From ae4454f82cddbf0735778f89d91dab099060dea1 Mon Sep 17 00:00:00 2001 From: mhjensen Date: Sun, 4 Nov 2018 21:43:08 +0100 Subject: [PATCH] updating sim, not yet ready --- doc/pub/svm/html/svm-bs.html | 79 ++++++++++- doc/pub/svm/html/svm-reveal.html | 89 +++++++++++- doc/pub/svm/html/svm-solarized.html | 76 +++++++++- doc/pub/svm/html/svm.html | 76 +++++++++- doc/pub/svm/ipynb/ipynb-svm-src.tar.gz | Bin 207 -> 205 bytes doc/pub/svm/ipynb/svm.ipynb | 158 +++++++++++++++++++++ doc/pub/svm/pdf/svm-beamer-handouts2x3.pdf | Bin 205590 -> 220999 bytes doc/pub/svm/pdf/svm-beamer.pdf | Bin 198391 -> 210376 bytes doc/pub/svm/pdf/svm-minted.pdf | Bin 275399 -> 280988 bytes doc/src/SupportVMachines/svm.do.txt | 70 +++++++++ 10 files changed, 540 insertions(+), 8 deletions(-) diff --git a/doc/pub/svm/html/svm-bs.html b/doc/pub/svm/html/svm-bs.html index 71a1a86a1..05d35bb72 100644 --- a/doc/pub/svm/html/svm-bs.html +++ b/doc/pub/svm/html/svm-bs.html @@ -49,7 +49,11 @@ Automatically generated HTML file from DocOnce source ('First attempt at a minimization approach', 2, None, '___sec6'), ('Solving the equations', 2, None, '___sec7'), ('A better approach', 2, None, '___sec8'), - ('Examples with kernels', 2, None, '___sec9')]} + ('A quick reminder on Lagrangian multipliers', + 2, + None, + '___sec9'), + ('Examples with kernels', 2, None, '___sec10')]} end of tocinfo --> @@ -96,7 +100,8 @@ MathJax.Hub.Config({
  • First attempt at a minimization approach
  • Solving the equations
  • A better approach
  • -
  • Examples with kernels
  • +
  • A quick reminder on Lagrangian multipliers
  • +
  • Examples with kernels
  • @@ -333,12 +338,80 @@ There are however problems with this approach, although it looks pretty straight

    A better approach

    + +

    A better approach is rather to try to define a large margin between the two classes (if they are well separated from the beginning). +

    +Thus, we wish to find a margin \( M \) with \( \hat{\beta} \) normalized to \( \vert\vert \hat{\beta}\vert\vert =1 \) subject to the condition +$$ +y_i(\beta_0+\beta_1 x_i) \geq M \forall i=1,2,\dots, n. +$$ + +All points are thus at a signed distance from the decision boundary defined by the line \( L \). The parameters \( \beta_0 \) and \( \beta_1 \) define this line. + +

    +We seek thus the largest value \( M \) defined by +$$ +\frac{1}{\vert \vert \hat{\beta}\vert\vert}y_i(\beta_0+\beta_1 x_) \geq M \forall i=1,2,\dots, n, +$$ + +or just +$$ +y_i(\beta_0+\beta_1 x_i) \geq M\vert \vert \hat{\beta}\vert\vert \forall i=1,2,\dots, n. +$$ + +If we scale the equation so that \( \vert \vert \hat{\beta}\vert\vert = 1/M \), we have to find the minimum of +\( \hat{\beta}^T\hat{\beta} \) subject to the condition +$$ +y_i(\beta_0+\beta_1 x_i) \geq 1 \forall i=1,2,\dots, n. +$$ +

    -

    Examples with kernels

    +

    A quick reminder on Lagrangian multipliers

    + +

    +Withe the above constraint, we introduce now the calculus of Lagrangian multiplier. +In order to solve the above problem, we define the following Lagrangian function to be minimized +$$ +L(\lambda)=\frac{1}{2}\hat{\beta}^T\hat{\beta}-\sum_{i=1}^n\lambda_i\left[y_i(\beta_0+\beta_1 x_i)-1\right], +$$ + +where \( \lambda_i \) is a so-called Lagrange multiplier subject to the condition \( \lambda_i \geq 0 \). + +

