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removing hat symbols

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Morten Hjorth-Jensen
2021-08-27 07:06:06 +02:00
parent 96e3d2b41e
commit d35fa373a3
11 changed files with 132 additions and 132 deletions
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doc/pub/week34/html/._week34-bs000.html
+1 -1
View File
@@ -384,7 +384,7 @@ MathJax.Hub.Config({
<center>[2] <b>Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University</b></center>
<br>
<p>
<center><h4>Aug 26, 2021</h4></center> <!-- date -->
<center><h4>Aug 27, 2021</h4></center> <!-- date -->
<br>
<p>
doc/pub/week34/html/._week34-bs035.html
+7 -7
View File
@@ -369,7 +369,7 @@ MathJax.Hub.Config({
<p>
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 \( \boldsymbol{A} \) as (recall that we user
lowercase letters for vectors and uppercase letters for matrices)
<p>
@@ -435,9 +435,9 @@ A <span style="color: #666666">=</span> np<span style="color: #666666">.</span>r
<p>
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
\( \boldsymbol{x}, \boldsymbol{y}, \boldsymbol{z} \) with \( n \) elements each. The covariance matrix is defined as
$$
\hat{\Sigma} = \begin{bmatrix} \sigma_{xx} & \sigma_{xy} & \sigma_{xz} \\
\boldsymbol{\Sigma} = \begin{bmatrix} \sigma_{xx} & \sigma_{xy} & \sigma_{xz} \\
\sigma_{yx} & \sigma_{yy} & \sigma_{yz} \\
\sigma_{zx} & \sigma_{zy} & \sigma_{zz}
\end{bmatrix},
@@ -449,9 +449,9 @@ $$
$$
The Numpy function <b>np.cov</b> 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 <b>np.vstack</b> 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 <b>np.vstack</b> function which takes each vector of dimension \( 1\times n \) and produces a \( 3\times n \) matrix \( \boldsymbol{W} \)
$$
\hat{W} = \begin{bmatrix} x_0 & y_0 & z_0 \\
\boldsymbol{W} = \begin{bmatrix} x_0 & y_0 & z_0 \\
x_1 & y_1 & z_1 \\
x_2 & y_2 & z_2 \\
\dots & \dots & \dots \\
@@ -462,8 +462,8 @@ $$
<p>
which in turn is converted into into the \( 3\times 3 \) covariance matrix
\( \hat{\Sigma} \) via the Numpy function <b>np.cov()</b>. We note that we can also calculate
the mean value of each set of samples \( \hat{x} \) etc using the Numpy
\( \boldsymbol{\Sigma} \) via the Numpy function <b>np.cov()</b>. We note that we can also calculate
the mean value of each set of samples \( \boldsymbol{x} \) etc using the Numpy
function <b>np.mean(x)</b>. We can also extract the eigenvalues of the
covariance matrix through the <b>np.linalg.eig()</b> function.
doc/pub/week34/html/._week34-bs038.html
+10 -10
View File
@@ -373,7 +373,7 @@ We start with perhaps our simplest possible example, using <b>Scikit-Learn</b> t
<p>
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 \( \boldsymbol{x} \) 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 \( y(x) \)
@@ -400,7 +400,7 @@ where \( N(0,1) \) represents random numbers generated by the normal
distribution. From <b>Scikit-Learn</b> we import then the
<b>LinearRegression</b> functionality and make a prediction \( \tilde{y} =
\alpha + \beta x \) using the function <b>fit(x,y)</b>. We call the set of
data \( (\hat{x},\hat{y}) \) for our training data. The Python package
data \( (\boldsymbol{x},\boldsymbol{y}) \) for our training data. The Python package
<b>scikit-learn</b> 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.
@@ -499,7 +499,7 @@ There are many ways to define the cost function. A simpler approach is to look a
the relative error (why would we prefer the MSE instead of the relative error?) as
$$
\epsilon_{\mathrm{relative}}= \frac{\vert \hat{y} -\hat{\tilde{y}}\vert}{\vert \hat{y}\vert}.
\epsilon_{\mathrm{relative}}= \frac{\vert \boldsymbol{y} -\boldsymbol{\tilde{y}}\vert}{\vert \boldsymbol{y}\vert}.
$$
<p>
@@ -581,7 +581,7 @@ plt<span style="color: #666666">.</span>show()
<p>
The function <b>coef</b> gives us the parameter \( \beta \) of our fit while <b>intercept</b> 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 <b>meansquarederror</b> gives us the mean square error, a risk metric corresponding to the expected value of the squared (quadratic) error or loss defined as
$$ MSE(\hat{y},\hat{\tilde{y}}) = \frac{1}{n}
$$ MSE(\boldsymbol{y},\boldsymbol{\tilde{y}}) = \frac{1}{n}
\sum_{i=0}^{n-1}(y_i-\tilde{y}_i)^2,
$$
@@ -595,16 +595,16 @@ The <b>r2score</b> 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 \( \boldsymbol{y} \),
disregarding the input features, would get a \( R^2 \) score of \( 0.0 \).
