lars/FYS-STK4155
1
0
Fork 0
Code Issues Pull Requests Actions Packages Projects Releases Wiki Activity

update week37, typos

Browse Source
This commit is contained in:
Morten Hjorth-Jensen
2021-09-17 11:55:22 +02:00
parent 5371b6a4ce
commit 2b525afb6a
10 changed files with 42 additions and 42 deletions
Download Patch File Download Diff File Expand all files Collapse all files
doc/pub/week37/html/._week37-bs024.html
+3 -3
View File
@@ -287,12 +287,12 @@ MathJax.Hub.Config({
<p>
Suppose we have a PDF \( p(x) \) from which we generate a series \( N \)
of averages \( \langle x_i \rangle \). Each mean value \( \langle x_i \rangle \)
of averages \( \mathbb{E}[x_i] \). Each mean value \( \mathbb{E}[x_i] \)
is viewed as the average of a specific measurement, e.g., throwing
dice 100 times and then taking the average value, or producing a certain
amount of random numbers.
For notational ease, we set \( \langle x_i \rangle=x_i \) in the discussion
which follows.
For notational ease, we set \( \mathbb{E}[x_i]=x_i \) in the discussion
which follows. We do the same for \( \mathbb{E}[z]=z \).
<p>
If we compute the mean \( z \) of \( m \) such mean values \( x_i \)
doc/pub/week37/html/._week37-bs028.html
+1 -1
View File
@@ -304,7 +304,7 @@ $$
<p>
The latter is true only if the average value is known exactly. This is obtained in the limit
\( m\rightarrow \infty \) only. Because the mean and the variance are measured quantities we obtain
the familiar expression in statistics
the familiar expression in statistics (the so-called Bessel correction)
$$
\sigma_m\approx
\frac{\sigma}{\sqrt{m-1}}.
doc/pub/week37/html/._week37-bs037.html
+2 -2
View File
@@ -289,8 +289,8 @@ MathJax.Hub.Config({
We will discuss the bias-variance tradeoff in the context of
continuous predictions such as regression. However, many of the
intuitions and ideas discussed here also carry over to classification
tasks. Consider a dataset \( \mathcal{L} \) consisting of the data
\( \mathbf{X}_\mathcal{L}=\{(y_j, \boldsymbol{x}_j), j=0\ldots n-1\} \).
tasks. Consider a dataset \( \mathcal{D} \) consisting of the data
\( \mathbf{X}_\mathcal{D}=\{(y_j, \boldsymbol{x}_j), j=0\ldots n-1\} \).
<p>
Let us assume that the true data is generated from a noisy model
doc/pub/week37/html/._week37-bs044.html
+1 -1
View File
@@ -301,7 +301,7 @@ $$
<ul>
<li> Evaluate the prediction performance of these models on the test set by \( \log\{L[y_i, \boldsymbol{X}_{i, \ast}; \boldsymbol{\beta}_{-i}(\lambda), \boldsymbol{\sigma}_{-i}^2(\lambda)]\} \). Or, by the prediction error \( |y_i - \boldsymbol{X}_{i, \ast} \boldsymbol{\beta}_{-i}(\lambda)| \), the relative error, the error squared or the R2 score function.</li>
<li> Evaluate the prediction performance of these models on the test set by \( [y_i, \boldsymbol{X}_{i, \ast}; \boldsymbol{\beta}_{-i}(\lambda), \boldsymbol{\sigma}_{-i}^2(\lambda)] \). Or, by the prediction error \( |y_i - \boldsymbol{X}_{i, \ast} \boldsymbol{\beta}_{-i}(\lambda)| \), the relative error, the error squared or the R2 score function.</li>
<li> Repeat the first three steps such that each sample plays the role of the test set once.</li>
<li> Average the prediction performances of the test sets at each grid point of the penalty bias/parameter. It is an estimate of the prediction performance of the model corresponding to this value of the penalty parameter on novel data. It is defined as</li>
</ul>
doc/pub/week37/html/week37-reveal.html
+7 -7
View File
@@ -906,12 +906,12 @@ Before we proceed however, we need to remind ourselves about a central theorem i
<p>
Suppose we have a PDF \( p(x) \) from which we generate a series \( N \)
of averages \( \langle x_i \rangle \). Each mean value \( \langle x_i \rangle \)
of averages \( \mathbb{E}[x_i] \). Each mean value \( \mathbb{E}[x_i] \)
is viewed as the average of a specific measurement, e.g., throwing
dice 100 times and then taking the average value, or producing a certain
amount of random numbers.
