more cleaning up

doc/pub/DimRed/html/._DimRed-bs018.html

@@ -193,14 +193,14 @@ each with dimension \( \boldsymbol{x}\in {\mathbb{R}}^{n} \).
 <p>
 We assume also that we have an orthogonal transformation \( \boldsymbol{W}\in {\mathbb{R}}^{p\times p} \).  We define the reconstruction error (which is similar to the mean squared error we have seen before) as
 $$
-J(\boldsymbol{W},\boldsymbol{Z}) = \frac{1}{p}\sum_i (\boldsymbol{x}_i - \overline{\boldsymbol{x}_i})^2,
+J(\boldsymbol{W},\boldsymbol{Z}) = \frac{1}{n}\sum_i (\boldsymbol{x}_i - \overline{\boldsymbol{x}_i})^2,
 $$
 
 with \( \overline{\boldsymbol{x}_i} = \boldsymbol{W}\boldsymbol{z}_i \), where \( \boldsymbol{z}_i \) is a row vector with dimension \( {\mathbb{R}}^{n} \) of the matrix
-\( \boldsymbol{Z}\in {\mathbb{R}}^{p\times n} \).  When doing PCA we want to reduce this dimensionality.
+\( \boldsymbol{Z}\in{\mathbb{R}}^{p\times n} \).  When doing PCA we want to reduce this dimensionality.
 
 <p>
-The PCA theorem states that minimizing the above reconstruction error corresponds to setting \( \boldsymbol{W}=\boldsymbol{S} \), the orthogonal matrix which diagonalizes the empirical covariance(correlation) matrix. The optimal low-dimensional encoding of the data is then given by a set of vectors \( l \) $\boldsymbol{z}_i$, with \( l < < p \), defined by the orthogonal projection of the data onto the columns spanned by the eigenvectors of the covariance(correlations matrix).
+The PCA theorem states that minimizing the above reconstruction error corresponds to setting \( \boldsymbol{W}=\boldsymbol{S} \), the orthogonal matrix which diagonalizes the empirical covariance(correlation) matrix. The optimal low-dimensional encoding of the data is then given by a set of vectors \( \boldsymbol{z}_i \) with at most \( l \) vectors, with \( l < < p \), defined by the orthogonal projection of the data onto the columns spanned by the eigenvectors of the covariance(correlations matrix).
 
 <p>
 <p>

doc/pub/DimRed/html/._DimRed-bs019.html

@@ -189,12 +189,12 @@ MathJax.Hub.Config({
 <p>
 To show the PCA theorem let us start with the assumption that there is one vector \( \boldsymbol{w}_0 \) which corresponds to a solution which minimized the reconstruction error \( J \). This is an orthogonal vector. It means that we now approximate the reconstruction error in terms of \( \boldsymbol{w}_0 \) and \( \boldsymbol{z}_0 \) as
 $$
-J(\boldsymbol{w}_0,\boldsymbol{z}_0)= \frac{1}{p}\sum_i (\boldsymbol{x}_i - z_{i0}\boldsymbol{w}_0)^2=\frac{1}{p}\sum_i (\boldsymbol{x}_i^T\boldsymbol{x}_i - 2z_{i0}\boldsymbol{w}_0^T\boldsymbol{x}_i+z_{i0}^2\boldsymbol{w}_0^T\boldsymbol{w}_0),
+J(\boldsymbol{w}_0,\boldsymbol{z}_0)= \frac{1}{n}\sum_i (\boldsymbol{x}_i - z_{i0}\boldsymbol{w}_0)^2=\frac{1}{n}\sum_i (\boldsymbol{x}_i^T\boldsymbol{x}_i - 2z_{i0}\boldsymbol{w}_0^T\boldsymbol{x}_i+z_{i0}^2\boldsymbol{w}_0^T\boldsymbol{w}_0),
 $$
 
 which we can rewrite due to the orthogonality of \( \boldsymbol{w}_i \) as
 $$
-J(\boldsymbol{w}_0,\boldsymbol{z}_0)=\frac{1}{p}\sum_i (\boldsymbol{x}_i^T\boldsymbol{x}_i - 2z_{i0}\boldsymbol{w}_0^T\boldsymbol{x}_i+z_{i0}^2).
+J(\boldsymbol{w}_0,\boldsymbol{z}_0)=\frac{1}{n}\sum_i (\boldsymbol{x}_i^T\boldsymbol{x}_i - 2z_{i0}\boldsymbol{w}_0^T\boldsymbol{x}_i+z_{i0}^2).
 $$
 
 Minimizing \( J \) with respect to the unknown parameters \( z_{0i} \) we obtain that

doc/pub/DimRed/html/._DimRed-bs020.html

@@ -187,15 +187,15 @@ MathJax.Hub.Config({
 <h2 id="___sec19" class="anchor">PCA Proof continued </h2>
 
 <p>
-We have now found the unknown parameters \( z_{i0} \). These correspond to the project coordinates and we can write
+We have now found the unknown parameters \( z_{i0} \). These correspond to the projected coordinates and we can write
 $$
-J(\boldsymbol{w}_0)= \frac{1}{p}\sum_i (\boldsymbol{x}_i^T\boldsymbol{x}_i - z_{i0}^2)=\mathrm{const}-\frac{1}{p}\sum_i z_{i0}^2.
+J(\boldsymbol{w}_0)= \frac{1}{p}\sum_i (\boldsymbol{x}_i^T\boldsymbol{x}_i - z_{i0}^2)=\mathrm{const}-\frac{1}{n}\sum_i z_{i0}^2.
 $$
 
