updating week 46
This commit is contained in:
mhjensen
@@ -42,39 +42,41 @@ Automatically generated HTML file from DocOnce source
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<!-- tocinfo
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{'highest level': 2,
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'sections': [('Overview of week 46', 2, None, '___sec0'),
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('Support Vector Machines, overarching aims', 2, None, '___sec1'),
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('Hyperplanes and all that', 2, None, '___sec2'),
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('What is a hyperplane?', 2, None, '___sec3'),
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('A $p$-dimensional space of features', 2, None, '___sec4'),
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('The two-dimensional case', 2, None, '___sec5'),
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('Getting into the details', 2, None, '___sec6'),
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('First attempt at a minimization approach', 2, None, '___sec7'),
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('Solving the equations', 2, None, '___sec8'),
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('Code Example', 2, None, '___sec9'),
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('Problems with the Simpler Approach', 2, None, '___sec10'),
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('A better approach', 2, None, '___sec11'),
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('Thursday', 2, None, '___sec1'),
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('Friday', 2, None, '___sec2'),
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('Support Vector Machines, overarching aims', 2, None, '___sec3'),
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('Hyperplanes and all that', 2, None, '___sec4'),
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('What is a hyperplane?', 2, None, '___sec5'),
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('A $p$-dimensional space of features', 2, None, '___sec6'),
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('The two-dimensional case', 2, None, '___sec7'),
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('Getting into the details', 2, None, '___sec8'),
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('First attempt at a minimization approach', 2, None, '___sec9'),
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('Solving the equations', 2, None, '___sec10'),
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('Code Example', 2, None, '___sec11'),
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('Problems with the Simpler Approach', 2, None, '___sec12'),
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('A better approach', 2, None, '___sec13'),
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('A quick Reminder on Lagrangian Multipliers',
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2,
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None,
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'___sec12'),
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('Adding the Multiplier', 2, None, '___sec13'),
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('Setting up the Problem', 2, None, '___sec14'),
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('The problem to solve', 2, None, '___sec15'),
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('The last steps', 2, None, '___sec16'),
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('A soft classifier', 2, None, '___sec17'),
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('Soft optmization problem', 2, None, '___sec18'),
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('Kernels and non-linearity', 2, None, '___sec19'),
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('The equations', 2, None, '___sec20'),
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('The problem to solve', 2, None, '___sec21'),
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("Different kernels and Mercer's theorem", 2, None, '___sec22'),
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('The moons example', 2, None, '___sec23'),
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'___sec14'),
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('Adding the Multiplier', 2, None, '___sec15'),
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('Setting up the Problem', 2, None, '___sec16'),
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('The problem to solve', 2, None, '___sec17'),
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('The last steps', 2, None, '___sec18'),
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('A soft classifier', 2, None, '___sec19'),
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('Soft optmization problem', 2, None, '___sec20'),
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('Kernels and non-linearity', 2, None, '___sec21'),
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('The equations', 2, None, '___sec22'),
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('The problem to solve', 2, None, '___sec23'),
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("Different kernels and Mercer's theorem", 2, None, '___sec24'),
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('The moons example', 2, None, '___sec25'),
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('Mathematical optimization of convex functions',
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2,
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None,
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'___sec24'),
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('How do we solve these problems?', 2, None, '___sec25'),
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('A simple example', 2, None, '___sec26'),
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('Back to the more realistic cases', 2, None, '___sec27')]}
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'___sec26'),
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('How do we solve these problems?', 2, None, '___sec27'),
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('A simple example', 2, None, '___sec28'),
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('Back to the more realistic cases', 2, None, '___sec29')]}
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end of tocinfo -->
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<body>
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@@ -113,33 +115,35 @@ MathJax.Hub.Config({
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<a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
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<ul class="dropdown-menu">