    +Taking the derivatives with respect to \( \beta_0 \) and \( \beta_1 \) we obtain +$$ +\frac{\partial L}{\partial \beta_0} = -\sum_{i} \lambda_iy_i=0, +$$ + +and +$$ +\frac{\partial L}{\partial \beta_1} = \beta_1-\sum_{i} \lambda_iy_ix_i=0. +$$ + +Inserting these constraints into the equation for \( L \) we obtain +$$ +L(\lambda)=\sum_i\lambda_i-\frac{1}{2}\sum_{ij}^n\lambda_i\lambda_jy_iy_jx_i^Tx_j, +$$ + +subject to the constraints \( \lambda_i\geq 0 \) and \( \sum_i\lambda_iy_i=0 \). +We must in addition satisfy the Koriush-Kuhn-Tucker (KKT) condition +$$ +\lambda_i\left[y_i(\beta_0+\beta_1 x_i) -1\right] \forall i=1,2,\dots, n. +$$ + + +

      +
    1. If \( \lambda_i > 0 \), then \( y_i(\beta_0+\beta_1 x_i)=1 \) and we say that \( x_i \) is on the boundary of the slab.
    2. +
    3. If \( y_i(\beta_0+\beta_1 x_i)> 1 \), we say \( x_i \) is not on the boundary and we set \( \lambda_i=0 \).
    4. +
    + + + +

    Examples with kernels

    diff --git a/doc/pub/svm/html/svm-reveal.html b/doc/pub/svm/html/svm-reveal.html index addb45114..11fba8a0b 100644 --- a/doc/pub/svm/html/svm-reveal.html +++ b/doc/pub/svm/html/svm-reveal.html @@ -378,12 +378,99 @@ There are however problems with this approach, although it looks pretty straight

    A better approach

    + +

    A better approach is rather to try to define a large margin between the two classes (if they are well separated from the beginning). + +

    +Thus, we wish to find a margin \( M \) with \( \hat{\beta} \) normalized to \( \vert\vert \hat{\beta}\vert\vert =1 \) subject to the condition +

     
    +$$ +y_i(\beta_0+\beta_1 x_i) \geq M \forall i=1,2,\dots, n. +$$ +

     
    + +All points are thus at a signed distance from the decision boundary defined by the line \( L \). The parameters \( \beta_0 \) and \( \beta_1 \) define this line. + +

    +We seek thus the largest value \( M \) defined by +

     
    +$$ +\frac{1}{\vert \vert \hat{\beta}\vert\vert}y_i(\beta_0+\beta_1 x_) \geq M \forall i=1,2,\dots, n, +$$ +

     
    + +or just +

     
    +$$ +y_i(\beta_0+\beta_1 x_i) \geq M\vert \vert \hat{\beta}\vert\vert \forall i=1,2,\dots, n. +$$ +

     
    + +If we scale the equation so that \( \vert \vert \hat{\beta}\vert\vert = 1/M \), we have to find the minimum of +\( \hat{\beta}^T\hat{\beta} \) subject to the condition +

     
    +$$ +y_i(\beta_0+\beta_1 x_i) \geq 1 \forall i=1,2,\dots, n. +$$ +

     

    -

    Examples with kernels

    +

    A quick reminder on Lagrangian multipliers

    + +

    +Withe the above constraint, we introduce now the calculus of Lagrangian multiplier. +In order to solve the above problem, we define the following Lagrangian function to be minimized +

     
    +$$ +L(\lambda)=\frac{1}{2}\hat{\beta}^T\hat{\beta}-\sum_{i=1}^n\lambda_i\left[y_i(\beta_0+\beta_1 x_i)-1\right], +$$ +

     
    + +where \( \lambda_i \) is a so-called Lagrange multiplier subject to the condition \( \lambda_i \geq 0 \). + +

    +Taking the derivatives with respect to \( \beta_0 \) and \( \beta_1 \) we obtain +

     
    +$$ +\frac{\partial L}{\partial \beta_0} = -\sum_{i} \lambda_iy_i=0, +$$ +

     
    + +and +

     
    +$$ +\frac{\partial L}{\partial \beta_1} = \beta_1-\sum_{i} \lambda_iy_ix_i=0. +$$ +

     
    + +Inserting these constraints into the equation for \( L \) we obtain +

     
    +$$ +L(\lambda)=\sum_i\lambda_i-\frac{1}{2}\sum_{ij}^n\lambda_i\lambda_jy_iy_jx_i^Tx_j, +$$ +

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

     
    +$$ +\lambda_i\left[y_i(\beta_0+\beta_1 x_i) -1\right] \forall i=1,2,\dots, n. +$$ +

     
    + + +

      +

    1. If \( \lambda_i > 0 \), then \( y_i(\beta_0+\beta_1 x_i)=1 \) and we say that \( x_i \) is on the boundary of the slab.
    2. +

