diff --git a/docs/03-Linear-Methods-for-Regression/3.7-Multiple-Outcome-Shrinkage-and-Selection.md b/docs/03-Linear-Methods-for-Regression/3.7-Multiple-Outcome-Shrinkage-and-Selection.md index f55df27c54..f5d812df5e 100644 --- a/docs/03-Linear-Methods-for-Regression/3.7-Multiple-Outcome-Shrinkage-and-Selection.md +++ b/docs/03-Linear-Methods-for-Regression/3.7-Multiple-Outcome-Shrinkage-and-Selection.md @@ -37,7 +37,7 @@ $$ !!! info "weiya 注:Ex. 3.20" 已解决,详见 [Issue 172: Ex. 3.20](https://github.com/szcf-weiya/ESL-CN/issues/172),欢迎讨论交流! -**降秩回归 (reduced-rank regression)** (Izenman, 1975[^1]; van der Merwe and Zidek, 1980[^2]) 采用显式地合并信息的回归模型来形式化这个方法.给定误差协方差 $\Cov(\varepsilon)=\bSigma$,求解下列带约束的多元回归问题: +**降秩回归 (reduced-rank regression)** (Izenman, 1975[^1]; van der Merwe and Zidek, 1980[^2]) 采用显式地合并信息的回归模型来 *正式化 (formalize)* 这个方法.给定误差协方差 $\Cov(\varepsilon)=\bSigma$,求解下列带约束的多元回归问题: $$ \hat\B^{rr}(m) = \underset{\rank(\B)=m}{\argmin}\sum_{i=1}^N(y_i-\B^Tx_i)^T\bSigma^{-1}(y_i-\B^Tx_i)\,.\tag{3.68}\label{3.68} @@ -74,7 +74,7 @@ $$ !!! info "weiya 注:Ex. 3.22" 已解决,详见 [Issue 175: Ex. 3.22](https://github.com/szcf-weiya/ESL-CN/issues/175),欢迎交流讨论! -降秩回归通过截断 CCA 从响应变量中借来了优点.Breiman and Friedman (1997)[^3] 探索了 $\X$ 和 $\Y$ 间典则变量的逐步收缩,这是光滑版本的降秩回归.他们的方法有如下形式(与 $\eqref{3.69}$ 对应) +降秩回归通过截断 CCA 从响应变量中借来了优势变量.Breiman and Friedman (1997)[^3] 探索了 $\X$ 和 $\Y$ 间典则变量的逐步收缩,这是光滑版本的降秩回归.他们的方法有如下形式(与 \eqref{3.69} 对应) $$ \hat\B^{c+w} = \hat\B\U\bLambda\U^{-1}\,,\tag{3.72} @@ -100,7 +100,15 @@ $$ \hat\Y^{\mathrm{ridge},c+w}=\A_\lambda\Y\S^{c+w}\,,\tag{3.75} $$ -其中 $\A_\lambda = \X(\X^T\X+\lambda\I)^{-1}\X^T$ 是岭回归收缩算子,和 (3.46) 一样.他们的文章及其讨论有更多细节. +其中 $\A_\lambda = \X(\X^T\X+\lambda\I)^{-1}\X^T$ 是岭回归收缩算子,和 \eqref{3.46} 一样.他们的文章及其讨论有更多细节. + +!!! note "Recall" + $$ + \begin{align} + \mathbf{X}\hat{\beta}^{ls}&=\mathbf{X(X^TX)^{-1}X^Ty}\notag\\ + &=\mathbf{UU^Ty}\tag{3.46}\label{3.46} + \end{align} + $$ [^1]: Izenman, A. (1975). Reduced-rank regression for the multivariate linear model, Journal of Multivariate Analysis 5: 248–264. [^2]: van der Merwe, A. and Zidek, J. (1980). Multivariate regression analysis and canonical variates, The Canadian Journal of Statistics 8: 27–39.