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Some Improvements on Good Lattice Point Sets.

Yu-Xuan Lin1,2, Tian-Yu Yan3, Kai-Tai Fang2,4

  • 1Research Center for Frontier Fundamental Studies, Zhejiang Lab, Kechuang Avenue, Zhongtai Sub-District, Yuhang District, Hangzhou 311121, China.

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Summary

Generalized good lattice point (GGLP) sets, derived from good lattice point (GLP) sets via linear-level permutations, enhance space-filling properties. These improved GLP sets demonstrate superior performance in applications like computer experiments.

Keywords:
Frobenius distanceKriging modelKullback–Leibler divergenceentropygeneralized good lattice point setgood lattice point setlinear level permutationmax-min distancemixture discrepancyrepresentative pointsthreshold accepting algorithm

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Area of Science:

  • Number theory
  • Applied mathematics
  • Computational statistics

Background:

  • Good lattice point (GLP) sets are number-theoretic methods valued for their space-filling properties.
  • Enhancing GLP sets is crucial for applications requiring better distribution, especially with large datasets.
  • Existing GLP sets are widely used but have room for improvement in space-filling capabilities.

Purpose of the Study:

  • To introduce and evaluate Generalized Good Lattice Point (GGLP) sets.
  • To assess the impact of linear-level permutations on GLP set properties.
  • To improve the space-filling characteristics of GLP sets using novel construction methods.

Main Methods:

  • Kullback-Leibler (KL) divergence was used to measure GLP and GGLP set distributions.
  • Linear-level permutation was applied to GLP sets to create GGLP sets.
  • A threshold-accepting algorithm and Frobenius distance were employed for constructing and evaluating large-sized GGLP sets.

Main Results:

  • GGLP sets, created by linear-level permutation, do not reduce the maximin distance criterion.
  • GGLP sets show enhanced space-filling properties compared to initial GLP sets.
  • KL divergence analysis indicates GGLP sets maintain distribution similarity, especially for large datasets.

Conclusions:

  • GGLP sets offer improved space-filling properties over traditional GLP sets.
  • The construction methods, including threshold-accepting algorithms, are effective for generating superior lattice point sets.
  • GGLP sets demonstrate practical advantages in applications such as computer experiments and selecting representative points.