Operational Bayesian GLS Regression for Regional Hydrologic Analyses

Abstract

This paper develops the quasi‐analytic Bayesian analysis of the generalized least squares (GLS) (B‐GLS) model introduced by Reis et al. (2005, https://doi.org/10.1029/2004WR003445) into an operational and statistically comprehensive GLS regional hydrologic regression methodology to estimate flood quantiles, regional shape parameters, low flows, and other statistics with spatially correlated flow records. New GLS regression diagnostic statistics include a Bayesian plausibility value, pseudo adjusted R2, pseudo analysis of variance table, and two diagnostic error variance ratios. Traditional leverage and influence are extended to identify rogue observations, address lack of fit, and support gauge network design and region‐of‐influence regression. Formulas are derived for the Bayesian computation of estimators, standard errors, and diagnostic statistics. The use of B‐GLS and the new diagnostic statistics are illustrated with a regional log‐space skew regression analysis for the Piedmont region in the Southeastern U.S. A comparison of ordinary, weighted, and GLS analyses documents the advantages of the Bayesian estimator over the method‐of‐moment estimator of model error variance introduced by Stedinger and Tasker (1985, https://doi.org/10.1029/WR021i009p01421). Of the three types of analyses, only GLS considers the covariance among concurrent flows. The example demonstrates that GLS regional skewness models can be highly accurate when correctly analyzed: The B‐GLS average variance of prediction is 0.090 for South Carolina (92 stations), whereas a traditional ordinary least squares analysis published by the U.S. Geological Survey yielded 0.193 (Feaster & Tasker, 2002, https://doi.org/10.3133/wri024140). B‐GLS provides a statistical valid framework for the rigorous analysis of spatially correlated hydrologic data, allowing for the estimation of parameters and their actual precision and computation of several diagnostic statistics, as well as correctly attributing variability to the three key sources: time sampling error, model error, and signal.

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