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<!-- navigation toc: --> <li><a href="._week46-bs001.html#___sec0" style="font-size: 80%;">Overview of week 46</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs002.html#___sec1" style="font-size: 80%;">Support Vector Machines, overarching aims</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs003.html#___sec2" style="font-size: 80%;">Hyperplanes and all that</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs004.html#___sec3" style="font-size: 80%;">What is a hyperplane?</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs005.html#___sec4" style="font-size: 80%;">A \( p \)-dimensional space of features</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs006.html#___sec5" style="font-size: 80%;">The two-dimensional case</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs007.html#___sec6" style="font-size: 80%;">Getting into the details</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs008.html#___sec7" style="font-size: 80%;">First attempt at a minimization approach</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs009.html#___sec8" style="font-size: 80%;">Solving the equations</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs010.html#___sec9" style="font-size: 80%;">Code Example</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs011.html#___sec10" style="font-size: 80%;">Problems with the Simpler Approach</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs012.html#___sec11" style="font-size: 80%;">A better approach</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs013.html#___sec12" style="font-size: 80%;">A quick Reminder on Lagrangian Multipliers</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs014.html#___sec13" style="font-size: 80%;">Adding the Multiplier</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs015.html#___sec14" style="font-size: 80%;">Setting up the Problem</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs016.html#___sec15" style="font-size: 80%;">The problem to solve</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs017.html#___sec16" style="font-size: 80%;">The last steps</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs018.html#___sec17" style="font-size: 80%;">A soft classifier</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs019.html#___sec18" style="font-size: 80%;">Soft optmization problem</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs020.html#___sec19" style="font-size: 80%;">Kernels and non-linearity</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs021.html#___sec20" style="font-size: 80%;">The equations</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs022.html#___sec21" style="font-size: 80%;">The problem to solve</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs023.html#___sec22" style="font-size: 80%;">Different kernels and Mercer's theorem</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs024.html#___sec23" style="font-size: 80%;">The moons example</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs025.html#___sec24" style="font-size: 80%;">Mathematical optimization of convex functions</a></li>
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<!-- navigation toc: --> <li><a href="#___sec25" style="font-size: 80%;">How do we solve these problems?</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs027.html#___sec26" style="font-size: 80%;">A simple example</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs028.html#___sec27" style="font-size: 80%;">Back to the more realistic cases</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs002.html#___sec1" style="font-size: 80%;">Thursday</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs003.html#___sec2" style="font-size: 80%;">Friday</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs004.html#___sec3" style="font-size: 80%;">Support Vector Machines, overarching aims</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs005.html#___sec4" style="font-size: 80%;">Hyperplanes and all that</a></li>
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||||
<!-- navigation toc: --> <li><a href="._week46-bs006.html#___sec5" style="font-size: 80%;">What is a hyperplane?</a></li>
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||||
<!-- navigation toc: --> <li><a href="._week46-bs007.html#___sec6" style="font-size: 80%;">A \( p \)-dimensional space of features</a></li>
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||||
<!-- navigation toc: --> <li><a href="._week46-bs008.html#___sec7" style="font-size: 80%;">The two-dimensional case</a></li>
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||||
<!-- navigation toc: --> <li><a href="._week46-bs009.html#___sec8" style="font-size: 80%;">Getting into the details</a></li>
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||||
<!-- navigation toc: --> <li><a href="._week46-bs010.html#___sec9" style="font-size: 80%;">First attempt at a minimization approach</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs011.html#___sec10" style="font-size: 80%;">Solving the equations</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs012.html#___sec11" style="font-size: 80%;">Code Example</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs013.html#___sec12" style="font-size: 80%;">Problems with the Simpler Approach</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs014.html#___sec13" style="font-size: 80%;">A better approach</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs015.html#___sec14" style="font-size: 80%;">A quick Reminder on Lagrangian Multipliers</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs016.html#___sec15" style="font-size: 80%;">Adding the Multiplier</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs017.html#___sec16" style="font-size: 80%;">Setting up the Problem</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs018.html#___sec17" style="font-size: 80%;">The problem to solve</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs019.html#___sec18" style="font-size: 80%;">The last steps</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs020.html#___sec19" style="font-size: 80%;">A soft classifier</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs021.html#___sec20" style="font-size: 80%;">Soft optmization problem</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs022.html#___sec21" style="font-size: 80%;">Kernels and non-linearity</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs023.html#___sec22" style="font-size: 80%;">The equations</a></li>
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||||
<!-- navigation toc: --> <li><a href="._week46-bs024.html#___sec23" style="font-size: 80%;">The problem to solve</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs025.html#___sec24" style="font-size: 80%;">Different kernels and Mercer's theorem</a></li>
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<!-- navigation toc: --> <li><a href="#___sec25" style="font-size: 80%;">The moons example</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs027.html#___sec26" style="font-size: 80%;">Mathematical optimization of convex functions</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs028.html#___sec27" style="font-size: 80%;">How do we solve these problems?</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs029.html#___sec28" style="font-size: 80%;">A simple example</a></li>
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<!-- navigation toc: --> <li><a href="._week46-bs030.html#___sec29" style="font-size: 80%;">Back to the more realistic cases</a></li>
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</ul>
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</li>
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@@ -155,29 +159,199 @@ MathJax.Hub.Config({
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<a name="part0026"></a>
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<!-- !split -->
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<h2 id="___sec25" class="anchor">How do we solve these problems? </h2>
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<p>
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If we use Python as programming language and wish to venture beyond
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<b>scikit-learn</b>, <b>tensorflow</b> and similar software which makes our
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lives so much easier, we need to dive into the wonderful world of
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quadratic programming. We can, if we wish, solve the minimization
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problem using say standard gradient methods or conjugate gradient
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methods. However, these methods tend to exhibit a rather slow
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converge. So, welcome to the promised land of quadratic programming.