    3. If \( y_i(\beta_0+\beta_1 x_i)> 1 \), we say \( x_i \) is not on the boundary and we set \( \lambda_i=0 \).
    4. +
    +
    + + +
    +

    Examples with kernels

    diff --git a/doc/pub/svm/html/svm-solarized.html b/doc/pub/svm/html/svm-solarized.html index 69804bbc9..04539f684 100644 --- a/doc/pub/svm/html/svm-solarized.html +++ b/doc/pub/svm/html/svm-solarized.html @@ -43,7 +43,11 @@ div { text-align: justify; text-justify: inter-word; } ('First attempt at a minimization approach', 2, None, '___sec6'), ('Solving the equations', 2, None, '___sec7'), ('A better approach', 2, None, '___sec8'), - ('Examples with kernels', 2, None, '___sec9')]} + ('A quick reminder on Lagrangian multipliers', + 2, + None, + '___sec9'), + ('Examples with kernels', 2, None, '___sec10')]} end of tocinfo --> @@ -285,12 +289,80 @@ There are however problems with this approach, although it looks pretty straight









    A better approach

    + +

    A better approach is rather to try to define a large margin between the two classes (if they are well separated from the beginning). +

    +Thus, we wish to find a margin \( M \) with \( \hat{\beta} \) normalized to \( \vert\vert \hat{\beta}\vert\vert =1 \) subject to the condition +$$ +y_i(\beta_0+\beta_1 x_i) \geq M \forall i=1,2,\dots, n. +$$ + +All points are thus at a signed distance from the decision boundary defined by the line \( L \). The parameters \( \beta_0 \) and \( \beta_1 \) define this line. + +

    +We seek thus the largest value \( M \) defined by +$$ +\frac{1}{\vert \vert \hat{\beta}\vert\vert}y_i(\beta_0+\beta_1 x_) \geq M \forall i=1,2,\dots, n, +$$ + +or just +$$ +y_i(\beta_0+\beta_1 x_i) \geq M\vert \vert \hat{\beta}\vert\vert \forall i=1,2,\dots, n. +$$ + +If we scale the equation so that \( \vert \vert \hat{\beta}\vert\vert = 1/M \), we have to find the minimum of +\( \hat{\beta}^T\hat{\beta} \) subject to the condition +$$ +y_i(\beta_0+\beta_1 x_i) \geq 1 \forall i=1,2,\dots, n. +$$ +











    -

    Examples with kernels

    +

    A quick reminder on Lagrangian multipliers

    + +

    +Withe the above constraint, we introduce now the calculus of Lagrangian multiplier. +In order to solve the above problem, we define the following Lagrangian function to be minimized +$$ +L(\lambda)=\frac{1}{2}\hat{\beta}^T\hat{\beta}-\sum_{i=1}^n\lambda_i\left[y_i(\beta_0+\beta_1 x_i)-1\right], +$$ + +where \( \lambda_i \) is a so-called Lagrange multiplier subject to the condition \( \lambda_i \geq 0 \). + +

    +Taking the derivatives with respect to \( \beta_0 \) and \( \beta_1 \) we obtain +$$ +\frac{\partial L}{\partial \beta_0} = -\sum_{i} \lambda_iy_i=0, +$$ + +and +$$ +\frac{\partial L}{\partial \beta_1} = \beta_1-\sum_{i} \lambda_iy_ix_i=0. +$$ + +Inserting these constraints into the equation for \( L \) we obtain +$$ +L(\lambda)=\sum_i\lambda_i-\frac{1}{2}\sum_{ij}^n\lambda_i\lambda_jy_iy_jx_i^Tx_j, +$$ + +subject to the constraints \( \lambda_i\geq 0 \) and \( \sum_i\lambda_iy_i=0 \). +We must in addition satisfy the Koriush-Kuhn-Tucker (KKT) condition +$$ +\lambda_i\left[y_i(\beta_0+\beta_1 x_i) -1\right] \forall i=1,2,\dots, n. +$$ + + +

      +
    1. If \( \lambda_i > 0 \), then \( y_i(\beta_0+\beta_1 x_i)=1 \) and we say that \( x_i \) is on the boundary of the slab.
    2. +
    3. If \( y_i(\beta_0+\beta_1 x_i)> 1 \), we say \( x_i \) is not on the boundary and we set \( \lambda_i=0 \).
    4. +
    + +









