Extrapolation of Threshold-Limited Null Measurement Frequencies

TL;DR

This paper introduces a method to extrapolate the distribution of pathogen levels below a measurable threshold, based on the assumption of exponential decay of pulses, resulting in a power law distribution for low amplitudes.

Contribution

It presents a novel extrapolation technique for threshold-limited measurements assuming exponential decay, enabling estimation of unmeasured low-level pathogen contributions.

Findings

Exponential decay assumption leads to power law distribution for low amplitudes.

The method allows estimation of the unmeasured portion of pathogen levels below the detection threshold.

Provides a simple extrapolation procedure for threshold-limited data.

Abstract

The total measurable level of a pathogen is due to many sources, which produce a variety of pulses, overlapping in time, that rise suddenly and then decay. What is measured is the level of the total contribution of the sources at a given time. But since we are only capable of measuring the total level above some threshold $x_0$, we would like to predict the distribution below this level. Our principal model assumption is that of the asymptotic exponential decay of all pulses. We show that this implies a power law distribution for the frequencies of low amplitude observations. As a consequence, there is a simple extrapolation procedure for carrying the data to the region below $x_0$.