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<p>
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The functions we need are contained in the quadratic programming package <b>CVXOPT</b> and we need to import it together with <b>numpy</b> as
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<h2 id="___sec25" class="anchor">The moons example </h2>
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<p>
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<!-- code=python (!bc pycod) typeset with pygments style "default" -->
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<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span>
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<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">cvxopt</span>
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</pre></div>
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<p>
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This will make our life much easier. You don't need t write your own optimizer.
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<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">__future__</span> <span style="color: #008000; font-weight: bold">import</span> division, print_function, unicode_literals
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<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
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np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>seed(<span style="color: #666666">42</span>)
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<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib</span>
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<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
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plt<span style="color: #666666">.</span>rcParams[<span style="color: #BA2121">'axes.labelsize'</span>] <span style="color: #666666">=</span> <span style="color: #666666">14</span>
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plt<span style="color: #666666">.</span>rcParams[<span style="color: #BA2121">'xtick.labelsize'</span>] <span style="color: #666666">=</span> <span style="color: #666666">12</span>
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plt<span style="color: #666666">.</span>rcParams[<span style="color: #BA2121">'ytick.labelsize'</span>] <span style="color: #666666">=</span> <span style="color: #666666">12</span>
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<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.svm</span> <span style="color: #008000; font-weight: bold">import</span> SVC
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<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn</span> <span style="color: #008000; font-weight: bold">import</span> datasets
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<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.pipeline</span> <span style="color: #008000; font-weight: bold">import</span> Pipeline
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<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.preprocessing</span> <span style="color: #008000; font-weight: bold">import</span> StandardScaler
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<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.svm</span> <span style="color: #008000; font-weight: bold">import</span> LinearSVC
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<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.datasets</span> <span style="color: #008000; font-weight: bold">import</span> make_moons
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X, y <span style="color: #666666">=</span> make_moons(n_samples<span style="color: #666666">=100</span>, noise<span style="color: #666666">=0.15</span>, random_state<span style="color: #666666">=42</span>)
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<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">plot_dataset</span>(X, y, axes):
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plt<span style="color: #666666">.</span>plot(X[:, <span style="color: #666666">0</span>][y<span style="color: #666666">==0</span>], X[:, <span style="color: #666666">1</span>][y<span style="color: #666666">==0</span>], <span style="color: #BA2121">"bs"</span>)
