The investigator studies two areas. The first is the large-sample theory of penalized splines. The second is Bayesian calibration and uncertainty analysis for computational expensive models. The theory for univariate penalized splines is generalized to quadratic and higher degree splines and to higher order difference penalties. The univariate theory is extended to additive models and to bivariate regression. Penalized splines methods are developed for deconvolution with heteroscedastic measurement error. Bayesian calibration and uncertainty analysis is studied in the cases of spatial-temporal correlations, measurement error in environmental inputs, and multiple responses.
Splines are mathematical tools for defining curves and surfaces. Splines are used, for example, to describe the shape of an automobile body in computer-assisted design. In statistics, splines are used to describe curved relationships between variables. For example, a recent study of the relationship between blood-lead concentration and IQ in children used splines and found an unexpected and important result. The dose-response curve is steepest at low doses, meaning that, at low doses, IQ declines with increasing lead concentrations more rapidly than previously realized. This unanticipated finding implies that, if environmental lead concentrations are reduced, then the intellectual development of children will be improved by an amount exceeding what was previously thought. Splines are particularly useful in this type of study because they can be combined with adjustments for maternal IQ and other factors and for correlations between multiple IQ measurements on a single subject. The investigator studies splines to improve the precision of estimation and to extend the range of applicability. Calibration of complex models is used in a variety of applications including, for example, petroleum exploration and management of watersheds. The PI and his colleagues study the Cannonsville watershed, a source of drinking water for New York City. Calibration of a model means using data to estimate unknown parameters. Uncertainty analysis measures the precision of the estimates. Calibration and uncertainty analysis is needed so that these models can be used for management. For example, the model for the Cannonsville Reservoir helps NYC manage the watershed to maintain water so that expensive filtration systems are not needed.
