minor typo

doc/pub/week35/html/._week35-bs000.html

@@ -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

@@ -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

@@ -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

@@ -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

@@ -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

@@ -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/src/week35/week35.do.txt

@@ -1721,15 +1721,19 @@ and using our SVD decomposition of $\bm{X}$ we have
 \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
 \[
-\tilde{y}_{\mathrm{OLS}}=\bm{U}\bm{U}^T\bm{y}=\sum_{i=0}^{n-1}\bm{u}_i\bm{u}^T_i\bm{y},
+\tilde{y}_{\mathrm{OLS}}=\bm{U}\bm{U}^T\bm{y}=\sum_{i=0}^{p-1}\bm{u}_i\bm{u}^T_i\bm{y},
 \]
 !et
 
-It means that the ordinary least square model (with the optimal parameters) $\bm{\tilde{y}}$, corresponds to an orthogonal transformation of the output (or target) vector $\bm{y}$ by the vectors of the matrix $\bm{U}$.
+It means that the ordinary least square model (with the optimal
+parameters) $\bm{\tilde{y}}$, corresponds to an orthogonal
+transformation of the output (or target) vector $\bm{y}$ by the
+vectors of the matrix $\bm{U}$. Note that the summation ends at $p-1$,
+that is $\bm{\tilde{y}}\ne \bm{y}$.
 
 !split
 ===== Further properties (important for our analyses later) =====