling method when estimating statistical quantities like means or standard deviations:

• Far fewer samples are needed to reach the same precision,

• Precision is far better with the same number of samples.

Here are the results of a comparison of the three sampling methods. They have been used to compute the median of the Rosenbrock function:b(X1,X2)=100(X2-X1²)²+(1-X1)²

Number of runs	% error in median estimation
	OLH	LH	RG
8	8,574386	21,987999	37,513075
13	5,178056	15,078734	37,371111
21	3,46702	13,538944	26,82003
34	2,555625	10,619336	16,939982
55	2,018984	2,018984	2,0189849
89	1,422516	5,447641	11,035489
144	1,157871	4,672613	9,833102
233	0,733929	4,101952	6,480452
377	---	3,112742	5,664913
610	---	2,137434	4,672426
987	---	1,678378	3,692437
1597	---	1,53707	2,957906
2584	---	1,149349	1,958579
4181	---	0,877801	1,504924

We see that an LH with 144 runs has the same precision as a random generator with 610 runs. Results with OLH are even more reliable. An OLH with 55 points has a better precision than an LH with 610 and than a random generator with 2000.

Drawbacks of OLH

The complexity of the optimization algorithm used for OLH generation increases drastically with the number of variables or runs. This explains why OLH generation can be very time-consuming in some situations. To cope with this issue, an OLH manager is provided in Simcenter Amesim (see hereafter). It allows you to save previously-obtained OLH samples in order to reuse them without needing to regenerate them each time a Monte-Carlo analysis is performed.