Optimal and Robust Designs for Active Learning and Regression Analysis

我的研究计划旨在发展构建回归的统计实验的最优和稳健设计以及为主动学习选择训练数据的理论和方法。我的计划包括以下研究问题，我打算在未来五年内研究这些问题。 (I)广义线性混合模型的稳健设计(GLMM) 基于我以前在这个方向上的研究，我将继续致力于GLMM的健壮设计，但不精确。经常遇到的GLMM离职将被考虑。我打算开发一种针对这些离职的健壮设计流程。我计划基于指数分布推导出一类一般GLMM的序贯设计。改进和新的方法有望更好地平衡计算复杂性和设计效率。 (Ii)复合分位数回归(CQR)的最优稳健设计。 CQR以最小的效率折衷增强了健壮性。我打算分两个阶段来构建CQR的最优稳健设计：首先，我将考虑等权重的CQR，并推导出关于假设的CQR模型的信任度最小化指定损失函数的稳健设计。然后，我将引入加权CQR，目的是确定使估计偏差最小的最优权重，并进一步构造最优设计，以使估计效率最大化。 (3)采用阶跃应力加载方案的加速寿命试验(ALT)的优化和稳健设计。 基于我对恒定应力和简单阶跃应力ALT最优/稳健设计的研究结果，我将继续为进一步的阶跃应力加载ALT构建最优和稳健设计。将强调设计的稳健性，以防止可能的模型偏离，包括生活压力关系的不精确性和寿命分布的不确定性。我将把先前的研究扩展到涉及异方差、多因素、多阶跃压力负荷和/或半参数ALT模型的情景。 (4)用于线性回归和分类的最优和稳健的主动学习。 我将为线性回归和分类问题的主动学习中的训练数据选择构建最优和稳健的设计。最优主动学习往往基于假设的线性回归模型，且精度不高，因此有必要进行粗暴处理。我还打算开发最优的和稳健的主动学习过程用于分类，以便识别一个学习的判别函数，该函数可以尽可能准确地对剩余的输入数据点进行分类。由于这些判别函数取决于假设的输入分布中的参数，因此我们将使用两阶段设计或序贯设计。 此外，当需要最优规划或最好的数据选择时，我将继续开发和应用各种实际应用的优化和稳健设计方法。

My research program aims to develop both theory and methods of constructing optimal and robust designs of statistical experiments for regression, and of training data selection for active learning. My program consists of the following research problems that I intend to investigate for the next five years. (i) Robust designs for generalized linear mixed models (GLMM). Building on my previous research in this direction, I will continue to work on robust designs for GLMMs with imprecision. Frequently encountered GLMM departures will be considered. I intend to develop a design process that is robust against these departures. I plan to derive sequential designs for a general class of GLMM based on exponential dispersion distributions. Improvement and new methods are expected to better balance the computation complexity and design efficiencies. (ii) Optimal and robust designs for composite quantile regression (CQR). CQR enhances the robustness with least trade-off in efficiency. I intend to construct optimal and robust designs for CQR with two phases: First, I will consider equally weighted CQR and derive robust designs that minimize a specified loss function with respect to the degree of belief on the assumed CQR model. I will then introduce weighted CQR and aim to identify the optimal weights that minimize the estimation bias and further construct optimal designs in order to maximize estimation efficiency. (iii) Optimal and robust designs for accelerated life testing (ALT), incorporating step-stress loading schemes. Building on my research results obtained for optimal/robust constant-stress and simple step-stress ALT designs, I will continue to construct optimal and robust designs for ALT further with step-stress loading. Robustness of the designs will be emphasized to protect possible model departures, including imprecision in life-stress relationship and uncertainty in lifetime distribution. I will extend the previous study to the scenarios that involve heteroscedasticity, multiple factors, multiple step-stress loading, and/or semi-parametric ALT models. (iv) Optimal and robust active learning for linear regression and classification. I will construct optimal and robust designs of training data selection in active learning for both linear regression and classification problems. Optimal active learning is often based on an assumed linear regression model with imprecision, so robustification will be necessary. I also intend to develop optimal and robust active learning process for classification in order to identify a learned discriminant function which can classify the remaining input data points as accurately as possible. As such discriminant functions depend on the parameters in the assumed input distributions, we will use two-stage or sequential designs. Further, I will continue developing and applying the methods of optimal and robust design for various practical applications when optimal planning or finest data selection is needed.