<p>
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 \( \tilde{\boldsymbol{y}}_i \) 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
$$
R^2(\hat{y}, \tilde{\hat{y}}) = 1 - \frac{\sum_{i=0}^{n - 1} (y_i - \tilde{y}_i)^2}{\sum_{i=0}^{n - 1} (y_i - \bar{y})^2},
R^2(\boldsymbol{y}, \tilde{\boldsymbol{y}}) = 1 - \frac{\sum_{i=0}^{n - 1} (y_i - \tilde{y}_i)^2}{\sum_{i=0}^{n - 1} (y_i - \bar{y})^2},
$$
where we have defined the mean value of \( \hat{y} \) as
where we have defined the mean value of \( \boldsymbol{y} \) as
$$
\bar{y} = \frac{1}{n} \sum_{i=0}^{n - 1} y_i.
$$
@@ -613,13 +613,13 @@ Another quantity taht we will meet again in our discussions of regression analys
the mean absolute error (MAE), a risk metric corresponding to the expected value of the absolute error loss or what we call the \( l1 \)-norm loss. In our discussion above we presented the relative error.
The MAE is defined as follows
$$
\text{MAE}(\hat{y}, \hat{\tilde{y}}) = \frac{1}{n} \sum_{i=0}^{n-1} \left| y_i - \tilde{y}_i \right|.
\text{MAE}(\boldsymbol{y}, \boldsymbol{\tilde{y}}) = \frac{1}{n} \sum_{i=0}^{n-1} \left| y_i - \tilde{y}_i \right|.
$$
We present the
squared logarithmic (quadratic) error
$$
\text{MSLE}(\hat{y}, \hat{\tilde{y}}) = \frac{1}{n} \sum_{i=0}^{n - 1} (\log_e (1 + y_i) - \log_e (1 + \tilde{y}_i) )^2,
\text{MSLE}(\boldsymbol{y}, \boldsymbol{\tilde{y}}) = \frac{1}{n} \sum_{i=0}^{n - 1} (\log_e (1 + y_i) - \log_e (1 + \tilde{y}_i) )^2,
$$
<p>
doc/pub/week34/html/._week34-bs066.html
+4 -4
View File
@@ -473,17 +473,17 @@ y <span style="color: #666666">=</span> <span style="color: #666666">2.0+5*</spa
<li> Using scikit-learn, compute also the mean square error, a risk metric corresponding to the expected value of the squared (quadratic) error defined as</li>
</ol>
$$ MSE(\hat{y},\hat{\tilde{y}}) = \frac{1}{n}
$$ MSE(\boldsymbol{y},\boldsymbol{\tilde{y}}) = \frac{1}{n}
\sum_{i=0}^{n-1}(y_i-\tilde{y}_i)^2,
$$
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 \( \tilde{\boldsymbol{y}}_i \) 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
$$
R^2(\hat{y}, \tilde{\hat{y}}) = 1 - \frac{\sum_{i=0}^{n - 1} (y_i - \tilde{y}_i)^2}{\sum_{i=0}^{n - 1} (y_i - \bar{y})^2},
R^2(\boldsymbol{y}, \tilde{\boldsymbol{y}}) = 1 - \frac{\sum_{i=0}^{n - 1} (y_i - \tilde{y}_i)^2}{\sum_{i=0}^{n - 1} (y_i - \bar{y})^2},
$$
where we have defined the mean value of \( \hat{y} \) as
where we have defined the mean value of \( \boldsymbol{y} \) as
$$
\bar{y} = \frac{1}{n} \sum_{i=0}^{n - 1} y_i.
$$
doc/pub/week34/html/week34-bs.html
+1 -1
View File
@@ -384,7 +384,7 @@ MathJax.Hub.Config({
<center>[2] <b>Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University</b></center>
<br>
<p>
<center><h4>Aug 26, 2021</h4></center> <!-- date -->
<center><h4>Aug 27, 2021</h4></center> <!-- date -->
<br>
<p>
doc/pub/week34/html/week34-reveal.html
+22 -22
View File
@@ -148,7 +148,7 @@ MathJax.Hub.Config({
<center>[2] <b>Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University</b></center>
<br>
<p>&nbsp;<br>
<center><h4>Aug 26, 2021</h4></center> <!-- date -->
<center><h4>Aug 27, 2021</h4></center> <!-- date -->
<br>
<p>
@@ -1170,7 +1170,7 @@ x = np.log(np.array([<span style="color: #B452CD">4.0</span>, <span style="color
<p>
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 \( \boldsymbol{A} \) as (recall that we user
lowercase letters for vectors and uppercase letters for matrices)
<p>
@@ -1236,10 +1236,10 @@ A = np.random.rand(n, n)
<p>
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
\( \boldsymbol{x}, \boldsymbol{y}, \boldsymbol{z} \) with \( n \) elements each. The covariance matrix is defined as
<p>&nbsp;<br>
$$
\hat{\Sigma} = \begin{bmatrix} \sigma_{xx} & \sigma_{xy} & \sigma_{xz} \\
\boldsymbol{\Sigma} = \begin{bmatrix} \sigma_{xx} & \sigma_{xy} & \sigma_{xz} \\
\sigma_{yx} & \sigma_{yy} & \sigma_{yz} \\
\sigma_{zx} & \sigma_{zy} & \sigma_{zz}
\end{bmatrix},
@@ -1254,10 +1254,10 @@ $$
<p>&nbsp;<br>
The Numpy function <b>np.cov</b> 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 <b>np.vstack</b> 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 <b>np.vstack</b> function which takes each vector of dimension \( 1\times n \) and produces a \( 3\times n \) matrix \( \boldsymbol{W} \)
<p>&nbsp;<br>
$$
\hat{W} = \begin{bmatrix} x_0 & y_0 & z_0 \\
\boldsymbol{W} = \begin{bmatrix} x_0 & y_0 & z_0 \\
x_1 & y_1 & z_1 \\
x_2 & y_2 & z_2 \\
\dots & \dots & \dots \\
@@ -1269,8 +1269,8 @@ $$
<p>
which in turn is converted into into the \( 3\times 3 \) covariance matrix
\( \hat{\Sigma} \) via the Numpy function <b>np.cov()</b>. We note that we can also calculate
the mean value of each set of samples \( \hat{x} \) etc using the Numpy
\( \boldsymbol{\Sigma} \) via the Numpy function <b>np.cov()</b>. We note that we can also calculate
the mean value of each set of samples \( \boldsymbol{x} \) etc using the Numpy
function <b>np.mean(x)</b>. We can also extract the eigenvalues of the
covariance matrix through the <b>np.linalg.eig()</b> function.
