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<h3>Contents</h3>
<ul>
<li><a class="reference internal" href="#">3.2. Support Vector Machines</a><ul>
<li><a class="reference internal" href="#classification">3.2.1. Classification</a></li>
<li><a class="reference internal" href="#regression">3.2.2. Regression</a></li>
<li><a class="reference internal" href="#density-estimation-outliers-detection">3.2.3. Density estimation, outliers detection</a></li>
<li><a class="reference internal" href="#support-vector-machines-for-sparse-data">3.2.4. Support Vector machines for sparse data</a></li>
<li><a class="reference internal" href="#complexity">3.2.5. Complexity</a></li>
<li><a class="reference internal" href="#tips-on-practical-use">3.2.6. Tips on Practical Use</a></li>
<li><a class="reference internal" href="#kernel-functions">3.2.7. Kernel functions</a><ul>
<li><a class="reference internal" href="#custom-kernels">3.2.7.1. Custom Kernels</a><ul>
<li><a class="reference internal" href="#using-python-functions-as-kernels">3.2.7.1.1. Using python functions as kernels</a></li>
<li><a class="reference internal" href="#using-the-gram-matrix">3.2.7.1.2. Using the Gram matrix</a></li>
</ul>
</li>
</ul>
</li>
<li><a class="reference internal" href="#mathematical-formulation">3.2.8. Mathematical formulation</a><ul>
<li><a class="reference internal" href="#svc">3.2.8.1. SVC</a></li>
<li><a class="reference internal" href="#nusvc">3.2.8.2. NuSVC</a></li>
</ul>
</li>
<li><a class="reference internal" href="#frequently-asked-questions">3.2.9. Frequently Asked Questions</a></li>
<li><a class="reference internal" href="#implementation-details">3.2.10. Implementation details</a></li>
</ul>
</li>
</ul>
</div>
<div class="content">
<div class="documentwrapper">
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<div class="body">
<div class="section" id="support-vector-machines">
<span id="svm"></span><h1>3.2. Support Vector Machines<a class="headerlink" href="#support-vector-machines" title="Permalink to this headline">¶</a></h1>
<p><strong>Support vector machines (SVMs)</strong> are a set of supervised learning
methods used for <a class="reference internal" href="#svm-classification"><em>classification</em></a>,
<a class="reference internal" href="#svm-regression"><em>regression</em></a> and <a class="reference internal" href="#svm-outlier-detection"><em>outliers detection</em></a>.</p>
<p>The advantages of Support Vector Machines are:</p>
<blockquote>
<ul class="simple">
<li>Effective in high dimensional spaces.</li>
<li>Still effective in cases where number of dimensions is greater
than the number of samples.</li>
<li>Uses a subset of training points in the decision function (called
support vectors), so it is also memory efficient.</li>
<li>Versatile: different <a class="reference internal" href="#svm-kernels"><em>Kernel functions</em></a> can be
specified for the decision function. Common kernels are
provided, but it is also possible to specify custom kernels.</li>
</ul>
</blockquote>
<p>The disadvantages of Support Vector Machines include:</p>
<blockquote>
<ul class="simple">
<li>If the number of features is much greater than the number of
samples, the method is likely to give poor performances.</li>
<li>SVMs do not directly provide probability estimates, these are
calculated using five-fold cross-validation, and thus
performance can suffer.</li>
</ul>
</blockquote>
<div class="section" id="classification">
<span id="svm-classification"></span><h2>3.2.1. Classification<a class="headerlink" href="#classification" title="Permalink to this headline">¶</a></h2>
<p>Suppose some given data points each belonging to one of N classes, and
the goal is to decide which class a new data point will be in. This
problem is called classification, and can be solved with SVMs using
<em>Support Vector Classifiers</em>, SVC. The
classes that perform this task are <tt class="xref py py-class docutils literal"><span class="pre">SVC</span></tt>, <tt class="xref py py-class docutils literal"><span class="pre">NuSVC</span></tt> and
<tt class="xref py py-class docutils literal"><span class="pre">LinearSVC</span></tt>.</p>
