updating typos

doc/src/week37/exercisesweek37.do.txt

@@ -6,14 +6,24 @@ DATE: Deadline is Sunday September 17 at midnight
 =====  Overarching aims of the exercises this week =====
 
 
-This exercise deals with various mean values and variances in  linear regression method (here it may be useful to look up chapter 3, equation (3.8) of "Trevor Hastie, Robert Tibshirani, Jerome H. Friedman, The Elements of Statistical Learning, Springer":"https://www.springer.com/gp/book/9780387848570"). The exercise is also a part of project 1 and can be reused in the theory part of the project.
+This exercise deals with various mean values and variances in linear
+regression method (here it may be useful to look up chapter 3,
+equation (3.8) of "Trevor Hastie, Robert Tibshirani, Jerome
+H. Friedman, The Elements of Statistical Learning,
+Springer":"https://www.springer.com/gp/book/9780387848570"). The
+exercise is also a part of project 1 and can be reused in the theory
+part of the project.
 
-For more discussions on Ridge regression and calculation of expectation values, "Wessel van Wieringen's":"https://arxiv.org/abs/1509.09169" article is highly recommended.
+For more discussions on Ridge regression and calculation of
+expectation values, "Wessel van
+Wieringen's":"https://arxiv.org/abs/1509.09169" article is highly
+recommended.
 
 
-The assumption we have made is 
-that there exists a continuous function $f(\bm{x})$ and  a normal distributed error $\bm{\varepsilon}\sim N(0, \sigma^2)$
-which describes our data
+The assumption we have made is that there exists a continuous function
+$f(\bm{x})$ and a normal distributed error $\bm{\varepsilon}\sim N(0,
+\sigma^2)$ which describes our data
+
 !bt
 \[
 \bm{y} = f(\bm{x})+\bm{\varepsilon}
@@ -70,7 +80,7 @@ A given parameter $\beta_j$ is given by the diagonal matrix element of the above
 Show that
 !bt
 \[
-\mathbb{E} \big[ \bm{\beta}^{\mathrm{Ridge}} \big]=(\mathbf{X}^{T} \mathbf{X} + \lambda \mathbf{I}_{pp})^{-1} (\mathbf{X}^{\top} \mathbf{X})\
+\mathbb{E} \big[ \hat{\bm{\beta}}^{\mathrm{Ridge}} \big]=(\mathbf{X}^{T} \mathbf{X} + \lambda \mathbf{I}_{pp})^{-1} (\mathbf{X}^{\top} \mathbf{X})\
 \bm{\beta}^{\mathrm{OLS}}.
 \]
 !et
@@ -82,7 +92,7 @@ Show also that the variance is
 
 !bt
 \[
-\mbox{Var}[\bm{\beta}^{\mathrm{Ridge}}]=\sigma^2[  \mathbf{X}^{T} \mathbf{X} + \lambda \mathbf{I} ]^{-1}  \mathbf{X}^{T}\mathbf{X} \{ [  \mathbf{X}^{\top} \mathbf{X} + \lambda \mathbf{I} ]^{-1}\}^{T},
+\mbox{Var}[\hat{\bm{\beta}}^{\mathrm{Ridge}}]=\sigma^2[  \mathbf{X}^{T} \mathbf{X} + \lambda \mathbf{I} ]^{-1}  \mathbf{X}^{T}\mathbf{X} \{ [  \mathbf{X}^{\top} \mathbf{X} + \lambda \mathbf{I} ]^{-1}\}^{T},
 \]
 !et
 and it is easy to see that if the parameter $\lambda$ goes to infinity then the variance of Ridge parameters $\bm{\beta}$ goes to zero.