    + +

    Examples with kernels

    diff --git a/doc/pub/svm/html/svm.html b/doc/pub/svm/html/svm.html index f5fb48306..19369e449 100644 --- a/doc/pub/svm/html/svm.html +++ b/doc/pub/svm/html/svm.html @@ -48,7 +48,11 @@ div { text-align: justify; text-justify: inter-word; } ('First attempt at a minimization approach', 2, None, '___sec6'), ('Solving the equations', 2, None, '___sec7'), ('A better approach', 2, None, '___sec8'), - ('Examples with kernels', 2, None, '___sec9')]} + ('A quick reminder on Lagrangian multipliers', + 2, + None, + '___sec9'), + ('Examples with kernels', 2, None, '___sec10')]} end of tocinfo --> @@ -290,12 +294,80 @@ There are however problems with this approach, although it looks pretty straight









    A better approach

    + +

    A better approach is rather to try to define a large margin between the two classes (if they are well separated from the beginning). +

    +Thus, we wish to find a margin \( M \) with \( \hat{\beta} \) normalized to \( \vert\vert \hat{\beta}\vert\vert =1 \) subject to the condition +$$ +y_i(\beta_0+\beta_1 x_i) \geq M \forall i=1,2,\dots, n. +$$ + +All points are thus at a signed distance from the decision boundary defined by the line \( L \). The parameters \( \beta_0 \) and \( \beta_1 \) define this line. + +

    +We seek thus the largest value \( M \) defined by +$$ +\frac{1}{\vert \vert \hat{\beta}\vert\vert}y_i(\beta_0+\beta_1 x_) \geq M \forall i=1,2,\dots, n, +$$ + +or just +$$ +y_i(\beta_0+\beta_1 x_i) \geq M\vert \vert \hat{\beta}\vert\vert \forall i=1,2,\dots, n. +$$ + +If we scale the equation so that \( \vert \vert \hat{\beta}\vert\vert = 1/M \), we have to find the minimum of +\( \hat{\beta}^T\hat{\beta} \) subject to the condition +$$ +y_i(\beta_0+\beta_1 x_i) \geq 1 \forall i=1,2,\dots, n. +$$ +











    -

    Examples with kernels

    +

    A quick reminder on Lagrangian multipliers

    + +

    +Withe the above constraint, we introduce now the calculus of Lagrangian multiplier. +In order to solve the above problem, we define the following Lagrangian function to be minimized +$$ +L(\lambda)=\frac{1}{2}\hat{\beta}^T\hat{\beta}-\sum_{i=1}^n\lambda_i\left[y_i(\beta_0+\beta_1 x_i)-1\right], +$$ + +where \( \lambda_i \) is a so-called Lagrange multiplier subject to the condition \( \lambda_i \geq 0 \). + +

    +Taking the derivatives with respect to \( \beta_0 \) and \( \beta_1 \) we obtain +$$ +\frac{\partial L}{\partial \beta_0} = -\sum_{i} \lambda_iy_i=0, +$$ + +and +$$ +\frac{\partial L}{\partial \beta_1} = \beta_1-\sum_{i} \lambda_iy_ix_i=0. +$$ + +Inserting these constraints into the equation for \( L \) we obtain +$$ +L(\lambda)=\sum_i\lambda_i-\frac{1}{2}\sum_{ij}^n\lambda_i\lambda_jy_iy_jx_i^Tx_j, +$$ + +subject to the constraints \( \lambda_i\geq 0 \) and \( \sum_i\lambda_iy_i=0 \). +We must in addition satisfy the Koriush-Kuhn-Tucker (KKT) condition +$$ +\lambda_i\left[y_i(\beta_0+\beta_1 x_i) -1\right] \forall i=1,2,\dots, n. +$$ + + +

      +
    1. If \( \lambda_i > 0 \), then \( y_i(\beta_0+\beta_1 x_i)=1 \) and we say that \( x_i \) is on the boundary of the slab.
    2. +
    3. If \( y_i(\beta_0+\beta_1 x_i)> 1 \), we say \( x_i \) is not on the boundary and we set \( \lambda_i=0 \).
    4. +
    + +









