From 871338800bc7a6a0c190e8d2444ff83db7ae6ac2 Mon Sep 17 00:00:00 2001 From: mhjensen Date: Thu, 24 Oct 2019 06:12:37 +0200 Subject: [PATCH] fixing minor typos --- doc/pub/DimRed/html/._DimRed-bs019.html | 4 +-- doc/pub/DimRed/html/._DimRed-bs021.html | 22 +++++++++++++--- doc/pub/DimRed/html/DimRed-reveal.html | 26 +++++++++++++++---- doc/pub/DimRed/html/DimRed-solarized.html | 26 +++++++++++++++---- doc/pub/DimRed/html/DimRed.html | 26 +++++++++++++++---- doc/pub/DimRed/ipynb/DimRed.ipynb | 24 +++++++++++++---- doc/pub/DimRed/ipynb/ipynb-DimRed-src.tar.gz | Bin 191 -> 190 bytes doc/pub/DimRed/pdf/DimRed-minted.pdf | Bin 285672 -> 253634 bytes doc/src/DimRed/DimRed.do.txt | 25 ++++++++++++++---- 9 files changed, 123 insertions(+), 30 deletions(-) diff --git a/doc/pub/DimRed/html/._DimRed-bs019.html b/doc/pub/DimRed/html/._DimRed-bs019.html index 92de15a5a..e4746e5e3 100644 --- a/doc/pub/DimRed/html/._DimRed-bs019.html +++ b/doc/pub/DimRed/html/._DimRed-bs019.html @@ -189,12 +189,12 @@ MathJax.Hub.Config({

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 $$ -J(\boldsymbol{w}_0,\boldsymbol{z}_0)= \frac{1}{p}\sum_i (\boldsymbol{x}_i - z_{i0}\boldsymbol{w}_0)^2=\frac{1}{p}\sum_i (\boldsymbol{x}_^T\boldsymbol{x}_i - 2z_{i0}\boldsymbol{w}_0^T\boldsymbol{x}_i+z_{i0}^2\boldsymbol{w}_0^T\boldsymbol{w}_0), +J(\boldsymbol{w}_0,\boldsymbol{z}_0)= \frac{1}{p}\sum_i (\boldsymbol{x}_i - z_{i0}\boldsymbol{w}_0)^2=\frac{1}{p}\sum_i (\boldsymbol{x}_i^T\boldsymbol{x}_i - 2z_{i0}\boldsymbol{w}_0^T\boldsymbol{x}_i+z_{i0}^2\boldsymbol{w}_0^T\boldsymbol{w}_0), $$ which we can rewrite due to the orthogonality of \( \boldsymbol{w}_i \) as $$ -J(\boldsymbol{w}_0,\boldsymbol{z}_0)=\frac{1}{p}\sum_i (\boldsymbol{x}_^T\boldsymbol{x}_i - 2z_{i0}\boldsymbol{w}_0^T\boldsymbol{x}_i+z_{i0}^2). +J(\boldsymbol{w}_0,\boldsymbol{z}_0)=\frac{1}{p}\sum_i (\boldsymbol{x}_i^T\boldsymbol{x}_i - 2z_{i0}\boldsymbol{w}_0^T\boldsymbol{x}_i+z_{i0}^2). $$ Minimizing \( J \) with respect to the unknown parameters \( z_{0i} \) we obtain that diff --git a/doc/pub/DimRed/html/._DimRed-bs021.html b/doc/pub/DimRed/html/._DimRed-bs021.html index 8d010f1b0..9fae4423f 100644 --- a/doc/pub/DimRed/html/._DimRed-bs021.html +++ b/doc/pub/DimRed/html/._DimRed-bs021.html @@ -191,7 +191,7 @@ We could trivially maximize the variance of the projection (and thereby minimize the error in the reconstruction function) by letting the norm-2 of \( \boldsymbol{w}_0 \) go to infinity. However, this norm since we want the matrix \( \boldsymbol{W} \) to be an orthogonal matrix, is constrained by -\( $\vert\vert \boldsymbol{w}_0 \vert\vert_2^2=1 \). Imposing this condition via a +\( \vert\vert \boldsymbol{w}_0 \vert\vert_2^2=1 \). Imposing this condition via a Lagrange multiplier we can then in turn maximize $$ @@ -211,10 +211,26 @@ $$ The direction that maximizes the variance (or minimizes the construction error) is an eigenvector of the covariance matrix! If we left multiply with \( \boldsymbol{w}_0^T \) we have the variance of the projected data is $$ -\boldsymbol{w}_^T\boldsymbol{C}[\boldsymbol{x}]\boldsymbol{w}_0=\lambda_0. +\boldsymbol{w}_0^T\boldsymbol{C}[\boldsymbol{x}]\boldsymbol{w}_0=\lambda_0. $$ -If we want to maximize the variance (minimize the construction error) we simply pick the eigenvector of the covariance matrix with the largest eigenvalue. This establishes the link between the minimization of the reconstruction function \( J \) in terms of an orthogonal matrix and the maximization of the variance and thereby the covariance of our observations encoded in the design/feature matrix \( \boldsymbol{X} \). The proof for the other eigenvectors \( \boldsymbol{w}_1,\boldsymbol{w}_2,\dots \) cna be established by applying the above arguments and using the fact that basis of eigenvectors is orthogonal, see Murphy chapter 12.2. The discussion in chapter 12.2 of Murphy's text has also a nice link with the Singular Value Decomposition theorem. For categorical data, see chapter 12.4 and discussion therein. +