@@ -1457,7 +1457,7 @@ We start with perhaps our simplest possible example, using <b>Scikit-Learn</b> t
<p>
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 \( \boldsymbol{x} \) 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 \( y(x) \)
@@ -1486,7 +1486,7 @@ where \( N(0,1) \) represents random numbers generated by the normal
distribution. From <b>Scikit-Learn</b> we import then the
<b>LinearRegression</b> functionality and make a prediction \( \tilde{y} =
\alpha + \beta x \) using the function <b>fit(x,y)</b>. We call the set of
data \( (\hat{x},\hat{y}) \) for our training data. The Python package
data \( (\boldsymbol{x},\boldsymbol{y}) \) for our training data. The Python package
<b>scikit-learn</b> 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.
@@ -1590,7 +1590,7 @@ the relative error (why would we prefer the MSE instead of the relative error?)
<p>&nbsp;<br>
$$
\epsilon_{\mathrm{relative}}= \frac{\vert \hat{y} -\hat{\tilde{y}}\vert}{\vert \hat{y}\vert}.
\epsilon_{\mathrm{relative}}= \frac{\vert \boldsymbol{y} -\boldsymbol{\tilde{y}}\vert}{\vert \boldsymbol{y}\vert}.
$$
<p>&nbsp;<br>
@@ -1674,7 +1674,7 @@ plt.show()
The function <b>coef</b> gives us the parameter \( \beta \) of our fit while <b>intercept</b> 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 <b>meansquarederror</b> gives us the mean square error, a risk metric corresponding to the expected value of the squared (quadratic) error or loss defined as
<p>&nbsp;<br>
$$ MSE(\hat{y},\hat{\tilde{y}}) = \frac{1}{n}
$$ MSE(\boldsymbol{y},\boldsymbol{\tilde{y}}) = \frac{1}{n}
\sum_{i=0}^{n-1}(y_i-\tilde{y}_i)^2,
$$
<p>&nbsp;<br>
@@ -1689,18 +1689,18 @@ The <b>r2score</b> 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 \( \boldsymbol{y} \),
disregarding the input features, would get a \( R^2 \) score of \( 0.0 \).
<p>
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 \( \tilde{\boldsymbol{y}}_i \) 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
<p>&nbsp;<br>
$$
R^2(\hat{y}, \tilde{\hat{y}}) = 1 - \frac{\sum_{i=0}^{n - 1} (y_i - \tilde{y}_i)^2}{\sum_{i=0}^{n - 1} (y_i - \bar{y})^2},
R^2(\boldsymbol{y}, \tilde{\boldsymbol{y}}) = 1 - \frac{\sum_{i=0}^{n - 1} (y_i - \tilde{y}_i)^2}{\sum_{i=0}^{n - 1} (y_i - \bar{y})^2},
$$
<p>&nbsp;<br>
where we have defined the mean value of \( \hat{y} \) as
where we have defined the mean value of \( \boldsymbol{y} \) as
<p>&nbsp;<br>
$$
\bar{y} = \frac{1}{n} \sum_{i=0}^{n - 1} y_i.
@@ -1712,7 +1712,7 @@ Another quantity taht we will meet again in our discussions of regression analys
The MAE is defined as follows
<p>&nbsp;<br>
$$
\text{MAE}(\hat{y}, \hat{\tilde{y}}) = \frac{1}{n} \sum_{i=0}^{n-1} \left| y_i - \tilde{y}_i \right|.
\text{MAE}(\boldsymbol{y}, \boldsymbol{\tilde{y}}) = \frac{1}{n} \sum_{i=0}^{n-1} \left| y_i - \tilde{y}_i \right|.
$$
<p>&nbsp;<br>
@@ -1720,7 +1720,7 @@ We present the
squared logarithmic (quadratic) error
<p>&nbsp;<br>
$$
\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}(\boldsymbol{y}, \boldsymbol{\tilde{y}}) = \frac{1}{n} \sum_{i=0}^{n - 1} (\log_e (1 + y_i) - \log_e (1 + \tilde{y}_i) )^2,
$$
<p>&nbsp;<br>
@@ -3406,20 +3406,20 @@ y = <span style="color: #B452CD">2.0</span>+<span style="color: #B452CD">5</span
<p><li> Using scikit-learn, compute also the mean square error, a risk metric corresponding to the expected value of the squared (quadratic) error defined as</li>
</ol>
<p>&nbsp;<br>
$$ MSE(\hat{y},\hat{\tilde{y}}) = \frac{1}{n}
$$ MSE(\boldsymbol{y},\boldsymbol{\tilde{y}}) = \frac{1}{n}
\sum_{i=0}^{n-1}(y_i-\tilde{y}_i)^2,
$$
<p>&nbsp;<br>
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 \( \tilde{\boldsymbol{y}}_i \) 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
<p>&nbsp;<br>
$$
R^2(\hat{y}, \tilde{\hat{y}}) = 1 - \frac{\sum_{i=0}^{n - 1} (y_i - \tilde{y}_i)^2}{\sum_{i=0}^{n - 1} (y_i - \bar{y})^2},
R^2(\boldsymbol{y}, \tilde{\boldsymbol{y}}) = 1 - \frac{\sum_{i=0}^{n - 1} (y_i - \tilde{y}_i)^2}{\sum_{i=0}^{n - 1} (y_i - \bar{y})^2},
$$
<p>&nbsp;<br>
where we have defined the mean value of \( \hat{y} \) as
where we have defined the mean value of \( \boldsymbol{y} \) as
<p>&nbsp;<br>
$$
\bar{y} = \frac{1}{n} \sum_{i=0}^{n - 1} y_i.