 <p>
-We can show that the variance of the project coordinates defined by \( \boldsymbol{w}_0^T\boldsymbol{x}_i \) are given by
+We can show that the variance of the projected coordinates defined by \( \boldsymbol{w}_0^T\boldsymbol{x}_i \) are given by
 $$
-\mathrm{var}[\boldsymbol{w}_0^T\boldsymbol{x}_i] = \frac{1}{p}\sum_i z_{i0}^2,
+\mathrm{var}[\boldsymbol{w}_0^T\boldsymbol{x}_i] = \frac{1}{n}\sum_i z_{i0}^2,
 $$
 
 since the expectation value of
@@ -208,12 +208,12 @@ where we have used the fact that our data are centered.
 <p>
 Recalling our definition of the covariance as 
 $$
-\boldsymbol{C}[\boldsymbol{x}] = \frac{1}{n}\boldsymbol{X}\boldsymbol{X}^T= \mathbb{E}[\boldsymbol{X}\boldsymbol{X}^T],
+\boldsymbol{C}[\boldsymbol{x}] = \frac{1}{n}\boldsymbol{X}\boldsymbol{X}^T=\mathbb{E}[\boldsymbol{X}\boldsymbol{X}^T],
 $$
 
 we have thus that
 $$
-\mathrm{var}[\boldsymbol{w}_0^T\boldsymbol{x}_i] = \frac{1}{p}\sum_i z_{i0}^2=\boldsymbol{w}_0^T\boldsymbol{C}[\boldsymbol{x}]\boldsymbol{w}_0.
+\mathrm{var}[\boldsymbol{w}_0^T\boldsymbol{x}_i] = \frac{1}{n}\sum_i z_{i0}^2=\boldsymbol{w}_0^T\boldsymbol{C}[\boldsymbol{x}]\boldsymbol{w}_0.
 $$
 
 <p>

doc/pub/DimRed/html/DimRed-reveal.html

@@ -999,15 +999,15 @@ each with dimension \( \boldsymbol{x}\in {\mathbb{R}}^{n} \).
 We assume also that we have an orthogonal transformation \( \boldsymbol{W}\in {\mathbb{R}}^{p\times p} \).  We define the reconstruction error (which is similar to the mean squared error we have seen before) as
 <p>&nbsp;<br>
 $$
-J(\boldsymbol{W},\boldsymbol{Z}) = \frac{1}{p}\sum_i (\boldsymbol{x}_i - \overline{\boldsymbol{x}_i})^2,
+J(\boldsymbol{W},\boldsymbol{Z}) = \frac{1}{n}\sum_i (\boldsymbol{x}_i - \overline{\boldsymbol{x}_i})^2,
 $$
 <p>&nbsp;<br>
 
 with \( \overline{\boldsymbol{x}_i} = \boldsymbol{W}\boldsymbol{z}_i \), where \( \boldsymbol{z}_i \) is a row vector with dimension \( {\mathbb{R}}^{n} \) of the matrix
-\( \boldsymbol{Z}\in {\mathbb{R}}^{p\times n} \).  When doing PCA we want to reduce this dimensionality.
+\( \boldsymbol{Z}\in{\mathbb{R}}^{p\times n} \).  When doing PCA we want to reduce this dimensionality.
 
 <p>
-The PCA theorem states that minimizing the above reconstruction error corresponds to setting \( \boldsymbol{W}=\boldsymbol{S} \), the orthogonal matrix which diagonalizes the empirical covariance(correlation) matrix. The optimal low-dimensional encoding of the data is then given by a set of vectors \( l \) $\boldsymbol{z}_i$, with \( l < < p \), defined by the orthogonal projection of the data onto the columns spanned by the eigenvectors of the covariance(correlations matrix).
+The PCA theorem states that minimizing the above reconstruction error corresponds to setting \( \boldsymbol{W}=\boldsymbol{S} \), the orthogonal matrix which diagonalizes the empirical covariance(correlation) matrix. The optimal low-dimensional encoding of the data is then given by a set of vectors \( \boldsymbol{z}_i \) with at most \( l \) vectors, with \( l < < p \), defined by the orthogonal projection of the data onto the columns spanned by the eigenvectors of the covariance(correlations matrix).
 </section>
 
 
@@ -1018,14 +1018,14 @@ The PCA theorem states that minimizing the above reconstruction error correspond
 To show the PCA theorem let us start with the assumption that there is one vector \( \boldsymbol{w}_0 \) which corresponds to a solution which minimized the reconstruction error \( J \). This is an orthogonal vector. It means that we now approximate the reconstruction error in terms of \( \boldsymbol{w}_0 \) and \( \boldsymbol{z}_0 \) as
 <p>&nbsp;<br>
 $$
-J(\boldsymbol{w}_0,\boldsymbol{z}_0)= \frac{1}{p}\sum_i (\boldsymbol{x}_i - z_{i0}\boldsymbol{w}_0)^2=\frac{1}{p}\sum_i (\boldsymbol{x}_i^T\boldsymbol{x}_i - 2z_{i0}\boldsymbol{w}_0^T\boldsymbol{x}_i+z_{i0}^2\boldsymbol{w}_0^T\boldsymbol{w}_0),
+J(\boldsymbol{w}_0,\boldsymbol{z}_0)= \frac{1}{n}\sum_i (\boldsymbol{x}_i - z_{i0}\boldsymbol{w}_0)^2=\frac{1}{n}\sum_i (\boldsymbol{x}_i^T\boldsymbol{x}_i - 2z_{i0}\boldsymbol{w}_0^T\boldsymbol{x}_i+z_{i0}^2\boldsymbol{w}_0^T\boldsymbol{w}_0),
 $$
 <p>&nbsp;<br>
 