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plt<span style="color: #666666">.</span>plot(X[:, <span style="color: #666666">0</span>][y<span style="color: #666666">==1</span>], X[:, <span style="color: #666666">1</span>][y<span style="color: #666666">==1</span>], <span style="color: #BA2121">"g^"</span>)
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plt<span style="color: #666666">.</span>axis(axes)
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plt<span style="color: #666666">.</span>grid(<span style="color: #008000; font-weight: bold">True</span>, which<span style="color: #666666">=</span><span style="color: #BA2121">'both'</span>)
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plt<span style="color: #666666">.</span>xlabel(<span style="color: #BA2121">r"$x_1$"</span>, fontsize<span style="color: #666666">=20</span>)
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plt<span style="color: #666666">.</span>ylabel(<span style="color: #BA2121">r"$x_2$"</span>, fontsize<span style="color: #666666">=20</span>, rotation<span style="color: #666666">=0</span>)
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plot_dataset(X, y, [<span style="color: #666666">-1.5</span>, <span style="color: #666666">2.5</span>, <span style="color: #666666">-1</span>, <span style="color: #666666">1.5</span>])
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plt<span style="color: #666666">.</span>show()
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<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.datasets</span> <span style="color: #008000; font-weight: bold">import</span> make_moons
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<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.pipeline</span> <span style="color: #008000; font-weight: bold">import</span> Pipeline
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<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.preprocessing</span> <span style="color: #008000; font-weight: bold">import</span> PolynomialFeatures
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polynomial_svm_clf <span style="color: #666666">=</span> Pipeline([
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(<span style="color: #BA2121">"poly_features"</span>, PolynomialFeatures(degree<span style="color: #666666">=3</span>)),
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(<span style="color: #BA2121">"scaler"</span>, StandardScaler()),
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(<span style="color: #BA2121">"svm_clf"</span>, LinearSVC(C<span style="color: #666666">=10</span>, loss<span style="color: #666666">=</span><span style="color: #BA2121">"hinge"</span>, random_state<span style="color: #666666">=42</span>))
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])
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polynomial_svm_clf<span style="color: #666666">.</span>fit(X, y)
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<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">plot_predictions</span>(clf, axes):
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x0s <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linspace(axes[<span style="color: #666666">0</span>], axes[<span style="color: #666666">1</span>], <span style="color: #666666">100</span>)
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x1s <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linspace(axes[<span style="color: #666666">2</span>], axes[<span style="color: #666666">3</span>], <span style="color: #666666">100</span>)
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x0, x1 <span style="color: #666666">=</span> np<span style="color: #666666">.</span>meshgrid(x0s, x1s)
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X <span style="color: #666666">=</span> np<span style="color: #666666">.</span>c_[x0<span style="color: #666666">.</span>ravel(), x1<span style="color: #666666">.</span>ravel()]
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y_pred <span style="color: #666666">=</span> clf<span style="color: #666666">.</span>predict(X)<span style="color: #666666">.</span>reshape(x0<span style="color: #666666">.</span>shape)
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y_decision <span style="color: #666666">=</span> clf<span style="color: #666666">.</span>decision_function(X)<span style="color: #666666">.</span>reshape(x0<span style="color: #666666">.</span>shape)
|
||||