doc/pub/week34/html/week34-solarized.html
+22 -22
View File
@@ -300,7 +300,7 @@ MathJax.Hub.Config({
<center>[2] <b>Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University</b></center>
<br>
<p>
<center><h4>Aug 26, 2021</h4></center> <!-- date -->
<center><h4>Aug 27, 2021</h4></center> <!-- date -->
<br>
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
@@ -1280,7 +1280,7 @@ x = np.log(np.array([<span style="color: #B452CD">4.0</span>, <span style="color
<p>
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 \( \boldsymbol{A} \) as (recall that we user
lowercase letters for vectors and uppercase letters for matrices)
<p>
@@ -1346,9 +1346,9 @@ A = np.random.rand(n, n)
<p>
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
\( \boldsymbol{x}, \boldsymbol{y}, \boldsymbol{z} \) with \( n \) elements each. The covariance matrix is defined as
$$
\hat{\Sigma} = \begin{bmatrix} \sigma_{xx} & \sigma_{xy} & \sigma_{xz} \\
\boldsymbol{\Sigma} = \begin{bmatrix} \sigma_{xx} & \sigma_{xy} & \sigma_{xz} \\
\sigma_{yx} & \sigma_{yy} & \sigma_{yz} \\
\sigma_{zx} & \sigma_{zy} & \sigma_{zz}
\end{bmatrix},
@@ -1360,9 +1360,9 @@ $$
$$
The Numpy function <b>np.cov</b> 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 <b>np.vstack</b> 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 <b>np.vstack</b> function which takes each vector of dimension \( 1\times n \) and produces a \( 3\times n \) matrix \( \boldsymbol{W} \)
$$
\hat{W} = \begin{bmatrix} x_0 & y_0 & z_0 \\
\boldsymbol{W} = \begin{bmatrix} x_0 & y_0 & z_0 \\
x_1 & y_1 & z_1 \\
x_2 & y_2 & z_2 \\
\dots & \dots & \dots \\
@@ -1373,8 +1373,8 @@ $$
<p>
which in turn is converted into into the \( 3\times 3 \) covariance matrix
\( \hat{\Sigma} \) via the Numpy function <b>np.cov()</b>. We note that we can also calculate
the mean value of each set of samples \( \hat{x} \) etc using the Numpy
\( \boldsymbol{\Sigma} \) via the Numpy function <b>np.cov()</b>. We note that we can also calculate
the mean value of each set of samples \( \boldsymbol{x} \) etc using the Numpy
function <b>np.mean(x)</b>. We can also extract the eigenvalues of the
covariance matrix through the <b>np.linalg.eig()</b> function.
@@ -1559,7 +1559,7 @@ We start with perhaps our simplest possible example, using <b>Scikit-Learn</b> t
<p>
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 \( \boldsymbol{x} \) 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 \( y(x) \)
@@ -1586,7 +1586,7 @@ where \( N(0,1) \) represents random numbers generated by the normal
distribution. From <b>Scikit-Learn</b> we import then the
<b>LinearRegression</b> functionality and make a prediction \( \tilde{y} =
\alpha + \beta x \) using the function <b>fit(x,y)</b>. We call the set of
data \( (\hat{x},\hat{y}) \) for our training data. The Python package
data \( (\boldsymbol{x},\boldsymbol{y}) \) for our training data. The Python package
<b>scikit-learn</b> 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.
@@ -1685,7 +1685,7 @@ There are many ways to define the cost function. A simpler approach is to look a
the relative error (why would we prefer the MSE instead of the relative error?) as
$$
\epsilon_{\mathrm{relative}}= \frac{\vert \hat{y} -\hat{\tilde{y}}\vert}{\vert \hat{y}\vert}.
\epsilon_{\mathrm{relative}}= \frac{\vert \boldsymbol{y} -\boldsymbol{\tilde{y}}\vert}{\vert \boldsymbol{y}\vert}.
$$
<p>
@@ -1767,7 +1767,7 @@ plt.show()
<p>
The function <b>coef</b> gives us the parameter \( \beta \) of our fit while <b>intercept</b> 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 <b>meansquarederror</b> gives us the mean square error, a risk metric corresponding to the expected value of the squared (quadratic) error or loss defined as
$$ MSE(\hat{y},\hat{\tilde{y}}) = \frac{1}{n}
$$ MSE(\boldsymbol{y},\boldsymbol{\tilde{y}}) = \frac{1}{n}
\sum_{i=0}^{n-1}(y_i-\tilde{y}_i)^2,
$$
@@ -1781,16 +1781,16 @@ The <b>r2score</b> 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 \( \boldsymbol{y} \),
disregarding the input features, would get a \( R^2 \) score of \( 0.0 \).
