Bayesian Gls Regression Leverage And Influence For Regionalization Of Hydrologic Statistics

The research presented in this dissertation develops new statistical techniques for estimating regional relationships of hydrologic statistics. These techniques include extensions of the quasi-analytic Bayesian Generalizes Least Squares (B-GLS) framework presented in Reis et al. [2005] and further developed by Gruber et al. [2007] and Gruber and Stedinger [2008]. Recent extensions include a Pseudo [R squared sub delta] and pseudo analysis of variance table, plus a range of model performance, diagnostic and goodness-of-fit statistic. This dissertation develops a more stable Bayesian WLS/GLS procedure with the corresponding measures of precision and model performance. Special attention is given to model performance criteria, and the meaning of and insight provided by alternative measures of leverage and influence. Examples address development of regional skewness coefficients to improve flood frequency analysis in the United States. Large cross-correlations between annual peak discharges, coupled with relatively small model error variances, present difficulties in regional GLS skewness analyses. The B-GLS framework seeks to exploit the cross-correlations among the sample skewness estimates to obtain the best possible estimates of the model parameters. However, if the cross-correlations are large, the GLS estimators can become relatively complicated as a result of the effort to find the most efficient estimator of the parameters. Unfortunately, it appears that the precision of the cross-correlation estimates between any two particular sites is not of sufficient precision to justify the seemingly incorrect weights (both positive and negative) that the B-GLS analysis generates. Thus, an alternate regression procedure using both Weighted Least Squares (WLS) and GLS is developed so that the regional skewness analysis can provide both stable and defensible results. This alternate regression framework, is applied to two different data sets from different parts of the United States: the State of California and the Southeastern United States, to develop regional skewness estimators for flood frequency analysis. In addition, special attention is devoted to comparing and developing leverage and influence diagnostics statistics for GLS and WLS/GLS analyses, which can be used to identify rogue observations and to effectively address lack-of-fit when estimating hydrologic statistics.

The research presented in this thesis develops new statistical techniques for estimating regional skewness coefficients to improve flood frequency analysis in the United States. Flood frequency guidelines for the United States, specified in Bulletin 17B, recommend fitting the log-Pearson Type III (LP3) distribution to the series of annual flood maxima, in which the third moment of the distribution, the skewness coefficient , is combined with a regional skewness coefficient to improve its precision. The research presented here extends the quasi-analytic Bayesian analysis of the Generalized Least Squares (GLS) regional hydrologic regression framework introduced by Reis et al. [2005] to more accurately and precisely estimate regional skewness coefficients. Specifically, formulas derived within a Bayesian regression framework for the computation of estimators, standard errors, and diagnostic statistics are provided by Reis [2005] and Reis et al. [2005]. Diagnostic statistics further developed here include a Bayesian plausibility value, pseudo adjusted R-squared, pseudo-Analysis of Variance table, two diagnostic error variance ratios, as well as leverage and influence metrics. In addition, this research also develops a new influence diagnostic statistic which, in conjunction with the Bayesian extension of GLS leverage and influence metrics, can be used to better identify rogue observations and to effectively address lack-of-fit when estimating skewness coefficients. Currently, Bulletin 17B allows for regional skew values to be obtained from the skew map included with the Bulletin. As it is over 30 years old, the regional skew values from the Bulletin 17B skew map do not reflect annual maximum data acquired since 1976. This increase in available data, along with advances in computing power to support the Bayesian GLS regional hydrologic regression framework, allow for a much more precise estimate of the regional skewness coefficient for use in flood frequency analysis. This research employs the Bayesian GLS regression framework to estimate regional log-space skewness coefficients for three data sets: the Illinois River basin, the state of South Carolina, and the Southeastern United States. Bulletin 17B allows for the generation of skew prediction equations as an alternative method for determining regional skew coefficients when the mean squared error of the equations is smaller than reported from the Bulletin's skew map. These skew prediction equations can be generated using Ordinary Least Squares analysis, Weighted Least Squares analysis, Generalized Least Squares analysis employing the method of moment model-error-variance estimator introduced by Stedinger and Tasker [1985, 1986ab], or the new Bayesian GLS estimator. The advantages of using the Bayesian GLS estimation technique to determine a skew prediction equation are demonstrated here in the Illinois River basin and the state of South Carolina studies. To correctly analyze the Southeastern United States data set, methods are developed for identifying and screening redundant sites corresponding to nested watersheds with similar drainage areas. Special attention is devoted to developing an improved cross-correlation model of annual peak flows. The Bayesian GLS analysis using 342 stations from the Southeastern U.S. results in a highly accurate, constant regional skew model, with an average variance of prediction equal to 0.14. More complex models which include regional information and basin characteristics as additional regression parameters result in very little improvement. The application of the Bayesian estimator in the Southeastern study generates improved results over the mean square error of 0.30 reported for the Bulletin 17B regional map skew.

