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typos in week 35

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Morten Hjorth-Jensen
2022-09-05 21:35:25 +02:00
parent 09cc50ce2f
commit 6a08df1fba
9 changed files with 433 additions and 488 deletions
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doc/pub/week35/html/._week35-bs000.html
+1 -1
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@@ -404,7 +404,7 @@ MathJax.Hub.Config({
</center>
<br>
<center>
<h4>Sep 2, 2022</h4>
<h4>Sep 5, 2022</h4>
</center> <!-- date -->
<br>
doc/pub/week35/html/._week35-bs050.html
+2 -8
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@@ -402,12 +402,6 @@ $$
<p>We define \( \boldsymbol{\Sigma}^T\boldsymbol{\Sigma}=\tilde{\boldsymbol{\Sigma}}^2 \) which is a diagonal matrix containing only the singular values squared. It has dimensionality \( p \times p \).</p>
<p>This means, using the orthogonality of \( \boldsymbol{V} \), that we get</p>
$$
\boldsymbol{X}^T\boldsymbol{X}=\tilde{\boldsymbol{\Sigma}}^2.
$$
<p>We can now insert the result for the matrix \( \boldsymbol{X}^T\boldsymbol{X} \) into our equation for ordinary least squares where</p>
$$
@@ -417,10 +411,10 @@ $$
<p>and using our SVD decomposition of \( \boldsymbol{X} \) we have</p>
$$
\tilde{y}_{\mathrm{OLS}}=\boldsymbol{U}\boldsymbol{\Sigma}\boldsymbol{V}^T\tilde{\boldsymbol{\Sigma}}^{-2}\boldsymbol{V}\boldsymbol{\Sigma}^T\boldsymbol{U}^T\boldsymbol{y},
\tilde{y}_{\mathrm{OLS}}=\boldsymbol{U}\boldsymbol{\Sigma}\boldsymbol{V}^T\left(\boldsymbol{V}\tilde{\boldsymbol{\Sigma}}^{2}(\boldsymbol{V}^T\right)^{-1}\boldsymbol{V}\boldsymbol{\Sigma}^T\boldsymbol{U}^T\boldsymbol{y},
$$
<p>which gives us, using the orthogonality of the matrices \( \boldsymbol{U} \) and \( \boldsymbol{V} \),</p>
<p>which gives us, using the orthogonality of the matrices \( \boldsymbol{U} \) and \( \boldsymbol{V} \),,</p>
$$
\tilde{y}_{\mathrm{OLS}}=\boldsymbol{U}\boldsymbol{U}^T\boldsymbol{y}=\sum_{i=0}^{p-1}\boldsymbol{u}_i\boldsymbol{u}^T_j\boldsymbol{y},
doc/pub/week35/html/week35-bs.html
+1 -1
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@@ -404,7 +404,7 @@ MathJax.Hub.Config({
</center>
<br>
<center>
<h4>Sep 2, 2022</h4>
<h4>Sep 5, 2022</h4>
</center> <!-- date -->
<br>
doc/pub/week35/html/week35-reveal.html
+3 -11
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@@ -184,7 +184,7 @@ MathJax.Hub.Config({
</center>
<br>
<center>
<h4>Sep 2, 2022</h4>
<h4>Sep 5, 2022</h4>
</center> <!-- date -->
<br>
@@ -2538,14 +2538,6 @@ $$
<p>We define \( \boldsymbol{\Sigma}^T\boldsymbol{\Sigma}=\tilde{\boldsymbol{\Sigma}}^2 \) which is a diagonal matrix containing only the singular values squared. It has dimensionality \( p \times p \).</p>
<p>This means, using the orthogonality of \( \boldsymbol{V} \), that we get</p>
<p>&nbsp;<br>
$$
\boldsymbol{X}^T\boldsymbol{X}=\tilde{\boldsymbol{\Sigma}}^2.