<p>
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 \( \tilde{\boldsymbol{y}}_i \) 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
$$
R^2(\hat{y}, \tilde{\hat{y}}) = 1 - \frac{\sum_{i=0}^{n - 1} (y_i - \tilde{y}_i)^2}{\sum_{i=0}^{n - 1} (y_i - \bar{y})^2},
R^2(\boldsymbol{y}, \tilde{\boldsymbol{y}}) = 1 - \frac{\sum_{i=0}^{n - 1} (y_i - \tilde{y}_i)^2}{\sum_{i=0}^{n - 1} (y_i - \bar{y})^2},
$$
where we have defined the mean value of \( \hat{y} \) as
where we have defined the mean value of \( \boldsymbol{y} \) as
$$
\bar{y} = \frac{1}{n} \sum_{i=0}^{n - 1} y_i.
$$
@@ -1799,13 +1799,13 @@ Another quantity taht we will meet again in our discussions of regression analys
the mean absolute error (MAE), a risk metric corresponding to the expected value of the absolute error loss or what we call the \( l1 \)-norm loss. In our discussion above we presented the relative error.
The MAE is defined as follows
$$
\text{MAE}(\hat{y}, \hat{\tilde{y}}) = \frac{1}{n} \sum_{i=0}^{n-1} \left| y_i - \tilde{y}_i \right|.
\text{MAE}(\boldsymbol{y}, \boldsymbol{\tilde{y}}) = \frac{1}{n} \sum_{i=0}^{n-1} \left| y_i - \tilde{y}_i \right|.
$$
We present the
squared logarithmic (quadratic) error
$$
\text{MSLE}(\hat{y}, \hat{\tilde{y}}) = \frac{1}{n} \sum_{i=0}^{n - 1} (\log_e (1 + y_i) - \log_e (1 + \tilde{y}_i) )^2,
\text{MSLE}(\boldsymbol{y}, \boldsymbol{\tilde{y}}) = \frac{1}{n} \sum_{i=0}^{n - 1} (\log_e (1 + y_i) - \log_e (1 + \tilde{y}_i) )^2,
$$
<p>
@@ -3394,17 +3394,17 @@ y = <span style="color: #B452CD">2.0</span>+<span style="color: #B452CD">5</span
<li> Using scikit-learn, compute also the mean square error, a risk metric corresponding to the expected value of the squared (quadratic) error defined as</li>
</ol>
$$ MSE(\hat{y},\hat{\tilde{y}}) = \frac{1}{n}
$$ MSE(\boldsymbol{y},\boldsymbol{\tilde{y}}) = \frac{1}{n}
\sum_{i=0}^{n-1}(y_i-\tilde{y}_i)^2,
$$
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 \( \tilde{\boldsymbol{y}}_i \) 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
$$
R^2(\hat{y}, \tilde{\hat{y}}) = 1 - \frac{\sum_{i=0}^{n - 1} (y_i - \tilde{y}_i)^2}{\sum_{i=0}^{n - 1} (y_i - \bar{y})^2},
R^2(\boldsymbol{y}, \tilde{\boldsymbol{y}}) = 1 - \frac{\sum_{i=0}^{n - 1} (y_i - \tilde{y}_i)^2}{\sum_{i=0}^{n - 1} (y_i - \bar{y})^2},
$$
where we have defined the mean value of \( \hat{y} \) as
where we have defined the mean value of \( \boldsymbol{y} \) as
$$
\bar{y} = \frac{1}{n} \sum_{i=0}^{n - 1} y_i.
$$
doc/pub/week34/html/week34.html
+22 -22
View File
@@ -305,7 +305,7 @@ MathJax.Hub.Config({
<center>[2] <b>Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University</b></center>
<br>
<p>
<center><h4>Aug 26, 2021</h4></center> <!-- date -->
<center><h4>Aug 27, 2021</h4></center> <!-- date -->
<br>
<p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
@@ -1285,7 +1285,7 @@ x <span style="color: #666666">=</span> np<span style="color: #666666">.</span>l
<p>
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 \( \boldsymbol{A} \) as (recall that we user
lowercase letters for vectors and uppercase letters for matrices)
<p>
@@ -1351,9 +1351,9 @@ A <span style="color: #666666">=</span> np<span style="color: #666666">.</span>r
<p>
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
\( \boldsymbol{x}, \boldsymbol{y}, \boldsymbol{z} \) with \( n \) elements each. The covariance matrix is defined as
$$
\hat{\Sigma} = \begin{bmatrix} \sigma_{xx} & \sigma_{xy} & \sigma_{xz} \\
\boldsymbol{\Sigma} = \begin{bmatrix} \sigma_{xx} & \sigma_{xy} & \sigma_{xz} \\
\sigma_{yx} & \sigma_{yy} & \sigma_{yz} \\
\sigma_{zx} & \sigma_{zy} & \sigma_{zz}
\end{bmatrix},
@@ -1365,9 +1365,9 @@ $$
$$
The Numpy function <b>np.cov</b> 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 <b>np.vstack</b> 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 <b>np.vstack</b> function which takes each vector of dimension \( 1\times n \) and produces a \( 3\times n \) matrix \( \boldsymbol{W} \)
$$
\hat{W} = \begin{bmatrix} x_0 & y_0 & z_0 \\
\boldsymbol{W} = \begin{bmatrix} x_0 & y_0 & z_0 \\
x_1 & y_1 & z_1 \\
x_2 & y_2 & z_2 \\
\dots & \dots & \dots \\
@@ -1378,8 +1378,8 @@ $$
<p>
which in turn is converted into into the \( 3\times 3 \) covariance matrix
\( \hat{\Sigma} \) via the Numpy function <b>np.cov()</b>. We note that we can also calculate
the mean value of each set of samples \( \hat{x} \) etc using the Numpy
\( \boldsymbol{\Sigma} \) via the Numpy function <b>np.cov()</b>. We note that we can also calculate
the mean value of each set of samples \( \boldsymbol{x} \) etc using the Numpy
function <b>np.mean(x)</b>. We can also extract the eigenvalues of the
covariance matrix through the <b>np.linalg.eig()</b> function.