<p><tt class="xref py py-class docutils literal"><span class="pre">SVC</span></tt> and <tt class="xref py py-class docutils literal"><span class="pre">NuSVC</span></tt> are similar methods, but accept
slightly different sets of parameters and have different mathematical
formulations (see section <a class="reference internal" href="#svm-mathematical-formulation"><em>Mathematical formulation</em></a>). On the
other hand, <tt class="xref py py-class docutils literal"><span class="pre">LinearSVC</span></tt> is another implementation of SVC
optimized in the case of a linear kernel. Note that <tt class="xref py py-class docutils literal"><span class="pre">LinearSVC</span></tt>
does not accept keyword ‘kernel’, as this is assumed to be linear. It
also lacks some of the members of SVC and NuSVC, like support_.</p>
<div class="figure align-center">
<a class="reference external image-reference" href="../auto_examples/svm/plot_iris.html"><img alt="auto_examples/svm/images/plot_iris.png" src="auto_examples/svm/images/plot_iris.png" /></a>
</div>
<p>As other classifiers, SVC and NuSVC have to be fitted with two arrays:
an array X of size [m_samples, n_features] holding the training
samples, and an array Y of size [n_samples] holding the target values
(class labels) for the training samples:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">scikits.learn</span> <span class="kn">import</span> <span class="n">svm</span>
<span class="gp">>>> </span><span class="n">X</span> <span class="o">=</span> <span class="p">[[</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">],</span> <span class="p">[</span><span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">]]</span>
<span class="gp">>>> </span><span class="n">Y</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span>
<span class="gp">>>> </span><span class="n">clf</span> <span class="o">=</span> <span class="n">svm</span><span class="o">.</span><span class="n">SVC</span><span class="p">()</span>
<span class="gp">>>> </span><span class="n">clf</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">Y</span><span class="p">)</span>
<span class="go">SVC(kernel='rbf', C=1.0, probability=False, degree=3, coef0=0.0, eps=0.001,</span>
<span class="go"> cache_size=100.0, shrinking=True, gamma=0.5)</span>
</pre></div>
</div>
<p>After being fitted, the model can then be used to predict new values:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="n">clf</span><span class="o">.</span><span class="n">predict</span><span class="p">([[</span><span class="mf">2.</span><span class="p">,</span> <span class="mf">2.</span><span class="p">]])</span>
<span class="go">array([ 1.])</span>
</pre></div>
</div>
<p>SVMs perform classification as a function of some subset of the
training data, called the support vectors. These vectors can be
accessed in member <cite>support_</cite>:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="n">clf</span><span class="o">.</span><span class="n">support_</span>
<span class="go">array([0, 1], dtype=int32)</span>
</pre></div>
</div>
<p>Member <cite>n_support_</cite> holds the number of support vectors for each class:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="n">clf</span><span class="o">.</span><span class="n">n_support_</span>
<span class="go">array([1, 1], dtype=int32)</span>
</pre></div>
</div>
<div class="topic">
<p class="topic-title first">Examples:</p>
<ul class="simple">
<li><a class="reference internal" href="../auto_examples/svm/plot_iris.html#example-svm-plot-iris-py"><em>Plot different SVM classifiers in the iris dataset</em></a>,</li>
<li><a class="reference internal" href="../auto_examples/svm/plot_separating_hyperplane.html#example-svm-plot-separating-hyperplane-py"><em>SVM: Maximum margin separating hyperplane</em></a>,</li>
<li><a class="reference internal" href="../auto_examples/svm/plot_svm_anova.html#example-svm-plot-svm-anova-py"><em>SVM-Anova: SVM with univariate feature selection</em></a>,</li>
<li><a class="reference internal" href="../auto_examples/svm/plot_svm_nonlinear.html#example-svm-plot-svm-nonlinear-py"><em>Non-linear SVM</em></a></li>
</ul>
</div>
</div>