 which we can rewrite due to the orthogonality of \( \boldsymbol{w}_i \) as
 <p>&nbsp;<br>
 $$
-J(\boldsymbol{w}_0,\boldsymbol{z}_0)=\frac{1}{p}\sum_i (\boldsymbol{x}_i^T\boldsymbol{x}_i - 2z_{i0}\boldsymbol{w}_0^T\boldsymbol{x}_i+z_{i0}^2).
+J(\boldsymbol{w}_0,\boldsymbol{z}_0)=\frac{1}{n}\sum_i (\boldsymbol{x}_i^T\boldsymbol{x}_i - 2z_{i0}\boldsymbol{w}_0^T\boldsymbol{x}_i+z_{i0}^2).
 $$
 <p>&nbsp;<br>
 
@@ -1044,18 +1044,18 @@ where the vectors on the rhs are known.
 <h2 id="___sec19">PCA Proof continued </h2>
 
 <p>
-We have now found the unknown parameters \( z_{i0} \). These correspond to the project coordinates and we can write
+We have now found the unknown parameters \( z_{i0} \). These correspond to the projected coordinates and we can write
 <p>&nbsp;<br>
 $$
-J(\boldsymbol{w}_0)= \frac{1}{p}\sum_i (\boldsymbol{x}_i^T\boldsymbol{x}_i - z_{i0}^2)=\mathrm{const}-\frac{1}{p}\sum_i z_{i0}^2.
+J(\boldsymbol{w}_0)= \frac{1}{p}\sum_i (\boldsymbol{x}_i^T\boldsymbol{x}_i - z_{i0}^2)=\mathrm{const}-\frac{1}{n}\sum_i z_{i0}^2.
 $$
 <p>&nbsp;<br>
 
 <p>
-We can show that the variance of the project coordinates defined by \( \boldsymbol{w}_0^T\boldsymbol{x}_i \) are given by
+We can show that the variance of the projected coordinates defined by \( \boldsymbol{w}_0^T\boldsymbol{x}_i \) are given by
 <p>&nbsp;<br>
 $$
-\mathrm{var}[\boldsymbol{w}_0^T\boldsymbol{x}_i] = \frac{1}{p}\sum_i z_{i0}^2,
+\mathrm{var}[\boldsymbol{w}_0^T\boldsymbol{x}_i] = \frac{1}{n}\sum_i z_{i0}^2,
 $$
 <p>&nbsp;<br>
 
@@ -1072,14 +1072,14 @@ where we have used the fact that our data are centered.
 Recalling our definition of the covariance as 
 <p>&nbsp;<br>
 $$
-\boldsymbol{C}[\boldsymbol{x}] = \frac{1}{n}\boldsymbol{X}\boldsymbol{X}^T= \mathbb{E}[\boldsymbol{X}\boldsymbol{X}^T],
+\boldsymbol{C}[\boldsymbol{x}] = \frac{1}{n}\boldsymbol{X}\boldsymbol{X}^T=\mathbb{E}[\boldsymbol{X}\boldsymbol{X}^T],
 $$
 <p>&nbsp;<br>
 
 we have thus that
 <p>&nbsp;<br>
 $$
-\mathrm{var}[\boldsymbol{w}_0^T\boldsymbol{x}_i] = \frac{1}{p}\sum_i z_{i0}^2=\boldsymbol{w}_0^T\boldsymbol{C}[\boldsymbol{x}]\boldsymbol{w}_0.
+\mathrm{var}[\boldsymbol{w}_0^T\boldsymbol{x}_i] = \frac{1}{n}\sum_i z_{i0}^2=\boldsymbol{w}_0^T\boldsymbol{C}[\boldsymbol{x}]\boldsymbol{w}_0.
 $$
 <p>&nbsp;<br>
 

doc/pub/DimRed/html/DimRed-solarized.html

@@ -964,14 +964,14 @@ each with dimension \( \boldsymbol{x}\in {\mathbb{R}}^{n} \).
 <p>
 We assume also that we have an orthogonal transformation \( \boldsymbol{W}\in {\mathbb{R}}^{p\times p} \).  We define the reconstruction error (which is similar to the mean squared error we have seen before) as
 $$
-J(\boldsymbol{W},\boldsymbol{Z}) = \frac{1}{p}\sum_i (\boldsymbol{x}_i - \overline{\boldsymbol{x}_i})^2,
+J(\boldsymbol{W},\boldsymbol{Z}) = \frac{1}{n}\sum_i (\boldsymbol{x}_i - \overline{\boldsymbol{x}_i})^2,
 $$
 
 with \( \overline{\boldsymbol{x}_i} = \boldsymbol{W}\boldsymbol{z}_i \), where \( \boldsymbol{z}_i \) is a row vector with dimension \( {\mathbb{R}}^{n} \) of the matrix
-\( \boldsymbol{Z}\in {\mathbb{R}}^{p\times n} \).  When doing PCA we want to reduce this dimensionality.
+\( \boldsymbol{Z}\in{\mathbb{R}}^{p\times n} \).  When doing PCA we want to reduce this dimensionality.
 