plt<span style="color: #666666">.</span>contourf(x0, x1, y_pred, cmap<span style="color: #666666">=</span>plt<span style="color: #666666">.</span>cm<span style="color: #666666">.</span>brg, alpha<span style="color: #666666">=0.2</span>)
|
||||
plt<span style="color: #666666">.</span>contourf(x0, x1, y_decision, cmap<span style="color: #666666">=</span>plt<span style="color: #666666">.</span>cm<span style="color: #666666">.</span>brg, alpha<span style="color: #666666">=0.1</span>)
|
||||
|
||||
plot_predictions(polynomial_svm_clf, [<span style="color: #666666">-1.5</span>, <span style="color: #666666">2.5</span>, <span style="color: #666666">-1</span>, <span style="color: #666666">1.5</span>])
|
||||
plot_dataset(X, y, [<span style="color: #666666">-1.5</span>, <span style="color: #666666">2.5</span>, <span style="color: #666666">-1</span>, <span style="color: #666666">1.5</span>])
|
||||
|
||||
plt<span style="color: #666666">.</span>show()
|
||||
|
||||
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.svm</span> <span style="color: #008000; font-weight: bold">import</span> SVC
|
||||
|
||||
poly_kernel_svm_clf <span style="color: #666666">=</span> Pipeline([
|
||||
(<span style="color: #BA2121">"scaler"</span>, StandardScaler()),
|
||||
(<span style="color: #BA2121">"svm_clf"</span>, SVC(kernel<span style="color: #666666">=</span><span style="color: #BA2121">"poly"</span>, degree<span style="color: #666666">=3</span>, coef0<span style="color: #666666">=1</span>, C<span style="color: #666666">=5</span>))
|
||||
])
|
||||
poly_kernel_svm_clf<span style="color: #666666">.</span>fit(X, y)
|
||||
|
||||
poly100_kernel_svm_clf <span style="color: #666666">=</span> Pipeline([
|
||||
(<span style="color: #BA2121">"scaler"</span>, StandardScaler()),
|
||||
(<span style="color: #BA2121">"svm_clf"</span>, SVC(kernel<span style="color: #666666">=</span><span style="color: #BA2121">"poly"</span>, degree<span style="color: #666666">=10</span>, coef0<span style="color: #666666">=100</span>, C<span style="color: #666666">=5</span>))
|
||||
])
|
||||
poly100_kernel_svm_clf<span style="color: #666666">.</span>fit(X, y)
|
||||
|
||||
plt<span style="color: #666666">.</span>figure(figsize<span style="color: #666666">=</span>(<span style="color: #666666">11</span>, <span style="color: #666666">4</span>))
|
||||
|
||||
plt<span style="color: #666666">.</span>subplot(<span style="color: #666666">121</span>)
|
||||
plot_predictions(poly_kernel_svm_clf, [<span style="color: #666666">-1.5</span>, <span style="color: #666666">2.5</span>, <span style="color: #666666">-1</span>, <span style="color: #666666">1.5</span>])
|
||||
plot_dataset(X, y, [<span style="color: #666666">-1.5</span>, <span style="color: #666666">2.5</span>, <span style="color: #666666">-1</span>, <span style="color: #666666">1.5</span>])
|
||||
plt<span style="color: #666666">.</span>title(<span style="color: #BA2121">r"$d=3, r=1, C=5$"</span>, fontsize<span style="color: #666666">=18</span>)
|
||||
|
||||
plt<span style="color: #666666">.</span>subplot(<span style="color: #666666">122</span>)
|
||||
plot_predictions(poly100_kernel_svm_clf, [<span style="color: #666666">-1.5</span>, <span style="color: #666666">2.5</span>, <span style="color: #666666">-1</span>, <span style="color: #666666">1.5</span>])
|
||||
plot_dataset(X, y, [<span style="color: #666666">-1.5</span>, <span style="color: #666666">2.5</span>, <span style="color: #666666">-1</span>, <span style="color: #666666">1.5</span>])
|
||||
plt<span style="color: #666666">.</span>title(<span style="color: #BA2121">r"$d=10, r=100, C=5$"</span>, fontsize<span style="color: #666666">=18</span>)
|
||||
|
||||
plt<span style="color: #666666">.</span>show()
|
||||
|
||||
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">gaussian_rbf</span>(x, landmark, gamma):
|
||||
<span style="color: #008000; font-weight: bold">return</span> np<span style="color: #666666">.</span>exp(<span style="color: #666666">-</span>gamma <span style="color: #666666">*</span> np<span style="color: #666666">.</span>linalg<span style="color: #666666">.</span>norm(x <span style="color: #666666">-</span> landmark, axis<span style="color: #666666">=1</span>)<span style="color: #666666">**2</span>)
|
||||
|
||||
gamma <span style="color: #666666">=</span> <span style="color: #666666">0.3</span>
|
||||
|
||||
x1s <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linspace(<span style="color: #666666">-4.5</span>, <span style="color: #666666">4.5</span>, <span style="color: #666666">200</span>)<span style="color: #666666">.</span>reshape(<span style="color: #666666">-1</span>, <span style="color: #666666">1</span>)
