Parameter estimation in a subcritical percolation model with colouring

TL;DR

This paper develops a Monte Carlo-based method to estimate parameters in a coloured percolation model on lattices, with applications to microfluidic device contamination analysis, proving strong consistency and demonstrating effectiveness through simulations.

Contribution

It introduces a novel parameter estimation approach for a coloured percolation model using simulated moments, with proven consistency and practical validation.

Findings

The method is strongly consistent under certain conditions.

Simulation results demonstrate accurate parameter recovery.

Application to microfluidic contamination estimation shows practical utility.

Abstract

In the bond percolation model on a lattice, we colour vertices with $n_c$ colours independently at random according to Bernoulli distributions. A vertex can receive multiple colours and each of these colours is individually observable. The colours colour the entire component into which they fall. Our goal is to estimate the $n_c +1$ parameters of the model: the probabilities of colouring of single vertices and the probability with which an edge is open. The input data is the configuration of colours once the complete components have been coloured, without the information which vertices were originally coloured or which edges are open. We use a Monte Carlo method, the method of simulated moments to achieve this goal. We prove that this method is a strongly consistent estimator by proving a uniform strong law of large numbers for the vertices' weakly dependent colour values. We evaluate the method in computer tests. The motivating application is cross-contamination rate estimation for digital PCR in lab-on-a-chip microfluidic devices.