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correction erreur sur un slide (dérivée en fn de beta)

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Pierre-Edouard Portier 2022-03-14 13:29:43 +01:00
parent 44a8555598
commit c2c2f5d64b
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02_moindres_carres_slides.Rmd
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@ -36,7 +36,7 @@ $$
# Moindres carrés
- Annuler la dérivée selon $\mathbf{X}$ de la fonction convexe $\|\mathbf{X}\boldsymbol\beta-\mathbf{y}\|^2_2$
- Annuler la dérivée selon $\boldsymbol\beta$ de la fonction convexe $\|\mathbf{X}\boldsymbol\beta-\mathbf{y}\|^2_2$
- $\|\mathbf{X}\boldsymbol\beta-\mathbf{y}\|^2_2 = \boldsymbol\beta^T\mathbf{X}^T\mathbf{X}\boldsymbol\beta - 2\boldsymbol\beta^T\mathbf{X}^T\mathbf{y} + \mathbf{y}^T\mathbf{y}$
- $\mathbf{0} = 2\mathbf{X}^T\mathbf{X}\boldsymbol\beta - 2\mathbf{X}^T\mathbf{y} \quad \equiv \quad \mathbf{X}^T\mathbf{X} \boldsymbol\beta = \mathbf{X}^T\mathbf{y}$
- $\hat{\boldsymbol\beta} = \left(\mathbf{X}^T\mathbf{X}\right)^{-1}\mathbf{X}^T\mathbf{y}$