<div class="section" id="regression">
<span id="svm-regression"></span><h2>3.2.2. Regression<a class="headerlink" href="#regression" title="Permalink to this headline">¶</a></h2>
<p>The method of Support Vector Classification can be extended to solve
regression problems. This method is called Support Vector Regression.</p>
<p>The model produced by support vector classification (as described
above) depends only on a subset of the training data, because the cost
function for building the model does not care about training points
that lie beyond the margin. Analogously, the model produced by Support
Vector Regression depends only on a subset of the training data,
because the cost function for building the model ignores any training
data close to the model prediction.</p>
<p>There are two flavors of Support Vector Regression: <tt class="xref py py-class docutils literal"><span class="pre">SVR</span></tt> and
<tt class="xref py py-class docutils literal"><span class="pre">NuSVR</span></tt>.</p>
<p>As with classification classes, the fit method will take as
argument vectors X, y, only that in this case y is expected to have
floating point values instead of integer values.</p>
<div class="topic">
<p class="topic-title first">Examples:</p>
<ul class="simple">
<li><a class="reference internal" href="../auto_examples/svm/plot_svm_regression.html#example-svm-plot-svm-regression-py"><em>Support Vector Regression (SVR) using linear and non-linear kernels</em></a></li>
</ul>
</div>
</div>
<div class="section" id="density-estimation-outliers-detection">
<span id="svm-outlier-detection"></span><h2>3.2.3. Density estimation, outliers detection<a class="headerlink" href="#density-estimation-outliers-detection" title="Permalink to this headline">¶</a></h2>
<p>One-class SVM is used for outliers detection, that is, given a set of
samples, it will detect the soft boundary of that set so as to
classify new points as belonging to that set or not. The class that
implements this is called <tt class="xref py py-class docutils literal"><span class="pre">OneClassSVM</span></tt></p>
<p>In this case, as it is a type of unsupervised learning, the fit method
will only take as input an array X, as there are no class labels.</p>
<div class="figure align-center">
<a class="reference external image-reference" href="../auto_examples/svm/plot_oneclass.html"><img alt="auto_examples/svm/images/plot_oneclass.png" src="auto_examples/svm/images/plot_oneclass.png" /></a>
</div>
<div class="topic">
<p class="topic-title first">Examples:</p>
<ul class="simple">
<li><a class="reference internal" href="../auto_examples/svm/plot_oneclass.html#example-svm-plot-oneclass-py"><em>One-class SVM with non-linear kernel (RBF)</em></a></li>
</ul>
</div>
</div>
<div class="section" id="support-vector-machines-for-sparse-data">
<h2>3.2.4. Support Vector machines for sparse data<a class="headerlink" href="#support-vector-machines-for-sparse-data" title="Permalink to this headline">¶</a></h2>
<p>There is support for sparse data given in any matrix in a format
supported by scipy.sparse. Classes have the same name, just prefixed
by the <cite>sparse</cite> namespace, and take the same arguments, with the
exception of training and test data, which is expected to be in a
matrix format defined in scipy.sparse.</p>
<p>For maximum efficiency, use the CSR matrix format as defined in
<a class="reference external" href="http://docs.scipy.org/doc/scipy/reference/generated/scipy.sparse.csr_matrix.html">scipy.sparse.csr_matrix</a>.</p>
<p>Implemented classes are <tt class="xref py py-class docutils literal"><span class="pre">SVC</span></tt>, <tt class="xref py py-class docutils literal"><span class="pre">NuSVC</span></tt>,
<tt class="xref py py-class docutils literal"><span class="pre">SVR</span></tt>, <tt class="xref py py-class docutils literal"><span class="pre">NuSVR</span></tt>, <tt class="xref py py-class docutils literal"><span class="pre">OneClassSVM</span></tt>,
<tt class="xref py py-class docutils literal"><span class="pre">LinearSVC</span></tt>.</p>
</div>
<div class="section" id="complexity">
<h2>3.2.5. Complexity<a class="headerlink" href="#complexity" title="Permalink to this headline">¶</a></h2>