+If we want to maximize the variance (minimize the construction error) +we simply pick the eigenvector of the covariance matrix with the +largest eigenvalue. This establishes the link between the minimization +of the reconstruction function \( J \) in terms of an orthogonal matrix +and the maximization of the variance and thereby the covariance of our +observations encoded in the design/feature matrix \( \boldsymbol{X} \). + +

+The proof +for the other eigenvectors \( \boldsymbol{w}_1,\boldsymbol{w}_2,\dots \) can be +established by applying the above arguments and using the fact that +our basis of eigenvectors is orthogonal, see Murphy chapter +12.2. The +discussion in chapter 12.2 of Murphy's text has also a nice link with +the Singular Value Decomposition theorem. For categorical data, see +chapter 12.4 and discussion therein.

diff --git a/doc/pub/DimRed/html/DimRed-reveal.html b/doc/pub/DimRed/html/DimRed-reveal.html index 217576ba5..fb83652a5 100644 --- a/doc/pub/DimRed/html/DimRed-reveal.html +++ b/doc/pub/DimRed/html/DimRed-reveal.html @@ -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

 
$$ -J(\boldsymbol{w}_0,\boldsymbol{z}_0)= \frac{1}{p}\sum_i (\boldsymbol{x}_i - z_{i0}\boldsymbol{w}_0)^2=\frac{1}{p}\sum_i (\boldsymbol{x}_^T\boldsymbol{x}_i - 2z_{i0}\boldsymbol{w}_0^T\boldsymbol{x}_i+z_{i0}^2\boldsymbol{w}_0^T\boldsymbol{w}_0), +J(\boldsymbol{w}_0,\boldsymbol{z}_0)= \frac{1}{p}\sum_i (\boldsymbol{x}_i - z_{i0}\boldsymbol{w}_0)^2=\frac{1}{p}\sum_i (\boldsymbol{x}_i^T\boldsymbol{x}_i - 2z_{i0}\boldsymbol{w}_0^T\boldsymbol{x}_i+z_{i0}^2\boldsymbol{w}_0^T\boldsymbol{w}_0), $$

 
which we can rewrite due to the orthogonality of \( \boldsymbol{w}_i \) as

 
$$ -J(\boldsymbol{w}_0,\boldsymbol{z}_0)=\frac{1}{p}\sum_i (\boldsymbol{x}_^T\boldsymbol{x}_i - 2z_{i0}\boldsymbol{w}_0^T\boldsymbol{x}_i+z_{i0}^2). +J(\boldsymbol{w}_0,\boldsymbol{z}_0)=\frac{1}{p}\sum_i (\boldsymbol{x}_i^T\boldsymbol{x}_i - 2z_{i0}\boldsymbol{w}_0^T\boldsymbol{x}_i+z_{i0}^2). $$

