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Article

Multivariate Tail Probabilities: Predicting Regional Pertussis Cases in Washington State

1
Department of Mathematics and Institute for Systems Research, University of Maryland, College Park, MD 20742, USA
2
Public Health Dynamics Laboratory, Department of Biostatistics, Graduate School of Public Health, University of Pittsburgh, Pittsburgh, PA 15261, USA
3
Health Analytics Network, Pittsburgh, PA 15237, USA
*
Authors to whom correspondence should be addressed.
Academic Editors: José A. Tenreiro Machado and Dimitri Volchenkov
Entropy 2021, 23(6), 675; https://doi.org/10.3390/e23060675
Received: 7 May 2021 / Revised: 23 May 2021 / Accepted: 23 May 2021 / Published: 27 May 2021
(This article belongs to the Special Issue Modeling and Forecasting of Rare and Extreme Events)
In disease modeling, a key statistical problem is the estimation of lower and upper tail probabilities of health events from given data sets of small size and limited range. Assuming such constraints, we describe a computational framework for the systematic fusion of observations from multiple sources to compute tail probabilities that could not be obtained otherwise due to a lack of lower or upper tail data. The estimation of multivariate lower and upper tail probabilities from a given small reference data set that lacks complete information about such tail data is addressed in terms of pertussis case count data. Fusion of data from multiple sources in conjunction with the density ratio model is used to give probability estimates that are non-obtainable from the empirical distribution. Based on a density ratio model with variable tilts, we first present a univariate fit and, subsequently, improve it with a multivariate extension. In the multivariate analysis, we selected the best model in terms of the Akaike Information Criterion (AIC). Regional prediction, in Washington state, of the number of pertussis cases is approached by providing joint probabilities using fused data from several relatively small samples following the selected density ratio model. The model is validated by a graphical goodness-of-fit plot comparing the estimated reference distribution obtained from the fused data with that of the empirical distribution obtained from the reference sample only. View Full-Text
Keywords: disease outbreak; density ratio model; variable tilt; model selection; goodness-of-fit; data fusion disease outbreak; density ratio model; variable tilt; model selection; goodness-of-fit; data fusion
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MDPI and ACS Style

Zhang, X.; Pyne, S.; Kedem, B. Multivariate Tail Probabilities: Predicting Regional Pertussis Cases in Washington State. Entropy 2021, 23, 675. https://doi.org/10.3390/e23060675

AMA Style

Zhang X, Pyne S, Kedem B. Multivariate Tail Probabilities: Predicting Regional Pertussis Cases in Washington State. Entropy. 2021; 23(6):675. https://doi.org/10.3390/e23060675

Chicago/Turabian Style

Zhang, Xuze, Saumyadipta Pyne, and Benjamin Kedem. 2021. "Multivariate Tail Probabilities: Predicting Regional Pertussis Cases in Washington State" Entropy 23, no. 6: 675. https://doi.org/10.3390/e23060675

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