 <p>
-The PCA theorem states that minimizing the above reconstruction error corresponds to setting \( \boldsymbol{W}=\boldsymbol{S} \), the orthogonal matrix which diagonalizes the empirical covariance(correlation) matrix. The optimal low-dimensional encoding of the data is then given by a set of vectors \( l \) $\boldsymbol{z}_i$, with \( l < < p \), defined by the orthogonal projection of the data onto the columns spanned by the eigenvectors of the covariance(correlations matrix).
+The PCA theorem states that minimizing the above reconstruction error corresponds to setting \( \boldsymbol{W}=\boldsymbol{S} \), the orthogonal matrix which diagonalizes the empirical covariance(correlation) matrix. The optimal low-dimensional encoding of the data is then given by a set of vectors \( \boldsymbol{z}_i \) with at most \( l \) vectors, with \( l < < p \), defined by the orthogonal projection of the data onto the columns spanned by the eigenvectors of the covariance(correlations matrix).
 
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
@@ -981,12 +981,12 @@ The PCA theorem states that minimizing the above reconstruction error correspond
 <p>
 To show the PCA theorem let us start with the assumption that there is one vector \( \boldsymbol{w}_0 \) which corresponds to a solution which minimized the reconstruction error \( J \). This is an orthogonal vector. It means that we now approximate the reconstruction error in terms of \( \boldsymbol{w}_0 \) and \( \boldsymbol{z}_0 \) as
 $$
-J(\boldsymbol{w}_0,\boldsymbol{z}_0)= \frac{1}{p}\sum_i (\boldsymbol{x}_i - z_{i0}\boldsymbol{w}_0)^2=\frac{1}{p}\sum_i (\boldsymbol{x}_i^T\boldsymbol{x}_i - 2z_{i0}\boldsymbol{w}_0^T\boldsymbol{x}_i+z_{i0}^2\boldsymbol{w}_0^T\boldsymbol{w}_0),
+J(\boldsymbol{w}_0,\boldsymbol{z}_0)= \frac{1}{n}\sum_i (\boldsymbol{x}_i - z_{i0}\boldsymbol{w}_0)^2=\frac{1}{n}\sum_i (\boldsymbol{x}_i^T\boldsymbol{x}_i - 2z_{i0}\boldsymbol{w}_0^T\boldsymbol{x}_i+z_{i0}^2\boldsymbol{w}_0^T\boldsymbol{w}_0),
 $$
 
 which we can rewrite due to the orthogonality of \( \boldsymbol{w}_i \) as
 $$
-J(\boldsymbol{w}_0,\boldsymbol{z}_0)=\frac{1}{p}\sum_i (\boldsymbol{x}_i^T\boldsymbol{x}_i - 2z_{i0}\boldsymbol{w}_0^T\boldsymbol{x}_i+z_{i0}^2).
+J(\boldsymbol{w}_0,\boldsymbol{z}_0)=\frac{1}{n}\sum_i (\boldsymbol{x}_i^T\boldsymbol{x}_i - 2z_{i0}\boldsymbol{w}_0^T\boldsymbol{x}_i+z_{i0}^2).
 $$
 
 Minimizing \( J \) with respect to the unknown parameters \( z_{0i} \) we obtain that
@@ -1002,15 +1002,15 @@ where the vectors on the rhs are known.
 <h2 id="___sec19">PCA Proof continued </h2>
 
 <p>
-We have now found the unknown parameters \( z_{i0} \). These correspond to the project coordinates and we can write
+We have now found the unknown parameters \( z_{i0} \). These correspond to the projected coordinates and we can write
 $$
-J(\boldsymbol{w}_0)= \frac{1}{p}\sum_i (\boldsymbol{x}_i^T\boldsymbol{x}_i - z_{i0}^2)=\mathrm{const}-\frac{1}{p}\sum_i z_{i0}^2.
+J(\boldsymbol{w}_0)= \frac{1}{p}\sum_i (\boldsymbol{x}_i^T\boldsymbol{x}_i - z_{i0}^2)=\mathrm{const}-\frac{1}{n}\sum_i z_{i0}^2.
 $$
 
 <p>
-We can show that the variance of the project coordinates defined by \( \boldsymbol{w}_0^T\boldsymbol{x}_i \) are given by
+We can show that the variance of the projected coordinates defined by \( \boldsymbol{w}_0^T\boldsymbol{x}_i \) are given by
 $$
-\mathrm{var}[\boldsymbol{w}_0^T\boldsymbol{x}_i] = \frac{1}{p}\sum_i z_{i0}^2,
+\mathrm{var}[\boldsymbol{w}_0^T\boldsymbol{x}_i] = \frac{1}{n}\sum_i z_{i0}^2,
 $$
 
 since the expectation value of
@@ -1023,12 +1023,12 @@ where we have used the fact that our data are centered.
 <p>
 Recalling our definition of the covariance as 
 $$
-\boldsymbol{C}[\boldsymbol{x}] = \frac{1}{n}\boldsymbol{X}\boldsymbol{X}^T= \mathbb{E}[\boldsymbol{X}\boldsymbol{X}^T],
+\boldsymbol{C}[\boldsymbol{x}] = \frac{1}{n}\boldsymbol{X}\boldsymbol{X}^T=\mathbb{E}[\boldsymbol{X}\boldsymbol{X}^T],
 $$
 
 we have thus that
 $$
-\mathrm{var}[\boldsymbol{w}_0^T\boldsymbol{x}_i] = \frac{1}{p}\sum_i z_{i0}^2=\boldsymbol{w}_0^T\boldsymbol{C}[\boldsymbol{x}]\boldsymbol{w}_0.
+\mathrm{var}[\boldsymbol{w}_0^T\boldsymbol{x}_i] = \frac{1}{n}\sum_i z_{i0}^2=\boldsymbol{w}_0^T\boldsymbol{C}[\boldsymbol{x}]\boldsymbol{w}_0.
 $$
 
