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updating week 46

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mhjensen
2020-11-09 23:21:39 +01:00
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@@ -42,39 +42,41 @@ Automatically generated HTML file from DocOnce source
<!-- tocinfo
{'highest level': 2,
'sections': [('Overview of week 46', 2, None, '___sec0'),
('Support Vector Machines, overarching aims', 2, None, '___sec1'),
('Hyperplanes and all that', 2, None, '___sec2'),
('What is a hyperplane?', 2, None, '___sec3'),
('A $p$-dimensional space of features', 2, None, '___sec4'),
('The two-dimensional case', 2, None, '___sec5'),
('Getting into the details', 2, None, '___sec6'),
('First attempt at a minimization approach', 2, None, '___sec7'),
('Solving the equations', 2, None, '___sec8'),
('Code Example', 2, None, '___sec9'),
('Problems with the Simpler Approach', 2, None, '___sec10'),
('A better approach', 2, None, '___sec11'),
('Thursday', 2, None, '___sec1'),
('Friday', 2, None, '___sec2'),
('Support Vector Machines, overarching aims', 2, None, '___sec3'),
('Hyperplanes and all that', 2, None, '___sec4'),
('What is a hyperplane?', 2, None, '___sec5'),
('A $p$-dimensional space of features', 2, None, '___sec6'),
('The two-dimensional case', 2, None, '___sec7'),
('Getting into the details', 2, None, '___sec8'),
('First attempt at a minimization approach', 2, None, '___sec9'),
('Solving the equations', 2, None, '___sec10'),
('Code Example', 2, None, '___sec11'),
('Problems with the Simpler Approach', 2, None, '___sec12'),
('A better approach', 2, None, '___sec13'),
('A quick Reminder on Lagrangian Multipliers',
2,
None,
'___sec12'),
('Adding the Multiplier', 2, None, '___sec13'),
('Setting up the Problem', 2, None, '___sec14'),
('The problem to solve', 2, None, '___sec15'),
('The last steps', 2, None, '___sec16'),
('A soft classifier', 2, None, '___sec17'),
('Soft optmization problem', 2, None, '___sec18'),
('Kernels and non-linearity', 2, None, '___sec19'),
('The equations', 2, None, '___sec20'),
('The problem to solve', 2, None, '___sec21'),
("Different kernels and Mercer's theorem", 2, None, '___sec22'),
('The moons example', 2, None, '___sec23'),
'___sec14'),
('Adding the Multiplier', 2, None, '___sec15'),
('Setting up the Problem', 2, None, '___sec16'),
('The problem to solve', 2, None, '___sec17'),
('The last steps', 2, None, '___sec18'),
('A soft classifier', 2, None, '___sec19'),
('Soft optmization problem', 2, None, '___sec20'),
('Kernels and non-linearity', 2, None, '___sec21'),
('The equations', 2, None, '___sec22'),
('The problem to solve', 2, None, '___sec23'),
("Different kernels and Mercer's theorem", 2, None, '___sec24'),
('The moons example', 2, None, '___sec25'),
('Mathematical optimization of convex functions',
2,
None,
'___sec24'),
('How do we solve these problems?', 2, None, '___sec25'),
('A simple example', 2, None, '___sec26'),
('Back to the more realistic cases', 2, None, '___sec27')]}
'___sec26'),
('How do we solve these problems?', 2, None, '___sec27'),
('A simple example', 2, None, '___sec28'),
('Back to the more realistic cases', 2, None, '___sec29')]}
end of tocinfo -->
<body>
@@ -113,33 +115,35 @@ MathJax.Hub.Config({
<a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
<ul class="dropdown-menu">
<!-- navigation toc: --> <li><a href="._week46-bs001.html#___sec0" style="font-size: 80%;">Overview of week 46</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs002.html#___sec1" style="font-size: 80%;">Support Vector Machines, overarching aims</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs003.html#___sec2" style="font-size: 80%;">Hyperplanes and all that</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs004.html#___sec3" style="font-size: 80%;">What is a hyperplane?</a></li>
<!-- navigation toc: --> <li><a href="#___sec4" style="font-size: 80%;">A \( p \)-dimensional space of features</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs006.html#___sec5" style="font-size: 80%;">The two-dimensional case</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs007.html#___sec6" style="font-size: 80%;">Getting into the details</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs008.html#___sec7" style="font-size: 80%;">First attempt at a minimization approach</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs009.html#___sec8" style="font-size: 80%;">Solving the equations</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs010.html#___sec9" style="font-size: 80%;">Code Example</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs011.html#___sec10" style="font-size: 80%;">Problems with the Simpler Approach</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs012.html#___sec11" style="font-size: 80%;">A better approach</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs013.html#___sec12" style="font-size: 80%;">A quick Reminder on Lagrangian Multipliers</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs014.html#___sec13" style="font-size: 80%;">Adding the Multiplier</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs015.html#___sec14" style="font-size: 80%;">Setting up the Problem</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs016.html#___sec15" style="font-size: 80%;">The problem to solve</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs017.html#___sec16" style="font-size: 