@@ -1564,7 +1564,7 @@ We start with perhaps our simplest possible example, using <b>Scikit-Learn</b> t
<p>
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 \( \boldsymbol{x} \) 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 \( y(x) \)
@@ -1591,7 +1591,7 @@ where \( N(0,1) \) represents random numbers generated by the normal
distribution. From <b>Scikit-Learn</b> we import then the
<b>LinearRegression</b> functionality and make a prediction \( \tilde{y} =
\alpha + \beta x \) using the function <b>fit(x,y)</b>. We call the set of
data \( (\hat{x},\hat{y}) \) for our training data. The Python package
data \( (\boldsymbol{x},\boldsymbol{y}) \) for our training data. The Python package
<b>scikit-learn</b> 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.
@@ -1690,7 +1690,7 @@ There are many ways to define the cost function. A simpler approach is to look a
the relative error (why would we prefer the MSE instead of the relative error?) as
$$
\epsilon_{\mathrm{relative}}= \frac{\vert \hat{y} -\hat{\tilde{y}}\vert}{\vert \hat{y}\vert}.
\epsilon_{\mathrm{relative}}= \frac{\vert \boldsymbol{y} -\boldsymbol{\tilde{y}}\vert}{\vert \boldsymbol{y}\vert}.
$$
<p>
@@ -1772,7 +1772,7 @@ plt<span style="color: #666666">.</span>show()
<p>
The function <b>coef</b> gives us the parameter \( \beta \) of our fit while <b>intercept</b> 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 <b>meansquarederror</b> gives us the mean square error, a risk metric corresponding to the expected value of the squared (quadratic) error or loss defined as
$$ MSE(\hat{y},\hat{\tilde{y}}) = \frac{1}{n}
$$ MSE(\boldsymbol{y},\boldsymbol{\tilde{y}}) = \frac{1}{n}
\sum_{i=0}^{n-1}(y_i-\tilde{y}_i)^2,
$$
@@ -1786,16 +1786,16 @@ The <b>r2score</b> 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 \( \boldsymbol{y} \),
disregarding the input features, would get a \( R^2 \) score of \( 0.0 \).
<p>
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 \( \tilde{\boldsymbol{y}}_i \) 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
$$
R^2(\hat{y}, \tilde{\hat{y}}) = 1 - \frac{\sum_{i=0}^{n - 1} (y_i - \tilde{y}_i)^2}{\sum_{i=0}^{n - 1} (y_i - \bar{y})^2},
R^2(\boldsymbol{y}, \tilde{\boldsymbol{y}}) = 1 - \frac{\sum_{i=0}^{n - 1} (y_i - \tilde{y}_i)^2}{\sum_{i=0}^{n - 1} (y_i - \bar{y})^2},
$$
where we have defined the mean value of \( \hat{y} \) as
where we have defined the mean value of \( \boldsymbol{y} \) as
$$
\bar{y} = \frac{1}{n} \sum_{i=0}^{n - 1} y_i.
$$
@@ -1804,13 +1804,13 @@ Another quantity taht we will meet again in our discussions of regression analys
the mean absolute error (MAE), a risk metric corresponding to the expected value of the absolute error loss or what we call the \( l1 \)-norm loss. In our discussion above we presented the relative error.
The MAE is defined as follows
$$
\text{MAE}(\hat{y}, \hat{\tilde{y}}) = \frac{1}{n} \sum_{i=0}^{n-1} \left| y_i - \tilde{y}_i \right|.
\text{MAE}(\boldsymbol{y}, \boldsymbol{\tilde{y}}) = \frac{1}{n} \sum_{i=0}^{n-1} \left| y_i - \tilde{y}_i \right|.
$$
We present the
squared logarithmic (quadratic) error
$$
\text{MSLE}(\hat{y}, \hat{\tilde{y}}) = \frac{1}{n} \sum_{i=0}^{n - 1} (\log_e (1 + y_i) - \log_e (1 + \tilde{y}_i) )^2,
\text{MSLE}(\boldsymbol{y}, \boldsymbol{\tilde{y}}) = \frac{1}{n} \sum_{i=0}^{n - 1} (\log_e (1 + y_i) - \log_e (1 + \tilde{y}_i) )^2,
$$
<p>
@@ -3399,17 +3399,17 @@ y <span style="color: #666666">=</span> <span style="color: #666666">2.0+5*</spa
<li> Using scikit-learn, compute also the mean square error, a risk metric corresponding to the expected value of the squared (quadratic) error defined as</li>
</ol>
$$ MSE(\hat{y},\hat{\tilde{y}}) = \frac{1}{n}
$$ MSE(\boldsymbol{y},\boldsymbol{\tilde{y}}) = \frac{1}{n}
\sum_{i=0}^{n-1}(y_i-\tilde{y}_i)^2,
$$
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 \( \tilde{\boldsymbol{y}}_i \) 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
$$
R^2(\hat{y}, \tilde{\hat{y}}) = 1 - \frac{\sum_{i=0}^{n - 1} (y_i - \tilde{y}_i)^2}{\sum_{i=0}^{n - 1} (y_i - \bar{y})^2},
R^2(\boldsymbol{y}, \tilde{\boldsymbol{y}}) = 1 - \frac{\sum_{i=0}^{n - 1} (y_i - \tilde{y}_i)^2}{\sum_{i=0}^{n - 1} (y_i - \bar{y})^2},
$$
where we have defined the mean value of \( \hat{y} \) as
where we have defined the mean value of \( \boldsymbol{y} \) as
$$
\bar{y} = \frac{1}{n} \sum_{i=0}^{n - 1} y_i.
