Some contributions to Latin hypercube designs, irregular region smoothing and uncertainty quantification

This thesis consists of three parts. In part I, we propose a general framework for slicing Latin hypercube designs in computer experiments and focus on a special case called doubly sliced Latin hypercube design (DSLHD). In part II, a completely-data-driven smoothing technique is proposed for irregular region smoothing and its properties are investigated. Part III deals with uncertainty quantification of energy assessment in the field of building technology.;Part I comprises the first two chapters. In Chapter 1, a new class of designs called doubly sliced Latin hypercube design is proposed for running computer experiments. A DSLHD is a special LHD which can be partitioned into slices of smaller LHDs, each of which can be further partitioned into even smaller LHDs. Both doubly sliced Latin hypercube designs and sliced Latin hypercube designs are special cases of multi-layer sliced Latin hypercube designs, which correspond to two layers and one layer respectively.;In Chapter 2, we introduce a new procedure for batch sequential design based on DSLHD. The proposed procedure uses a slice of a DSLHD as the batch at each iteration. Since each slice of a DSLHD is an LHD, it is ensured that each batch has good space-filling properties. Batch sequential sampling are sometimes necessary because the turnaround time is shorter. I also incorporate some other criteria such as maximin distance criterion ([27]) to search for optimal batches at each iteration. The proposed procedure utilizes exchange algorithms to search for optimal batches within the DSLHD framework, which ensures the quality of the batches and is computationally efficient.;Part II of the thesis comprises Chapters 3 and 4. Smoothing or functional estimation is one of the most studied topics in statistics. It consists of many popular techniques as special cases, such as linear regression, smoothing splines, neural networks, etc. In this part of the thesis, I focus on the smoothing splines framework. A typical problem formulation in smoothing splines is to optimize an objective function which is a tradeoff between goodness-of-fit to the data and the roughness of the fitted function. The state-of-the art procedure to solve this problem is to assume the true function belongs to a functional class and convert the infinite dimensional functional estimation problem to a quadratic programming problem.;In Chapter 3, I focus on one-dimensional input and show that our new approach has exactly the same theoretical performance as the natural cubic splines. Specifically, our estimator achieves optimal convergence rate for nonparametric regression and achieves the sharp bound for minimax risk. All the derivations are based on the assumption of equally spaced design.;In Chapter 4, we derive more general theoretical properties of our estimator. With the same regularity conditions on the boundary as smoothing splines and some analytical assumptions on the underlying function, we show that our estimator enjoys all the nice theoretical properties that smoothing splines do. The derivation of the theoretical properties relies mainly on the connection between continuous semi-norms and discrete semi-norms in Sobolev space. More interestingly, we show through numerical experiments that our method is comparable to soap film smoothing ([72]) for irregular region smoothing and much better than thin plate splines.;Chapter 5 belongs to Part III of this thesis. In this part, my focus is on uncertainty quantification of two microclimate parameters in building technology: (1) local wind speed, (2) wind pressure coefficient. We first use design of experiments to collect data for statistical analysis. Statistical models are then built to connect the standard model and the meso-scale model. The explicit form of statistical models facilitate the improvement of standard models in the current simulation tools such as "EnergyPlus". (Abstract shortened by UMI.).