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minor typo

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This commit is contained in:
Morten Hjorth-Jensen
2022-09-08 14:30:23 +02:00
parent 44e11e9c2a
commit fa06cccfd8
10 changed files with 1398 additions and 1417 deletions
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doc/pub/week35/html/._week35-bs000.html
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@@ -303,7 +303,7 @@ MathJax.Hub.Config({
</center>
<br>
<center>
<h4>Sep 7, 2022</h4>
<h4>Sep 8, 2022</h4>
</center> <!-- date -->
<br>
</p>
doc/pub/week35/html/._week35-bs050.html
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@@ -316,13 +316,18 @@ $$
\tilde{y}_{\mathrm{OLS}}=\boldsymbol{U}\boldsymbol{\Sigma}\boldsymbol{V}^T\left(\boldsymbol{V}\tilde{\boldsymbol{\Sigma}}^{2}(\boldsymbol{V}^T\right)^{-1}\boldsymbol{V}\boldsymbol{\Sigma}^T\boldsymbol{U}^T\boldsymbol{y},
$$
<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>
$$
\tilde{y}_{\mathrm{OLS}}=\boldsymbol{U}\boldsymbol{U}^T\boldsymbol{y}=\sum_{i=0}^{n-1}\boldsymbol{u}_i\boldsymbol{u}^T_i\boldsymbol{y},
\tilde{y}_{\mathrm{OLS}}=\boldsymbol{U}\boldsymbol{U}^T\boldsymbol{y}=\sum_{i=0}^{p-1}\boldsymbol{u}_i\boldsymbol{u}^T_i\boldsymbol{y},
$$
<p>It means that the ordinary least square model (with the optimal parameters) \( \boldsymbol{\tilde{y}} \), corresponds to an orthogonal transformation of the output (or target) vector \( \boldsymbol{y} \) by the vectors of the matrix \( \boldsymbol{U} \).</p>
<p>It means that the ordinary least square model (with the optimal
parameters) \( \boldsymbol{\tilde{y}} \), corresponds to an orthogonal
transformation of the output (or target) vector \( \boldsymbol{y} \) by the
vectors of the matrix \( \boldsymbol{U} \). Note that the summation ends at \( p-1 \),
that is \( \boldsymbol{\tilde{y}}\ne \boldsymbol{y} \).
</p>
<p>
<!-- navigation buttons at the bottom of the page -->
doc/pub/week35/html/week35-bs.html
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@@ -303,7 +303,7 @@ MathJax.Hub.Config({
</center>
<br>
<center>
<h4>Sep 7, 2022</h4>
<h4>Sep 8, 2022</h4>
</center> <!-- date -->
<br>
</p>
doc/pub/week35/html/week35-reveal.html
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@@ -181,7 +181,7 @@ MathJax.Hub.Config({
</center>
<br>
<center>
<h4>Sep 7, 2022</h4>
<h4>Sep 8, 2022</h4>
</center> <!-- date -->
<br>
</p>
@@ -2552,15 +2552,20 @@ $$
$$
<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>
$$
\tilde{y}_{\mathrm{OLS}}=\boldsymbol{U}\boldsymbol{U}^T\boldsymbol{y}=\sum_{i=0}^{n-1}\boldsymbol{u}_i\boldsymbol{u}^T_i\boldsymbol{y},
\tilde{y}_{\mathrm{OLS}}=\boldsymbol{U}\boldsymbol{U}^T\boldsymbol{y}=\sum_{i=0}^{p-1}\boldsymbol{u}_i\boldsymbol{u}^T_i\boldsymbol{y},
$$
<p>&nbsp;<br>
<p>It means that the ordinary least square model (with the optimal parameters) \( \boldsymbol{\tilde{y}} \), corresponds to an orthogonal transformation of the output (or target) vector \( \boldsymbol{y} \) by the vectors of the matrix \( \boldsymbol{U} \).</p>
<p>It means that the ordinary least square model (with the optimal
parameters) \( \boldsymbol{\tilde{y}} \), corresponds to an orthogonal
transformation of the output (or target) vector \( \boldsymbol{y} \) by the
vectors of the matrix \( \boldsymbol{U} \). Note that the summation ends at \( p-1 \),
that is \( \boldsymbol{\tilde{y}}\ne \boldsymbol{y} \).
