more typos

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

@@ -2138,14 +2138,14 @@ $$
 $$
 <p>&nbsp;<br>
 
-We define $\boldsymbol{\Sigma}^T\boldsymbol{\sigma}=\boldsymbol{D}^2, where \( \boldsymbol{D} \) is a diagonal matrix containing only the singular values squared. It has dimensionality \( p \times p \).
+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>
 This means, using the orthogonality of \( \boldsymbol{V} \),  that we get
 
 <p>&nbsp;<br>
 $$
-\boldsymbol{X}^T\boldsymbol{X}=\boldsymbol{D}^2.
+\boldsymbol{X}^T\boldsymbol{X}=\tilde{\boldsymbol{\Sigma}}^2.
 $$
 <p>&nbsp;<br>
 
@@ -2162,7 +2162,7 @@ and using our SVD decomposition of \( \boldsymbol{X} \) we have
 
 <p>&nbsp;<br>
 $$
-\tilde{y}_{\mathrm{OLS}}=\boldsymbol{U}\boldsymbol{\Sigma}\boldsymbol{V}^T\boldsymbol{D}^{-2}\boldsymbol{V}\boldsymbol{\Sigma}^T\boldsymbol{U}^T\boldsymbol{y},
+\tilde{y}_{\mathrm{OLS}}=\boldsymbol{U}\boldsymbol{\Sigma}\boldsymbol{V}^T\tilde{\boldsymbol{\Sigma}}^{-2}\boldsymbol{V}\boldsymbol{\Sigma}^T\boldsymbol{U}^T\boldsymbol{y},
 $$
 <p>&nbsp;<br>
 
@@ -2182,7 +2182,7 @@ $$
 $$
 <p>&nbsp;<br>
 
-that belong to \( i>p-1 \), give all zeros when we perform the multiplications. This means the sum above has non-zero elements only up to \( i=p-1 \). This corresponds also to the number of singular values (these are all non-zero).
+that belong to \( i>p-1 \), result in only  zeros when we perform the multiplications. This means that the sum above has non-zero elements only up to \( i=p-1 \). This corresponds also to the number of singular values (these are all non-zero).
 
 <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 \( \boldsymbol{U} \).

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

@@ -2199,13 +2199,13 @@ $$
 \boldsymbol{X}^T\boldsymbol{X}=\boldsymbol{V}\boldsymbol{\Sigma}^T\boldsymbol{\Sigma}\boldsymbol{V}^T.
 $$
 
-We define $\boldsymbol{\Sigma}^T\boldsymbol{\sigma}=\boldsymbol{D}^2, where \( \boldsymbol{D} \) is a diagonal matrix containing only the singular values squared. It has dimensionality \( p \times p \).
+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>
 This means, using the orthogonality of \( \boldsymbol{V} \),  that we get
 
 $$
-\boldsymbol{X}^T\boldsymbol{X}=\boldsymbol{D}^2.
+\boldsymbol{X}^T\boldsymbol{X}=\tilde{\boldsymbol{\Sigma}}^2.
 $$
 
 <p>
@@ -2218,7 +2218,7 @@ $$
 and using our SVD decomposition of \( \boldsymbol{X} \) we have
 
 $$
-\tilde{y}_{\mathrm{OLS}}=\boldsymbol{U}\boldsymbol{\Sigma}\boldsymbol{V}^T\boldsymbol{D}^{-2}\boldsymbol{V}\boldsymbol{\Sigma}^T\boldsymbol{U}^T\boldsymbol{y},
+\tilde{y}_{\mathrm{OLS}}=\boldsymbol{U}\boldsymbol{\Sigma}\boldsymbol{V}^T\tilde{\boldsymbol{\Sigma}}^{-2}\boldsymbol{V}\boldsymbol{\Sigma}^T\boldsymbol{U}^T\boldsymbol{y},
 $$
 
 which gives us, using the orthogonality of the matrices \( \boldsymbol{U} \) and \( \boldsymbol{V} \),
@@ -2233,7 +2233,7 @@ $$
 \boldsymbol{U}=[\boldsymbol{u}_0,\boldsymbol{u}_1,\dots,\boldsymbol{u}_{n-1}],
 $$
 
-that belong to \( i>p-1 \), give all zeros when we perform the multiplications. This means the sum above has non-zero elements only up to \( i=p-1 \). This corresponds also to the number of singular values (these are all non-zero).
+that belong to \( i>p-1 \), result in only  zeros when we perform the multiplications. This means that the sum above has non-zero elements only up to \( i=p-1 \). This corresponds also to the number of singular values (these are all non-zero).
 
 <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 \( \boldsymbol{U} \).

doc/pub/week35/html/week35.html

@@ -2204,13 +2204,13 @@ $$
 \boldsymbol{X}^T\boldsymbol{X}=\boldsymbol{V}\boldsymbol{\Sigma}^T\boldsymbol{\Sigma}\boldsymbol{V}^T.
 $$
 
-We define $\boldsymbol{\Sigma}^T\boldsymbol{\sigma}=\boldsymbol{D}^2, where \( \boldsymbol{D} \) is a diagonal matrix containing only the singular values squared. It has dimensionality \( p \times p \).
+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>
 This means, using the orthogonality of \( \boldsymbol{V} \),  that we get
 
 $$
-\boldsymbol{X}^T\boldsymbol{X}=\boldsymbol{D}^2.
+\boldsymbol{X}^T\boldsymbol{X}=\tilde{\boldsymbol{\Sigma}}^2.
 $$
 
