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The inner partial least square: An exploration of the “necessary” dimension reduction

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  • Yin, Yunjian
  • Liu, Lan

Abstract

The partial least square (PLS) algorithm retains the combinations of predictors that maximize the covariance with the outcome. Cook et al. (2013) showed that PLS results in a predictor envelope, which is the smallest reducing subspace of predictors’ covariance that contains the coefficient. However, PLS and predictor envelope both target at a space that contains the regression coefficients and therefore they may sometimes be too conservative to reduce the dimension of the predictors. In this paper, we propose a new method that may improve the estimation efficiency of regression coefficients when both PLS and predictor envelope fail to do so. Specifically, our method results in the largest reducing subspace of predictors’ covariance that is contained in the coefficient matrix space. Interestingly, the moment based algorithm of our proposed method can be achieved by changing the max in PLS to min. We define the modified PLS as the inner PLS and the resulting space as the inner predictor envelope space. We provide the theoretical properties of our proposed methods as well as demonstrate their use in China Health and Nutrition Survey.

Suggested Citation

  • Yin, Yunjian & Liu, Lan, 2024. "The inner partial least square: An exploration of the “necessary” dimension reduction," Journal of Multivariate Analysis, Elsevier, vol. 204(C).
    • Handle: RePEc:eee:jmvana:v:204:y:2024:i:c:s0047259x24000630
    DOI: 10.1016/j.jmva.2024.105356
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    References listed on IDEAS

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    1. Zhihua Su & R. Dennis Cook, 2012. "Inner envelopes: efficient estimation in multivariate linear regression," Biometrika, Biometrika Trust, vol. 99(3), pages 687-702.
    2. Li, Ying & Udén, Peter & von Rosen, Dietrich, 2015. "A two-step estimation method for grouped data with connections to the extended growth curve model and partial least squares regression," Journal of Multivariate Analysis, Elsevier, vol. 139(C), pages 347-359.
    3. Z. Su & G. Zhu & X. Chen & Y. Yang, 2016. "Sparse envelope model: efficient estimation and response variable selection in multivariate linear regression," Biometrika, Biometrika Trust, vol. 103(3), pages 579-593.
    4. Marco Singer & Tatyana Krivobokova & Axel Munk & Bert de Groot, 2016. "Partial least squares for dependent data," Biometrika, Biometrika Trust, vol. 103(2), pages 351-362.
    5. Cook, R. Dennis & Forzani, Liliana & Liu, Lan, 2023. "Partial least squares for simultaneous reduction of response and predictor vectors in regression," Journal of Multivariate Analysis, Elsevier, vol. 196(C).
    6. Cook, R. Dennis & Forzani, Liliana & Su, Zhihua, 2016. "A note on fast envelope estimation," Journal of Multivariate Analysis, Elsevier, vol. 150(C), pages 42-54.
    7. R. D. Cook & I. S. Helland & Z. Su, 2013. "Envelopes and partial least squares regression," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 75(5), pages 851-877, November.
    8. Yuexiao Dong & Bing Li, 2010. "Dimension reduction for non-elliptically distributed predictors: second-order methods," Biometrika, Biometrika Trust, vol. 97(2), pages 279-294.
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