# LHS in LHS: A new expansion strategy for Latin hypercube sampling in simulation design

## TL;DR

This paper introduces LHS in LHS, an expansion algorithm for Latin Hypercube Sampling that allows adding samples to existing designs while maintaining their properties, along with a new metric to measure design quality.

## Contribution

It proposes a novel expansion strategy for LHS that preserves its properties and introduces a new metric for assessing LHS quality, implemented in a Python package.

## Key findings

- The algorithm effectively adds samples to existing LHS designs.
- The new metric quantifies deviation from ideal LHS properties.
- Implementation is available as an open-source Python package.

## Abstract

Latin Hypercube Sampling (LHS) is a prominent tool in simulation design, with a variety of applications in high-dimensional and computationally expensive problems. LHS allows for various optimization strategies, most notably to ensure space-filling properties. However, LHS is a single-stage algorithm that requires a priori knowledge of the targeted sample size. In this work, we present LHS in LHS, a new expansion algorithm for LHS that enables the addition of new samples to an existing LHS-distributed set while (approximately) preserving its properties. In summary, the algorithm identifies regions of the parameter space that are far from the initial set, draws a new LHS within those regions, and then merges it with the original samples. As a by-product, we introduce a new metric, the LHS degree, which quantifies the deviation of a given design from an LHS distribution. Our public implementation is distributed via the Python package expandLHS.