From 44ca45170b25963d452f80e7bd018a55cc15baac Mon Sep 17 00:00:00 2001 From: mhjensen Date: Mon, 5 Nov 2018 14:19:55 +0100 Subject: [PATCH] small typo before jumping into kernels --- doc/pub/svm/html/._svm-bs009.html | 8 ++++---- doc/pub/svm/html/._svm-bs012.html | 2 +- doc/pub/svm/html/._svm-bs016.html | 8 ++++---- doc/pub/svm/html/svm-reveal.html | 18 +++++++++--------- doc/pub/svm/html/svm-solarized.html | 18 +++++++++--------- doc/pub/svm/html/svm.html | 18 +++++++++--------- doc/pub/svm/ipynb/ipynb-svm-src.tar.gz | Bin 207 -> 206 bytes doc/pub/svm/ipynb/svm.ipynb | 16 ++++++++-------- doc/pub/svm/pdf/svm-minted.pdf | Bin 231268 -> 231283 bytes doc/src/SupportVMachines/svm.do.txt | 18 +++++++++--------- 10 files changed, 53 insertions(+), 53 deletions(-) diff --git a/doc/pub/svm/html/._svm-bs009.html b/doc/pub/svm/html/._svm-bs009.html index 2d4c4bada..501fff25e 100644 --- a/doc/pub/svm/html/._svm-bs009.html +++ b/doc/pub/svm/html/._svm-bs009.html @@ -140,7 +140,7 @@ Thus, we wish to find a margin \( M \) with \( \hat{w} \) normalized to \( \vert\vert \hat{w}\vert\vert =1 \) subject to the condition $$ -y_i(\hat{w}^T\hat{x}_i+b) \geq M \forall i=1,2,\dots, p. +y_i(\hat{w}^T\hat{x}_i+b) \geq M \hspace{0.1cm}\forall i=1,2,\dots, p. $$ All points are thus at a signed distance from the decision boundary defined by the line \( L \). The parameters \( b \) and \( w_1 \) and \( w_2 \) define this line. @@ -148,18 +148,18 @@ All points are thus at a signed distance from the decision boundary defined by t

We seek thus the largest value \( M \) defined by $$ -\frac{1}{\vert \vert \hat{w}\vert\vert}y_i(\hat{w}^T\hat{x}_i+b) \geq M \forall i=1,2,\dots, n, +\frac{1}{\vert \vert \hat{w}\vert\vert}y_i(\hat{w}^T\hat{x}_i+b) \geq M \hspace{0.1cm}\forall i=1,2,\dots, n, $$ or just $$ -y_i(\hat{w}^T\hat{x}_i+b) \geq M\vert \vert \hat{w}\vert\vert \forall i. +y_i(\hat{w}^T\hat{x}_i+b) \geq M\vert \vert \hat{w}\vert\vert \hspace{0.1cm}\forall i. $$ If we scale the equation so that \( \vert \vert \hat{w}\vert\vert = 1/M \), we have to find the minimum of \( \hat{w}^T\hat{w}=\vert \vert \hat{w}\vert\vert \) (the norm) subject to the condition $$ -y_i(\hat{w}^T\hat{x}_i+b) \geq 1 \forall i. +y_i(\hat{w}^T\hat{x}_i+b) \geq 1 \hspace{0.1cm}\forall i. $$