 
@@ -1096,7 +1096,7 @@ We could trivially maximize the variance of the projection (and thereby minimize the error in the reconstruction function) by letting the norm-2 of \( \boldsymbol{w}_0 \) go to infinity. However, this norm since we want the matrix \( \boldsymbol{W} \) to be an orthogonal matrix, is constrained by -\( $\vert\vert \boldsymbol{w}_0 \vert\vert_2^2=1 \). Imposing this condition via a +\( \vert\vert \boldsymbol{w}_0 \vert\vert_2^2=1 \). Imposing this condition via a Lagrange multiplier we can then in turn maximize

 
@@ -1123,11 +1123,27 @@ $$ The direction that maximizes the variance (or minimizes the construction error) is an eigenvector of the covariance matrix! If we left multiply with \( \boldsymbol{w}_0^T \) we have the variance of the projected data is

 
$$ -\boldsymbol{w}_^T\boldsymbol{C}[\boldsymbol{x}]\boldsymbol{w}_0=\lambda_0. +\boldsymbol{w}_0^T\boldsymbol{C}[\boldsymbol{x}]\boldsymbol{w}_0=\lambda_0. $$

 
-If we want to maximize the variance (minimize the construction error) we simply pick the eigenvector of the covariance matrix with the largest eigenvalue. This establishes the link between the minimization of the reconstruction function \( J \) in terms of an orthogonal matrix and the maximization of the variance and thereby the covariance of our observations encoded in the design/feature matrix \( \boldsymbol{X} \). The proof for the other eigenvectors \( \boldsymbol{w}_1,\boldsymbol{w}_2,\dots \) cna be established by applying the above arguments and using the fact that basis of eigenvectors is orthogonal, see Murphy chapter 12.2. The discussion in chapter 12.2 of Murphy's text has also a nice link with the Singular Value Decomposition theorem. For categorical data, see chapter 12.4 and discussion therein. +

+If we want to maximize the variance (minimize the construction error) +we simply pick the eigenvector of the covariance matrix with the +largest eigenvalue. This establishes the link between the minimization +of the reconstruction function \( J \) in terms of an orthogonal matrix +and the maximization of the variance and thereby the covariance of our +observations encoded in the design/feature matrix \( \boldsymbol{X} \). + +

