Fourier Methods for Estimating the Central Subspace and the Central Mean Subspace in Regression

Abstract

In high dimensional regression, it is important to estimate the central and central mean subspaces, to which the projections of the predictors preserve sufficient information about the response and the mean response, respectively. Using the Fourier transform, we have derived the candidate matrices whose column spaces recover the central and central mean subspaces exhaustively. Under the normality assumption of the predictors, explicit estimates of the central and central mean subspaces are derived. Bootstrap procedures are used for determining dimensionality and choosing tuning parameters. Simulation results and an application to a real data are reported. Our methods demonstrate competitive performance compared to SIR, SAVE and other existing methods. The approach proposed in the paper provides a novel view on sufficient dimension reduction and may lead to more powerful tools in the future.

On Directional Regression for Dimension Reduction

“…For example, the number of slices (SIR, SAVE, and SR), the percentage of empirical distribution (SCR), the (σ 2 T , σ 2 W ) value (Fourier), and also the bandwidths (MAVE and SR). To evaluate the sensitivity of those methods with respect to the tuning parameter specification, the following simulation configurations are considered: (1) The number of slices for SIR, SAVE, and SR is fixed to be 5 or 10; (2) The percentage of empirical directions used by SCR is given by 5% or 10% (Li et al, 2005); (3) For the Fourier method, we fix σ 2 T = 1.0 but σ 2 W = 5% or 10% (Zhu and Zeng, 2006); (4) Lastly, the Gaussian kernel is used for MAVE and SR.…”

On aggregate dimension reduction

“…To estimate the structural dimension d, we adopt the bootstrap procedure proposed in Ye and Weiss (2003) and Zhu and Zeng (2006). Let Finally, we choose the number of observations in each neighborhood to be 2p ≤ k ≤ 4p.…”

“…Figure 6 shows the dimension variability plots (Zhu and Zeng, 2006) for models 1-4. As expected, large variability showed up when d * > d. Out of 100 samples with n = 400 and p = 10, the accuracy of correctly estimated d is 99%, 94%, 99% and 84%…”

A Note on Bootstrapping in Sufficient Dimension Reduction

“…Recent usage of bootstrapping in SDR context can be found in Zhu and Zeng (2006), Yoo (2011) andYoo (2013b). Bootstrapping is well established in sufficient dimension reduction; however numerical studies for the dimension estimation and comparison with the permutation test have not yet been well studied.…”