# Construction of Uniform Designs on Arbitrary Domains by Inverse Rosenblatt Transformation

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## Abstract

The uniform design proposed by Fang [6] and Wang and Fang [17] has become an important class of designs for both traditional industrial experiments and modern computer experiments. There exist established theory and methods for constructing uniform designs on hypercube domains, while the uniform design construction on arbitrary domains remains a challenging problem. In this paper, we propose a deterministic construction method through inverse Rosenblatt transformation, as a general approach to convert the uniformly designed points from the unit hypercubes to arbitrary domains. To evaluate the constructed designs, we employ the central composite discrepancy as a uniformity measure suitable for irregular domains. The proposed method is demonstrated with a class of flexible regions, constrained and manifold domains, and the geographical domain with very irregular boundary. The new construction results are shown competitive to traditional stochastic representation and acceptance-rejection methods.

## Notes

### Acknowledgements

This work was supported by National Natural Science Foundation of China (11871288) and Natural Science Foundation of Tianjin (19JCZDJC31100).

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