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Power laws (PLs) have been found to describe a wide variety of natural (physical, biological, astronomic, meteorological, and geological) and man-made (social, financial, and computational) phenomena over a wide range of magnitudes, although their underlying mechanisms are not always clear. In statistics, PL distribution is often found to fit data exceptionally well when the normal (Gaussian) distribution fails. Nevertheless, predicting PL phenomena is notoriously difficult because of some of its idiosyncratic properties, such as lack of well-defined average value and potentially unbounded variance. Taylor's power law (TPL) is a PL first discovered to characterize the spatial and/or temporal distribution of biological populations. It has also been extended to describe the spatiotemporal heterogeneities (distributions) of human microbiomes and other natural and artificial systems, such as fitness distribution in computational (artificial) intelligence. The PL with exponential cutoff (PLEC) is a variant of power-law function that tapers off the exponential growth of power-law function ultimately and can be particularly useful for certain predictive problems, such as biodiversity estimation and turning-point prediction for Coronavirus Diease-2019 (COVID-19) infection/fatality. Here, we propose coupling (integration) of TPL and PLEC to offer a methodology for quantifying the uncertainty in certain estimation (prediction) problems that can be modeled with PLs. The coupling takes advantage of variance prediction using TPL and asymptote estimation using PLEC and delivers CI for the asymptote. We demonstrate the integrated approach to the estimation of potential (dark) biodiversity of the American gut microbiome (AGM) and the turning point of COVID-19 fatality. We expect this integrative approach should have wide applications given duel (contesting) relationship between PL and normal statistical distributions. Compared with the worldwide COVID-19 fatality number on January 24th, 2022 (when this paper is online), the error rate of the prediction with our coupled power laws, made in the May 2021 (based on the fatality data then alone), is approximately 7% only. It also predicted that the turning (inflection) point of the worldwide COVID-19 fatality would not occur until the July of 2022, which contrasts with a recent prediction made by Murray on January 19th of 2022, who suggested that the “end of the pandemic is near” by March 2022.

Coupling Power Laws Offers a Powerful Modeling Approach to Certain Prediction/Estimation Problems With Quantified Uncertainty