updating regression slides

doc/src/Regression/Regression.do.txt

@@ -1606,6 +1606,15 @@ Going back to our OLS equation we have
 We will come back to this expression when we discuss Ridge regression. 
 
 
+$$ \tilde{y}^{OLS}={\bf X}\hat{\beta}^{OLS}=\sum_{j=1}^p {\bf u}_j{\bf u}_j^T{\bf y}$$ and for Ridge we have 
+
+$$ \tilde{y}^{Ridge}={\bf X}\hat{\beta}^{Ridge}=\sum_{j=1}^p {\bf u}_j\frac{\sigma_j^2}{\sigma_j^2+\lambda}{\bf u}_j^T{\bf y}$$ . 
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+It is indeed the economy-sized SVD, note the summation runs up tp $$p$$ only and not $$n$$. 
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+Here we have that $${\bf X} = {\bf U}{\bf \Sigma}{\bf V}^T$$, with $$\Sigma$$ being an $$ n\times p$$ matrix and $${\bf V}$$ being a $$ p\times p$$ matrix. We also have assumed here that $$ n > p$$. 
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+
 !split
 ===== Ridge and LASSO Regression =====