AUTHOR=Ma Zhanshan (Sam) 

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

JOURNAL=Frontiers in Applied Mathematics and Statistics

VOLUME=Volume 8 - 2022

YEAR=2022

URL=https://www.frontiersin.org/journals/applied-mathematics-and-statistics/articles/10.3389/fams.2022.801830

DOI=10.3389/fams.2022.801830

ISSN=2297-4687

ABSTRACT=Power laws have been found to describe a wide variety of natural (physical, biological, astronomic, meteorological, geological) and man-made (social, financial, computational) phenomena over a wide range of magnitudes, although their underlying mechanisms are not always clear. In statistics, power law distribution is often found to fit data exceptionally well when the normal (Gaussian) distribution fails. Nevertheless, predicting power law phenomena is notoriously difficult because some of its idiosyncratic properties such as lack of well-defined average value, and potentially unbounded variance. TPL (Taylor’s power law) is a power law 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 power law 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 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 power laws. The coupling takes advantages of variance prediction using TPL and asymptote estimation using PLEC, and delivers confidence interval for the asymptote. We demonstrate the integrated approach to the estimation of potential (dark) biodiversity of the American gut microbiome (AGM) and turning point of COVID-19 fatality.  We expect this integrative approach should have wide applications given the duel relationship between power law and normal statistical distributions.