For notational ease, we set \( \langle x_i \rangle=x_i \) in the discussion
which follows.
For notational ease, we set \( \mathbb{E}[x_i]=x_i \) in the discussion
which follows. We do the same for \( \mathbb{E}[z]=z \).
<p>
If we compute the mean \( z \) of \( m \) such mean values \( x_i \)
@@ -1044,7 +1044,7 @@ $$
<p>
The latter is true only if the average value is known exactly. This is obtained in the limit
\( m\rightarrow \infty \) only. Because the mean and the variance are measured quantities we obtain
the familiar expression in statistics
the familiar expression in statistics (the so-called Bessel correction)
<p>&nbsp;<br>
$$
\sigma_m\approx
@@ -1282,8 +1282,8 @@ plt.show()
We will discuss the bias-variance tradeoff in the context of
continuous predictions such as regression. However, many of the
intuitions and ideas discussed here also carry over to classification
tasks. Consider a dataset \( \mathcal{L} \) consisting of the data
\( \mathbf{X}_\mathcal{L}=\{(y_j, \boldsymbol{x}_j), j=0\ldots n-1\} \).
tasks. Consider a dataset \( \mathcal{D} \) consisting of the data
\( \mathbf{X}_\mathcal{D}=\{(y_j, \boldsymbol{x}_j), j=0\ldots n-1\} \).
<p>
Let us assume that the true data is generated from a noisy model
@@ -1637,7 +1637,7 @@ $$
<ul>
<p><li> Evaluate the prediction performance of these models on the test set by \( \log\{L[y_i, \boldsymbol{X}_{i, \ast}; \boldsymbol{\beta}_{-i}(\lambda), \boldsymbol{\sigma}_{-i}^2(\lambda)]\} \). Or, by the prediction error \( |y_i - \boldsymbol{X}_{i, \ast} \boldsymbol{\beta}_{-i}(\lambda)| \), the relative error, the error squared or the R2 score function.</li>
<p><li> Evaluate the prediction performance of these models on the test set by \( [y_i, \boldsymbol{X}_{i, \ast}; \boldsymbol{\beta}_{-i}(\lambda), \boldsymbol{\sigma}_{-i}^2(\lambda)] \). Or, by the prediction error \( |y_i - \boldsymbol{X}_{i, \ast} \boldsymbol{\beta}_{-i}(\lambda)| \), the relative error, the error squared or the R2 score function.</li>
<p><li> Repeat the first three steps such that each sample plays the role of the test set once.</li>
<p><li> Average the prediction performances of the test sets at each grid point of the penalty bias/parameter. It is an estimate of the prediction performance of the model corresponding to this value of the penalty parameter on novel data. It is defined as</li>
</ul>
doc/pub/week37/html/week37-solarized.html
+7 -7
View File
@@ -949,12 +949,12 @@ Before we proceed however, we need to remind ourselves about a central theorem i
<p>
Suppose we have a PDF \( p(x) \) from which we generate a series \( N \)
of averages \( \langle x_i \rangle \). Each mean value \( \langle x_i \rangle \)
of averages \( \mathbb{E}[x_i] \). Each mean value \( \mathbb{E}[x_i] \)
is viewed as the average of a specific measurement, e.g., throwing
dice 100 times and then taking the average value, or producing a certain
amount of random numbers.
For notational ease, we set \( \langle x_i \rangle=x_i \) in the discussion
which follows.
For notational ease, we set \( \mathbb{E}[x_i]=x_i \) in the discussion
which follows. We do the same for \( \mathbb{E}[z]=z \).