</p>
</section>
<section>
doc/pub/week35/html/week35-solarized.html
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@@ -327,7 +327,7 @@ MathJax.Hub.Config({
</center>
<br>
<center>
<h4>Sep 7, 2022</h4>
<h4>Sep 8, 2022</h4>
</center> <!-- date -->
<br>
</p>
@@ -2523,13 +2523,18 @@ $$
\tilde{y}_{\mathrm{OLS}}=\boldsymbol{U}\boldsymbol{\Sigma}\boldsymbol{V}^T\left(\boldsymbol{V}\tilde{\boldsymbol{\Sigma}}^{2}(\boldsymbol{V}^T\right)^{-1}\boldsymbol{V}\boldsymbol{\Sigma}^T\boldsymbol{U}^T\boldsymbol{y},
$$
<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>
$$
\tilde{y}_{\mathrm{OLS}}=\boldsymbol{U}\boldsymbol{U}^T\boldsymbol{y}=\sum_{i=0}^{n-1}\boldsymbol{u}_i\boldsymbol{u}^T_i\boldsymbol{y},
\tilde{y}_{\mathrm{OLS}}=\boldsymbol{U}\boldsymbol{U}^T\boldsymbol{y}=\sum_{i=0}^{p-1}\boldsymbol{u}_i\boldsymbol{u}^T_i\boldsymbol{y},
$$
<p>It means that the ordinary least square model (with the optimal parameters) \( \boldsymbol{\tilde{y}} \), corresponds to an orthogonal transformation of the output (or target) vector \( \boldsymbol{y} \) by the vectors of the matrix \( \boldsymbol{U} \).</p>
<p>It means that the ordinary least square model (with the optimal
parameters) \( \boldsymbol{\tilde{y}} \), corresponds to an orthogonal
transformation of the output (or target) vector \( \boldsymbol{y} \) by the
vectors of the matrix \( \boldsymbol{U} \). Note that the summation ends at \( p-1 \),
that is \( \boldsymbol{\tilde{y}}\ne \boldsymbol{y} \).
</p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="further-properties-important-for-our-analyses-later">Further properties (important for our analyses later) </h2>
doc/pub/week35/html/week35.html
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@@ -404,7 +404,7 @@ MathJax.Hub.Config({
</center>
<br>
<center>
<h4>Sep 7, 2022</h4>
<h4>Sep 8, 2022</h4>
</center> <!-- date -->
<br>
</p>
@@ -2600,13 +2600,18 @@ $$
\tilde{y}_{\mathrm{OLS}}=\boldsymbol{U}\boldsymbol{\Sigma}\boldsymbol{V}^T\left(\boldsymbol{V}\tilde{\boldsymbol{\Sigma}}^{2}(\boldsymbol{V}^T\right)^{-1}\boldsymbol{V}\boldsymbol{\Sigma}^T\boldsymbol{U}^T\boldsymbol{y},
$$
<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>
$$
\tilde{y}_{\mathrm{OLS}}=\boldsymbol{U}\boldsymbol{U}^T\boldsymbol{y}=\sum_{i=0}^{n-1}\boldsymbol{u}_i\boldsymbol{u}^T_i\boldsymbol{y},
\tilde{y}_{\mathrm{OLS}}=\boldsymbol{U}\boldsymbol{U}^T\boldsymbol{y}=\sum_{i=0}^{p-1}\boldsymbol{u}_i\boldsymbol{u}^T_i\boldsymbol{y},
$$
<p>It means that the ordinary least square model (with the optimal parameters) \( \boldsymbol{\tilde{y}} \), corresponds to an orthogonal transformation of the output (or target) vector \( \boldsymbol{y} \) by the vectors of the matrix \( \boldsymbol{U} \).</p>
<p>It means that the ordinary least square model (with the optimal
parameters) \( \boldsymbol{\tilde{y}} \), corresponds to an orthogonal
transformation of the output (or target) vector \( \boldsymbol{y} \) by the
vectors of the matrix \( \boldsymbol{U} \). Note that the summation ends at \( p-1 \),
that is \( \boldsymbol{\tilde{y}}\ne \boldsymbol{y} \).
</p>
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
<h2 id="further-properties-important-for-our-analyses-later">Further properties (important for our analyses later) </h2>
doc/pub/week35/ipynb/ipynb-week35-src.tar.gz
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doc/pub/week35/ipynb/week35.ipynb
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doc/pub/week36/ipynb/week36.ipynb
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