update

doc/pub/week35/html/._week35-bs045.html

@@ -350,16 +350,17 @@ $$
 <p>which gives us, using the orthogonality of the matrix \( \boldsymbol{V} \),</p>
 
 $$
-\tilde{y}_{\mathrm{OLS}}=\boldsymbol{U}\boldsymbol{U}^T\boldsymbol{y}=\sum_{i=0}^{p-1}\boldsymbol{u}_i\boldsymbol{u}^T_i\boldsymbol{y},
+\tilde{y}_{\mathrm{OLS}}=\sum_{i=0}^{p-1}\boldsymbol{u}_i\boldsymbol{u}^T_i\boldsymbol{y},
 $$
 
+<p>which is not the same as \( \tilde{y}_{\mathrm{OLS}}=\boldsymbol{U}\boldsymbol{U}^T\boldsymbol{y} \), which due to the orthogonality of \( \boldsymbol{U} \) would have given us that the model equals the output.</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 \( \boldsymbol{U} \). <b>Note that the summation ends at</b>
 \( p-1 \), that is \( \boldsymbol{\tilde{y}}\ne \boldsymbol{y} \). We can thus not use the
-orthogonality relation for the matrix \( \boldsymbol{U} \). This can already be
-when we multiply the matrices \( \boldsymbol{\Sigma}^T\boldsymbol{U}^T \).
+orthogonality relation for the matrix \( \boldsymbol{U} \). 
 </p>
 
 <p>

doc/pub/week35/html/week35-reveal.html

@@ -2189,17 +2189,18 @@ $$
 
 <p>&nbsp;<br>
 $$
-\tilde{y}_{\mathrm{OLS}}=\boldsymbol{U}\boldsymbol{U}^T\boldsymbol{y}=\sum_{i=0}^{p-1}\boldsymbol{u}_i\boldsymbol{u}^T_i\boldsymbol{y},
+\tilde{y}_{\mathrm{OLS}}=\sum_{i=0}^{p-1}\boldsymbol{u}_i\boldsymbol{u}^T_i\boldsymbol{y},
 $$
 <p>&nbsp;<br>
 
+<p>which is not the same as \( \tilde{y}_{\mathrm{OLS}}=\boldsymbol{U}\boldsymbol{U}^T\boldsymbol{y} \), which due to the orthogonality of \( \boldsymbol{U} \) would have given us that the model equals the output.</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 \( \boldsymbol{U} \). <b>Note that the summation ends at</b>
 \( p-1 \), that is \( \boldsymbol{\tilde{y}}\ne \boldsymbol{y} \). We can thus not use the
-orthogonality relation for the matrix \( \boldsymbol{U} \). This can already be
-when we multiply the matrices \( \boldsymbol{\Sigma}^T\boldsymbol{U}^T \).
+orthogonality relation for the matrix \( \boldsymbol{U} \). 
 </p>
 </section>
 

doc/pub/week35/html/week35-solarized.html

@@ -2062,16 +2062,17 @@ $$
 <p>which gives us, using the orthogonality of the matrix \( \boldsymbol{V} \),</p>
 
 $$
-\tilde{y}_{\mathrm{OLS}}=\boldsymbol{U}\boldsymbol{U}^T\boldsymbol{y}=\sum_{i=0}^{p-1}\boldsymbol{u}_i\boldsymbol{u}^T_i\boldsymbol{y},
+\tilde{y}_{\mathrm{OLS}}=\sum_{i=0}^{p-1}\boldsymbol{u}_i\boldsymbol{u}^T_i\boldsymbol{y},
 $$
 
+<p>which is not the same as \( \tilde{y}_{\mathrm{OLS}}=\boldsymbol{U}\boldsymbol{U}^T\boldsymbol{y} \), which due to the orthogonality of \( \boldsymbol{U} \) would have given us that the model equals the output.</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 \( \boldsymbol{U} \). <b>Note that the summation ends at</b>
 \( p-1 \), that is \( \boldsymbol{\tilde{y}}\ne \boldsymbol{y} \). We can thus not use the
-orthogonality relation for the matrix \( \boldsymbol{U} \). This can already be
-when we multiply the matrices \( \boldsymbol{\Sigma}^T\boldsymbol{U}^T \).
+orthogonality relation for the matrix \( \boldsymbol{U} \). 
 </p>
 
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>

doc/pub/week35/html/week35.html

@@ -2139,16 +2139,17 @@ $$
 <p>which gives us, using the orthogonality of the matrix \( \boldsymbol{V} \),</p>
 
 $$
-\tilde{y}_{\mathrm{OLS}}=\boldsymbol{U}\boldsymbol{U}^T\boldsymbol{y}=\sum_{i=0}^{p-1}\boldsymbol{u}_i\boldsymbol{u}^T_i\boldsymbol{y},
+\tilde{y}_{\mathrm{OLS}}=\sum_{i=0}^{p-1}\boldsymbol{u}_i\boldsymbol{u}^T_i\boldsymbol{y},
 $$
 
+<p>which is not the same as \( \tilde{y}_{\mathrm{OLS}}=\boldsymbol{U}\boldsymbol{U}^T\boldsymbol{y} \), which due to the orthogonality of \( \boldsymbol{U} \) would have given us that the model equals the output.</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 \( \boldsymbol{U} \). <b>Note that the summation ends at</b>
 \( p-1 \), that is \( \boldsymbol{\tilde{y}}\ne \boldsymbol{y} \). We can thus not use the
-orthogonality relation for the matrix \( \boldsymbol{U} \). This can already be
-when we multiply the matrices \( \boldsymbol{\Sigma}^T\boldsymbol{U}^T \).
+orthogonality relation for the matrix \( \boldsymbol{U} \). 
 </p>
 
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>

doc/src/week35/week35.do.txt

@@ -1630,17 +1630,17 @@ which gives us, using the orthogonality of the matrix $\bm{V}$,
 
 !bt
 \[
-\tilde{y}_{\mathrm{OLS}}=\bm{U}\bm{U}^T\bm{y}=\sum_{i=0}^{p-1}\bm{u}_i\bm{u}^T_i\bm{y},
+\tilde{y}_{\mathrm{OLS}}=\sum_{i=0}^{p-1}\bm{u}_i\bm{u}^T_i\bm{y},
 \]
 !et
+which is not the same as $\tilde{y}_{\mathrm{OLS}}=\bm{U}\bm{U}^T\bm{y}$, which due to the orthogonality of $\bm{U}$ would have given us that the model equals the output.
 
 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 $\bm{\tilde{y}}\ne \bm{y}$. We can thus not use the
-orthogonality relation for the matrix $\bm{U}$. This can already be
-when we multiply the matrices $\bm{\Sigma}^T\bm{U}^T$.
+orthogonality relation for the matrix $\bm{U}$. 
 
 !split
 ===== Further properties (important for our analyses later) =====