 <p>
@@ -2223,7 +2223,7 @@ $$
 and using our SVD decomposition of \( \boldsymbol{X} \) we have
 
 $$
-\tilde{y}_{\mathrm{OLS}}=\boldsymbol{U}\boldsymbol{\Sigma}\boldsymbol{V}^T\boldsymbol{D}^{-2}\boldsymbol{V}\boldsymbol{\Sigma}^T\boldsymbol{U}^T\boldsymbol{y},
+\tilde{y}_{\mathrm{OLS}}=\boldsymbol{U}\boldsymbol{\Sigma}\boldsymbol{V}^T\tilde{\boldsymbol{\Sigma}}^{-2}\boldsymbol{V}\boldsymbol{\Sigma}^T\boldsymbol{U}^T\boldsymbol{y},
 $$
 
 which gives us, using the orthogonality of the matrices \( \boldsymbol{U} \) and \( \boldsymbol{V} \),
@@ -2238,7 +2238,7 @@ $$
 \boldsymbol{U}=[\boldsymbol{u}_0,\boldsymbol{u}_1,\dots,\boldsymbol{u}_{n-1}],
 $$
 
-that belong to \( i>p-1 \), give all zeros when we perform the multiplications. This means the sum above has non-zero elements only up to \( i=p-1 \). This corresponds also to the number of singular values (these are all non-zero).
+that belong to \( i>p-1 \), result in only  zeros when we perform the multiplications. This means that the sum above has non-zero elements only up to \( i=p-1 \). This corresponds also to the number of singular values (these are all non-zero).
 
 <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 \( \boldsymbol{U} \).

doc/pub/week35/ipynb/week35.ipynb

@@ -2665,7 +2665,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "We define $\\boldsymbol{\\Sigma}^T\\boldsymbol{\\sigma}=\\boldsymbol{D}^2, where $\\boldsymbol{D}$ is a diagonal matrix containing only the singular values squared. It has dimensionality $p \\times p$.\n",
+    "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 $p \\times p$.\n",
     "\n",
     "This means, using the orthogonality of $\\boldsymbol{V}$,  that we get"
    ]
@@ -2675,7 +2675,7 @@
    "metadata": {},
    "source": [
     "$$\n",
-    "\\boldsymbol{X}^T\\boldsymbol{X}=\\boldsymbol{D}^2.\n",
+    "\\boldsymbol{X}^T\\boldsymbol{X}=\\tilde{\\boldsymbol{\\Sigma}}^2.\n",
     "$$"
    ]
   },
@@ -2707,7 +2707,7 @@
    "metadata": {},
    "source": [
     "$$\n",
-    "\\tilde{y}_{\\mathrm{OLS}}=\\boldsymbol{U}\\boldsymbol{\\Sigma}\\boldsymbol{V}^T\\boldsymbol{D}^{-2}\\boldsymbol{V}\\boldsymbol{\\Sigma}^T\\boldsymbol{U}^T\\boldsymbol{y},\n",
+    "\\tilde{y}_{\\mathrm{OLS}}=\\boldsymbol{U}\\boldsymbol{\\Sigma}\\boldsymbol{V}^T\\tilde{\\boldsymbol{\\Sigma}}^{-2}\\boldsymbol{V}\\boldsymbol{\\Sigma}^T\\boldsymbol{U}^T\\boldsymbol{y},\n",
     "$$"
    ]
   },
@@ -2747,7 +2747,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "that belong to $i>p-1$, give all zeros when we perform the multiplications. This means the sum above has non-zero elements only up to $i=p-1$. This corresponds also to the number of singular values (these are all non-zero).\n",
+    "that belong to $i>p-1$, result in only  zeros when we perform the multiplications. This means that the sum above has non-zero elements only up to $i=p-1$. This corresponds also to the number of singular values (these are all non-zero).\n",
     "\n",
     "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 $\\boldsymbol{U}$.\n",
     "\n",

doc/src/week35/week35.do.txt

@@ -1695,13 +1695,13 @@ and using the orthogonality of the matrix $\bm{U}$ we have
 \bm{X}^T\bm{X}=\bm{V}\bm{\Sigma}^T\bm{\Sigma}\bm{V}^T.
 \]
 !et
-We define $\bm{\Sigma}^T\bm{\sigma}=\bm{D}^2, where $\bm{D}$ is a diagonal matrix containing only the singular values squared. It has dimensionality $p \times p$.
+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
 \[
-\bm{X}^T\bm{X}=\bm{D}^2.
+\bm{X}^T\bm{X}=\tilde{\bm{\Sigma}}^2.
 \]
 !et
 
@@ -1716,7 +1716,7 @@ and using our SVD decomposition of $\bm{X}$ we have
 
 !bt
 \[
-\tilde{y}_{\mathrm{OLS}}=\bm{U}\bm{\Sigma}\bm{V}^T\bm{D}^{-2}\bm{V}\bm{\Sigma}^T\bm{U}^T\bm{y},
+\tilde{y}_{\mathrm{OLS}}=\bm{U}\bm{\Sigma}\bm{V}^T\tilde{\bm{\Sigma}}^{-2}\bm{V}\bm{\Sigma}^T\bm{U}^T\bm{y},
 \]
 !et
 which gives us, using the orthogonality of the matrices $\bm{U}$ and $\bm{V}$,
@@ -1733,7 +1733,7 @@ Note here that when we perform the  multiplication of the various matrices, the
 \bm{U}=[\bm{u}_0,\bm{u}_1,\dots,\bm{u}_{n-1}],
 \]
 !et
-that belong to $i>p-1$, give all zeros when we perform the multiplications. This means the sum above has non-zero elements only up to $i=p-1$. This corresponds also to the number of singular values (these are all non-zero).
+that belong to $i>p-1$, result in only  zeros when we perform the multiplications. This means that the sum above has non-zero elements only up to $i=p-1$. This corresponds also to the number of singular values (these are all non-zero).
 
 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 $\bm{U}$.