<p>
If we compute the mean \( z \) of \( m \) such mean values \( x_i \)
@@ -1068,7 +1068,7 @@ $$
<p>
The latter is true only if the average value is known exactly. This is obtained in the limit
\( m\rightarrow \infty \) only. Because the mean and the variance are measured quantities we obtain
the familiar expression in statistics
the familiar expression in statistics (the so-called Bessel correction)
$$
\sigma_m\approx
\frac{\sigma}{\sqrt{m-1}}.
@@ -1299,8 +1299,8 @@ plt.show()
We will discuss the bias-variance tradeoff in the context of
continuous predictions such as regression. However, many of the
intuitions and ideas discussed here also carry over to classification
tasks. Consider a dataset \( \mathcal{L} \) consisting of the data
\( \mathbf{X}_\mathcal{L}=\{(y_j, \boldsymbol{x}_j), j=0\ldots n-1\} \).
tasks. Consider a dataset \( \mathcal{D} \) consisting of the data
\( \mathbf{X}_\mathcal{D}=\{(y_j, \boldsymbol{x}_j), j=0\ldots n-1\} \).
<p>
Let us assume that the true data is generated from a noisy model
@@ -1638,7 +1638,7 @@ $$
<ul>
<li> Evaluate the prediction performance of these models on the test set by \( \log\{L[y_i, \boldsymbol{X}_{i, \ast}; \boldsymbol{\beta}_{-i}(\lambda), \boldsymbol{\sigma}_{-i}^2(\lambda)]\} \). Or, by the prediction error \( |y_i - \boldsymbol{X}_{i, \ast} \boldsymbol{\beta}_{-i}(\lambda)| \), the relative error, the error squared or the R2 score function.</li>
<li> Evaluate the prediction performance of these models on the test set by \( [y_i, \boldsymbol{X}_{i, \ast}; \boldsymbol{\beta}_{-i}(\lambda), \boldsymbol{\sigma}_{-i}^2(\lambda)] \). Or, by the prediction error \( |y_i - \boldsymbol{X}_{i, \ast} \boldsymbol{\beta}_{-i}(\lambda)| \), the relative error, the error squared or the R2 score function.</li>
<li> Repeat the first three steps such that each sample plays the role of the test set once.</li>
<li> Average the prediction performances of the test sets at each grid point of the penalty bias/parameter. It is an estimate of the prediction performance of the model corresponding to this value of the penalty parameter on novel data. It is defined as</li>
</ul>
doc/pub/week37/html/week37.html
+7 -7
View File
@@ -954,12 +954,12 @@ Before we proceed however, we need to remind ourselves about a central theorem i
<p>
Suppose we have a PDF \( p(x) \) from which we generate a series \( N \)
of averages \( \langle x_i \rangle \). Each mean value \( \langle x_i \rangle \)
of averages \( \mathbb{E}[x_i] \). Each mean value \( \mathbb{E}[x_i] \)
is viewed as the average of a specific measurement, e.g., throwing
dice 100 times and then taking the average value, or producing a certain
amount of random numbers.
For notational ease, we set \( \langle x_i \rangle=x_i \) in the discussion
which follows.
For notational ease, we set \( \mathbb{E}[x_i]=x_i \) in the discussion
which follows. We do the same for \( \mathbb{E}[z]=z \).
<p>
If we compute the mean \( z \) of \( m \) such mean values \( x_i \)
@@ -1073,7 +1073,7 @@ $$
<p>
The latter is true only if the average value is known exactly. This is obtained in the limit
\( m\rightarrow \infty \) only. Because the mean and the variance are measured quantities we obtain
the familiar expression in statistics
the familiar expression in statistics (the so-called Bessel correction)
$$
\sigma_m\approx
\frac{\sigma}{\sqrt{m-1}}.