diff --git a/doc/pub/svm/html/._svm-bs012.html b/doc/pub/svm/html/._svm-bs012.html index 08e2127ab..5c23450ec 100644 --- a/doc/pub/svm/html/._svm-bs012.html +++ b/doc/pub/svm/html/._svm-bs012.html @@ -156,7 +156,7 @@ $$ subject to the constraints \( \lambda_i\geq 0 \) and \( \sum_i\lambda_iy_i=0 \). We must in addition satisfy the Karush-Kuhn-Tucker (KKT) condition $$ -\lambda_i\left[y_i(\hat{w}^T\hat{x}_i+b) -1\right] \forall i. +\lambda_i\left[y_i(\hat{w}^T\hat{x}_i+b) -1\right] \hspace{0.1cm}\forall i. $$ diff --git a/doc/pub/svm/html/._svm-bs016.html b/doc/pub/svm/html/._svm-bs016.html index 4bc13115c..56a68c5ae 100644 --- a/doc/pub/svm/html/._svm-bs016.html +++ b/doc/pub/svm/html/._svm-bs016.html @@ -139,7 +139,7 @@ $$ subject to $$ -y_i(\hat{w}^T\hat{x}_i+b)=1-\xi_i \forall i, +y_i(\hat{w}^T\hat{x}_i+b)=1-\xi_i \hspace{0.1cm}\forall i, $$ with the requirement \( \xi_i\geq 0 \). @@ -157,7 +157,7 @@ $$ and $$ -\lambda_i = C-\gamma_i \forall i. +\lambda_i = C-\gamma_i \hspace{0.1cm}\forall i. $$ Inserting these constraints into the equation for \( {\cal L} \) we obtain the same equation as before @@ -168,7 +168,7 @@ $$ but now subject to the constraints \( \lambda_i\geq 0 \), \( \sum_i\lambda_iy_i=0 \) and $ 0 \leq \lambda_i \leq C$. We must in addition satisfy the Karush-Kuhn-Tucker condition which now reads $$ -\lambda_i\left[y_i(\hat{w}^T\hat{x}_i+b) -(1-\xi_)\right]=0 \forall i, +\lambda_i\left[y_i(\hat{w}^T\hat{x}_i+b) -(1-\xi_)\right]=0 \hspace{0.1cm}\forall i, $$ $$ @@ -177,7 +177,7 @@ $$ and $$ -y_i(\hat{w}^T\hat{x}_i+b) -(1-\xi_) \geq 0 \forall i. +y_i(\hat{w}^T\hat{x}_i+b) -(1-\xi_) \geq 0 \hspace{0.1cm}\forall i. $$

diff --git a/doc/pub/svm/html/svm-reveal.html b/doc/pub/svm/html/svm-reveal.html index 199fc4e85..001aab8ba 100644 --- a/doc/pub/svm/html/svm-reveal.html +++ b/doc/pub/svm/html/svm-reveal.html @@ -416,7 +416,7 @@ Thus, we wish to find a margin \( M \) with \( \hat{w} \) normalized to

 
$$ -y_i(\hat{w}^T\hat{x}_i+b) \geq M \forall i=1,2,\dots, p. +y_i(\hat{w}^T\hat{x}_i+b) \geq M \hspace{0.1cm}\forall i=1,2,\dots, p. $$

 
@@ -426,14 +426,14 @@ All points are thus at a signed distance from the decision boundary defined by t We seek thus the largest value \( M \) defined by

 
$$ -\frac{1}{\vert \vert \hat{w}\vert\vert}y_i(\hat{w}^T\hat{x}_i+b) \geq M \forall i=1,2,\dots, n, +\frac{1}{\vert \vert \hat{w}\vert\vert}y_i(\hat{w}^T\hat{x}_i+b) \geq M \hspace{0.1cm}\forall i=1,2,\dots, n, $$

 
or just

 
$$ -y_i(\hat{w}^T\hat{x}_i+b) \geq M\vert \vert \hat{w}\vert\vert \forall i. +y_i(\hat{w}^T\hat{x}_i+b) \geq M\vert \vert \hat{w}\vert\vert \hspace{0.1cm}\forall i. $$

 
@@ -441,7 +441,7 @@ If we scale the equation so that \( \vert \vert \hat{w}\vert\vert = 1/M \), we h \( \hat{w}^T\hat{w}=\vert \vert \hat{w}\vert\vert \) (the norm) subject to the condition

 
$$ -y_i(\hat{w}^T\hat{x}_i+b) \geq 1 \forall i. +y_i(\hat{w}^T\hat{x}_i+b) \geq 1 \hspace{0.1cm}\forall i. $$