|
||||
x2s <span style="color: #666666">=</span> gaussian_rbf(x1s, <span style="color: #666666">-2</span>, gamma)
|
||||
x3s <span style="color: #666666">=</span> gaussian_rbf(x1s, <span style="color: #666666">1</span>, gamma)
|
||||
|
||||
XK <span style="color: #666666">=</span> np<span style="color: #666666">.</span>c_[gaussian_rbf(X1D, <span style="color: #666666">-2</span>, gamma), gaussian_rbf(X1D, <span style="color: #666666">1</span>, gamma)]
|
||||
yk <span style="color: #666666">=</span> np<span style="color: #666666">.</span>array([<span style="color: #666666">0</span>, <span style="color: #666666">0</span>, <span style="color: #666666">1</span>, <span style="color: #666666">1</span>, <span style="color: #666666">1</span>, <span style="color: #666666">1</span>, <span style="color: #666666">1</span>, <span style="color: #666666">0</span>, <span style="color: #666666">0</span>])
|
||||
|
||||
plt<span style="color: #666666">.</span>figure(figsize<span style="color: #666666">=</span>(<span style="color: #666666">11</span>, <span style="color: #666666">4</span>))
|
||||
|
||||
plt<span style="color: #666666">.</span>subplot(<span style="color: #666666">121</span>)
|
||||
plt<span style="color: #666666">.</span>grid(<span style="color: #008000; font-weight: bold">True</span>, which<span style="color: #666666">=</span><span style="color: #BA2121">'both'</span>)
|
||||
plt<span style="color: #666666">.</span>axhline(y<span style="color: #666666">=0</span>, color<span style="color: #666666">=</span><span style="color: #BA2121">'k'</span>)
|
||||
plt<span style="color: #666666">.</span>scatter(x<span style="color: #666666">=</span>[<span style="color: #666666">-2</span>, <span style="color: #666666">1</span>], y<span style="color: #666666">=</span>[<span style="color: #666666">0</span>, <span style="color: #666666">0</span>], s<span style="color: #666666">=150</span>, alpha<span style="color: #666666">=0.5</span>, c<span style="color: #666666">=</span><span style="color: #BA2121">"red"</span>)
|
||||
plt<span style="color: #666666">.</span>plot(X1D[:, <span style="color: #666666">0</span>][yk<span style="color: #666666">==0</span>], np<span style="color: #666666">.</span>zeros(<span style="color: #666666">4</span>), <span style="color: #BA2121">"bs"</span>)
|
||||
plt<span style="color: #666666">.</span>plot(X1D[:, <span style="color: #666666">0</span>][yk<span style="color: #666666">==1</span>], np<span style="color: #666666">.</span>zeros(<span style="color: #666666">5</span>), <span style="color: #BA2121">"g^"</span>)
|
||||
plt<span style="color: #666666">.</span>plot(x1s, x2s, <span style="color: #BA2121">"g--"</span>)
|
||||
plt<span style="color: #666666">.</span>plot(x1s, x3s, <span style="color: #BA2121">"b:"</span>)
|
||||
plt<span style="color: #666666">.</span>gca()<span style="color: #666666">.</span>get_yaxis()<span style="color: #666666">.</span>set_ticks([<span style="color: #666666">0</span>, <span style="color: #666666">0.25</span>, <span style="color: #666666">0.5</span>, <span style="color: #666666">0.75</span>, <span style="color: #666666">1</span>])
|
||||
plt<span style="color: #666666">.</span>xlabel(<span style="color: #BA2121">r"$x_1$"</span>, fontsize<span style="color: #666666">=20</span>)
|
||||
plt<span style="color: #666666">.</span>ylabel(<span style="color: #BA2121">r"Similarity"</span>, fontsize<span style="color: #666666">=14</span>)
|
||||
plt<span style="color: #666666">.</span>annotate(<span style="color: #BA2121">r'$\mathbf</span><span style="color: #BB6688; font-weight: bold">{x}</span><span style="color: #BA2121">$'</span>,
|
||||
xy<span style="color: #666666">=</span>(X1D[<span style="color: #666666">3</span>, <span style="color: #666666">0</span>], <span style="color: #666666">0</span>),
|
||||
xytext<span style="color: #666666">=</span>(<span style="color: #666666">-0.5</span>, <span style="color: #666666">0.20</span>),
|
||||
ha<span style="color: #666666">=</span><span style="color: #BA2121">"center"</span>,
|
||||
arrowprops<span style="color: #666666">=</span><span style="color: #008000">dict</span>(facecolor<span style="color: #666666">=</span><span style="color: #BA2121">'black'</span>, shrink<span style="color: #666666">=0.1</span>),
|
||||
fontsize<span style="color: #666666">=18</span>,