+The proof +for the other eigenvectors \( \boldsymbol{w}_1,\boldsymbol{w}_2,\dots \) can be +established by applying the above arguments and using the fact that +our basis of eigenvectors is orthogonal, see Murphy chapter +12.2. The +discussion in chapter 12.2 of Murphy's text has also a nice link with +the Singular Value Decomposition theorem. For categorical data, see +chapter 12.4 and discussion therein. diff --git a/doc/pub/DimRed/html/DimRed-solarized.html b/doc/pub/DimRed/html/DimRed-solarized.html index e9cc33dae..d48c7a463 100644 --- a/doc/pub/DimRed/html/DimRed-solarized.html +++ b/doc/pub/DimRed/html/DimRed-solarized.html @@ -981,12 +981,12 @@ 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 $$ -J(\boldsymbol{w}_0,\boldsymbol{z}_0)= \frac{1}{p}\sum_i (\boldsymbol{x}_i - z_{i0}\boldsymbol{w}_0)^2=\frac{1}{p}\sum_i (\boldsymbol{x}_^T\boldsymbol{x}_i - 2z_{i0}\boldsymbol{w}_0^T\boldsymbol{x}_i+z_{i0}^2\boldsymbol{w}_0^T\boldsymbol{w}_0), +J(\boldsymbol{w}_0,\boldsymbol{z}_0)= \frac{1}{p}\sum_i (\boldsymbol{x}_i - z_{i0}\boldsymbol{w}_0)^2=\frac{1}{p}\sum_i (\boldsymbol{x}_i^T\boldsymbol{x}_i - 2z_{i0}\boldsymbol{w}_0^T\boldsymbol{x}_i+z_{i0}^2\boldsymbol{w}_0^T\boldsymbol{w}_0), $$ which we can rewrite due to the orthogonality of \( \boldsymbol{w}_i \) as $$ -J(\boldsymbol{w}_0,\boldsymbol{z}_0)=\frac{1}{p}\sum_i (\boldsymbol{x}_^T\boldsymbol{x}_i - 2z_{i0}\boldsymbol{w}_0^T\boldsymbol{x}_i+z_{i0}^2). +J(\boldsymbol{w}_0,\boldsymbol{z}_0)=\frac{1}{p}\sum_i (\boldsymbol{x}_i^T\boldsymbol{x}_i - 2z_{i0}\boldsymbol{w}_0^T\boldsymbol{x}_i+z_{i0}^2). $$ Minimizing \( J \) with respect to the unknown parameters \( z_{0i} \) we obtain that @@ -1044,7 +1044,7 @@ We could trivially maximize the variance of the projection (and thereby minimize the error in the reconstruction function) by letting the norm-2 of \( \boldsymbol{w}_0 \) go to infinity. However, this norm since we want the matrix \( \boldsymbol{W} \) to be an orthogonal matrix, is constrained by -\( $\vert\vert \boldsymbol{w}_0 \vert\vert_2^2=1 \). Imposing this condition via a +\( \vert\vert \boldsymbol{w}_0 \vert\vert_2^2=1 \). Imposing this condition via a Lagrange multiplier we can then in turn maximize $$ @@ -1064,10 +1064,26 @@ $$ The direction that maximizes the variance (or minimizes the construction error) is an eigenvector of the covariance matrix! If we left multiply with \( \boldsymbol{w}_0^T \) we have the variance of the projected data is $$ -\boldsymbol{w}_^T\boldsymbol{C}[\boldsymbol{x}]\boldsymbol{w}_0=\lambda_0. +\boldsymbol{w}_0^T\boldsymbol{C}[\boldsymbol{x}]\boldsymbol{w}_0=\lambda_0. $$ -If we want to maximize the variance (minimize the construction error) we simply pick the eigenvector of the covariance matrix with the largest eigenvalue. This establishes the link between the minimization of the reconstruction function \( J \) in terms of an orthogonal matrix and the maximization of the variance and thereby the covariance of our observations encoded in the design/feature matrix \( \boldsymbol{X} \). The proof for the other eigenvectors \( \boldsymbol{w}_1,\boldsymbol{w}_2,\dots \) cna be established by applying the above arguments and using the fact that basis of eigenvectors is orthogonal, see Murphy chapter 12.2. The discussion in chapter 12.2 of Murphy's text has also a nice link with the Singular Value Decomposition theorem. For categorical data, see chapter 12.4 and discussion therein. +

+If we want to maximize the variance (minimize the construction error) +we simply pick the eigenvector of the covariance matrix with the +largest eigenvalue. This establishes the link between the minimization +of the reconstruction function \( J \) in terms of an orthogonal matrix +and the maximization of the variance and thereby the covariance of our +observations encoded in the design/feature matrix \( \boldsymbol{X} \). + +

+The proof +for the other eigenvectors \( \boldsymbol{w}_1,\boldsymbol{w}_2,\dots \) can be +established by applying the above arguments and using the fact that +our basis of eigenvectors is orthogonal, see Murphy chapter +12.2. The +discussion in chapter 12.2 of Murphy's text has also a nice link with +the Singular Value Decomposition theorem. For categorical data, see +chapter 12.4 and discussion therein.