 
@@ -604,7 +604,7 @@ subject to the constraints \( \lambda_i\geq 0 \) and \( \sum_i\lambda_iy_i=0 \). We must in addition satisfy the Karush-Kuhn-Tucker (KKT) condition

 
$$ -\lambda_i\left[y_i(\hat{w}^T\hat{x}_i+b) -1\right] \forall i. +\lambda_i\left[y_i(\hat{w}^T\hat{x}_i+b) -1\right] \hspace{0.1cm}\forall i. $$

 
@@ -737,7 +737,7 @@ $$ subject to

 
$$ -y_i(\hat{w}^T\hat{x}_i+b)=1-\xi_i \forall i, +y_i(\hat{w}^T\hat{x}_i+b)=1-\xi_i \hspace{0.1cm}\forall i, $$

 
@@ -761,7 +761,7 @@ $$ and

 
$$ -\lambda_i = C-\gamma_i \forall i. +\lambda_i = C-\gamma_i \hspace{0.1cm}\forall i. $$

 
@@ -776,7 +776,7 @@ but now subject to the constraints \( \lambda_i\geq 0 \), \( \sum_i\lambda_iy_i= We must in addition satisfy the Karush-Kuhn-Tucker condition which now reads

 
$$ -\lambda_i\left[y_i(\hat{w}^T\hat{x}_i+b) -(1-\xi_)\right]=0 \forall i, +\lambda_i\left[y_i(\hat{w}^T\hat{x}_i+b) -(1-\xi_)\right]=0 \hspace{0.1cm}\forall i, $$

 
@@ -789,7 +789,7 @@ $$ and

 
$$ -y_i(\hat{w}^T\hat{x}_i+b) -(1-\xi_) \geq 0 \forall i. +y_i(\hat{w}^T\hat{x}_i+b) -(1-\xi_) \geq 0 \hspace{0.1cm}\forall i. $$

 
diff --git a/doc/pub/svm/html/svm-solarized.html b/doc/pub/svm/html/svm-solarized.html index 362def158..143a1f4e1 100644 --- a/doc/pub/svm/html/svm-solarized.html +++ b/doc/pub/svm/html/svm-solarized.html @@ -328,7 +328,7 @@ Thus, we wish to find a margin \( M \) with \( \hat{w} \) normalized to \( \vert\vert \hat{w}\vert\vert =1 \) subject to the condition $$ -y_i(\hat{w}^T\hat{x}_i+b) \geq M \forall i=1,2,\dots, p. +y_i(\hat{w}^T\hat{x}_i+b) \geq M \hspace{0.1cm}\forall i=1,2,\dots, p. $$ All points are thus at a signed distance from the decision boundary defined by the line \( L \). The parameters \( b \) and \( w_1 \) and \( w_2 \) define this line. @@ -336,18 +336,18 @@ All points are thus at a signed distance from the decision boundary defined by t

We seek thus the largest value \( M \) defined by $$ -\frac{1}{\vert \vert \hat{w}\vert\vert}y_i(\hat{w}^T\hat{x}_i+b) \geq M \forall i=1,2,\dots, n, +\frac{1}{\vert \vert \hat{w}\vert\vert}y_i(\hat{w}^T\hat{x}_i+b) \geq M \hspace{0.1cm}\forall i=1,2,\dots, n, $$ or just $$ -y_i(\hat{w}^T\hat{x}_i+b) \geq M\vert \vert \hat{w}\vert\vert \forall i. +y_i(\hat{w}^T\hat{x}_i+b) \geq M\vert \vert \hat{w}\vert\vert \hspace{0.1cm}\forall i. $$ If we scale the equation so that \( \vert \vert \hat{w}\vert\vert = 1/M \), we have to find the minimum of \( \hat{w}^T\hat{w}=\vert \vert \hat{w}\vert\vert \) (the norm) subject to the condition $$ -y_i(\hat{w}^T\hat{x}_i+b) \geq 1 \forall i. +y_i(\hat{w}^T\hat{x}_i+b) \geq 1 \hspace{0.1cm}\forall i. $$