$$
<p>&nbsp;<br>
<p>We can now insert the result for the matrix \( \boldsymbol{X}^T\boldsymbol{X} \) into our equation for ordinary least squares where</p>
<p>&nbsp;<br>
@@ -2558,11 +2550,11 @@ $$
<p>&nbsp;<br>
$$
\tilde{y}_{\mathrm{OLS}}=\boldsymbol{U}\boldsymbol{\Sigma}\boldsymbol{V}^T\tilde{\boldsymbol{\Sigma}}^{-2}\boldsymbol{V}\boldsymbol{\Sigma}^T\boldsymbol{U}^T\boldsymbol{y},
\tilde{y}_{\mathrm{OLS}}=\boldsymbol{U}\boldsymbol{\Sigma}\boldsymbol{V}^T\left(\boldsymbol{V}\tilde{\boldsymbol{\Sigma}}^{2}(\boldsymbol{V}^T\right)^{-1}\boldsymbol{V}\boldsymbol{\Sigma}^T\boldsymbol{U}^T\boldsymbol{y},
$$
<p>&nbsp;<br>
<p>which gives us, using the orthogonality of the matrices \( \boldsymbol{U} \) and \( \boldsymbol{V} \),</p>
<p>which gives us, using the orthogonality of the matrices \( \boldsymbol{U} \) and \( \boldsymbol{V} \),,</p>
<p>&nbsp;<br>
$$
doc/pub/week35/html/week35-solarized.html
+3 -9
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@@ -330,7 +330,7 @@ MathJax.Hub.Config({
</center>
<br>
<center>
<h4>Sep 2, 2022</h4>
<h4>Sep 5, 2022</h4>
</center> <!-- date -->
<br>
@@ -2513,12 +2513,6 @@ $$
<p>We define \( \boldsymbol{\Sigma}^T\boldsymbol{\Sigma}=\tilde{\boldsymbol{\Sigma}}^2 \) which is a diagonal matrix containing only the singular values squared. It has dimensionality \( p \times p \).</p>
<p>This means, using the orthogonality of \( \boldsymbol{V} \), that we get</p>
$$
\boldsymbol{X}^T\boldsymbol{X}=\tilde{\boldsymbol{\Sigma}}^2.
$$
<p>We can now insert the result for the matrix \( \boldsymbol{X}^T\boldsymbol{X} \) into our equation for ordinary least squares where</p>
$$
@@ -2528,10 +2522,10 @@ $$
<p>and using our SVD decomposition of \( \boldsymbol{X} \) we have</p>
$$
\tilde{y}_{\mathrm{OLS}}=\boldsymbol{U}\boldsymbol{\Sigma}\boldsymbol{V}^T\tilde{\boldsymbol{\Sigma}}^{-2}\boldsymbol{V}\boldsymbol{\Sigma}^T\boldsymbol{U}^T\boldsymbol{y},
\tilde{y}_{\mathrm{OLS}}=\boldsymbol{U}\boldsymbol{\Sigma}\boldsymbol{V}^T\left(\boldsymbol{V}\tilde{\boldsymbol{\Sigma}}^{2}(\boldsymbol{V}^T\right)^{-1}\boldsymbol{V}\boldsymbol{\Sigma}^T\boldsymbol{U}^T\boldsymbol{y},
$$
<p>which gives us, using the orthogonality of the matrices \( \boldsymbol{U} \) and \( \boldsymbol{V} \),</p>
<p>which gives us, using the orthogonality of the matrices \( \boldsymbol{U} \) and \( \boldsymbol{V} \),,</p>
$$
\tilde{y}_{\mathrm{OLS}}=\boldsymbol{U}\boldsymbol{U}^T\boldsymbol{y}=\sum_{i=0}^{p-1}\boldsymbol{u}_i\boldsymbol{u}^T_j\boldsymbol{y},
doc/pub/week35/html/week35.html
+3 -9
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@@ -407,7 +407,7 @@ MathJax.Hub.Config({
</center>
<br>
<center>
<h4>Sep 2, 2022</h4>
<h4>Sep 5, 2022</h4>
</center> <!-- date -->
<br>
@@ -2590,12 +2590,6 @@ $$
<p>We define \( \boldsymbol{\Sigma}^T\boldsymbol{\Sigma}=\tilde{\boldsymbol{\Sigma}}^2 \) which is a diagonal matrix containing only the singular values squared. It has dimensionality \( p \times p \).</p>
<p>This means, using the orthogonality of \( \boldsymbol{V} \), that we get</p>
$$
\boldsymbol{X}^T\boldsymbol{X}=\tilde{\boldsymbol{\Sigma}}^2.
