Asymptotic Theory of Penalized Splines and Calibration of Computationally Expensive Models

研究人员研究了两个领域。第一个是惩罚样条的大样本理论。第二个是计算昂贵的模型的贝叶斯校准和不确定性分析。将一元罚样条理论推广到二次、高阶样条和高阶差分罚样条。将单变量理论推广到可加模型和二元回归。本文提出了一种用于具有异方差测量误差反褶积的惩罚样条方法。贝叶斯校准和不确定性分析的情况下，时空相关性，测量误差的环境输入，和多个响应进行了研究。样条是定义曲线和曲面的数学工具。例如，在计算机辅助设计中，样条曲线被用来描述汽车车身的形状。在统计学中，样条曲线用于描述变量之间的曲线关系。例如，最近的一项关于儿童血铅浓度和智商之间关系的研究使用了样条函数，并发现了一个意想不到的重要结果。剂量-反应曲线在低剂量时最陡，这意味着，在低剂量时，智商随着铅浓度的增加而下降的速度比以前认识到的要快。这一出乎意料的发现意味着，如果环境中的铅浓度降低，那么儿童的智力发育将得到改善，其程度将超过以前的想象。样条曲线在这类研究中特别有用，因为它们可以与母亲智商和其他因素的调整以及单个受试者多项智商测量之间的相关性相结合。研究者研究样条函数以提高估计精度和扩大适用范围。复杂模型的校准用于各种应用，包括石油勘探和流域管理。PI和他的同事们研究了坎农斯维尔分水岭，这是纽约市的饮用水源。模型的校准意味着使用数据来估计未知参数。不确定性分析测量估计的精度。需要校准和不确定性分析，以便这些模型可以用于管理。例如，Cannonsville水库的模型帮助纽约市管理流域以保持水资源，因此不需要昂贵的过滤系统。

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 splinesis 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 anduncertainty 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 therelationship 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 thistype 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 complexmodels 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.