@@ -476,7 +476,7 @@ $$ subject to the constraints \( \lambda_i\geq 0 \) and \( \sum_i\lambda_iy_i=0 \). We must in addition satisfy the Karush-Kuhn-Tucker (KKT) condition $$ -\lambda_i\left[y_i(\hat{w}^T\hat{x}_i+b) -1\right] \forall i. +\lambda_i\left[y_i(\hat{w}^T\hat{x}_i+b) -1\right] \hspace{0.1cm}\forall i. $$ @@ -588,7 +588,7 @@ $$ subject to $$ -y_i(\hat{w}^T\hat{x}_i+b)=1-\xi_i \forall i, +y_i(\hat{w}^T\hat{x}_i+b)=1-\xi_i \hspace{0.1cm}\forall i, $$ with the requirement \( \xi_i\geq 0 \). @@ -606,7 +606,7 @@ $$ and $$ -\lambda_i = C-\gamma_i \forall i. +\lambda_i = C-\gamma_i \hspace{0.1cm}\forall i. $$ Inserting these constraints into the equation for \( {\cal L} \) we obtain the same equation as before @@ -617,7 +617,7 @@ $$ but now subject to the constraints \( \lambda_i\geq 0 \), \( \sum_i\lambda_iy_i=0 \) and $ 0 \leq \lambda_i \leq C$. We must in addition satisfy the Karush-Kuhn-Tucker condition which now reads $$ -\lambda_i\left[y_i(\hat{w}^T\hat{x}_i+b) -(1-\xi_)\right]=0 \forall i, +\lambda_i\left[y_i(\hat{w}^T\hat{x}_i+b) -(1-\xi_)\right]=0 \hspace{0.1cm}\forall i, $$ $$ @@ -626,7 +626,7 @@ $$ and $$ -y_i(\hat{w}^T\hat{x}_i+b) -(1-\xi_) \geq 0 \forall i. +y_i(\hat{w}^T\hat{x}_i+b) -(1-\xi_) \geq 0 \hspace{0.1cm}\forall i. $$

diff --git a/doc/pub/svm/html/svm.html b/doc/pub/svm/html/svm.html index 36137b5c6..3909d9ce0 100644 --- a/doc/pub/svm/html/svm.html +++ b/doc/pub/svm/html/svm.html @@ -333,7 +333,7 @@ Thus, we wish to find a margin \( M \) with \( \hat{w} \) normalized to \( \vert\vert \hat{w}\vert\vert =1 \) subject to the condition $$ -y_i(\hat{w}^T\hat{x}_i+b) \geq M \forall i=1,2,\dots, p. +y_i(\hat{w}^T\hat{x}_i+b) \geq M \hspace{0.1cm}\forall i=1,2,\dots, p. $$ All points are thus at a signed distance from the decision boundary defined by the line \( L \). The parameters \( b \) and \( w_1 \) and \( w_2 \) define this line. @@ -341,18 +341,18 @@ All points are thus at a signed distance from the decision boundary defined by t

We seek thus the largest value \( M \) defined by $$ -\frac{1}{\vert \vert \hat{w}\vert\vert}y_i(\hat{w}^T\hat{x}_i+b) \geq M \forall i=1,2,\dots, n, +\frac{1}{\vert \vert \hat{w}\vert\vert}y_i(\hat{w}^T\hat{x}_i+b) \geq M \hspace{0.1cm}\forall i=1,2,\dots, n, $$ or just $$ -y_i(\hat{w}^T\hat{x}_i+b) \geq M\vert \vert \hat{w}\vert\vert \forall i. +y_i(\hat{w}^T\hat{x}_i+b) \geq M\vert \vert \hat{w}\vert\vert \hspace{0.1cm}\forall i. $$ If we scale the equation so that \( \vert \vert \hat{w}\vert\vert = 1/M \), we have to find the minimum of \( \hat{w}^T\hat{w}=\vert \vert \hat{w}\vert\vert \) (the norm) subject to the condition $$ -y_i(\hat{w}^T\hat{x}_i+b) \geq 1 \forall i. +y_i(\hat{w}^T\hat{x}_i+b) \geq 1 \hspace{0.1cm}\forall i. $$

