Tutorial: Methodologies for sufficient dimension reduction in regression

Abstract

In the paper, as a sequence of the first tutorial, we discuss sufficient dimension reduction methodologies used to estimate central subspace (sliced inverse regression, sliced average variance estimation), central mean subspace (ordinary least square, principal Hessian direction, iterative Hessian transformation), and central k thmoment subspace (covariance method). Large-sample tests to determine the structural dimensions of the three target subspaces are well derived in most of the methodologies; however, a permutation test (which does not require large-sample distributions) is introduced. The test can be applied to the methodologies discussed in the paper. Theoretical relationships among the sufficient dimension reduction methodologies are also investigated and real data analysis is presented for illustration purposes. A seeded dimension reduction approach is then introduced for the methodologies to apply to large p small n regressions.

Fused inverse regression with multi-dimensional responses

“…On the other hand, in Figure 8, it is observed that the second sufficient predictors from KIR and FHIR with 3, 6 and 9 clusters are highly correlated to each other, so we expect that the structural dimension determination for KIR and FHIR should be, at least, two following the same rationale in Yoo (2018). To investigate this, a permutation dimension test (Yin and Bura, 2006;Yoo, 2016b) were conducted for FHIR and FpSIR, and weighted χ 2 test for KIR (Setoji and Cook, 2004) starting H 0 : d = 0 with nominal level 5%. If H 0 : d = 0 is not rejected, increment d by 1 and redo the test.…”

“…Letting Σ = cov(X), Li (1991) showed that Yoo (2016b). In practice, E(Z|Y) is restored instead of E(X|Y).…”

Iterative projection of sliced inverse regression with fused approach

“…In SDR context, it is normally assumed that S(E(Z|Y)) = S Z|Y . For more about the two conditions mentioned here, readers are recommended to read Yoo (2016aYoo ( , 2016b. Hereafter without mentioning clearly, we assume that S(E(Z|Y)) = S Y|Z .…”

On hierarchical clustering in sufficient dimension reduction

“…The response is sliced for each category or to have equal numbers of observations (although it is not strictly required). Readers are recommended to read Yoo (2016aYoo ( , 2016b for further insights about SDR and details on SIR and SAVE.…”