80%;">The last steps</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs018.html#___sec17" style="font-size: 80%;">A soft classifier</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs019.html#___sec18" style="font-size: 80%;">Soft optmization problem</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs020.html#___sec19" style="font-size: 80%;">Kernels and non-linearity</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs021.html#___sec20" style="font-size: 80%;">The equations</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs022.html#___sec21" style="font-size: 80%;">The problem to solve</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs023.html#___sec22" style="font-size: 80%;">Different kernels and Mercer's theorem</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs024.html#___sec23" style="font-size: 80%;">The moons example</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs025.html#___sec24" style="font-size: 80%;">Mathematical optimization of convex functions</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs026.html#___sec25" style="font-size: 80%;">How do we solve these problems?</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs027.html#___sec26" style="font-size: 80%;">A simple example</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs028.html#___sec27" style="font-size: 80%;">Back to the more realistic cases</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs002.html#___sec1" style="font-size: 80%;">Thursday</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs003.html#___sec2" style="font-size: 80%;">Friday</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs004.html#___sec3" style="font-size: 80%;">Support Vector Machines, overarching aims</a></li>
<!-- navigation toc: --> <li><a href="#___sec4" style="font-size: 80%;">Hyperplanes and all that</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs006.html#___sec5" style="font-size: 80%;">What is a hyperplane?</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs007.html#___sec6" style="font-size: 80%;">A \( p \)-dimensional space of features</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs008.html#___sec7" style="font-size: 80%;">The two-dimensional case</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs009.html#___sec8" style="font-size: 80%;">Getting into the details</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs010.html#___sec9" style="font-size: 80%;">First attempt at a minimization approach</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs011.html#___sec10" style="font-size: 80%;">Solving the equations</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs012.html#___sec11" style="font-size: 80%;">Code Example</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs013.html#___sec12" style="font-size: 80%;">Problems with the Simpler Approach</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs014.html#___sec13" style="font-size: 80%;">A better approach</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs015.html#___sec14" style="font-size: 80%;">A quick Reminder on Lagrangian Multipliers</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs016.html#___sec15" style="font-size: 80%;">Adding the Multiplier</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs017.html#___sec16" style="font-size: 80%;">Setting up the Problem</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs018.html#___sec17" style="font-size: 80%;">The problem to solve</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs019.html#___sec18" style="font-size: 80%;">The last steps</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs020.html#___sec19" style="font-size: 80%;">A soft classifier</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs021.html#___sec20" style="font-size: 80%;">Soft optmization problem</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs022.html#___sec21" style="font-size: 80%;">Kernels and non-linearity</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs023.html#___sec22" style="font-size: 80%;">The equations</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs024.html#___sec23" style="font-size: 80%;">The problem to solve</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs025.html#___sec24" style="font-size: 80%;">Different kernels and Mercer's theorem</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs026.html#___sec25" style="font-size: 80%;">The moons example</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs027.html#___sec26" style="font-size: 80%;">Mathematical optimization of convex functions</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs028.html#___sec27" style="font-size: 80%;">How do we solve these problems?</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs029.html#___sec28" style="font-size: 80%;">A simple example</a></li>
<!-- navigation toc: --> <li><a href="._week46-bs030.html#___sec29" style="font-size: 80%;">Back to the more realistic cases</a></li>
</ul>
</li>
@@ -155,45 +159,90 @@ MathJax.Hub.Config({
<a name="part0005"></a>
<!-- !split -->
<h2 id="___sec4" class="anchor">A \( p \)-dimensional space of features </h2>
<h2 id="___sec4" class="anchor">Hyperplanes and all that </h2>
<p>
We limit ourselves to two classes of outputs \( y_i \) and assign these classes the values \( y_i = \pm 1 \).