<p>Support Vector Machines are powerful tools, but their compute and
storage requirements increase rapidly with the number of training
vectors. The core of an SVM is a quadratic programming problem (QP),
separating support vectors from the rest of the training data. The QP
solver used by this <a class="reference external" href="http://www.csie.ntu.edu.tw/~cjlin/libsvm/">libsvm</a>-based implementation scales between
<span class="math">O(n_{features} \times n_{samples}^2)</span> and
<span class="math">O(n_{features} \times n_{samples}^3)</span> depending on how efficiently
the <a class="reference external" href="http://www.csie.ntu.edu.tw/~cjlin/libsvm/">libsvm</a> cache is used in practice (dataset dependent). If the data
is very sparse <span class="math">n_{features}</span> should be replaced by the average number
of non-zero features in a sample vector.</p>
<p>Also note that for the linear case, the algorithm used in
<tt class="xref py py-class docutils literal"><span class="pre">LinearSVC</span></tt> by the <a class="reference external" href="http://www.csie.ntu.edu.tw/~cjlin/liblinear/">liblinear</a> implementation is much more
efficient than its <a class="reference external" href="http://www.csie.ntu.edu.tw/~cjlin/libsvm/">libsvm</a>-based <tt class="xref py py-class docutils literal"><span class="pre">SVC</span></tt> counterpart and can
scale almost linearly to millions of samples and/or features.</p>
</div>
<div class="section" id="tips-on-practical-use">
<h2>3.2.6. Tips on Practical Use<a class="headerlink" href="#tips-on-practical-use" title="Permalink to this headline">¶</a></h2>
<blockquote>
<ul class="simple">
<li>Support Vector Machine algorithms are not scale invariant, so it
is highly recommended to scale your data. For example, scale each
attribute on the input vector X to [0,1] or [-1,+1], or standardize
it to have mean 0 and variance 1. Note that the <em>same</em> scaling
must be applied to the test vector to obtain meaningful
results. See <a class="reference external" href="https://sourceforge.net/apps/trac/scikit-learn/wiki/CookBook">The CookBook</a>
for some examples on scaling.</li>
<li>Parameter nu in NuSVC/OneClassSVM/NuSVR approximates the fraction
of training errors and support vectors.</li>
<li>If data for classification are unbalanced (e.g. many positive and
few negative), set class_weight=’auto’ and/or try different
penalty parameters C.</li>
<li>Specify larger cache size (keyword cache) for huge problems.</li>
<li>The underlying <tt class="xref py py-class docutils literal"><span class="pre">LinearSVC</span></tt> implementation uses a random
number generator to select features when fitting the model. It is
thus not uncommon, to have slightly different results for the same
input data. If that happens, try with a smaller eps parameter.</li>
</ul>
</blockquote>
</div>
<div class="section" id="kernel-functions">
<span id="svm-kernels"></span><h2>3.2.7. Kernel functions<a class="headerlink" href="#kernel-functions" title="Permalink to this headline">¶</a></h2>
<p>The <em>kernel function</em> can be any of the following:</p>
<blockquote>
<ul class="simple">
<li>linear: <span class="math"><x_i, x_j'></span>.</li>
<li>polynomial: <span class="math">(\gamma <x, x'> + r)^d</span>. d is specified by
keyword <cite>degree</cite>.</li>
<li>rbf (<span class="math">exp(-\gamma |x-x'|^2), \gamma > 0</span>). <span class="math">\gamma</span> is
specified by keyword gamma.</li>
<li>sigmoid (<span class="math">tanh(<x_i,x_j> + r)</span>).</li>
</ul>
</blockquote>
<p>Different kernels are specified by keyword kernel at initialization:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="n">linear_svc</span> <span class="o">=</span> <span class="n">svm</span><span class="o">.</span><span class="n">SVC</span><span class="p">(</span><span class="n">kernel</span><span class="o">=</span><span class="s">'linear'</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">linear_svc</span><span class="o">.</span><span class="n">kernel</span>
<span class="go">'linear'</span>