$$
doc/pub/week34/ipynb/ipynb-week34-src.tar.gz
BIN
View File
Binary file not shown.
doc/pub/week34/ipynb/week34.ipynb
+22 -22
View File
@@ -10,7 +10,7 @@
"<!-- Author: --> \n",
"**Morten Hjorth-Jensen**, Department of Physics, University of Oslo and Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University\n",
"\n",
"Date: **Aug 26, 2021**\n",
"Date: **Aug 27, 2021**\n",
"\n",
"Copyright 1999-2021, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license\n",
"\n",
@@ -994,7 +994,7 @@
"## Matrices in Python\n",
"\n",
"Having defined vectors, we are now ready to try out matrices. We can\n",
"define a $3 \\times 3 $ real matrix $\\hat{A}$ as (recall that we user\n",
"define a $3 \\times 3 $ real matrix $\\boldsymbol{A}$ as (recall that we user\n",
"lowercase letters for vectors and uppercase letters for matrices)"
]
},
@@ -1131,7 +1131,7 @@
"source": [
"As we will see throughout these lectures, there are several extremely useful functionalities in Numpy.\n",
"As an example, consider the discussion of the covariance matrix. Suppose we have defined three vectors\n",
"$\\hat{x}, \\hat{y}, \\hat{z}$ with $n$ elements each. The covariance matrix is defined as"
"$\\boldsymbol{x}, \\boldsymbol{y}, \\boldsymbol{z}$ with $n$ elements each. The covariance matrix is defined as"
]
},
{
@@ -1139,7 +1139,7 @@
"metadata": {},
"source": [
"$$\n",
"\\hat{\\Sigma} = \\begin{bmatrix} \\sigma_{xx} & \\sigma_{xy} & \\sigma_{xz} \\\\\n",
"\\boldsymbol{\\Sigma} = \\begin{bmatrix} \\sigma_{xx} & \\sigma_{xy} & \\sigma_{xz} \\\\\n",
" \\sigma_{yx} & \\sigma_{yy} & \\sigma_{yz} \\\\\n",
" \\sigma_{zx} & \\sigma_{zy} & \\sigma_{zz} \n",
" \\end{bmatrix},\n",
@@ -1167,7 +1167,7 @@
"metadata": {},
"source": [
"The Numpy function **np.cov** calculates the covariance elements using the factor $1/(n-1)$ instead of $1/n$ since it assumes we do not have the exact mean values. \n",
"The following simple function uses the **np.vstack** function which takes each vector of dimension $1\\times n$ and produces a $3\\times n$ matrix $\\hat{W}$"
"The following simple function uses the **np.vstack** function which takes each vector of dimension $1\\times n$ and produces a $3\\times n$ matrix $\\boldsymbol{W}$"
]
},
{
@@ -1175,7 +1175,7 @@
"metadata": {},
"source": [
"$$\n",
"\\hat{W} = \\begin{bmatrix} x_0 & y_0 & z_0 \\\\\n",
"\\boldsymbol{W} = \\begin{bmatrix} x_0 & y_0 & z_0 \\\\\n",
" x_1 & y_1 & z_1 \\\\\n",
" x_2 & y_2 & z_2 \\\\\n",
" \\dots & \\dots & \\dots \\\\\n",
@@ -1190,8 +1190,8 @@
"metadata": {},
"source": [
"which in turn is converted into into the $3\\times 3$ covariance matrix\n",
"$\\hat{\\Sigma}$ via the Numpy function **np.cov()**. We note that we can also calculate\n",
"the mean value of each set of samples $\\hat{x}$ etc using the Numpy\n",
"$\\boldsymbol{\\Sigma}$ via the Numpy function **np.cov()**. We note that we can also calculate\n",
"the mean value of each set of samples $\\boldsymbol{x}$ etc using the Numpy\n",
"function **np.mean(x)**. We can also extract the eigenvalues of the\n",
"covariance matrix through the **np.linalg.eig()** function."
