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 $$ -y_i(\hat{w}^T\hat{x}_i+b) -(1-\xi_) \geq 0 \forall i. +y_i(\hat{w}^T\hat{x}_i+b) -(1-\xi_) \geq 0 \hspace{0.1cm}\forall i. $$
diff --git a/doc/pub/svm/ipynb/ipynb-svm-src.tar.gz b/doc/pub/svm/ipynb/ipynb-svm-src.tar.gz index 0327bab80..527cc9070 100644 Binary files a/doc/pub/svm/ipynb/ipynb-svm-src.tar.gz and b/doc/pub/svm/ipynb/ipynb-svm-src.tar.gz differ diff --git a/doc/pub/svm/ipynb/svm.ipynb b/doc/pub/svm/ipynb/svm.ipynb index 303d9d19f..dfe41a617 100644 --- a/doc/pub/svm/ipynb/svm.ipynb +++ b/doc/pub/svm/ipynb/svm.ipynb @@ -359,7 +359,7 @@ "metadata": {}, "source": [ "$$\n", - "y_i(\\hat{w}^T\\hat{x}_i+b) \\geq M \\forall i=1,2,\\dots, p.\n", + "y_i(\\hat{w}^T\\hat{x}_i+b) \\geq M \\hspace{0.1cm}\\forall i=1,2,\\dots, p.\n", "$$" ] }, @@ -377,7 +377,7 @@ "metadata": {}, "source": [ "$$\n", - "\\frac{1}{\\vert \\vert \\hat{w}\\vert\\vert}y_i(\\hat{w}^T\\hat{x}_i+b) \\geq M \\forall i=1,2,\\dots, n,\n", + "\\frac{1}{\\vert \\vert \\hat{w}\\vert\\vert}y_i(\\hat{w}^T\\hat{x}_i+b) \\geq M \\hspace{0.1cm}\\forall i=1,2,\\dots, n,\n", "$$" ] }, @@ -393,7 +393,7 @@ "metadata": {}, "source": [ "$$\n", - "y_i(\\hat{w}^T\\hat{x}_i+b) \\geq M\\vert \\vert \\hat{w}\\vert\\vert \\forall i.\n", + "y_i(\\hat{w}^T\\hat{x}_i+b) \\geq M\\vert \\vert \\hat{w}\\vert\\vert \\hspace{0.1cm}\\forall i.\n", "$$" ] }, @@ -410,7 +410,7 @@ "metadata": {}, "source": [ "$$\n", - "y_i(\\hat{w}^T\\hat{x}_i+b) \\geq 1 \\forall i.\n", + "y_i(\\hat{w}^T\\hat{x}_i+b) \\geq 1 \\hspace{0.1cm}\\forall i.\n", "$$" ] }, @@ -711,7 +711,7 @@ "metadata": {}, "source": [ "$$\n", - "\\lambda_i\\left[y_i(\\hat{w}^T\\hat{x}_i+b) -1\\right] \\forall i.\n", + "\\lambda_i\\left[y_i(\\hat{w}^T\\hat{x}_i+b) -1\\right] \\hspace{0.1cm}\\forall i.\n", "$$" ] }, @@ -914,7 +914,7 @@ "metadata": {}, "source": [ "$$\n", - "y_i(\\hat{w}^T\\hat{x}_i+b)=1-\\xi_i \\forall i,\n", + "y_i(\\hat{w}^T\\hat{x}_i+b)=1-\\xi_i \\hspace{0.1cm}\\forall i,\n", "$$" ] }, @@ -964,7 +964,7 @@ "metadata": {}, "source": [ "$$\n", - "\\lambda_i = C-\\gamma_i \\forall i.\n", + "\\lambda_i = C-\\gamma_i \\hspace{0.1cm}\\forall i.\n", "$$" ] }, @@ -1037,7 +1037,7 @@ "metadata": {}, "source": [ "$$\n", - "y_i(\\hat{w}^T\\hat{x}_i+b) -(1-\\xi_) \\geq 0 \\forall i.\n", + "y_i(\\hat{w}^T\\hat{x}_i+b) -(1-\\xi_) \\geq 0 \\hspace{0.1cm}\\forall i.\n", "$$" ] }, diff --git a/doc/pub/svm/pdf/svm-minted.pdf b/doc/pub/svm/pdf/svm-minted.pdf index 024e20d01..e9ed6ca79 100644 Binary files a/doc/pub/svm/pdf/svm-minted.pdf and b/doc/pub/svm/pdf/svm-minted.pdf differ diff --git a/doc/src/SupportVMachines/svm.do.txt b/doc/src/SupportVMachines/svm.do.txt index 9b957a7f0..1c9a1c61a 100644 --- a/doc/src/SupportVMachines/svm.do.txt +++ b/doc/src/SupportVMachines/svm.do.txt @@ -215,7 +215,7 @@ $\vert\vert \hat{w}\vert\vert =1$ subject to the condition !bt \[ -y_i(\hat{w}^T\hat{x}_i+b) \geq M \forall i=1,2,\dots, p. +y_i(\hat{w}^T\hat{x}_i+b) \geq M \hspace{0.1cm}\forall i=1,2,\dots, p. \] !et 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. @@ -223,20 +223,20 @@ All points are thus at a signed distance from the decision boundary defined by t We seek thus the largest value $M$ defined by !bt \[ -\frac{1}{\vert \vert \hat{w}\vert\vert}y_i(\hat{w}^T\hat{x}_i+b) \geq M \forall i=1,2,\dots, n, +\frac{1}{\vert \vert \hat{w}\vert\vert}y_i(\hat{w}^T\hat{x}_i+b) \geq M \hspace{0.1cm}\forall i=1,2,\dots, n, \] !et or just !bt \[ -y_i(\hat{w}^T\hat{x}_i+b) \geq M\vert \vert \hat{w}\vert\vert \forall i. +y_i(\hat{w}^T\hat{x}_i+b) \geq M\vert \vert \hat{w}\vert\vert \hspace{0.1cm}\forall i. \] !et 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 !bt \[ -y_i(\hat{w}^T\hat{x}_i+b) \geq 1 \forall i. +y_i(\hat{w}^T\hat{x}_i+b) \geq 1 \hspace{0.1cm}\forall i. \] !et @@ -373,7 +373,7 @@ subject to the constraints $\lambda_i\geq 0$ and $\sum_i\lambda_iy_i=0$. We must in addition satisfy the "Karush-Kuhn-Tucker":"https://en.wikipedia.org/wiki/Karush%E2%80%93Kuhn%E2%80%93Tucker_conditions" (KKT) condition !bt \[ -\lambda_i\left[y_i(\hat{w}^T\hat{x}_i+b) -1\right] \forall i. +\lambda_i\left[y_i(\hat{w}^T\hat{x}_i+b) -1\right] \hspace{0.1cm}\forall i. \] !et o If $\lambda_i > 0$, then $y_i(\hat{w}^T\hat{x}_i+b)=1$ 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