@@ -1304,8 +1304,8 @@ plt<span style="color: #666666">.</span>show()
We will discuss the bias-variance tradeoff in the context of
continuous predictions such as regression. However, many of the
intuitions and ideas discussed here also carry over to classification
tasks. Consider a dataset \( \mathcal{L} \) consisting of the data
\( \mathbf{X}_\mathcal{L}=\{(y_j, \boldsymbol{x}_j), j=0\ldots n-1\} \).
tasks. Consider a dataset \( \mathcal{D} \) consisting of the data
\( \mathbf{X}_\mathcal{D}=\{(y_j, \boldsymbol{x}_j), j=0\ldots n-1\} \).
<p>
Let us assume that the true data is generated from a noisy model
@@ -1643,7 +1643,7 @@ $$
<ul>
<li> Evaluate the prediction performance of these models on the test set by \( \log\{L[y_i, \boldsymbol{X}_{i, \ast}; \boldsymbol{\beta}_{-i}(\lambda), \boldsymbol{\sigma}_{-i}^2(\lambda)]\} \). Or, by the prediction error \( |y_i - \boldsymbol{X}_{i, \ast} \boldsymbol{\beta}_{-i}(\lambda)| \), the relative error, the error squared or the R2 score function.</li>
<li> Evaluate the prediction performance of these models on the test set by \( [y_i, \boldsymbol{X}_{i, \ast}; \boldsymbol{\beta}_{-i}(\lambda), \boldsymbol{\sigma}_{-i}^2(\lambda)] \). Or, by the prediction error \( |y_i - \boldsymbol{X}_{i, \ast} \boldsymbol{\beta}_{-i}(\lambda)| \), the relative error, the error squared or the R2 score function.</li>
<li> Repeat the first three steps such that each sample plays the role of the test set once.</li>
<li> Average the prediction performances of the test sets at each grid point of the penalty bias/parameter. It is an estimate of the prediction performance of the model corresponding to this value of the penalty parameter on novel data. It is defined as</li>
</ul>
doc/pub/week37/ipynb/ipynb-week37-src.tar.gz
BIN
View File
Binary file not shown.
doc/pub/week37/ipynb/week37.ipynb
+7 -7
View File
@@ -890,12 +890,12 @@
"\n",
"\n",
"Suppose we have a PDF $p(x)$ from which we generate a series $N$\n",
"of averages $\\langle x_i \\rangle$. Each mean value $\\langle x_i \\rangle$\n",
"of averages $\\mathbb{E}[x_i]$. Each mean value $\\mathbb{E}[x_i]$\n",
"is viewed as the average of a specific measurement, e.g., throwing \n",
"dice 100 times and then taking the average value, or producing a certain\n",
"amount of random numbers. \n",
"For notational ease, we set $\\langle x_i \\rangle=x_i$ in the discussion\n",
"which follows. \n",
"For notational ease, we set $\\mathbb{E}[x_i]=x_i$ in the discussion\n",
"which follows. We do the same for $\\mathbb{E}[z]=z$.\n",
"\n",
"If we compute the mean $z$ of $m$ such mean values $x_i$"
]
@@ -1085,7 +1085,7 @@
"source": [
"The latter is true only if the average value is known exactly. This is obtained in the limit\n",
"$m\\rightarrow \\infty$ only. Because the mean and the variance are measured quantities we obtain \n",
"the familiar expression in statistics"
"the familiar expression in statistics (the so-called Bessel correction)"
]
},
{
@@ -1325,8 +1325,8 @@
"We will discuss the bias-variance tradeoff in the context of\n",
"continuous predictions such as regression. However, many of the\n",
"intuitions and ideas discussed here also carry over to classification\n",
"tasks. Consider a dataset $\\mathcal{L}$ consisting of the data\n",
"$\\mathbf{X}_\\mathcal{L}=\\{(y_j, \\boldsymbol{x}_j), j=0\\ldots n-1\\}$. \n",
"tasks. Consider a dataset $\\mathcal{D}$ consisting of the data\n",
"$\\mathbf{X}_\\mathcal{D}=\\{(y_j, \\boldsymbol{x}_j), j=0\\ldots n-1\\}$. \n",