diff --git a/doc/pub/DimRed/html/DimRed.html b/doc/pub/DimRed/html/DimRed.html index e3bc0b44a..effe9e5c9 100644 --- a/doc/pub/DimRed/html/DimRed.html +++ b/doc/pub/DimRed/html/DimRed.html @@ -986,12 +986,12 @@ 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 $$ -J(\boldsymbol{w}_0,\boldsymbol{z}_0)= \frac{1}{p}\sum_i (\boldsymbol{x}_i - z_{i0}\boldsymbol{w}_0)^2=\frac{1}{p}\sum_i (\boldsymbol{x}_^T\boldsymbol{x}_i - 2z_{i0}\boldsymbol{w}_0^T\boldsymbol{x}_i+z_{i0}^2\boldsymbol{w}_0^T\boldsymbol{w}_0), +J(\boldsymbol{w}_0,\boldsymbol{z}_0)= \frac{1}{p}\sum_i (\boldsymbol{x}_i - z_{i0}\boldsymbol{w}_0)^2=\frac{1}{p}\sum_i (\boldsymbol{x}_i^T\boldsymbol{x}_i - 2z_{i0}\boldsymbol{w}_0^T\boldsymbol{x}_i+z_{i0}^2\boldsymbol{w}_0^T\boldsymbol{w}_0), $$ which we can rewrite due to the orthogonality of \( \boldsymbol{w}_i \) as $$ -J(\boldsymbol{w}_0,\boldsymbol{z}_0)=\frac{1}{p}\sum_i (\boldsymbol{x}_^T\boldsymbol{x}_i - 2z_{i0}\boldsymbol{w}_0^T\boldsymbol{x}_i+z_{i0}^2). +J(\boldsymbol{w}_0,\boldsymbol{z}_0)=\frac{1}{p}\sum_i (\boldsymbol{x}_i^T\boldsymbol{x}_i - 2z_{i0}\boldsymbol{w}_0^T\boldsymbol{x}_i+z_{i0}^2). $$ Minimizing \( J \) with respect to the unknown parameters \( z_{0i} \) we obtain that @@ -1049,7 +1049,7 @@ We could trivially maximize the variance of the projection (and thereby minimize the error in the reconstruction function) by letting the norm-2 of \( \boldsymbol{w}_0 \) go to infinity. However, this norm since we want the matrix \( \boldsymbol{W} \) to be an orthogonal matrix, is constrained by -\( $\vert\vert \boldsymbol{w}_0 \vert\vert_2^2=1 \). Imposing this condition via a +\( \vert\vert \boldsymbol{w}_0 \vert\vert_2^2=1 \). Imposing this condition via a Lagrange multiplier we can then in turn maximize $$ @@ -1069,10 +1069,26 @@ $$ The direction that maximizes the variance (or minimizes the construction error) is an eigenvector of the covariance matrix! If we left multiply with \( \boldsymbol{w}_0^T \) we have the variance of the projected data is $$ -\boldsymbol{w}_^T\boldsymbol{C}[\boldsymbol{x}]\boldsymbol{w}_0=\lambda_0. +\boldsymbol{w}_0^T\boldsymbol{C}[\boldsymbol{x}]\boldsymbol{w}_0=\lambda_0. $$ -If we want to maximize the variance (minimize the construction error) we simply pick the eigenvector of the covariance matrix with the largest eigenvalue. This establishes the link between the minimization of the reconstruction function \( J \) in terms of an orthogonal matrix and the maximization of the variance and thereby the covariance of our observations encoded in the design/feature matrix \( \boldsymbol{X} \). The proof for the other eigenvectors \( \boldsymbol{w}_1,\boldsymbol{w}_2,\dots \) cna be established by applying the above arguments and using the fact that basis of eigenvectors is orthogonal, see Murphy chapter 12.2. The discussion in chapter 12.2 of Murphy's text has also a nice link with the Singular Value Decomposition theorem. For categorical data, see chapter 12.4 and discussion therein. +

