removing hat symbols

doc/src/week34/week34.do.txt

@@ -786,7 +786,7 @@ print(x.itemsize)
 ===== Matrices in Python =====
 
 Having defined vectors, we are now ready to try out matrices. We can
-define a $3 \times 3 $ real matrix $\hat{A}$ as (recall that we user
+define a $3 \times 3 $ real matrix $\bm{A}$ as (recall that we user
 lowercase letters for vectors and uppercase letters for matrices)
 
 !bc pycod
@@ -835,10 +835,10 @@ print(A)
 
 As we will see throughout these lectures, there are several extremely useful functionalities in Numpy.
 As an example, consider the discussion of the covariance matrix. Suppose we have defined three vectors
-$\hat{x}, \hat{y}, \hat{z}$ with $n$ elements each. The covariance matrix is defined as 
+$\bm{x}, \bm{y}, \bm{z}$ with $n$ elements each. The covariance matrix is defined as 
 !bt
 \[
-\hat{\Sigma} = \begin{bmatrix} \sigma_{xx} & \sigma_{xy} & \sigma_{xz} \\
+\bm{\Sigma} = \begin{bmatrix} \sigma_{xx} & \sigma_{xy} & \sigma_{xz} \\
                               \sigma_{yx} & \sigma_{yy} & \sigma_{yz} \\
                               \sigma_{zx} & \sigma_{zy} & \sigma_{zz} 
              \end{bmatrix},
@@ -851,10 +851,10 @@ where for example
 \]
 !et
 The Numpy function _np.cov_ calculates the covariance elements using the factor $1/(n-1)$ instead of $1/n$ since it assumes we do not have the exact mean values. 
-The following simple function uses the _np.vstack_ function which takes each vector of dimension $1\times n$ and produces a $3\times n$ matrix $\hat{W}$
+The following simple function uses the _np.vstack_ function which takes each vector of dimension $1\times n$ and produces a $3\times n$ matrix $\bm{W}$
 !bt
 \[
-\hat{W} = \begin{bmatrix} x_0 & y_0 & z_0 \\
+\bm{W} = \begin{bmatrix} x_0 & y_0 & z_0 \\
                           x_1 & y_1 & z_1 \\
                           x_2 & y_2 & z_2 \\
                           \dots & \dots & \dots \\
@@ -865,8 +865,8 @@ The following simple function uses the _np.vstack_ function which takes each vec
 !et
 
 which in turn is converted into into the $3\times 3$ covariance matrix
-$\hat{\Sigma}$ via the Numpy function _np.cov()_. We note that we can also calculate
-the mean value of each set of samples $\hat{x}$ etc using the Numpy
+$\bm{\Sigma}$ via the Numpy function _np.cov()_. We note that we can also calculate
+the mean value of each set of samples $\bm{x}$ etc using the Numpy
 function _np.mean(x)_. We can also extract the eigenvalues of the
 covariance matrix through the _np.linalg.eig()_ function.
 
@@ -1026,7 +1026,7 @@ We start with perhaps our simplest possible example, using _Scikit-Learn_ to per
 
 What follows is a simple Python code where we have defined a function
 $y$ in terms of the variable $x$. Both are defined as vectors with  $100$ entries. 
-The numbers in the vector $\hat{x}$ are given
+The numbers in the vector $\bm{x}$ are given
 by random numbers generated with a uniform distribution with entries
 $x_i \in [0,1]$ (more about probability distribution functions
 later). These values are then used to define a function $y(x)$
@@ -1054,7 +1054,7 @@ where $N(0,1)$ represents random numbers generated by the normal
 distribution.  From _Scikit-Learn_ we import then the
 _LinearRegression_ functionality and make a prediction $\tilde{y} =
 \alpha + \beta x$ using the function _fit(x,y)_. We call the set of
-data $(\hat{x},\hat{y})$ for our training data. The Python package
+data $(\bm{x},\bm{y})$ for our training data. The Python package
 _scikit-learn_ has also a functionality which extracts the above
 fitting parameters $\alpha$ and $\beta$ (see below). Later we will
 distinguish between training data and test data.
@@ -1149,7 +1149,7 @@ the relative error (why would we prefer the MSE instead of the relative error?)
 
