Parameter estimation in a subcritical percolation model with colouring. Issue 5 (4th July 2019)

• Record Type: Journal Article
• Authors: Beck, Felix; Mélykúti, Bence
• Is Part Of: Stochastics. Volume 91:Issue 5(2019)
• Page Start: 657
• Page End: 694
• Publication Date: 2019-07-04
• Subjects: Cross-contamination -- method of simulated moments -- microfluidics -- parameter estimation -- percolation -- strong law of large numbers with dependence -- uniform law of large numbers; Stochastic processes -- Periodicals; Probabilities -- Periodicals; 519.2
• DOI: 10.1080/17442508.2018.1539089
• Languages: English
• ISSNs: 1744-2508

Abstract

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. Under the unproven assumption of identifiability, we show that this method is a strongly consistent estimator by proving a uniform strong law of large numbers for the vertices' weakly dependent colour values. Our proof method quantifies dependence among the spatially arranged random variables using percolation theory: the FKG and BK inequalities, and the exponential decay of the cluster size distribution. We evaluate the method in computer tests. We have made our software publicly available. The motivating application is cross-contamination rate estimation for digital PCR in lab-on-a-chip microfluidic devices.