+If we want to maximize the variance (minimize the construction error) +we simply pick the eigenvector of the covariance matrix with the +largest eigenvalue. This establishes the link between the minimization +of the reconstruction function \( J \) in terms of an orthogonal matrix +and the maximization of the variance and thereby the covariance of our +observations encoded in the design/feature matrix \( \boldsymbol{X} \). + +

+The proof +for the other eigenvectors \( \boldsymbol{w}_1,\boldsymbol{w}_2,\dots \) can be +established by applying the above arguments and using the fact that +our basis of eigenvectors is orthogonal, see Murphy chapter +12.2. The +discussion in chapter 12.2 of Murphy's text has also a nice link with +the Singular Value Decomposition theorem. For categorical data, see +chapter 12.4 and discussion therein.











diff --git a/doc/pub/DimRed/ipynb/DimRed.ipynb b/doc/pub/DimRed/ipynb/DimRed.ipynb index 4ac228c70..24a0aba51 100644 --- a/doc/pub/DimRed/ipynb/DimRed.ipynb +++ b/doc/pub/DimRed/ipynb/DimRed.ipynb @@ -1089,7 +1089,7 @@ "metadata": {}, "source": [ "$$\n", - "J(\\boldsymbol{w}_0,\\boldsymbol{z}_0)= \\frac{1}{p}\\sum_i (\\boldsymbol{x}_i - z_{i0}\\boldsymbol{w}_0)^2=\\frac{1}{p}\\sum_i (\\boldsymbol{x}_^T\\boldsymbol{x}_i - 2z_{i0}\\boldsymbol{w}_0^T\\boldsymbol{x}_i+z_{i0}^2\\boldsymbol{w}_0^T\\boldsymbol{w}_0),\n", + "J(\\boldsymbol{w}_0,\\boldsymbol{z}_0)= \\frac{1}{p}\\sum_i (\\boldsymbol{x}_i - z_{i0}\\boldsymbol{w}_0)^2=\\frac{1}{p}\\sum_i (\\boldsymbol{x}_i^T\\boldsymbol{x}_i - 2z_{i0}\\boldsymbol{w}_0^T\\boldsymbol{x}_i+z_{i0}^2\\boldsymbol{w}_0^T\\boldsymbol{w}_0),\n", "$$" ] }, @@ -1105,7 +1105,7 @@ "metadata": {}, "source": [ "$$\n", - "J(\\boldsymbol{w}_0,\\boldsymbol{z}_0)=\\frac{1}{p}\\sum_i (\\boldsymbol{x}_^T\\boldsymbol{x}_i - 2z_{i0}\\boldsymbol{w}_0^T\\boldsymbol{x}_i+z_{i0}^2).\n", + "J(\\boldsymbol{w}_0,\\boldsymbol{z}_0)=\\frac{1}{p}\\sum_i (\\boldsymbol{x}_i^T\\boldsymbol{x}_i - 2z_{i0}\\boldsymbol{w}_0^T\\boldsymbol{x}_i+z_{i0}^2).\n", "$$" ] }, @@ -1224,7 +1224,7 @@ "thereby minimize the error in the reconstruction function) by letting\n", "the norm-2 of $\\boldsymbol{w}_0$ go to infinity. However, this norm since we\n", "want the matrix $\\boldsymbol{W}$ to be an orthogonal matrix, is constrained by\n", - "$$\\vert\\vert \\boldsymbol{w}_0 \\vert\\vert_2^2=1$. Imposing this condition via a\n", + "$\\vert\\vert \\boldsymbol{w}_0 \\vert\\vert_2^2=1$. Imposing this condition via a\n", "Lagrange multiplier we can then in turn maximize" ] }, @@ -1281,7 +1281,7 @@ "metadata": {}, "source": [ "$$\n", - "\\boldsymbol{w}_^T\\boldsymbol{C}[\\boldsymbol{x}]\\boldsymbol{w}_0=\\lambda_0.\n", + "\\boldsymbol{w}_0^T\\boldsymbol{C}[\\boldsymbol{x}]\\boldsymbol{w}_0=\\lambda_0.