 <p>

doc/pub/DimRed/html/DimRed.html

@@ -969,14 +969,14 @@ each with dimension \( \boldsymbol{x}\in {\mathbb{R}}^{n} \).
 <p>
 We assume also that we have an orthogonal transformation \( \boldsymbol{W}\in {\mathbb{R}}^{p\times p} \).  We define the reconstruction error (which is similar to the mean squared error we have seen before) as
 $$
-J(\boldsymbol{W},\boldsymbol{Z}) = \frac{1}{p}\sum_i (\boldsymbol{x}_i - \overline{\boldsymbol{x}_i})^2,
+J(\boldsymbol{W},\boldsymbol{Z}) = \frac{1}{n}\sum_i (\boldsymbol{x}_i - \overline{\boldsymbol{x}_i})^2,
 $$
 
 with \( \overline{\boldsymbol{x}_i} = \boldsymbol{W}\boldsymbol{z}_i \), where \( \boldsymbol{z}_i \) is a row vector with dimension \( {\mathbb{R}}^{n} \) of the matrix
-\( \boldsymbol{Z}\in {\mathbb{R}}^{p\times n} \).  When doing PCA we want to reduce this dimensionality.
+\( \boldsymbol{Z}\in{\mathbb{R}}^{p\times n} \).  When doing PCA we want to reduce this dimensionality.
 
 <p>
-The PCA theorem states that minimizing the above reconstruction error corresponds to setting \( \boldsymbol{W}=\boldsymbol{S} \), the orthogonal matrix which diagonalizes the empirical covariance(correlation) matrix. The optimal low-dimensional encoding of the data is then given by a set of vectors \( l \) $\boldsymbol{z}_i$, with \( l < < p \), defined by the orthogonal projection of the data onto the columns spanned by the eigenvectors of the covariance(correlations matrix).
+The PCA theorem states that minimizing the above reconstruction error corresponds to setting \( \boldsymbol{W}=\boldsymbol{S} \), the orthogonal matrix which diagonalizes the empirical covariance(correlation) matrix. The optimal low-dimensional encoding of the data is then given by a set of vectors \( \boldsymbol{z}_i \) with at most \( l \) vectors, with \( l < < p \), defined by the orthogonal projection of the data onto the columns spanned by the eigenvectors of the covariance(correlations matrix).
 
 <p>
 <!-- !split --><br><br><br><br><br><br><br><br><br><br>
@@ -986,12 +986,12 @@ The PCA theorem states that minimizing the above reconstruction error correspond
 <p>
 To show the PCA theorem let us start with the assumption that there is one vector \( \boldsymbol{w}_0 \) which corresponds to a solution which minimized the reconstruction error \( J \). This is an orthogonal vector. It means that we now approximate the reconstruction error in terms of \( \boldsymbol{w}_0 \) and \( \boldsymbol{z}_0 \) as
 $$
-J(\boldsymbol{w}_0,\boldsymbol{z}_0)= \frac{1}{p}\sum_i (\boldsymbol{x}_i - z_{i0}\boldsymbol{w}_0)^2=\frac{1}{p}\sum_i (\boldsymbol{x}_i^T\boldsymbol{x}_i - 2z_{i0}\boldsymbol{w}_0^T\boldsymbol{x}_i+z_{i0}^2\boldsymbol{w}_0^T\boldsymbol{w}_0),
+J(\boldsymbol{w}_0,\boldsymbol{z}_0)= \frac{1}{n}\sum_i (\boldsymbol{x}_i - z_{i0}\boldsymbol{w}_0)^2=\frac{1}{n}\sum_i (\boldsymbol{x}_i^T\boldsymbol{x}_i - 2z_{i0}\boldsymbol{w}_0^T\boldsymbol{x}_i+z_{i0}^2\boldsymbol{w}_0^T\boldsymbol{w}_0),
 $$
 
 which we can rewrite due to the orthogonality of \( \boldsymbol{w}_i \) as
 $$
-J(\boldsymbol{w}_0,\boldsymbol{z}_0)=\frac{1}{p}\sum_i (\boldsymbol{x}_i^T\boldsymbol{x}_i - 2z_{i0}\boldsymbol{w}_0^T\boldsymbol{x}_i+z_{i0}^2).
+J(\boldsymbol{w}_0,\boldsymbol{z}_0)=\frac{1}{n}\sum_i (\boldsymbol{x}_i^T\boldsymbol{x}_i - 2z_{i0}\boldsymbol{w}_0^T\boldsymbol{x}_i+z_{i0}^2).
 $$
 
 Minimizing \( J \) with respect to the unknown parameters \( z_{0i} \) we obtain that
@@ -1007,15 +1007,15 @@ where the vectors on the rhs are known.
 <h2 id="___sec19">PCA Proof continued </h2>
 
