Global Sensitivity Analysis For High-dimensional Complex Models

High-dimensional models are often used in biomedical, financial, economical and en-gineering fields and high dimensional data analysis is one of the key issues in modern statistical research. However, most of the existing methods are based on the assumptions of linearity and additivity but many models in real problems do not satisfy such assump-tions. Based on this background, Sobol, Saltelli and other researchers proposed "Global sensitivity analysis" to deal with the non-linearity, non-additivity and non-monotonic problems. First, they describe the uncertainty of the inputs by distribution functions and decompose the objective function into summands of different dimensions which are orthogonal with each other such that the variance of the output are decomposed into variances of different dimensions. Then they provide global sensitivity indices to mea-sure the importance of the inputs and their interactions to the output, and simplify the model by discarding the non-essential inputs. Computing global sensitivity indices is the crucial and also the difficulty in global sensitivity analysis and many methods have been proposed for this problem. However, these methods are time-consulting and can not es-timate the high-order indices. In this dissertation, we propose new methods for global sensitivity analysis in different situations such that the defects of the existing methods can be overcome. The contents of this dissertation are as follows:In Chapter2, we present global sensitivity analysis when the decomposition sum-mands of the objective function can be obtained. In this case, the calculation of global sensitivity indices translates to the integration of the multi-dimension complex functions. Existing methods for this problem is to use numerical integration through Monte Carlo or quasi-Monte Carlo sampling. However, due to the complexity of the objective function, obtaining an accuracy result needs too much time. To improve the computation efficiency, we bring orthogonal arrays, as sampling plans, to the calculation of global sensitivity in-dices and propose a new estimator for the multi-dimension integration. Then by sampling theory, we establish a connection between orthogonal array and Monte Carlo sampling, and obtain the unbiasedness and convergence of the new estimator. At the end of this chapter, we show the higher sampling efficiency of the new method by simulation.In Chapter3, we consider the estimation of global sensitivity indices when the ob-jective function is known but the decomposition summands can not be obtained. In this situation, the existing methods are to use the values of f(χ) directly to obtain the estima-tion of global sensitivity indices by computer experiments. However, these methods have two disadvantages:1. They cannot get the estimation of high order sensitivity indices.2. They estimate all the sensitivity indices together, which results in a waste of calculation amount. To overcome these disadvantages, we make a difference between the significant and non-significant indices and propose a two-step algorithm to get the estimation of sensitivity indices. This new algorithm not only considers any order sensitivity indices but also greatly improves the computation efficiency. We get the convergence of this new method by the results of Chapter2. At the end of the chapter, a comparison with the existing algorithms is presented by simulation.In Chapter4, we show some theoretical results for the estimation of the ANOVA model under nonparametric situation. First, through model translation, we obtain the lower bound of the risks of all the linear unbiased estimation for ΘMThen by using matrix image, we prove that orthogonal arrays can make the risk achieve the lower bound, that is, we prove the superiority of orthogonal arrays when estimating ΘM.Finally, we propose estimators for global sensitivity indices of nonparametric models and get some useful properties.In Chapter5, we present a new method for global sensitivity analysis when the model is non-parametric. The existing methods in this situation are all based on model fitting. However, nonparametric model fitting is a rather difficult procedure and most of the methods only consider interactions of order two. Moreover, all the model fitting methods are based on Monte Carlo technique which requires plenty of experimental times. The new method is proposed from another point of view. It gives up model fitting and works directly on the noisy observations obtained by DOE to estimate the sensitivity indices. There are two stages in this method. All the non-influential global sensitivity indices are identified in the first stage by using a "W statistic" algorithm and the significant indices are precisely estimated in the second stage. The efficiency of this method is illustrated by simulation and real data analysis.In the last chapter, we summarize the whole essay and present some envisages.