In a \( p \)-dimensional space of say \( p \) features we have a hyperplane defines as
$$
b+wx_1+w_2x_2+\dots +w_px_p=0.
$$
If we define a
matrix \( \boldsymbol{X}=\left[\boldsymbol{x}_1,\boldsymbol{x}_2,\dots, \boldsymbol{x}_p\right] \)
of dimension \( n\times p \), where \( n \) represents the observations for each feature and each vector \( x_i \) is a column vector of the matrix \( \boldsymbol{X} \),
$$
\boldsymbol{x}_i = \begin{bmatrix} x_{i1} \\ x_{i2} \\ \dots \\ \dots \\ x_{ip} \end{bmatrix}.
$$
If the above condition is not met for a given vector \( \boldsymbol{x}_i \) we have
$$
b+w_1x_{i1}+w_2x_{i2}+\dots +w_px_{ip} >0,
$$
if our output \( y_i=1 \).
In this case we say that \( \boldsymbol{x}_i \) lies on one of the sides of the hyperplane and if
$$
b+w_1x_{i1}+w_2x_{i2}+\dots +w_px_{ip} < 0,
$$
for the class of observations \( y_i=-1 \),
then \( \boldsymbol{x}_i \) lies on the other side.
The theory behind support vector machines (SVM hereafter) is based on
the mathematical description of so-called hyperplanes. Let us start
with a two-dimensional case. This will also allow us to introduce our
first SVM examples. These will be tailored to the case of two specific
classes, as displayed in the figure here based on the usage of the petal data.
<p>
Equivalently, for the two classes of observations we have
$$
y_i\left(b+w_1x_{i1}+w_2x_{i2}+\dots +w_px_{ip}\right) > 0.
$$
We assume here that our data set can be well separated into two
domains, where a straight line does the job in the separating the two
classes. Here the two classes are represented by either squares or
circles.
<p>
When we try to separate hyperplanes, if it exists, we can use it to construct a natural classifier: a test observation is assigned a given class depending on which side of the hyperplane it is located.
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
<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">sklearn</span> <span style="color: #008000; font-weight: bold">import</span> datasets
<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, LinearSVC
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.linear_model</span> <span style="color: #008000; font-weight: bold">import</span> SGDClassifier
<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
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib</span>
<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>
plt<span style="color: #666666">.</span>rcParams[<span style="color: #BA2121">&#39;axes.labelsize&#39;</span>] <span style="color: #666666">=</span> <span style="color: #666666">14</span>
plt<span style="color: #666666">.</span>rcParams[<span style="color: #BA2121">&#39;xtick.labelsize&#39;</span>] <span style="color: #666666">=</span> <span style="color: #666666">12</span>
plt<span style="color: #666666">.</span>rcParams[<span style="color: #BA2121">&#39;ytick.labelsize&#39;</span>] <span style="color: #666666">=</span> <span style="color: #666666">12</span>
iris <span style="color: #666666">=</span> datasets<span style="color: #666666">.</span>load_iris()
X <span style="color: #666666">=</span> iris[<span style="color: #BA2121">&quot;data&quot;</span>][:, (<span style="color: #666666">2</span>, <span style="color: #666666">3</span>)] <span style="color: #408080; font-style: italic"># petal length, petal width</span>
y <span style="color: #666666">=</span> iris[<span style="color: #BA2121">&quot;target&quot;</span>]
setosa_or_versicolor <span style="color: #666666">=</span> (y <span style="color: #666666">==</span> <span style="color: #666666">0</span>) <span style="color: #666666">|</span> (y <span style="color: #666666">==</span> <span style="color: #666666">1</span>)
X <span style="color: #666666">=</span> X[setosa_or_versicolor]
y <span style="color: #666666">=</span> y[setosa_or_versicolor]
C <span style="color: #666666">=</span> <span style="color: #666666">5</span>
alpha <span style="color: #666666">=</span> <span style="color: #666666">1</span> <span style="color: #666666">/</span> (C <span style="color: #666666">*</span> <span style="color: #008000">len</span>(X))
