The Optimal Arbitrary-Proportional Finite-Set-Partitioning

TL;DR

This paper introduces an optimal method for partitioning a finite set into subsets with arbitrary proportions, minimizing bias and applicable to problems like quota setting and resampling in particle filters.

Contribution

It proposes a novel scheme for finite-set partitioning that minimizes bias based on a new discrepancy metric, with theoretical proof and simulation validation.

Findings

The scheme achieves minimal bias according to the defined metric.

The method is proven to be optimal theoretically.

Simulation results confirm the scheme's effectiveness.

Abstract

This paper considers the arbitrary-proportional finite-set-partitioning problem which involves partitioning a finite set into multiple subsets with respect to arbitrary nonnegative proportions. This is the core art of many fundamental problems such as determining quotas for different individuals of different weights or sampling from a discrete-valued weighted sample set to get a new identically distributed but non-weighted sample set (e.g. the resampling needed in the particle filter). The challenge raises as the size of each subset must be an integer while its unbiased expectation is often not. To solve this problem, a metric (cost function) is defined on their discrepancies and correspondingly a solution is proposed to determine the sizes of each subsets, gaining the minimal bias. Theoretical proof and simulation demonstrations are provided to demonstrate the optimality of the scheme in the sense of the proposed metric.