"\n",
"Let us assume that the true data is generated from a noisy model"
]
@@ -1757,7 +1757,7 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"* Evaluate the prediction performance of these models on the test set by $\\log\\{L[y_i, \\boldsymbol{X}_{i, \\ast}; \\boldsymbol{\\beta}_{-i}(\\lambda), \\boldsymbol{\\sigma}_{-i}^2(\\lambda)]\\}$. Or, by the prediction error $|y_i - \\boldsymbol{X}_{i, \\ast} \\boldsymbol{\\beta}_{-i}(\\lambda)|$, the relative error, the error squared or the R2 score function.\n",
"* Evaluate the prediction performance of these models on the test set by $[y_i, \\boldsymbol{X}_{i, \\ast}; \\boldsymbol{\\beta}_{-i}(\\lambda), \\boldsymbol{\\sigma}_{-i}^2(\\lambda)]$. Or, by the prediction error $|y_i - \\boldsymbol{X}_{i, \\ast} \\boldsymbol{\\beta}_{-i}(\\lambda)|$, the relative error, the error squared or the R2 score function.\n",
"\n",
"* Repeat the first three steps such that each sample plays the role of the test set once.\n",
"\n",
doc/src/week37/week37.do.txt
+7 -7
View File
@@ -643,12 +643,12 @@ Before we proceed however, we need to remind ourselves about a central theorem i
Suppose we have a PDF $p(x)$ from which we generate a series $N$
of averages $\langle x_i \rangle$. Each mean value $\langle x_i \rangle$
of averages $\mathbb{E}[x_i]$. Each mean value $\mathbb{E}[x_i]$
is viewed as the average of a specific measurement, e.g., throwing
dice 100 times and then taking the average value, or producing a certain
amount of random numbers.
For notational ease, we set $\langle x_i \rangle=x_i$ in the discussion
which follows.
For notational ease, we set $\mathbb{E}[x_i]=x_i$ in the discussion
which follows. We do the same for $\mathbb{E}[z]=z$.
If we compute the mean $z$ of $m$ such mean values $x_i$
!bt
@@ -759,7 +759,7 @@ standard deviation, given by
The latter is true only if the average value is known exactly. This is obtained in the limit
$m\rightarrow \infty$ only. Because the mean and the variance are measured quantities we obtain
the familiar expression in statistics
the familiar expression in statistics (the so-called Bessel correction)
!bt
\[
\sigma_m\approx
@@ -962,8 +962,8 @@ plt.show()
We will discuss the bias-variance tradeoff in the context of
continuous predictions such as regression. However, many of the
intuitions and ideas discussed here also carry over to classification
tasks. Consider a dataset $\mathcal{L}$ consisting of the data
$\mathbf{X}_\mathcal{L}=\{(y_j, \boldsymbol{x}_j), j=0\ldots n-1\}$.
tasks. Consider a dataset $\mathcal{D}$ consisting of the data
$\mathbf{X}_\mathcal{D}=\{(y_j, \boldsymbol{x}_j), j=0\ldots n-1\}$.
Let us assume that the true data is generated from a noisy model
@@ -1295,7 +1295,7 @@ cross-validation (LOOCV).
\end{align*}
!et
* Evaluate the prediction performance of these models on the test set by $\log\{L[y_i, \bm{X}_{i, \ast}; \bm{\beta}_{-i}(\lambda), \bm{\sigma}_{-i}^2(\lambda)]\}$. Or, by the prediction error $|y_i - \bm{X}_{i, \ast} \bm{\beta}_{-i}(\lambda)|$, the relative error, the error squared or the R2 score function.
* Evaluate the prediction performance of these models on the test set by $[y_i, \bm{X}_{i, \ast}; \bm{\beta}_{-i}(\lambda), \bm{\sigma}_{-i}^2(\lambda)]$. Or, by the prediction error $|y_i - \bm{X}_{i, \ast} \bm{\beta}_{-i}(\lambda)|$, the relative error, the error squared or the R2 score function.
* Repeat the first three steps such that each sample plays the role of the test set once.