]
@@ -1472,7 +1472,7 @@
"\n",
"What follows is a simple Python code where we have defined a function\n",
"$y$ in terms of the variable $x$. Both are defined as vectors with $100$ entries. \n",
"The numbers in the vector $\\hat{x}$ are given\n",
"The numbers in the vector $\\boldsymbol{x}$ are given\n",
"by random numbers generated with a uniform distribution with entries\n",
"$x_i \\in [0,1]$ (more about probability distribution functions\n",
"later). These values are then used to define a function $y(x)$\n",
@@ -1508,7 +1508,7 @@
"distribution. From **Scikit-Learn** we import then the\n",
"**LinearRegression** functionality and make a prediction $\\tilde{y} =\n",
"\\alpha + \\beta x$ using the function **fit(x,y)**. We call the set of\n",
"data $(\\hat{x},\\hat{y})$ for our training data. The Python package\n",
"data $(\\boldsymbol{x},\\boldsymbol{y})$ for our training data. The Python package\n",
"**scikit-learn** has also a functionality which extracts the above\n",
"fitting parameters $\\alpha$ and $\\beta$ (see below). Later we will\n",
"distinguish between training data and test data.\n",
@@ -1638,7 +1638,7 @@
"metadata": {},
"source": [
"$$\n",
"\\epsilon_{\\mathrm{relative}}= \\frac{\\vert \\hat{y} -\\hat{\\tilde{y}}\\vert}{\\vert \\hat{y}\\vert}.\n",
"\\epsilon_{\\mathrm{relative}}= \\frac{\\vert \\boldsymbol{y} -\\boldsymbol{\\tilde{y}}\\vert}{\\vert \\boldsymbol{y}\\vert}.\n",
"$$"
]
},
@@ -1752,7 +1752,7 @@
"metadata": {},
"source": [
"$$\n",
"MSE(\\hat{y},\\hat{\\tilde{y}}) = \\frac{1}{n}\n",
"MSE(\\boldsymbol{y},\\boldsymbol{\\tilde{y}}) = \\frac{1}{n}\n",
"\\sum_{i=0}^{n-1}(y_i-\\tilde{y}_i)^2,\n",
"$$"
]
@@ -1769,10 +1769,10 @@
"determination. It provides a measure of how well future samples are\n",
"likely to be predicted by the model. Best possible score is 1.0 and it\n",
"can be negative (because the model can be arbitrarily worse). A\n",
"constant model that always predicts the expected value of $\\hat{y}$,\n",
"constant model that always predicts the expected value of $\\boldsymbol{y}$,\n",
"disregarding the input features, would get a $R^2$ score of $0.0$.\n",
"\n",
"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 $\\tilde{\\boldsymbol{y}}_i$ 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"
]
},
{
@@ -1780,7 +1780,7 @@
"metadata": {},
"source": [
"$$\n",
"R^2(\\hat{y}, \\tilde{\\hat{y}}) = 1 - \\frac{\\sum_{i=0}^{n - 1} (y_i - \\tilde{y}_i)^2}{\\sum_{i=0}^{n - 1} (y_i - \\bar{y})^2},\n",
"R^2(\\boldsymbol{y}, \\tilde{\\boldsymbol{y}}) = 1 - \\frac{\\sum_{i=0}^{n - 1} (y_i - \\tilde{y}_i)^2}{\\sum_{i=0}^{n - 1} (y_i - \\bar{y})^2},\n",
"$$"
]
},
@@ -1788,7 +1788,7 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"where we have defined the mean value of $\\hat{y}$ as"
"where we have defined the mean value of $\\boldsymbol{y}$ as"
]
},
{
@@ -1814,7 +1814,7 @@
"metadata": {},
"source": [
"$$\n",
"\\text{MAE}(\\hat{y}, \\hat{\\tilde{y}}) = \\frac{1}{n} \\sum_{i=0}^{n-1} \\left| y_i - \\tilde{y}_i \\right|.\n",
"\\text{MAE}(\\boldsymbol{y}, \\boldsymbol{\\tilde{y}}) = \\frac{1}{n} \\sum_{i=0}^{n-1} \\left| y_i - \\tilde{y}_i \\right|.\n",
"$$"
]
},
@@ -1831,7 +1831,7 @@
"metadata": {},
"source": [
"$$\n",
"\\text{MSLE}(\\hat{y}, \\hat{\\tilde{y}}) = \\frac{1}{n} \\sum_{i=0}^{n - 1} (\\log_e (1 + y_i) - \\log_e (1 + \\tilde{y}_i) )^2,\n",
"\\text{MSLE}(\\boldsymbol{y}, \\boldsymbol{\\tilde{y}}) = \\frac{1}{n} \\sum_{i=0}^{n - 1} (\\log_e (1 + y_i) - \\log_e (1 + \\tilde{y}_i) )^2,\n",
"$$"
]
},
@@ -4156,7 +4156,7 @@
"metadata": {},
"source": [
"$$\n",
"MSE(\\hat{y},\\hat{\\tilde{y}}) = \\frac{1}{n}\n",
"MSE(\\boldsymbol{y},\\boldsymbol{\\tilde{y}}) = \\frac{1}{n}\n",
"\\sum_{i=0}^{n-1}(y_i-\\tilde{y}_i)^2,\n",
"$$"
]
@@ -4166,7 +4166,7 @@
"metadata": {},
"source": [
"and the $R^2$ score function.\n",
"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 $\\tilde{\\boldsymbol{y}}_i$ 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"
]
},
{
@@ -4174,7 +4174,7 @@
"metadata": {},
"source": [
"$$\n",
"R^2(\\hat{y}, \\tilde{\\hat{y}}) = 1 - \\frac{\\sum_{i=0}^{n - 1} (y_i - \\tilde{y}_i)^2}{\\sum_{i=0}^{n - 1} (y_i - \\bar{y})^2},\n",
"R^2(\\boldsymbol{y}, \\tilde{\\boldsymbol{y}}) = 1 - \\frac{\\sum_{i=0}^{n - 1} (y_i - \\tilde{y}_i)^2}{\\sum_{i=0}^{n - 1} (y_i - \\bar{y})^2},\n",
"$$"
]
},
@@ -4182,7 +4182,7 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"where we have defined the mean value of $\\hat{y}$ as"
"where we have defined the mean value of $\\boldsymbol{y}$ as"
]
},
{
doc/src/week34/week34.do.txt
+21 -21
View File
@@ -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.