rewriting reg analysis

doc/src/Regression/Regression.do.txt

@@ -938,18 +938,22 @@ plt.show()
 !split
 ===== The singular value decompostion  =====
 !bblock
-How can we use the singular value decomposition to find the parameters $\beta_j$? More details will come. We first note that a general $m\times n$ matrix $\hat{A}$ can be written in terms of a diagonal matrix $\hat{\Sigma}$ of dimensionality $n\times n$ and two orthognal matrices $\hat{U}$ and $\hat{V}$, where the first has dimensionality $m \times n$ and the last dimensionality $n\times n$. We have then
+
+
+A general
+$m\times n$ matrix $\hat{A}$ can be written in terms of a diagonal
+matrix $\hat{D}$ of dimensionality $n\times n$ and two orthognal
+matrices $\hat{U}$ and $\hat{V}$, where the first has dimensionality
+$m \times m$ and the last dimensionality $n\times n$. 
+We have then 
 !bt
-\[
-\hat{A} = \hat{U}\hat{\Sigma}\hat{V}
-\]
+\[ 
+\hat{A} = \hat{U}\hat{D}\hat{V}^T 
+\] 
 !et 
 !eblock
 
 
-Add codes and discuss this in connection with lasso and ridge, show example where the standard inversion of a matrix fails and where SVD comes to rescue. Include first a discussion of the general SVD and then discuss the thin SVD.
-
-
 !split
 ===== Lasso and Ridge regression =====
 
@@ -957,7 +961,7 @@ Discuss the mathematics here
 
 
 !split
-===== Ridge and Lasso Regression =====
+===== Code examples for Ridge and Lasso Regression =====
 
 !bc pycod
 import matplotlib.pyplot as plt