    + +

    Examples with kernels

    diff --git a/doc/pub/svm/ipynb/ipynb-svm-src.tar.gz b/doc/pub/svm/ipynb/ipynb-svm-src.tar.gz index 8f47939d0fba4e248d2f73e1bea1e6c27dcb7a0e..dccb72d8fd9d88db3c5344414fcfcedda6170914 100644 GIT binary patch literal 205 zcmb2|=3qD!c|V$g`R)0GU55+=juoD_J8I+TnZ5RT>(>X40v}nGdd+rcFY*-m{cRi9 zyUe|vC;rt>@1Mz&m}dF+#~lC7C4SAz1HD31HEaD=eKZX%T)qC&o(VN3t9BhPTErDt zyO3?syzQ6PC;M2HtTu^1=KA>GoY%+9X0!jl<2SK-UuS6Y;xl5oS$l+^PmZkqulsS! z={NgIvsdoztc`E7l*u(%6sx^!UGAoPFXunB`;mKL_YB_nT22OJ@JC-CLX!kt^8wohI&cNLf64o0?hYrP|-? zxV|ni*F5_F{U^i224~hh+dTd0o)h;gzZiI}Of#Ao>^J+@oK<_Sq(@C(w0hOdYg5;i zs%B}tDpGk>s2(dDWL9Xgw0gT;WPNd-#O<7SZ+7lc+W*skPpRI?M_bRwet6`(?X`fN zifno7?eBHExf@ H4F(1PG>c 0$, then $y_i(\\beta_0+\\beta_1 x_i)=1$ and we say that $x_i$ is on the boundary of the slab.\n", + "\n", + "2. If $y_i(\\beta_0+\\beta_1 x_i)> 1$, we say $x_i$ is not on the boundary and we set $\\lambda_i=0$. \n", + "\n", "## Examples with kernels" ] }, diff --git a/doc/pub/svm/pdf/svm-beamer-handouts2x3.pdf b/doc/pub/svm/pdf/svm-beamer-handouts2x3.pdf index ec79cbc31e23241450ac06d4bc08b41198cddb2c..c7da9db4d1afac97ccf7be27e2740a9d5a4f63c7 100644 GIT binary patch delta 74225 zcmY)UV{o8N7d4DV6Wg|JOl(YS+n(50?1^pLwmq?Jb7Cjwe$P2?ea}~2)zz!IdslV; zSiSaMoiYMH+JjIR01Bz7V&}obns|##+*k_A3H`qq9QGCJe=#YlB*g#XSWMmjiyLoo zA0YpC4vv)l2Pk_yiV)C=IKT|Q`Ht}wfXgc{)j&qFicDq<77+r=a{FFnwODY63o9-a z52e$yKp{W?9%5`bW10O!4l~(kRc*Mvue`pB?7-ZXc&Ey@b)P&LGf{jPqEV$ja%qnu z-vf!y%=DJSwUJ^K$`*#>a2GWlgPN0Ce!wT#)gUNH!kUSkx<`g=|C${A!m%EwC|=%^ zX|A{EGZ?dJKI7-2!&kx1`S%uM;ueLs2yF07TaD9%q+I)9(l_zZ&r+Yoq|B^4Nq1!Y z0X|^8R(Q8xDX~8fEm4D7?SIcXPrdek&)rzhgbIqwMa)dBOw1&~&P>e3{r?}cauc&K zGdFHB`~Udgy@8zj|BDkN`RWq`X(&-xxrjM={x?iwJ`MU0*8gH~C2le+gR^n7CgvlP 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zaAOEL)4Sd$zFuivANJa;w(E|#u3PeX9X;`Ev1Za#t8EZJbVT2EE?@sW3U}rF9*#1Z z#?}~7Kg)vJwKBcyy!&@?$N;5k>Z=H7CTVDtWU>y%hU3jhkZCLX>he{Q>>SI7R8dUs7fDB2Cg1&Liru*9y{riv6E2JAmh&|qRx6C9QJ*T_l+qz|^I zv;KBRo(@V%;rK|ZvzXsuYB$x9R^kW-Izbk^flO-=C^f<8(I(idCirNDB+;5MU{BXi zfC8L0Q}n3%wn66S5(UAo`L*RJ;Px^a5CZ3V6c?#O-<5-g$&Kj0k9%3Tju+>w+nd30s*`XNxqA zCj~v!Y9D1Ss;d3em5Q_N+Les6>~CpmbUxp>>XhDxZN+lMR&CXC>$R&td%UUHRNHv; zT2&Kzu#nYQuK0g|R!B$&*>VOx=L~i}Q)c#Gr}Ti0ahd6E|D!(mkI#miyQ! 0$, then $y_i(\beta_0+\beta_1 x_i)=1$ and we say that $x_i$ is on the boundary of the slab. +o If $y_i(\beta_0+\beta_1 x_i)> 1$, we say $x_i$ is not on the boundary and we set $\lambda_i=0$. + + !split ===== Examples with kernels =====