Advances in calibration and interpolation: Censored and big data applications
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Advances of computing capability and increasing demand for analyzing data from complex systems in various engineering fields have made computer experiments an inevitable tool for exploring and optimizing systems. Physical experiments are very costly to conduct in many applications such as a cardiovascular system study or a rocket engine design. With the aid of high-performance computing, the cost for expensive physical experiments can be reduced dramatically by running the simulation codes on computers. Due to the deterministic nature of computer codes, Gaussian process model or kriging is widely used for interpolation and calibration. Chapter 1 of the thesis deals with model calibration using censored data. The purpose of model calibration is to use data from a physical experiment to adjust the computer model so that the predictions can become closer to reality. The classic Kennedy-O'Hagan approach is widely used for model calibration, which can account for the inadequacy of the computer model while simultaneously estimating the unknown calibration parameters. In many applications, the phenomenon of censoring occurs when the exact outcome of the physical experiment is not observed, but is only known to fall within a certain region. In such cases, the Kennedy-O'Hagan approach cannot be used directly, and we propose a method to incorporate the censoring information when performing model calibration. The method is applied to study the compression phenomenon of liquid inside a bottle. The results show significant improvement over the traditional calibration methods, especially when the number of censored observations is large. Chapter 2 proposes an interpolation technique which can be used with large and unstructured data. Kriging is widely used for interpolation of unstructured data because of its ability to produce confidence intervals for predictions. The model is fitted to the data using maximum likelihood or cross validation-based methods. Unfortunately, the fitting is expense for large data because one evaluation of the objective function requires $O(n^3)$ operations, where n is the size of the data. There exist other interpolation techniques such as inverse distance weighting (IDW), which doesn’t require any estimation and therefore can be easily used with large data. However, the performance of IDW can be significantly worse than kriging. In this chapter, we propose a kriging method that does not require any estimation from data and whose performance is much better than that of IDW. We also propose a novel approach to choose nuggets in kriging that can produce numerically stable results, which is important for applying the technique to unstructured data. A technique for adaptively choosing the kernels is also developed. Chapter 3 extends the automatic kriging proposed in Chapter 2 by exploiting the sequential nature of the adaptive modeling method. When more computing resource is available, we have the option to make estimates from adaptive nugget and adaptive kernel more accurate. A two-stage version of adaptive nugget predictor is proposed which is shown to outperform the state-of-the-art methods in terms of prediction accuracy. We also propose fast estimation techniques to improve the adaptive kernel predictor. The improved predictor is demonstrated to have enhanced stability and predictive performance over the traditional kriging method according to various simulation studies.