\n", "$$" ] }, @@ -1289,7 +1289,21 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "If we want to maximize the variance (minimize the construction error) we simply pick the eigenvector of the covariance matrix with the largest eigenvalue. This establishes the link between the minimization of the reconstruction function $J$ in terms of an orthogonal matrix and the maximization of the variance and thereby the covariance of our observations encoded in the design/feature matrix $\\boldsymbol{X}$. The proof for the other eigenvectors $\\boldsymbol{w}_1,\\boldsymbol{w}_2,\\dots$ cna be established by applying the above arguments and using the fact that basis of eigenvectors is orthogonal, see [Murphy chapter 12.2](https://mitpress.mit.edu/books/machine-learning-1). The discussion in chapter 12.2 of Murphy's text has also a nice link with the Singular Value Decomposition theorem. For categorical data, see chapter 12.4 and discussion therein. \n", + "If we want to maximize the variance (minimize the construction error)\n", + "we simply pick the eigenvector of the covariance matrix with the\n", + "largest eigenvalue. This establishes the link between the minimization\n", + "of the reconstruction function $J$ in terms of an orthogonal matrix\n", + "and the maximization of the variance and thereby the covariance of our\n", + "observations encoded in the design/feature matrix $\\boldsymbol{X}$.\n", + "\n", + "The proof\n", + "for the other eigenvectors $\\boldsymbol{w}_1,\\boldsymbol{w}_2,\\dots$ can be\n", + "established by applying the above arguments and using the fact that\n", + "our basis of eigenvectors is orthogonal, see [Murphy chapter\n", + "12.2](https://mitpress.mit.edu/books/machine-learning-1). The\n", + "discussion in chapter 12.2 of Murphy's text has also a nice link with\n", + "the Singular Value Decomposition theorem. For categorical data, see\n", + "chapter 12.4 and discussion therein.\n", "\n", "\n", "\n", diff --git a/doc/pub/DimRed/ipynb/ipynb-DimRed-src.tar.gz b/doc/pub/DimRed/ipynb/ipynb-DimRed-src.tar.gz index 6437b95f4d00cd6ade8467882bc2fd1fb0901519..e8237b747cae75a688a55bea30f7718d61da2cbf 100644 GIT binary patch literal 190 zcmV;v073sBiwFRwB(Yrp1MSaC3c@fD2H>uHia9|^nxw9UcHu&h;ssKY+Ne!xl7hXx zeSoeMH${Yeo1bBZVWup$`MyZ}-AAiI2xXMQl$(sriPEK>Vaxy%#%MaC86ZpuBN3qW zPI~E`=XN}$IZN%Nemggg)%Axx%Pa8AKXGi7gWX~9jnSZumz6OW$W>cPk*IFa6$m5U s)B-DSy|NOx9)Js>yfT_!uHia9|^+N7?9cHu&h;ssKYny5``l7hXx zeSoeMH${Yeo1bBZVWuj##lB4Z-AAiI2<3#pl$)H+iPEK>VN3yMl#(>XgknHA%P5Kf zt#{H(?>w{PDe^3|6Z-AkIMy^D_AIZ!GylXz&}uDmjuUE|K0lVz<-;$J@#1VIpd?Ex^sW9k41007w1S%?4t diff --git a/doc/pub/DimRed/pdf/DimRed-minted.pdf b/doc/pub/DimRed/pdf/DimRed-minted.pdf index 8b6bcbffa3587cefba9e13900ba33dfae50e9de4..3812702b561dbfdbb3dce13562d2bb7e590f2de5 100644 GIT binary patch delta 38597 zcmZs?Q*0wr$(C*|BYN#mb*vNbJm!r 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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 \[ -J(\bm{w}_0,\bm{z}_0)= \frac{1}{p}\sum_i (\bm{x}_i - z_{i0}\bm{w}_0)^2=\frac{1}{p}\sum_i (\bm{x}_^T\bm{x}_i - 2z_{i0}\bm{w}_0^T\bm{x}_i+z_{i0}^2\bm{w}_0^T\bm{w}_0), +J(\bm{w}_0,\bm{z}_0)= \frac{1}{p}\sum_i (\bm{x}_i - z_{i0}\bm{w}_0)^2=\frac{1}{p}\sum_i (\bm{x}_i^T\bm{x}_i - 2z_{i0}\bm{w}_0^T\bm{x}_i+z_{i0}^2\bm{w}_0^T\bm{w}_0), \] !et which we can rewrite due to the orthogonality of $\bm{w}_i$ as !bt \[ -J(\bm{w}_0,\bm{z}_0)=\frac{1}{p}\sum_i (\bm{x}_^T\bm{x}_i - 2z_{i0}\bm{w}_0^T\bm{x}_i+z_{i0}^2). +J(\bm{w}_0,\bm{z}_0)=\frac{1}{p}\sum_i (\bm{x}_i^T\bm{x}_i - 2z_{i0}\bm{w}_0^T\bm{x}_i+z_{i0}^2). \] !et Minimizing $J$ with respect to the unknown parameters $z_{0i}$ we obtain that @@ -830,7 +830,7 @@ We could trivially maximize the variance of the projection (and thereby minimize the error in the reconstruction function) by letting the norm-2 of $\bm{w}_0$ go to infinity. However, this norm since we want the matrix $\bm{W}$ to be an orthogonal matrix, is constrained by -$$\vert\vert \bm{w}_0 \vert\vert_2^2=1$. Imposing this condition via a +$\vert\vert \bm{w}_0 \vert\vert_2^2=1$. Imposing this condition via a Lagrange multiplier we can then in turn maximize !bt @@ -854,10 +854,25 @@ meaning that _The direction that maximizes the variance (or minimizes the construction error) is an eigenvector of the covariance matrix_! If we left multiply with $\bm{w}_0^T$ we have the variance of the projected data is !bt \[ -\bm{w}_^T\bm{C}[\bm{x}]\bm{w}_0=\lambda_0. +\bm{w}_0^T\bm{C}[\bm{x}]\bm{w}_0=\lambda_0. \] !et -If we want to maximize the variance (minimize the construction error) we simply pick the eigenvector of the covariance matrix with the largest eigenvalue. This establishes the link between the minimization of the reconstruction function $J$ in terms of an orthogonal matrix and the maximization of the variance and thereby the covariance of our observations encoded in the design/feature matrix $\bm{X}$. The proof for the other eigenvectors $\bm{w}_1,\bm{w}_2,\dots$ cna be established by applying the above arguments and using the fact that basis of eigenvectors is orthogonal, see "Murphy chapter 12.2":"https://mitpress.mit.edu/books/machine-learning-1". The discussion in chapter 12.2 of Murphy's text has also a nice link with the Singular Value Decomposition theorem. For categorical data, see chapter 12.4 and discussion therein. + +If we want to maximize the variance (minimize the construction error) +we simply pick the eigenvector of the covariance matrix with the +largest eigenvalue. This establishes the link between the minimization +of the reconstruction function $J$ in terms of an orthogonal matrix +and the maximization of the variance and thereby the covariance of our +observations encoded in the design/feature matrix $\bm{X}$. + +The proof +for the other eigenvectors $\bm{w}_1,\bm{w}_2,\dots$ can be +established by applying the above arguments and using the fact that +our basis of eigenvectors is orthogonal, see "Murphy chapter +12.2":"https://mitpress.mit.edu/books/machine-learning-1". The +discussion in chapter 12.2 of Murphy's text has also a nice link with +the Singular Value Decomposition theorem. For categorical data, see +chapter 12.4 and discussion therein.