Efficient Sparse Estimate of Sufficient Dimension Reduction in High Dimension

Sparse dimension reduction based on energy and ball statistics

“…To understand how the variables listed in Table 7 bear relation to the median house price per census tract, the central subspace can be estimated, which by definition (1) contains all information relevant to describe the dependent variable. However, in this analysis, one cannot rule out that not all of the predictors originally considered, should contribute the central subspace, as was surmised by Chen et al (2018) as well. These authors proceeded to apply sparse DCOV-SDR in order to obtain a sparse estimate of the central subspace and report the resulting coefficients.…”

“…Note that in practice, the dimension of the central subspace h needs to be estimated, and to that end the bootstrap method proposed in Sheng and Yin (2016) can be applied. As statistic P, we will consider the three options presented in the previous section, which lead to sufficient variable selection by distance covariance (DCOV-SDR, Chen et al, 2018), Martingale Difference Divergence (MDD-SDR) and Ball Covariance (BCOV-SDR), respectively, the latter two options being introduced in this paper. Note that the sparse sufficient dimension reduction techniques all intrinsically achieve variable selection, and for that purpose, will interchangeably be referred to as sparse variable selection as well, e.g.…”

“…It is practicable to fix the upper bound of iterations N to 200. The tolerance τ is set 0.001 as in Chen et al (2018) and the starting point B 0 is taken to be the nonsparse solution from (2) obtained in python through the direpack package (Menvouta et al, 2020).To choose the penalty parameter θ, values in [0, 0.5] spaced by 0.01 are considered in algorithm 1 and the solution to ( 14) is selected. When the structural dimension h is unknown, it can be estimated using the bootstrap method of Sheng and Yin (2016) with 200 replications.…”

“…The Boston Housing data are known to be skewed, particularly with respect to crime rate, which can be mitigated by removing observations greater than 3.2 for that variable, as was also suggested in Chen et al (2018). Upon trimming, 374 observations out of the original 506 census tracts, remain.…”

“…Several approaches have been published to achieve variable selection for SDR, either using hypothesis tests based on the estimated latent variables (Cook, 2004), called sufficient variable selection (SVS), or by a sure independence screening (Fan and Lv, 2008) procedure prior to the analysis. More recently, a sparse method to estimate the central subspace has been proposed, based on a criterion that maximizes distance covariance (Székely et al, 2007) to the dependent variable in the presence of a sparsity penalty (Chen et al, 2018). This method relies on fewer assumptions than its predecessors, and as it is based on a nonlinear measure of association, will achieve dimension reduction with respect to nonlinear dependencies.…”