 <p>
-We have now found the unknown parameters \( z_{i0} \). These correspond to the project coordinates and we can write
+We have now found the unknown parameters \( z_{i0} \). These correspond to the projected coordinates and we can write
 $$
-J(\boldsymbol{w}_0)= \frac{1}{p}\sum_i (\boldsymbol{x}_i^T\boldsymbol{x}_i - z_{i0}^2)=\mathrm{const}-\frac{1}{p}\sum_i z_{i0}^2.
+J(\boldsymbol{w}_0)= \frac{1}{p}\sum_i (\boldsymbol{x}_i^T\boldsymbol{x}_i - z_{i0}^2)=\mathrm{const}-\frac{1}{n}\sum_i z_{i0}^2.
 $$
 
 <p>
-We can show that the variance of the project coordinates defined by \( \boldsymbol{w}_0^T\boldsymbol{x}_i \) are given by
+We can show that the variance of the projected coordinates defined by \( \boldsymbol{w}_0^T\boldsymbol{x}_i \) are given by
 $$
-\mathrm{var}[\boldsymbol{w}_0^T\boldsymbol{x}_i] = \frac{1}{p}\sum_i z_{i0}^2,
+\mathrm{var}[\boldsymbol{w}_0^T\boldsymbol{x}_i] = \frac{1}{n}\sum_i z_{i0}^2,
 $$
 
 since the expectation value of
@@ -1028,12 +1028,12 @@ where we have used the fact that our data are centered.
 <p>
 Recalling our definition of the covariance as 
 $$
-\boldsymbol{C}[\boldsymbol{x}] = \frac{1}{n}\boldsymbol{X}\boldsymbol{X}^T= \mathbb{E}[\boldsymbol{X}\boldsymbol{X}^T],
+\boldsymbol{C}[\boldsymbol{x}] = \frac{1}{n}\boldsymbol{X}\boldsymbol{X}^T=\mathbb{E}[\boldsymbol{X}\boldsymbol{X}^T],
 $$
 
 we have thus that
 $$
-\mathrm{var}[\boldsymbol{w}_0^T\boldsymbol{x}_i] = \frac{1}{p}\sum_i z_{i0}^2=\boldsymbol{w}_0^T\boldsymbol{C}[\boldsymbol{x}]\boldsymbol{w}_0.
+\mathrm{var}[\boldsymbol{w}_0^T\boldsymbol{x}_i] = \frac{1}{n}\sum_i z_{i0}^2=\boldsymbol{w}_0^T\boldsymbol{C}[\boldsymbol{x}]\boldsymbol{w}_0.
 $$
 