$$
<p>We can now insert the result for the matrix \( \boldsymbol{X}^T\boldsymbol{X} \) into our equation for ordinary least squares where</p>
$$
@@ -2605,10 +2599,10 @@ $$
<p>and using our SVD decomposition of \( \boldsymbol{X} \) we have</p>
$$
\tilde{y}_{\mathrm{OLS}}=\boldsymbol{U}\boldsymbol{\Sigma}\boldsymbol{V}^T\tilde{\boldsymbol{\Sigma}}^{-2}\boldsymbol{V}\boldsymbol{\Sigma}^T\boldsymbol{U}^T\boldsymbol{y},
\tilde{y}_{\mathrm{OLS}}=\boldsymbol{U}\boldsymbol{\Sigma}\boldsymbol{V}^T\left(\boldsymbol{V}\tilde{\boldsymbol{\Sigma}}^{2}(\boldsymbol{V}^T\right)^{-1}\boldsymbol{V}\boldsymbol{\Sigma}^T\boldsymbol{U}^T\boldsymbol{y},
$$
<p>which gives us, using the orthogonality of the matrices \( \boldsymbol{U} \) and \( \boldsymbol{V} \),</p>
<p>which gives us, using the orthogonality of the matrices \( \boldsymbol{U} \) and \( \boldsymbol{V} \),,</p>
$$
\tilde{y}_{\mathrm{OLS}}=\boldsymbol{U}\boldsymbol{U}^T\boldsymbol{y}=\sum_{i=0}^{p-1}\boldsymbol{u}_i\boldsymbol{u}^T_j\boldsymbol{y},
doc/pub/week35/ipynb/ipynb-week35-src.tar.gz
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doc/pub/week35/ipynb/week35.ipynb
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doc/src/week35/week35.do.txt
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@@ -1706,13 +1706,6 @@ and using the orthogonality of the matrix $\bm{U}$ we have
!et
We define $\bm{\Sigma}^T\bm{\Sigma}=\tilde{\bm{\Sigma}}^2$ which 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
!bt
\[
\bm{X}^T\bm{X}=\tilde{\bm{\Sigma}}^2.
\]
!et
We can now insert the result for the matrix $\bm{X}^T\bm{X}$ into our equation for ordinary least squares where
@@ -1725,10 +1718,10 @@ and using our SVD decomposition of $\bm{X}$ we have
!bt
\[
\tilde{y}_{\mathrm{OLS}}=\bm{U}\bm{\Sigma}\bm{V}^T\tilde{\bm{\Sigma}}^{-2}\bm{V}\bm{\Sigma}^T\bm{U}^T\bm{y},
\tilde{y}_{\mathrm{OLS}}=\bm{U}\bm{\Sigma}\bm{V}^T\left(\bm{V}\tilde{\bm{\Sigma}}^{2}(\bm{V}^T\right)^{-1}\bm{V}\bm{\Sigma}^T\bm{U}^T\bm{y},
\]
!et
which gives us, using the orthogonality of the matrices $\bm{U}$ and $\bm{V}$,
which gives us, using the orthogonality of the matrices $\bm{U}$ and $\bm{V}$,,
!bt
\[