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Space-Filling DOEsThese designs use values of variables inside range instead of at boundariesLatin hypercubes, one popular space-filling DOE uses as many levels as design pointsSpace-filling term is appropriate for low dimensional spacesFor high-dimensional spaces, we cannot afford to fill the space.

Monte Carlo samplingSampling at random points.Given a region in design space, we can assign a uniform distribution to the region and sample points to generate DOEIt is likely, though, that some regions will be poorly sampledIn 5-dimensional space, with 32 sample points, what is the chance that all orthants will be occupied?(31/32)(30/32)(1/32)=1.8e-13.

Example of MC sampling

x=rand(20,2);subplot(2,2,1); plot(x(:,1), x(:,2), 'o');subplot(2,2,2); hist(x(:,2));subplot(2,2,3); hist(x(:,1));

Latin Hypercube samplingEach variable range divided into ny equal probability intervals. One point at each interval.

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Latin Hypercube definition matrixFor n points with m variables: m by n matrix, with each column a permutation of 1,,nExamples

Points are better distributed for each variable, but can still have holes in m-dimensional space.

Matlab lhsdesignx=lhsdesign(10,2);plot(x(:,1), x(:,2), 'o');xr=lhsdesign(10,2,'criterion','correlation');hold onplot(x(:,1), x(:,2), 'r+');r=corrcoef(x)r = 1.0000 0.7746 0.7746 1.0000 r=corrcoef(xr)r = 1.0000 0.0303 0.0303 1.0000

More iterationsx=lhsdesign(10,2,'iterations',5000);plot(x(:,1), x(:,2), 'o'); hold onxr=lhsdesign(10,2,'criterion','correlation','iterations',5000);plot(xr(:,1), xr(:,2), 'r+');>> r=corrcoef(x)r =1.0000 -0.4510 -0.4510 1.0000>> r=corrcoef(xr)r =1.0000 -0.0424 -0.0424 1.0000

Recall MC sampling

x=rand(20,2);subplot(2,2,1); plot(x(:,1), x(:,2), 'o');subplot(2,2,2); hist(x(:,2));subplot(2,2,3); hist(x(:,1));

Distribution of single variables with LHS

x=lhsdesign(20,2,'iteration',5000,'criterion','correlation')subplot(2,2,1); plot(x(:,1), x(:,2), 'o');subplot(2,2,2); hist(x(:,2));subplot(2,2,3); hist(x(:,1));

Empty spaceGoel, T., Haftka, R.T., Shyy, W., and Watson, L.T., (2008), Pitfalls of using a single criterion for selecting experimental designs, International Journal for Numerical Methods in Engineering, 75: 127 155.

Instead of maximizing minimum distance it seems that it would be better to minimize the volume of the largest void. Why dont we do that?

Figure 2. Illustration of the largest spherical empty space inside the three-dimensional design space (20 points): (a) D-optimal design and (b) LHS design.

Mixed designsD-optimal designs may leave much space inside.LHS designs may leave out the boundary and lead to large extrapolation errors.It may be desirable to combine the two.In low dimensional spaces you can add the vertices to LHS designs.In higher dimensional spaces you can generate a larger LHS design and choose a D-optimal subset.