On the extension of sliced average variance estimation to multivariate regression

Abstract

Many sufficient dimension reduction methods for univariate regression have been extended to multivariate regression. Sliced average variance estimation (SAVE) has the potential to recover more reductive information and recent development enables us to test the dimension and predictor effects with distributions commonly used in the literature. In this paper, we aim to extend the functionality of the SAVE to multivariate regression. Toward the goal, we propose three new methods. Numerical studies and real data analysis demonstrate that the proposed methods perform well.

Fused inverse regression with multi-dimensional responses

“…When Y is multi-dimensional, grouping by similarity can be done via clustering algorithms. Setodji and Cook (2004) and Yoo et al (2010) successfully replace the usual slicing scheme with the K-means clustering algorithm for SIR, called K-means inverse regression (KIR), and sliced average variance estimation, respectively. In a perspective of fusing, the K-means algorithm is not be effective according to Yoo et al (2020).…”

“…This relation was firstly observed and utilized by Yoo et al (2010), which proposed pooled sliced average variance estimation. It directly implies that combining all information on the central subspace of the coordinate regressions contains useful information on S Y|X .…”

“…So far, various SDR methodologies have been developed to estimate S Y|X in multivariate regression. Various SDR methodologies for multivariate regression have been proposed (Lee et al, 2019;Setodji and Cook, 2004;Yin and Bura, 2006;Yoo, 2008Yoo, , 2009Yoo and Cook, 2007;Yoo et al, 2010). In Setodji and Cook (2004) and Yoo et al (2010), an inverse regression approach, called K-means inverse regression (KIR) and K-means average variance estimation, is adopted, while the other methods combine the information from the coordinate regression of…”

On hierarchical clustering in sufficient dimension reduction

“…The number of observations for each cluster were (7, 27, 66), (76,10,14), and (19, 74, 7); consequently, we can see that the three results are different. This implies that, under the same number of clusters, the multivariate inverse regression methods by Setodji and Cook (2004) and Yoo et al (2010) possibly provides different results. However, it is not clear which result should be used.…”

“…To avoid this issue, the K-means clustering algorithm (KCA) has been adopted successfully in replacing hierarchical slicing. Setodji and Cook (2004) and Yoo et al (2010) use the KCA to categorize the multi-dimensional responses, and extended the direct applicability SIR and SAVE to multivariate regression. The clusters constructed by the KCA play the same role as slices.…”