update of week 35

doc/src/week35/week35.do.txt

@@ -1242,7 +1242,7 @@ We can then interpret our optimal model $\tilde{\bm{y}}$ as being represented  b
 We have defined the residual error as
 !bt
 \[
-\bm{\epsilon}=\bm{y}-\tilde{\bm{y}}=(\bm{I}-\bm{X}\left(\bm{X}^T\bm{X}\right^{-1}\bm{X}^T)\bm{y}.
+\bm{\epsilon}=\bm{y}-\tilde{\bm{y}}=\left[\bm{I}-\bm{X}\left(\bm{X}^T\bm{X}\right)^{-1}\bm{X}^T\right]\bm{y}.
 \]
 !et
 
@@ -1574,21 +1574,21 @@ More material will be added here, see handwritten notes also.   Note that this m
 
 
 !split
-===== Matheamtics of the SVD and implications =====
+===== Mathematics of the SVD and implications =====
 
 Let us take a closer look at the mathematics of the SVD and the various implications for machine learning studies.
 
 Our starting point is our design matrix $\bm{X}$ of dimension $n\times p$
 !bt
 \[
-\bm{X}=
-\begin{bmatrix}
-x_{0,0} & x_{0,1} & x_{0,2}& \dots & \dots & x_{0,p-1}\\
-x_{1,0} & x_{1,1} & x_{1,2& \dots & \dots & x_{1,p-1}\\
-x_{2,0} & x_{2,1} & x_{2,2}& \dots & \dots & x_{2,p-1}\\
-\dots& \dots &\dots & \dots & \dots &\dots\\
-x_{n-1,0} & x_{n-1,1} & x_{n-1,2} & \dots & \dots & x_{n-1,p-1}\\
-\end{bmatrix}
+\bm{X}=\begin{bmatrix}
+x_{0,0} & x_{0,1} & x_{0,2}& \dots & \dots x_{0,p-1}\\
+x_{1,0} & x_{1,1} & x_{1,2}& \dots & \dots x_{1,p-1}\\
+x_{2,0} & x_{2,1} & x_{2,2}& \dots & \dots x_{2,p-1}\\
+\dots & \dots & \dots & \dots \dots & \dots \\
+x_{n-2,0} & x_{n-2,1} & x_{n-2,2}& \dots & \dots x_{n-2,p-1}\\
+x_{n-1,0} & x_{n-1,1} & x_{n-1,2}& \dots & \dots x_{n-1,p-1}\\
+\end{bmatrix}.
 \]
 !et
 
@@ -1695,7 +1695,7 @@ and using the orthogonality of the matrix $\bm{U}$ we have
 \bm{X}^T\bm{X}=\bm{V}\bm{\Sigma}^T\bm{\Sigma}\bm{V}^T.
 \]
 !et
-We define $\bm{\Sigma}^T\bm{\sigma}=\bm{D}^2, where $\bm{D}$ is a diagonal matrix containing only the singular values squared. It has dimensionality $\p \times p$.
+We define $\bm{\Sigma}^T\bm{\sigma}=\bm{D}^2, where $\bm{D}$ is a diagonal matrix containing only the singular values squared. It has dimensionality $p \times p$.
 
 This means, using the orthogonality of $\bm{V}$,  that we get
 
@@ -1805,7 +1805,7 @@ function, that is we have
 
 !bt
 \[
-\frac{\partial}{\partial \bm{\beta}^T}\frac{\partial C(\bm{\beta})}{\partial \bm{\beta}} =\frac{2}{n}\bm{X}^T\bm{X}.  
+\frac{\partial^2 C(\bm{\beta})}{\partial \bm{\beta}^T\partial \bm{\beta}} =\frac{2}{n}\bm{X}^T\bm{X}.  
 \]
 !et
 This quantity defines was what is called the Hessian matrix (the second derivative of a function we want to optimize).
@@ -2190,7 +2190,7 @@ where the tilde-matrix $\tilde{\bm{\Sigma}}$ is a matrix of dimension $p\times p
 \[
 \tilde{\bm{\Sigma}}=\begin{bmatrix} \sigma_0 & 0 & 0 & \dots & 0 & 0 \\
                                     0 & \sigma_1 & 0 & \dots & 0 & 0 \\
-				    0 & 0 & \sigma_2 & \dots & 0 & 0 \
+				    0 & 0 & \sigma_2 & \dots & 0 & 0 \\
 				    0 & 0 & 0 & \dots & \sigma_{p-2} & 0 \\
 				    0 & 0 & 0 & \dots & 0 & \sigma_{p-1} \\
 \end{bmatrix},