@@ -481,7 +481,7 @@ $$ subject to the constraints \( \lambda_i\geq 0 \) and \( \sum_i\lambda_iy_i=0 \). We must in addition satisfy the Karush-Kuhn-Tucker (KKT) condition $$ -\lambda_i\left[y_i(\hat{w}^T\hat{x}_i+b) -1\right] \forall i. +\lambda_i\left[y_i(\hat{w}^T\hat{x}_i+b) -1\right] \hspace{0.1cm}\forall i. $$ @@ -593,7 +593,7 @@ $$ subject to $$ -y_i(\hat{w}^T\hat{x}_i+b)=1-\xi_i \forall i, +y_i(\hat{w}^T\hat{x}_i+b)=1-\xi_i \hspace{0.1cm}\forall i, $$ with the requirement \( \xi_i\geq 0 \). @@ -611,7 +611,7 @@ $$ and $$ -\lambda_i = C-\gamma_i \forall i. +\lambda_i = C-\gamma_i \hspace{0.1cm}\forall i. $$ Inserting these constraints into the equation for \( {\cal L} \) we obtain the same equation as before @@ -622,7 +622,7 @@ $$ but now subject to the constraints \( \lambda_i\geq 0 \), \( \sum_i\lambda_iy_i=0 \) and $ 0 \leq \lambda_i \leq C$. We must in addition satisfy the Karush-Kuhn-Tucker condition which now reads $$ -\lambda_i\left[y_i(\hat{w}^T\hat{x}_i+b) -(1-\xi_)\right]=0 \forall i, +\lambda_i\left[y_i(\hat{w}^T\hat{x}_i+b) -(1-\xi_)\right]=0 \hspace{0.1cm}\forall i, $$ $$ @@ -631,7 +631,7 @@ $$ and $$ -y_i(\hat{w}^T\hat{x}_i+b) -(1-\xi_) \geq 0 \forall i. +y_i(\hat{w}^T\hat{x}_i+b) -(1-\xi_) \geq 0 \hspace{0.1cm}\forall i. $$

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$y_i(\hat{w}^T\hat{x}_i+b)=1$ and we say that $x_i$ is on the boundary. @@ -478,7 +478,7 @@ This has in turn the consequences that we change our optmization problem to find subject to !bt \[ -y_i(\hat{w}^T\hat{x}_i+b)=1-\xi_i \forall i, +y_i(\hat{w}^T\hat{x}_i+b)=1-\xi_i \hspace{0.1cm}\forall i, \] !et with the requirement $\xi_i\geq 0$. @@ -498,7 +498,7 @@ and and !bt \[ -\lambda_i = C-\gamma_i \forall i. +\lambda_i = C-\gamma_i \hspace{0.1cm}\forall i. \] !et Inserting these constraints into the equation for ${\cal L}$ we obtain the same equation as before @@ -511,7 +511,7 @@ but now subject to the constraints $\lambda_i\geq 0$, $\sum_i\lambda_iy_i=0$ and We must in addition satisfy the Karush-Kuhn-Tucker condition which now reads !bt \[ -\lambda_i\left[y_i(\hat{w}^T\hat{x}_i+b) -(1-\xi_)\right]=0 \forall i, +\lambda_i\left[y_i(\hat{w}^T\hat{x}_i+b) -(1-\xi_)\right]=0 \hspace{0.1cm}\forall i, \] !et !bt @@ -522,7 +522,7 @@ We must in addition satisfy the Karush-Kuhn-Tucker condition which now reads and !bt \[ -y_i(\hat{w}^T\hat{x}_i+b) -(1-\xi_) \geq 0 \forall i. +y_i(\hat{w}^T\hat{x}_i+b) -(1-\xi_) \geq 0 \hspace{0.1cm}\forall i. \] !et