lin_clf <span style="color: #666666">=</span> LinearSVC(loss<span style="color: #666666">=</span><span style="color: #BA2121">&quot;hinge&quot;</span>, C<span style="color: #666666">=</span>C, random_state<span style="color: #666666">=42</span>)
svm_clf <span style="color: #666666">=</span> SVC(kernel<span style="color: #666666">=</span><span style="color: #BA2121">&quot;linear&quot;</span>, C<span style="color: #666666">=</span>C)
sgd_clf <span style="color: #666666">=</span> SGDClassifier(loss<span style="color: #666666">=</span><span style="color: #BA2121">&quot;hinge&quot;</span>, learning_rate<span style="color: #666666">=</span><span style="color: #BA2121">&quot;constant&quot;</span>, eta0<span style="color: #666666">=0.001</span>, alpha<span style="color: #666666">=</span>alpha,
max_iter<span style="color: #666666">=100000</span>, random_state<span style="color: #666666">=42</span>)
scaler <span style="color: #666666">=</span> StandardScaler()
X_scaled <span style="color: #666666">=</span> scaler<span style="color: #666666">.</span>fit_transform(X)
lin_clf<span style="color: #666666">.</span>fit(X_scaled, y)
svm_clf<span style="color: #666666">.</span>fit(X_scaled, y)
sgd_clf<span style="color: #666666">.</span>fit(X_scaled, y)
<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;LinearSVC: &quot;</span>, lin_clf<span style="color: #666666">.</span>intercept_, lin_clf<span style="color: #666666">.</span>coef_)
<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;SVC: &quot;</span>, svm_clf<span style="color: #666666">.</span>intercept_, svm_clf<span style="color: #666666">.</span>coef_)
<span style="color: #008000">print</span>(<span style="color: #BA2121">&quot;SGDClassifier(alpha=</span><span style="color: #BB6688; font-weight: bold">{:.5f}</span><span style="color: #BA2121">):&quot;</span><span style="color: #666666">.</span>format(sgd_clf<span style="color: #666666">.</span>alpha), sgd_clf<span style="color: #666666">.</span>intercept_, sgd_clf<span style="color: #666666">.</span>coef_)
<span style="color: #408080; font-style: italic"># Compute the slope and bias of each decision boundary</span>
w1 <span style="color: #666666">=</span> <span style="color: #666666">-</span>lin_clf<span style="color: #666666">.</span>coef_[<span style="color: #666666">0</span>, <span style="color: #666666">0</span>]<span style="color: #666666">/</span>lin_clf<span style="color: #666666">.</span>coef_[<span style="color: #666666">0</span>, <span style="color: #666666">1</span>]
b1 <span style="color: #666666">=</span> <span style="color: #666666">-</span>lin_clf<span style="color: #666666">.</span>intercept_[<span style="color: #666666">0</span>]<span style="color: #666666">/</span>lin_clf<span style="color: #666666">.</span>coef_[<span style="color: #666666">0</span>, <span style="color: #666666">1</span>]
w2 <span style="color: #666666">=</span> <span style="color: #666666">-</span>svm_clf<span style="color: #666666">.</span>coef_[<span style="color: #666666">0</span>, <span style="color: #666666">0</span>]<span style="color: #666666">/</span>svm_clf<span style="color: #666666">.</span>coef_[<span style="color: #666666">0</span>, <span style="color: #666666">1</span>]
b2 <span style="color: #666666">=</span> <span style="color: #666666">-</span>svm_clf<span style="color: #666666">.</span>intercept_[<span style="color: #666666">0</span>]<span style="color: #666666">/</span>svm_clf<span style="color: #666666">.</span>coef_[<span style="color: #666666">0</span>, <span style="color: #666666">1</span>]
w3 <span style="color: #666666">=</span> <span style="color: #666666">-</span>sgd_clf<span style="color: #666666">.</span>coef_[<span style="color: #666666">0</span>, <span style="color: #666666">0</span>]<span style="color: #666666">/</span>sgd_clf<span style="color: #666666">.</span>coef_[<span style="color: #666666">0</span>, <span style="color: #666666">1</span>]