 !bt
 \[
-\epsilon_{\mathrm{relative}}= \frac{\vert \hat{y} -\hat{\tilde{y}}\vert}{\vert \hat{y}\vert}.
+\epsilon_{\mathrm{relative}}= \frac{\vert \bm{y} -\bm{\tilde{y}}\vert}{\vert \bm{y}\vert}.
 \]
 !et
 
@@ -1225,7 +1225,7 @@ plt.show()
 The function _coef_ gives us the parameter $\beta$ of our fit while _intercept_ yields 
 $\alpha$. Depending on the constant in front of the normal distribution, we get values near or far from $\alpha =2$ and $\beta =5$. Try to play around with different parameters in front of the normal distribution. The function _meansquarederror_ gives us the mean square error, a risk metric corresponding to the expected value of the squared (quadratic) error or loss defined as
 !bt 
-\[ MSE(\hat{y},\hat{\tilde{y}}) = \frac{1}{n}
+\[ MSE(\bm{y},\bm{\tilde{y}}) = \frac{1}{n}
 \sum_{i=0}^{n-1}(y_i-\tilde{y}_i)^2, 
 \] 
 !et 
@@ -1238,16 +1238,16 @@ The _r2score_ function computes $R^2$, the coefficient of
 determination. It provides a measure of how well future samples are
 likely to be predicted by the model. Best possible score is 1.0 and it
 can be negative (because the model can be arbitrarily worse). A
-constant model that always predicts the expected value of $\hat{y}$,
+constant model that always predicts the expected value of $\bm{y}$,
 disregarding the input features, would get a $R^2$ score of $0.0$.
 
-If $\tilde{\hat{y}}_i$ is the predicted value of the $i-th$ sample and $y_i$ is the corresponding true value, then the score $R^2$ is defined as
+If $\tilde{\bm{y}}_i$ is the predicted value of the $i-th$ sample and $y_i$ is the corresponding true value, then the score $R^2$ is defined as
 !bt
 \[
-R^2(\hat{y}, \tilde{\hat{y}}) = 1 - \frac{\sum_{i=0}^{n - 1} (y_i - \tilde{y}_i)^2}{\sum_{i=0}^{n - 1} (y_i - \bar{y})^2},
+R^2(\bm{y}, \tilde{\bm{y}}) = 1 - \frac{\sum_{i=0}^{n - 1} (y_i - \tilde{y}_i)^2}{\sum_{i=0}^{n - 1} (y_i - \bar{y})^2},
 \]
 !et
-where we have defined the mean value  of $\hat{y}$ as
+where we have defined the mean value  of $\bm{y}$ as
 !bt
 \[
 \bar{y} =  \frac{1}{n} \sum_{i=0}^{n - 1} y_i.
@@ -1258,14 +1258,14 @@ Another quantity taht we will meet again in our discussions of regression analys
 The MAE is defined as follows
 !bt
 \[
-\text{MAE}(\hat{y}, \hat{\tilde{y}}) = \frac{1}{n} \sum_{i=0}^{n-1} \left| y_i - \tilde{y}_i \right|.
+\text{MAE}(\bm{y}, \bm{\tilde{y}}) = \frac{1}{n} \sum_{i=0}^{n-1} \left| y_i - \tilde{y}_i \right|.
 \]
 !et
 We present the 
 squared logarithmic (quadratic) error
 !bt
 \[
-\text{MSLE}(\hat{y}, \hat{\tilde{y}}) = \frac{1}{n} \sum_{i=0}^{n - 1} (\log_e (1 + y_i) - \log_e (1 + \tilde{y}_i) )^2,
+\text{MSLE}(\bm{y}, \bm{\tilde{y}}) = \frac{1}{n} \sum_{i=0}^{n - 1} (\log_e (1 + y_i) - \log_e (1 + \tilde{y}_i) )^2,
 \]
 !et
 
@@ -2723,18 +2723,18 @@ o Write your own code (following the examples under the "regression notes":"http
 o Use thereafter _scikit-learn_ (see again the examples in the regression slides) and compare with your own code.   
 o Using scikit-learn, compute also the mean square error, a risk metric corresponding to the expected value of the squared (quadratic) error defined as
 !bt 
-\[ MSE(\hat{y},\hat{\tilde{y}}) = \frac{1}{n}
+\[ MSE(\bm{y},\bm{\tilde{y}}) = \frac{1}{n}
 \sum_{i=0}^{n-1}(y_i-\tilde{y}_i)^2, 
 \] 
 !et 
 and the $R^2$ score function.
-If $\tilde{\hat{y}}_i$ is the predicted value of the $i-th$ sample and $y_i$ is the corresponding true value, then the score $R^2$ is defined as
+If $\tilde{\bm{y}}_i$ is the predicted value of the $i-th$ sample and $y_i$ is the corresponding true value, then the score $R^2$ is defined as
 !bt
 \[
-R^2(\hat{y}, \tilde{\hat{y}}) = 1 - \frac{\sum_{i=0}^{n - 1} (y_i - \tilde{y}_i)^2}{\sum_{i=0}^{n - 1} (y_i - \bar{y})^2},
+R^2(\bm{y}, \tilde{\bm{y}}) = 1 - \frac{\sum_{i=0}^{n - 1} (y_i - \tilde{y}_i)^2}{\sum_{i=0}^{n - 1} (y_i - \bar{y})^2},
 \]
 !et
-where we have defined the mean value  of $\hat{y}$ as
+where we have defined the mean value  of $\bm{y}$ as
 !bt
 \[
 \bar{y} =  \frac{1}{n} \sum_{i=0}^{n - 1} y_i.