<span class="gp">>>> </span><span class="n">rbf_svc</span> <span class="o">=</span> <span class="n">svm</span><span class="o">.</span><span class="n">SVC</span><span class="p">(</span><span class="n">kernel</span><span class="o">=</span><span class="s">'rbf'</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">rbf_svc</span><span class="o">.</span><span class="n">kernel</span>
<span class="go">'rbf'</span>
</pre></div>
</div>
<div class="section" id="custom-kernels">
<h3>3.2.7.1. Custom Kernels<a class="headerlink" href="#custom-kernels" title="Permalink to this headline">¶</a></h3>
<p>You can define your own kernels by either giving the kernel as a
python function or by precomputing the Gram matrix.</p>
<p>Classifiers with custom kernels behave the same way as any other
classifiers, except that:</p>
<blockquote>
<ul class="simple">
<li>Field <cite>support_vectors_</cite> is now empty, only indices of support
vectors are stored in <cite>support_</cite></li>
<li>A reference (and not a copy) of the first argument in the fit()
method is stored for future reference. If that array changes
between the use of fit() and predict() you will have unexpected
results.</li>
</ul>
</blockquote>
<div class="section" id="using-python-functions-as-kernels">
<h4>3.2.7.1.1. Using python functions as kernels<a class="headerlink" href="#using-python-functions-as-kernels" title="Permalink to this headline">¶</a></h4>
<p>You can also use your own defined kernels by passing a function to the
keyword <cite>kernel</cite> in the constructor.</p>
<p>Your kernel must take as arguments two matrices and return a third matrix.</p>
<p>The following code defines a linear kernel and creates a classifier
instance that will use that kernel:</p>
<div class="highlight-python"><div class="highlight"><pre><span class="gp">>>> </span><span class="kn">import</span> <span class="nn">numpy</span> <span class="kn">as</span> <span class="nn">np</span>
<span class="gp">>>> </span><span class="kn">from</span> <span class="nn">scikits.learn</span> <span class="kn">import</span> <span class="n">svm</span>
<span class="gp">>>> </span><span class="k">def</span> <span class="nf">my_kernel</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">):</span>
<span class="gp">... </span> <span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="o">.</span><span class="n">T</span><span class="p">)</span>
<span class="gp">...</span>
<span class="gp">>>> </span><span class="n">clf</span> <span class="o">=</span> <span class="n">svm</span><span class="o">.</span><span class="n">SVC</span><span class="p">(</span><span class="n">kernel</span><span class="o">=</span><span class="n">my_kernel</span><span class="p">)</span>
</pre></div>
</div>
</div>
<div class="section" id="using-the-gram-matrix">
<h4>3.2.7.1.2. Using the Gram matrix<a class="headerlink" href="#using-the-gram-matrix" title="Permalink to this headline">¶</a></h4>
<p>Set kernel=’precomputed’ and pass the Gram matrix instead of X in the
fit method.</p>
<div class="topic">
<p class="topic-title first">Examples:</p>
<ul class="simple">
<li><a class="reference internal" href="../auto_examples/svm/plot_custom_kernel.html#example-svm-plot-custom-kernel-py"><em>SVM with custom kernel</em></a>.</li>
</ul>
</div>
</div>
</div>
</div>
<div class="section" id="mathematical-formulation">
<span id="svm-mathematical-formulation"></span><h2>3.2.8. Mathematical formulation<a class="headerlink" href="#mathematical-formulation" title="Permalink to this headline">¶</a></h2>
<p>A support vector machine constructs a hyper-plane or set of hyper-planes
in a high or infinite dimensional space, which can be used for
classification, regression or other tasks. Intuitively, a good
separation is achieved by the hyper-plane that has the largest distance
to the nearest training data points of any class (so-called functional
margin), since in general the larger the margin the lower the
generalization error of the classifier.</p>
<div class="figure align-center">
<a class="reference internal image-reference" href="auto_examples/svm/images/plot_separating_hyperplane.png"><img alt="auto_examples/svm/images/plot_separating_hyperplane.png" src="auto_examples/svm/images/plot_separating_hyperplane.png" /></a>