Modelling Spatial and Spatial-Temporal Data: A Bayesian Approach is aimed at statisticians and quantitative social, economic and public health students and researchers who work with spatial and spatial-temporal data. It assumes a grounding in statistical theory up to the standard linear regression model. The book compares both hierarchical and spatial econometric modelling, providing both a reference and a teaching text with exercises in each chapter. The book provides a fully Bayesian, self-contained, treatment of the underlying statistical theory, with chapters dedicated to substantive applications. The book includes WinBUGS code and R code and all datasets are available online. Part I covers fundamental issues arising when modelling spatial and spatial-temporal data. Part II focuses on modelling cross-sectional spatial data and begins by describing exploratory methods that help guide the modelling process. There are then two theoretical chapters on Bayesian models and a chapter of applications. Two chapters follow on spatial econometric modelling, one describing different models, the other substantive applications. Part III discusses modelling spatial-temporal data, first introducing models for time series data. Exploratory methods for detecting different types of space-time interaction are presented followed by two chapters on the theory of space-time separable (without space-time interaction) and inseparable (with space-time interaction) models. An applications chapter includes: the evaluation of a policy intervention; analysing the temporal dynamics of crime hotspots; chronic disease surveillance; and testing for evidence of spatial spillovers in the spread of an infectious disease. A final chapter suggests some future directions and challenges.

INLA stands for Integrated Nested Laplace Approximations, which is a new method for fitting a broad class of Bayesian regression models. No samples of the posterior marginal distributions need to be drawn using INLA, so it is a computationally convenient alternative to Markov chain Monte Carlo (MCMC), the standard tool for Bayesian inference. Bayesian Regression Modeling with INLA covers a wide range of modern regression models and focuses on the INLA technique for building Bayesian models using real-world data and assessing their validity. A key theme throughout the book is that it makes sense to demonstrate the interplay of theory and practice with reproducible studies. Complete R commands are provided for each example, and a supporting website holds all of the data described in the book. An R package including the data and additional functions in the book is available to download. The book is aimed at readers who have a basic knowledge of statistical theory and Bayesian methodology. It gets readers up to date on the latest in Bayesian inference using INLA and prepares them for sophisticated, real-world work. Xiaofeng Wang is Professor of Medicine and Biostatistics at the Cleveland Clinic Lerner College of Medicine of Case Western Reserve University and a Full Staff in the Department of Quantitative Health Sciences at Cleveland Clinic. Yu Ryan Yue is Associate Professor of Statistics in the Paul H. Chook Department of Information Systems and Statistics at Baruch College, The City University of New York. Julian J. Faraway is Professor of Statistics in the Department of Mathematical Sciences at the University of Bath.

Emphasizes the interactive analysis of hydrological data made possible through the widespread availability of desktop computers. Demonstrates new techniques for assessing the adequacy and performance of hydrological models. Offers an in-depth discussion of examples drawn from numerous applications such as the analysis of river flow extremes, regionalization of flow characteristics, infiltration of water into soil profiles, overland flow studies and rainfall-runoff modelling.

This thesis focuses on development of a Bayesian methodology for analysis of regional Generalized Least Squares (GLS) regression models, and the use of regional regression models and imperfect historical and palaeoflood information to reduce the uncertainty in flood quantile estimators.