The input is the number of dimensions M and the parameter H.
The output is a point cloud W involving N points.
W is N uniformly distributed points with M dimensions on the unit hyperplane.
The relationship between M, H and point cloud size N is as follows.
In the case of M=3 dimensions, the results for cases where H is 1 to 6 are as follows.
Ex)
- [W,N] = SLD(M=3,H=2) -> W = [0 0 1; 0 .5 .5; 0 1 0; .5 0 .5; .5 .5 0; 1 0 0]; N = 6;
- [W,N] = SLD(M=5,H=2) -> W = [0 0 0 0 1; 0 0 0 .5 .5; 0 0 0 1 0; 0 0 .5 0 .5; 0 0 .5 .5 0; 0 0 1 0 0; 0 .5 0 0 .5; 0 .5 0 .5 0; 0 .5 .5 0 0; 0 1 0 0 0; .5 0 0 0 .5; .5 0 0 .5 0; .5 0 .5 0 0; .5 .5 0 0 0; 1 0 0 0 0]; N = 15;
Reference
- H. Scheffé, “Experiments with mixtures," Journal of the Royal Statistical Society: Series B (Methodological), Vol. 20, No. 2, pp. 344–360, 1958.
- I. Das and J. E. Dennis, “Normal-boundary intersection: A new method for generating the Pareto surface in nonlinear multicriteria optimization problems," SIAM Journal on Optimization, Vol. 8, No. 3, pp. 631–657, 1998.
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The tests for W are pitiful. I solved this for real, with some real checks in my scratch pad.
More specifically,
After the check of N, there is a brief, repeatable set of tests which can confirm if W is valid, without providing W in the test. I'm pretty sure any edge cases it misses would be harder to exploit than actually solving the stated problem.
assert(isequal(size(W),[N M]))
assert(all(ismembertol(W*H,0:H),'all'))
WNI=round(W*H); % pretty safe now
assert(all(sum(WNI,2)==H))
assert(isequal(size(unique(WNI,'rows')),[N,M]))
assert(isequal(unique(WNI,'rows'),WNI))