|
||||
)
|
||||
plt<span style="color: #666666">.</span>text(<span style="color: #666666">-2</span>, <span style="color: #666666">0.9</span>, <span style="color: #BA2121">"$x_2$"</span>, ha<span style="color: #666666">=</span><span style="color: #BA2121">"center"</span>, fontsize<span style="color: #666666">=20</span>)
|
||||
plt<span style="color: #666666">.</span>text(<span style="color: #666666">1</span>, <span style="color: #666666">0.9</span>, <span style="color: #BA2121">"$x_3$"</span>, ha<span style="color: #666666">=</span><span style="color: #BA2121">"center"</span>, fontsize<span style="color: #666666">=20</span>)
|
||||
plt<span style="color: #666666">.</span>axis([<span style="color: #666666">-4.5</span>, <span style="color: #666666">4.5</span>, <span style="color: #666666">-0.1</span>, <span style="color: #666666">1.1</span>])
|
||||
|
||||
plt<span style="color: #666666">.</span>subplot(<span style="color: #666666">122</span>)
|
||||
plt<span style="color: #666666">.</span>grid(<span style="color: #008000; font-weight: bold">True</span>, which<span style="color: #666666">=</span><span style="color: #BA2121">'both'</span>)
|
||||
plt<span style="color: #666666">.</span>axhline(y<span style="color: #666666">=0</span>, color<span style="color: #666666">=</span><span style="color: #BA2121">'k'</span>)
|
||||
plt<span style="color: #666666">.</span>axvline(x<span style="color: #666666">=0</span>, color<span style="color: #666666">=</span><span style="color: #BA2121">'k'</span>)
|
||||
plt<span style="color: #666666">.</span>plot(XK[:, <span style="color: #666666">0</span>][yk<span style="color: #666666">==0</span>], XK[:, <span style="color: #666666">1</span>][yk<span style="color: #666666">==0</span>], <span style="color: #BA2121">"bs"</span>)
|
||||
plt<span style="color: #666666">.</span>plot(XK[:, <span style="color: #666666">0</span>][yk<span style="color: #666666">==1</span>], XK[:, <span style="color: #666666">1</span>][yk<span style="color: #666666">==1</span>], <span style="color: #BA2121">"g^"</span>)
|
||||
plt<span style="color: #666666">.</span>xlabel(<span style="color: #BA2121">r"$x_2$"</span>, fontsize<span style="color: #666666">=20</span>)
|
||||
plt<span style="color: #666666">.</span>ylabel(<span style="color: #BA2121">r"$x_3$ "</span>, fontsize<span style="color: #666666">=20</span>, rotation<span style="color: #666666">=0</span>)
|
||||
plt<span style="color: #666666">.</span>annotate(<span style="color: #BA2121">r'$\phi\left(\mathbf</span><span style="color: #BB6688; font-weight: bold">{x}</span><span style="color: #BA2121">\right)$'</span>,
|
||||
xy<span style="color: #666666">=</span>(XK[<span style="color: #666666">3</span>, <span style="color: #666666">0</span>], XK[<span style="color: #666666">3</span>, <span style="color: #666666">1</span>]),
|
||||
xytext<span style="color: #666666">=</span>(<span style="color: #666666">0.65</span>, <span style="color: #666666">0.50</span>),
|
||||
ha<span style="color: #666666">=</span><span style="color: #BA2121">"center"</span>,
|
||||
arrowprops<span style="color: #666666">=</span><span style="color: #008000">dict</span>(facecolor<span style="color: #666666">=</span><span style="color: #BA2121">'black'</span>, shrink<span style="color: #666666">=0.1</span>),
|
||||
fontsize<span style="color: #666666">=18</span>,
|
||||
)
|
||||
plt<span style="color: #666666">.</span>plot([<span style="color: #666666">-0.1</span>, <span style="color: #666666">1.1</span>], [<span style="color: #666666">0.57</span>, <span style="color: #666666">-0.1</span>], <span style="color: #BA2121">"r--"</span>, linewidth<span style="color: #666666">=3</span>)
|
||||
plt<span style="color: #666666">.</span>axis([<span style="color: #666666">-0.1</span>, <span style="color: #666666">1.1</span>, <span style="color: #666666">-0.1</span>, <span style="color: #666666">1.1</span>])
|
||||
|
||||
plt<span style="color: #666666">.</span>subplots_adjust(right<span style="color: #666666">=1</span>)
|
||||
|
||||
plt<span style="color: #666666">.</span>show()
|
||||
|
||||
|
||||
x1_example <span style="color: #666666">=</span> X1D[<span style="color: #666666">3</span>, <span style="color: #666666">0</span>]
|
||||
<span style="color: #008000; font-weight: bold">for</span> landmark <span style="color: #AA22FF; font-weight: bold">in</span> (<span style="color: #666666">-2</span>, <span style="color: #666666">1</span>):