</div>
<div class="section" id="svc">
<h3>3.2.8.1. SVC<a class="headerlink" href="#svc" title="Permalink to this headline">¶</a></h3>
<p>Given training vectors <span class="math">x_i \in R^n</span>, i=1,..., l, in two
classes, and a vector <span class="math">y \in R^l</span> such that <span class="math">y_i \in {1,
-1}</span>, SVC solves the following primal problem:</p>
<div class="math">
<p><span class="math">\min_ {w, b, \zeta} \frac{1}{2} w^T w + C \sum_{i=1, l} \zeta_i
\textrm {subject to } & y_i (w^T \phi (x_i) + b) \geq 1 - \zeta_i,\\
& \zeta_i \geq 0, i=1, ..., l</span></p>
</div><p>Its dual is</p>
<div class="math">
<p><span class="math">\min_{\alpha} \frac{1}{2} \alpha^T Q \alpha - e^T \alpha
\textrm {subject to } & y^T \alpha = 0\\
& 0 \leq \alpha_i \leq C, i=1, ..., l</span></p>
</div><p>where <span class="math">e</span> is the vector of all ones, C > 0 is the upper bound, Q
is an l by l positive semidefinite matrix, <span class="math">Q_ij \equiv K(x_i,
x_j)</span> and <span class="math">\phi (x_i)^T \phi (x)</span> is the kernel. Here training
vectors are mapped into a higher (maybe infinite) dimensional space by
the function <span class="math">\phi</span></p>
<p>The decision function is:</p>
<div class="math">
<p><span class="math">sgn(\sum_{i=1}^l y_i \alpha_i K(x_i, x) + \rho)</span></p>
</div><p>This parameters can be accessed through the members <cite>dual_coef_</cite>
which holds the product <span class="math">y_i \alpha_i</span>, <cite>support_vectors_</cite> which
holds the support vectors, and <cite>intercept_</cite> which holds the independent
term <span class="math">-\rho</span> :</p>
<div class="topic">
<p class="topic-title first">References:</p>
<ul class="simple">
<li><a class="reference external" href="http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.17.7215">“Automatic Capacity Tuning of Very Large VC-dimension Classifiers”</a>
I Guyon, B Boser, V Vapnik - Advances in neural information
processing 1993,</li>
<li><a class="reference external" href="http://www.springerlink.com/content/k238jx04hm87j80g/">“Support-vector networks”</a>
C. Cortes, V. Vapnik, Machine Leaming, 20, 273-297 (1995)</li>
</ul>
</div>
</div>
<div class="section" id="nusvc">
<h3>3.2.8.2. NuSVC<a class="headerlink" href="#nusvc" title="Permalink to this headline">¶</a></h3>
<p>We introduce a new parameter <span class="math">\nu</span> which controls the number of
support vectors and training errors. The parameter <span class="math">\nu \in (0,
1]</span> is an upper bound on the fraction of training errors and a lower
bound of the fraction of support vectors.</p>
</div>
</div>
<div class="section" id="frequently-asked-questions">
<h2>3.2.9. Frequently Asked Questions<a class="headerlink" href="#frequently-asked-questions" title="Permalink to this headline">¶</a></h2>
<blockquote>
<ul>
<li><p class="first">Q: Can I get the indices of the support vectors instead of the
support vectors ?</p>
<p>A: The underlying C implementation does not provide this
information.</p>
</li>
</ul>
</blockquote>
</div>
<div class="section" id="implementation-details">
<h2>3.2.10. Implementation details<a class="headerlink" href="#implementation-details" title="Permalink to this headline">¶</a></h2>
<p>Internally, we use <a class="reference external" href="http://www.csie.ntu.edu.tw/~cjlin/libsvm/">libsvm</a> and <a class="reference external" href="http://www.csie.ntu.edu.tw/~cjlin/liblinear/">liblinear</a> to handle all
computations. These libraries are wrapped using C and Cython.</p>
<div class="topic">
<p class="topic-title first">References:</p>
<p>For a description of the implementation and details of the algorithms
used, please refer to</p>
<blockquote>
<ul class="simple">
<li><a class="reference external" href="http://www.csie.ntu.edu.tw/~cjlin/papers/libsvm.pdf">LIBSVM: a library for Support Vector Machines</a></li>
<li><a class="reference external" href="http://www.csie.ntu.edu.tw/~cjlin/liblinear/">LIBLINEAR – A Library for Large Linear Classification</a></li>
</ul>
</blockquote>
</div>
</div>
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