 <p>

doc/pub/DimRed/ipynb/DimRed.ipynb

@@ -1064,7 +1064,7 @@
    "metadata": {},
    "source": [
     "$$\n",
-    "J(\\boldsymbol{W},\\boldsymbol{Z}) = \\frac{1}{p}\\sum_i (\\boldsymbol{x}_i - \\overline{\\boldsymbol{x}_i})^2,\n",
+    "J(\\boldsymbol{W},\\boldsymbol{Z}) = \\frac{1}{n}\\sum_i (\\boldsymbol{x}_i - \\overline{\\boldsymbol{x}_i})^2,\n",
     "$$"
    ]
   },
@@ -1073,9 +1073,9 @@
    "metadata": {},
    "source": [
     "with $\\overline{\\boldsymbol{x}_i} = \\boldsymbol{W}\\boldsymbol{z}_i$, where $\\boldsymbol{z}_i$ is a row vector with dimension ${\\mathbb{R}}^{n}$ of the matrix\n",
-    "$\\boldsymbol{Z}\\in {\\mathbb{R}}^{p\\times n}$.  When doing PCA we want to reduce this dimensionality. \n",
+    "$\\boldsymbol{Z}\\in{\\mathbb{R}}^{p\\times n}$.  When doing PCA we want to reduce this dimensionality. \n",
     "\n",
-    "The PCA theorem states that minimizing the above reconstruction error corresponds to setting $\\boldsymbol{W}=\\boldsymbol{S}$, the orthogonal matrix which diagonalizes the empirical covariance(correlation) matrix. The optimal low-dimensional encoding of the data is then given by a set of vectors $l$ $\\boldsymbol{z}_i$, with $l << p$, defined by the orthogonal projection of the data onto the columns spanned by the eigenvectors of the covariance(correlations matrix).\n",
+    "The PCA theorem states that minimizing the above reconstruction error corresponds to setting $\\boldsymbol{W}=\\boldsymbol{S}$, the orthogonal matrix which diagonalizes the empirical covariance(correlation) matrix. The optimal low-dimensional encoding of the data is then given by a set of vectors $\\boldsymbol{z}_i$ with at most $l$ vectors, with $l << p$, defined by the orthogonal projection of the data onto the columns spanned by the eigenvectors of the covariance(correlations matrix).\n",
     "\n",
     "\n",
     "\n",
@@ -1089,7 +1089,7 @@
    "metadata": {},
    "source": [
     "$$\n",
-    "J(\\boldsymbol{w}_0,\\boldsymbol{z}_0)= \\frac{1}{p}\\sum_i (\\boldsymbol{x}_i - z_{i0}\\boldsymbol{w}_0)^2=\\frac{1}{p}\\sum_i (\\boldsymbol{x}_i^T\\boldsymbol{x}_i - 2z_{i0}\\boldsymbol{w}_0^T\\boldsymbol{x}_i+z_{i0}^2\\boldsymbol{w}_0^T\\boldsymbol{w}_0),\n",
+    "J(\\boldsymbol{w}_0,\\boldsymbol{z}_0)= \\frac{1}{n}\\sum_i (\\boldsymbol{x}_i - z_{i0}\\boldsymbol{w}_0)^2=\\frac{1}{n}\\sum_i (\\boldsymbol{x}_i^T\\boldsymbol{x}_i - 2z_{i0}\\boldsymbol{w}_0^T\\boldsymbol{x}_i+z_{i0}^2\\boldsymbol{w}_0^T\\boldsymbol{w}_0),\n",
     "$$"
    ]
   },
@@ -1105,7 +1105,7 @@
    "metadata": {},
    "source": [
     "$$\n",
-    "J(\\boldsymbol{w}_0,\\boldsymbol{z}_0)=\\frac{1}{p}\\sum_i (\\boldsymbol{x}_i^T\\boldsymbol{x}_i - 2z_{i0}\\boldsymbol{w}_0^T\\boldsymbol{x}_i+z_{i0}^2).\n",
+    "J(\\boldsymbol{w}_0,\\boldsymbol{z}_0)=\\frac{1}{n}\\sum_i (\\boldsymbol{x}_i^T\\boldsymbol{x}_i - 2z_{i0}\\boldsymbol{w}_0^T\\boldsymbol{x}_i+z_{i0}^2).\n",
     "$$"
    ]
   },
@@ -1134,7 +1134,7 @@
     "\n",
     "## PCA Proof continued\n",
     "\n",
-    "We have now found the unknown parameters $z_{i0}$. These correspond to the project coordinates and we can write"
+    "We have now found the unknown parameters $z_{i0}$. These correspond to the projected coordinates and we can write"
    ]
   },
   {
@@ -1142,7 +1142,7 @@
    "metadata": {},
    "source": [
     "$$\n",
-    "J(\\boldsymbol{w}_0)= \\frac{1}{p}\\sum_i (\\boldsymbol{x}_i^T\\boldsymbol{x}_i - z_{i0}^2)=\\mathrm{const}-\\frac{1}{p}\\sum_i z_{i0}^2.\n",
+    "J(\\boldsymbol{w}_0)= \\frac{1}{p}\\sum_i (\\boldsymbol{x}_i^T\\boldsymbol{x}_i - z_{i0}^2)=\\mathrm{const}-\\frac{1}{n}\\sum_i z_{i0}^2.\n",
     "$$"
    ]
   },
@@ -1150,7 +1150,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "We can show that the variance of the project coordinates defined by $\\boldsymbol{w}_0^T\\boldsymbol{x}_i$ are given by"
+    "We can show that the variance of the projected coordinates defined by $\\boldsymbol{w}_0^T\\boldsymbol{x}_i$ are given by"
    ]
   },
   {
@@ -1158,7 +1158,7 @@
    "metadata": {},
    "source": [
     "$$\n",
-    "\\mathrm{var}[\\boldsymbol{w}_0^T\\boldsymbol{x}_i] = \\frac{1}{p}\\sum_i z_{i0}^2,\n",
+    "\\mathrm{var}[\\boldsymbol{w}_0^T\\boldsymbol{x}_i] = \\frac{1}{n}\\sum_i z_{i0}^2,\n",
     "$$"
    ]
   },
@@ -1192,7 +1192,7 @@
    "metadata": {},
    "source": [
     "$$\n",
-    "\\boldsymbol{C}[\\boldsymbol{x}] = \\frac{1}{n}\\boldsymbol{X}\\boldsymbol{X}^T= \\mathbb{E}[\\boldsymbol{X}\\boldsymbol{X}^T],\n",
+    "\\boldsymbol{C}[\\boldsymbol{x}] = \\frac{1}{n}\\boldsymbol{X}\\boldsymbol{X}^T=\\mathbb{E}[\\boldsymbol{X}\\boldsymbol{X}^T],\n",
     "$$"
    ]
   },
@@ -1208,7 +1208,7 @@
    "metadata": {},
    "source": [
     "$$\n",
-    "\\mathrm{var}[\\boldsymbol{w}_0^T\\boldsymbol{x}_i] = \\frac{1}{p}\\sum_i z_{i0}^2=\\boldsymbol{w}_0^T\\boldsymbol{C}[\\boldsymbol{x}]\\boldsymbol{w}_0.\n",
+    "\\mathrm{var}[\\boldsymbol{w}_0^T\\boldsymbol{x}_i] = \\frac{1}{n}\\sum_i z_{i0}^2=\\boldsymbol{w}_0^T\\boldsymbol{C}[\\boldsymbol{x}]\\boldsymbol{w}_0.\n",
     "$$"
    ]
   },

doc/src/DimRed/DimRed.do.txt

@@ -750,13 +750,13 @@ each with dimension $\bm{x}\in {\mathbb{R}}^{n}$.
 We assume also that we have an orthogonal transformation $\bm{W}\in {\mathbb{R}}^{p\times p}$.  We define the reconstruction error (which is similar to the mean squared error we have seen before) as
 !bt
 \[
-J(\bm{W},\bm{Z}) = \frac{1}{p}\sum_i (\bm{x}_i - \overline{\bm{x}_i})^2,
+J(\bm{W},\bm{Z}) = \frac{1}{n}\sum_i (\bm{x}_i - \overline{\bm{x}_i})^2,
 \]
 !et
 with $\overline{\bm{x}_i} = \bm{W}\bm{z}_i$, where $\bm{z}_i$ is a row vector with dimension ${\mathbb{R}}^{n}$ of the matrix
-$\bm{Z}\in {\mathbb{R}}^{p\times n}$.  When doing PCA we want to reduce this dimensionality. 
+$\bm{Z}\in{\mathbb{R}}^{p\times n}$.  When doing PCA we want to reduce this dimensionality. 
 
