What question did this study set out to answer?

This research investigates the properties and construction of designs based on a centered version of $L_2$-discrepancy.

October 16, 2000Mathematics of Computation190 citationsOpen Access

Centered L₂-discrepancy of random sampling and Latin hypercubedesign, and construction of uniform designs

KFKai‐Tai Fang CMChang‐Xing Ma PWPeter Winker

Key Points

This research investigates the properties and construction of designs based on a centered version of $L_2$-discrepancy.
Analyzed properties of random and Latin hypercube designs
Derived theoretic expectation and variance for both design types
Performed optimization using the threshold accepting heuristic.
Latin hypercube designs demonstrate significantly lower expectation and variance than random designs.
In one dimension, the uniform design is equivalent to equidistant points.
Higher-dimensional low-discrepancy designs require explicit optimization for generation.

Abstract

In this paper properties and construction of designs under a centered version of the L₂-discrepancy are analyzed. The theoretic expectation and variance of this discrepancy are derived for random designs and Latin hypercube designs. The expectation and variance of Latin hypercube designs are significantly lower than that of random designs. While in dimension one the unique uniform design is also a set of equidistant points, low-discrepancy designs in higher dimension have to be generated by explicit optimization. Optimization is performed using the threshold accepting heuristic which produces low discrepancy designs compared to theoretic expectation and variance.

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