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Space-Filling DOEs These designs use values of variables inside range instead of at boundaries Latin hypercubes, one popular space- filling DOE uses as many levels as design points Space-filling term is appropriate for low dimensional spaces For high-dimensional spaces, we cannot afford to “fill” the space.

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Monte Carlo sampling Sampling at random points. Given a region in design space, we can assign a uniform distribution to the region and sample points to generate DOE It is likely, though, that some regions will be poorly sampled In 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.

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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));

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Latin Hypercube sampling Each variable range divided into n y equal probability intervals. One point at each interval. 12 23 34 41 55

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Latin Hypercube definition matrix For n points with m variables: m by n matrix, with each column a permutation of 1,…,n Examples Points are better distributed for each variable, but can still have holes in m-dimensional space.

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Matlab lhsdesign x=lhsdesign(10,2); plot(x(:,1), x(:,2), 'o'); xr=lhsdesign(10,2,'criterion','correlation'); hold on plot(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

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More iterations x=lhsdesign(10,2,'iterations',5000); plot(x(:,1), x(:,2), 'o'); hold on xr=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

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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));

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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));

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Empty space Goel, 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 don’t 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.

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Mixed designs D-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.

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