-The PCA theorem states that minimizing the above reconstruction error corresponds to setting $\bm{W}=\bm{S}$, the orthogonal matrix which diagonalizes the empirical covariance(correlation) matrix. The optimal low-dimensional encoding of the data is then given by a set of vectors $l$ $\bm{z}_i$, with $l << p$, defined by the orthogonal projection of the data onto the columns spanned by the eigenvectors of the covariance(correlations matrix).
+The PCA theorem states that minimizing the above reconstruction error corresponds to setting $\bm{W}=\bm{S}$, the orthogonal matrix which diagonalizes the empirical covariance(correlation) matrix. The optimal low-dimensional encoding of the data is then given by a set of vectors $\bm{z}_i$ with at most $l$ vectors, with $l << p$, defined by the orthogonal projection of the data onto the columns spanned by the eigenvectors of the covariance(correlations matrix).
 
 
 
@@ -766,13 +766,13 @@ The PCA theorem states that minimizing the above reconstruction error correspond
 To show the PCA theorem let us start with the assumption that there is one vector $\bm{w}_0$ which corresponds to a solution which minimized the reconstruction error $J$. This is an orthogonal vector. It means that we now approximate the reconstruction error in terms of $\bm{w}_0$ and $\bm{z}_0$ as
 !bt
 \[
-J(\bm{w}_0,\bm{z}_0)= \frac{1}{p}\sum_i (\bm{x}_i - z_{i0}\bm{w}_0)^2=\frac{1}{p}\sum_i (\bm{x}_i^T\bm{x}_i - 2z_{i0}\bm{w}_0^T\bm{x}_i+z_{i0}^2\bm{w}_0^T\bm{w}_0),
+J(\bm{w}_0,\bm{z}_0)= \frac{1}{n}\sum_i (\bm{x}_i - z_{i0}\bm{w}_0)^2=\frac{1}{n}\sum_i (\bm{x}_i^T\bm{x}_i - 2z_{i0}\bm{w}_0^T\bm{x}_i+z_{i0}^2\bm{w}_0^T\bm{w}_0),
 \]
 !et
 which we can rewrite due to the orthogonality of $\bm{w}_i$ as
 !bt
 \[
-J(\bm{w}_0,\bm{z}_0)=\frac{1}{p}\sum_i (\bm{x}_i^T\bm{x}_i - 2z_{i0}\bm{w}_0^T\bm{x}_i+z_{i0}^2).
+J(\bm{w}_0,\bm{z}_0)=\frac{1}{n}\sum_i (\bm{x}_i^T\bm{x}_i - 2z_{i0}\bm{w}_0^T\bm{x}_i+z_{i0}^2).
 \]
 !et
 Minimizing $J$ with respect to the unknown parameters $z_{0i}$ we obtain that
@@ -787,17 +787,17 @@ where the vectors on the rhs are known.
 !split
 ===== PCA Proof continued =====
 
-We have now found the unknown parameters $z_{i0}$. These correspond to the project coordinates and we can write
+We have now found the unknown parameters $z_{i0}$. These correspond to the projected coordinates and we can write
 !bt
 \[
-J(\bm{w}_0)= \frac{1}{p}\sum_i (\bm{x}_i^T\bm{x}_i - z_{i0}^2)=\mathrm{const}-\frac{1}{p}\sum_i z_{i0}^2.
+J(\bm{w}_0)= \frac{1}{p}\sum_i (\bm{x}_i^T\bm{x}_i - z_{i0}^2)=\mathrm{const}-\frac{1}{n}\sum_i z_{i0}^2.
 \]
 !et
 
-We can show that the variance of the project coordinates defined by $\bm{w}_0^T\bm{x}_i$ are given by
+We can show that the variance of the projected coordinates defined by $\bm{w}_0^T\bm{x}_i$ are given by
 !bt
 \[
-\mathrm{var}[\bm{w}_0^T\bm{x}_i] = \frac{1}{p}\sum_i z_{i0}^2,
+\mathrm{var}[\bm{w}_0^T\bm{x}_i] = \frac{1}{n}\sum_i z_{i0}^2,
 \]
 !et
 since the expectation value of
@@ -811,13 +811,13 @@ where we have used the fact that our data are centered.
 Recalling our definition of the covariance as 
 !bt
 \[
-\bm{C}[\bm{x}] = \frac{1}{n}\bm{X}\bm{X}^T= \mathbb{E}[\bm{X}\bm{X}^T],
+\bm{C}[\bm{x}] = \frac{1}{n}\bm{X}\bm{X}^T=\mathbb{E}[\bm{X}\bm{X}^T],
 \]
 !et
 we have thus that
 !bt
 \[
-\mathrm{var}[\bm{w}_0^T\bm{x}_i] = \frac{1}{p}\sum_i z_{i0}^2=\bm{w}_0^T\bm{C}[\bm{x}]\bm{w}_0.
+\mathrm{var}[\bm{w}_0^T\bm{x}_i] = \frac{1}{n}\sum_i z_{i0}^2=\bm{w}_0^T\bm{C}[\bm{x}]\bm{w}_0.
 \]
 !et