New developments of dimension reduction

Variable selection becomes more crucial than before, since high dimensional data are frequently seen in many research areas. Many model-based variable selection methods have been developed. However, the performance might be poor when the model is mis-specified. Sufficient dimension reduction (SDR, Li 1991; Cook 1998) provides a general framework for model-free variable selection methods. In this thesis, we first propose a novel model-free variable selection method to deal with multi-population data by incorporating the grouping information. Theoretical properties of our proposed method are also presented. Simulation studies show that our new method significantly improves the selection performance compared with those ignoring the grouping information. In the second part of this dissertation, we apply partial SDR method to conduct conditional model-free variable (feature) screening for ultra-high dimensional data, when researchers have prior information regarding the importance of certain predictors based on experience or previous investigations. Comparing to the state of art conditional screening method, conditional sure independence screening (CSIS; Barut, Fan and Verhasselt, 2016), our method greatly outperforms CSIS for nonlinear models. The sure screening consistency property of our proposed method is also established --Abstract, page iv