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@@ -37,7 +37,7 @@ $$ w=\cfrac{\boldsymbol{x}_{d}^T\boldsymbol{y}_{d}}{\boldsymbol{x}_d^T\boldsymbo
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$$ \cfrac{\partial E_{\hat{\boldsymbol w}}}{\partial \hat{\boldsymbol w}}=2\mathbf{X}^T(\mathbf{X}\hat{\boldsymbol w}-\boldsymbol{y}) $$
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$$ \cfrac{\partial E_{\hat{\boldsymbol w}}}{\partial \hat{\boldsymbol w}}=2\mathbf{X}^T(\mathbf{X}\hat{\boldsymbol w}-\boldsymbol{y}) $$
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-[推导]:将$ E_{\hat{\boldsymbol w}}=(\boldsymbol{y}-\boldsymbol{X}\hat{\boldsymbol w})^T(\boldsymbol{y}-\mathbf{X}\hat{\boldsymbol w}) $展开可得:
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+[推导]:将$ E_{\hat{\boldsymbol w}}=(\boldsymbol{y}-\mathbf{X}\hat{\boldsymbol w})^T(\boldsymbol{y}-\mathbf{X}\hat{\boldsymbol w}) $展开可得:
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$$ E_{\hat{\boldsymbol w}}= \boldsymbol{y}^T\boldsymbol{y}-\boldsymbol{y}^T\mathbf{X}\hat{\boldsymbol w}-\hat{\boldsymbol w}^T\mathbf{X}^T\boldsymbol{y}+\hat{\boldsymbol w}^T\mathbf{X}^T\mathbf{X}\hat{\boldsymbol w} $$
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$$ E_{\hat{\boldsymbol w}}= \boldsymbol{y}^T\boldsymbol{y}-\boldsymbol{y}^T\mathbf{X}\hat{\boldsymbol w}-\hat{\boldsymbol w}^T\mathbf{X}^T\boldsymbol{y}+\hat{\boldsymbol w}^T\mathbf{X}^T\mathbf{X}\hat{\boldsymbol w} $$
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对$ \hat{\boldsymbol w} $求导可得:
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对$ \hat{\boldsymbol w} $求导可得:
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$$ \cfrac{\partial E_{\hat{\boldsymbol w}}}{\partial \hat{\boldsymbol w}}= \cfrac{\partial \boldsymbol{y}^T\boldsymbol{y}}{\partial \hat{\boldsymbol w}}-\cfrac{\partial \boldsymbol{y}^T\mathbf{X}\hat{\boldsymbol w}}{\partial \hat{\boldsymbol w}}-\cfrac{\partial \hat{\boldsymbol w}^T\mathbf{X}^T\boldsymbol{y}}{\partial \hat{\boldsymbol w}}+\cfrac{\partial \hat{\boldsymbol w}^T\mathbf{X}^T\mathbf{X}\hat{\boldsymbol w}}{\partial \hat{\boldsymbol w}} $$
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$$ \cfrac{\partial E_{\hat{\boldsymbol w}}}{\partial \hat{\boldsymbol w}}= \cfrac{\partial \boldsymbol{y}^T\boldsymbol{y}}{\partial \hat{\boldsymbol w}}-\cfrac{\partial \boldsymbol{y}^T\mathbf{X}\hat{\boldsymbol w}}{\partial \hat{\boldsymbol w}}-\cfrac{\partial \hat{\boldsymbol w}^T\mathbf{X}^T\boldsymbol{y}}{\partial \hat{\boldsymbol w}}+\cfrac{\partial \hat{\boldsymbol w}^T\mathbf{X}^T\mathbf{X}\hat{\boldsymbol w}}{\partial \hat{\boldsymbol w}} $$
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