b3 <span style="color: #666666">=</span> <span style="color: #666666">-</span>sgd_clf<span style="color: #666666">.</span>intercept_[<span style="color: #666666">0</span>]<span style="color: #666666">/</span>sgd_clf<span style="color: #666666">.</span>coef_[<span style="color: #666666">0</span>, <span style="color: #666666">1</span>]
<span style="color: #408080; font-style: italic"># Transform the decision boundary lines back to the original scale</span>
line1 <span style="color: #666666">=</span> scaler<span style="color: #666666">.</span>inverse_transform([[<span style="color: #666666">-10</span>, <span style="color: #666666">-10</span> <span style="color: #666666">*</span> w1 <span style="color: #666666">+</span> b1], [<span style="color: #666666">10</span>, <span style="color: #666666">10</span> <span style="color: #666666">*</span> w1 <span style="color: #666666">+</span> b1]])
line2 <span style="color: #666666">=</span> scaler<span style="color: #666666">.</span>inverse_transform([[<span style="color: #666666">-10</span>, <span style="color: #666666">-10</span> <span style="color: #666666">*</span> w2 <span style="color: #666666">+</span> b2], [<span style="color: #666666">10</span>, <span style="color: #666666">10</span> <span style="color: #666666">*</span> w2 <span style="color: #666666">+</span> b2]])
line3 <span style="color: #666666">=</span> scaler<span style="color: #666666">.</span>inverse_transform([[<span style="color: #666666">-10</span>, <span style="color: #666666">-10</span> <span style="color: #666666">*</span> w3 <span style="color: #666666">+</span> b3], [<span style="color: #666666">10</span>, <span style="color: #666666">10</span> <span style="color: #666666">*</span> w3 <span style="color: #666666">+</span> b3]])
<span style="color: #408080; font-style: italic"># Plot all three decision boundaries</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>plot(line1[:, <span style="color: #666666">0</span>], line1[:, <span style="color: #666666">1</span>], <span style="color: #BA2121">&quot;k:&quot;</span>, label<span style="color: #666666">=</span><span style="color: #BA2121">&quot;LinearSVC&quot;</span>)
plt<span style="color: #666666">.</span>plot(line2[:, <span style="color: #666666">0</span>], line2[:, <span style="color: #666666">1</span>], <span style="color: #BA2121">&quot;b--&quot;</span>, linewidth<span style="color: #666666">=2</span>, label<span style="color: #666666">=</span><span style="color: #BA2121">&quot;SVC&quot;</span>)
plt<span style="color: #666666">.</span>plot(line3[:, <span style="color: #666666">0</span>], line3[:, <span style="color: #666666">1</span>], <span style="color: #BA2121">&quot;r-&quot;</span>, label<span style="color: #666666">=</span><span style="color: #BA2121">&quot;SGDClassifier&quot;</span>)
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">&quot;bs&quot;</span>) <span style="color: #408080; font-style: italic"># label=&quot;Iris-Versicolor&quot;</span>
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">&quot;yo&quot;</span>) <span style="color: #408080; font-style: italic"># label=&quot;Iris-Setosa&quot;</span>
plt<span style="color: #666666">.</span>xlabel(<span style="color: #BA2121">&quot;Petal length&quot;</span>, fontsize<span style="color: #666666">=14</span>)
plt<span style="color: #666666">.</span>ylabel(<span style="color: #BA2121">&quot;Petal width&quot;</span>, fontsize<span style="color: #666666">=14</span>)
plt<span style="color: #666666">.</span>legend(loc<span style="color: #666666">=</span><span style="color: #BA2121">&quot;upper center&quot;</span>, fontsize<span style="color: #666666">=14</span>)
plt<span style="color: #666666">.</span>axis([<span style="color: #666666">0</span>, <span style="color: #666666">5.5</span>, <span style="color: #666666">0</span>, <span style="color: #666666">2</span>])
plt<span style="color: #666666">.</span>show()
</pre></div>
<p>
<p>
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@@ -215,7 +264,7 @@ When we try to separate hyperplanes, if it exists, we can use it to construct a
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