|
||||
k <span style="color: #666666">=</span> gaussian_rbf(np<span style="color: #666666">.</span>array([[x1_example]]), np<span style="color: #666666">.</span>array([[landmark]]), gamma)
|
||||
<span style="color: #008000">print</span>(<span style="color: #BA2121">"Phi(</span><span style="color: #BB6688; font-weight: bold">{}</span><span style="color: #BA2121">, </span><span style="color: #BB6688; font-weight: bold">{}</span><span style="color: #BA2121">) = </span><span style="color: #BB6688; font-weight: bold">{}</span><span style="color: #BA2121">"</span><span style="color: #666666">.</span>format(x1_example, landmark, k))
|
||||
|
||||
rbf_kernel_svm_clf <span style="color: #666666">=</span> Pipeline([
|
||||
(<span style="color: #BA2121">"scaler"</span>, StandardScaler()),
|
||||
(<span style="color: #BA2121">"svm_clf"</span>, SVC(kernel<span style="color: #666666">=</span><span style="color: #BA2121">"rbf"</span>, gamma<span style="color: #666666">=5</span>, C<span style="color: #666666">=0.001</span>))
|
||||
])
|
||||
rbf_kernel_svm_clf<span style="color: #666666">.</span>fit(X, y)
|
||||
|
||||
|
||||
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.svm</span> <span style="color: #008000; font-weight: bold">import</span> SVC
|
||||
|
||||
gamma1, gamma2 <span style="color: #666666">=</span> <span style="color: #666666">0.1</span>, <span style="color: #666666">5</span>
|
||||
C1, C2 <span style="color: #666666">=</span> <span style="color: #666666">0.001</span>, <span style="color: #666666">1000</span>
|
||||
hyperparams <span style="color: #666666">=</span> (gamma1, C1), (gamma1, C2), (gamma2, C1), (gamma2, C2)
|
||||
|
||||
svm_clfs <span style="color: #666666">=</span> []
|
||||
<span style="color: #008000; font-weight: bold">for</span> gamma, C <span style="color: #AA22FF; font-weight: bold">in</span> hyperparams:
|
||||
rbf_kernel_svm_clf <span style="color: #666666">=</span> Pipeline([
|
||||
(<span style="color: #BA2121">"scaler"</span>, StandardScaler()),
|
||||
(<span style="color: #BA2121">"svm_clf"</span>, SVC(kernel<span style="color: #666666">=</span><span style="color: #BA2121">"rbf"</span>, gamma<span style="color: #666666">=</span>gamma, C<span style="color: #666666">=</span>C))
|
||||
])
|
||||
rbf_kernel_svm_clf<span style="color: #666666">.</span>fit(X, y)
|
||||
svm_clfs<span style="color: #666666">.</span>append(rbf_kernel_svm_clf)
|
||||
|
||||
plt<span style="color: #666666">.</span>figure(figsize<span style="color: #666666">=</span>(<span style="color: #666666">11</span>, <span style="color: #666666">7</span>))
|
||||
|
||||
<span style="color: #008000; font-weight: bold">for</span> i, svm_clf <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">enumerate</span>(svm_clfs):
|
||||
plt<span style="color: #666666">.</span>subplot(<span style="color: #666666">221</span> <span style="color: #666666">+</span> i)
|
||||
plot_predictions(svm_clf, [<span style="color: #666666">-1.5</span>, <span style="color: #666666">2.5</span>, <span style="color: #666666">-1</span>, <span style="color: #666666">1.5</span>])
|
||||
plot_dataset(X, y, [<span style="color: #666666">-1.5</span>, <span style="color: #666666">2.5</span>, <span style="color: #666666">-1</span>, <span style="color: #666666">1.5</span>])
|
||||
gamma, C <span style="color: #666666">=</span> hyperparams[i]
|
||||
plt<span style="color: #666666">.</span>title(<span style="color: #BA2121">r"$\gamma = </span><span style="color: #BB6688; font-weight: bold">{}</span><span style="color: #BA2121">, C = </span><span style="color: #BB6688; font-weight: bold">{}</span><span style="color: #BA2121">$"</span><span style="color: #666666">.</span>format(gamma, C), fontsize<span style="color: #666666">=16</span>)
|
||||
|
||||
plt<span style="color: #666666">.</span>show()
|
||||
</pre></div>
|
||||
<p>
|
||||
<p>
|
||||
<!-- navigation buttons at the bottom of the page -->
|
||||
@@ -196,6 +370,8 @@ This will make our life much easier. You don't need t write your own optimizer.
|
||||
<li class="active"><a href="._week46-bs026.html">27</a></li>
|
||||
<li><a href="._week46-bs027.html">28</a></li>
|
||||
<li><a href="._week46-bs028.html">29</a></li>
|
||||
<li><a href="._week46-bs029.html">30</a></li>
|
||||
<li><a href="._week46-bs030.html">31</a></li>
|
||||
<li><a href="._week46-bs027.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
Reference in New Issue
Block a user