在高維度資料的降維分析中，主要目的是將反應變數和大量的共變數之關係透過少數的共變數之線性組合來描述。其中一個重要的問題是確定這些和反應變數相關的共變數線性組合中有多少個線性獨立的組合，這即為充分降維(SDR)子空間的維度。在這個計畫中，我們將提出一個基於模擬的虛擬擴增變數來估計維度的辦法，可以被廣泛地運用在各種維度縮減方法上。這個辦法使用逐次檢定來估計維度，而每次檢定都是去比較原始共變數與虛擬共變數的信號強度。我們預期將證明在一個較弱的均勻方向條件下，我們的檢定統計量在虛無假設下會漸進地服從貝他分佈，因此會很容易校準。 ;In data analysis using dimension reduction methods, the main purpose is to summarize how the response is related to the covariates through a limited set of their linear combinations. One key issue is to determine the number of independent, relevant covariate combinations, which is the dimension of the sufficient dimension reduction (SDR) subspace. In this proposal, we will propose a broadly applicable approach to estimate the dimensionality of the SDR subspace, based on augmentation of the covariate set with simulated pseudo-covariates. The dimensionality is estimated using sequential testing, which compares the strength of the signal arising from the original covariates to that arising from the pseudo-covariates. We expect to show that under a weak uniform-direction condition, our test statistic follows a beta distribution asymptotically under the null hypothesis, and therefore is easily calibrated.
