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<li><a class="reference internal" href="#">Demo of HDBSCAN clustering algorithm</a><ul>
<li><a class="reference internal" href="#generate-sample-data">Generate sample data</a></li>
<li><a class="reference internal" href="#scale-invariance">Scale Invariance</a></li>
<li><a class="reference internal" href="#multi-scale-clustering">Multi-Scale Clustering</a></li>
<li><a class="reference internal" href="#hyperparameter-robustness">Hyperparameter Robustness</a><ul>
<li><a class="reference internal" href="#min-cluster-size"><code class="docutils literal notranslate"><span class="pre">min_cluster_size</span></code></a></li>
<li><a class="reference internal" href="#min-samples"><code class="docutils literal notranslate"><span class="pre">min_samples</span></code></a></li>
<li><a class="reference internal" href="#dbscan-clustering"><code class="docutils literal notranslate"><span class="pre">dbscan_clustering</span></code></a></li>
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<section class="sphx-glr-example-title" id="demo-of-hdbscan-clustering-algorithm">
<span id="sphx-glr-auto-examples-cluster-plot-hdbscan-py"></span><h1>Demo of HDBSCAN clustering algorithm<a class="headerlink" href="#demo-of-hdbscan-clustering-algorithm" title="Link to this heading">¶</a></h1>
<p>In this demo we will take a look at <a class="reference internal" href="../../modules/generated/sklearn.cluster.HDBSCAN.html#sklearn.cluster.HDBSCAN" title="sklearn.cluster.HDBSCAN"><code class="xref py py-class docutils literal notranslate"><span class="pre">cluster.HDBSCAN</span></code></a> from the
perspective of generalizing the <a class="reference internal" href="../../modules/generated/sklearn.cluster.DBSCAN.html#sklearn.cluster.DBSCAN" title="sklearn.cluster.DBSCAN"><code class="xref py py-class docutils literal notranslate"><span class="pre">cluster.DBSCAN</span></code></a> algorithm.
We’ll compare both algorithms on specific datasets. Finally we’ll evaluate
HDBSCAN’s sensitivity to certain hyperparameters.</p>
<p>We first define a couple utility functions for convenience.</p>
<div class="highlight-Python notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="kn">from</span> <span class="nn">sklearn.cluster</span> <span class="kn">import</span> <a href="../../modules/generated/sklearn.cluster.DBSCAN.html#sklearn.cluster.DBSCAN" title="sklearn.cluster.DBSCAN" class="sphx-glr-backref-module-sklearn-cluster sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">DBSCAN</span></a><span class="p">,</span> <a href="../../modules/generated/sklearn.cluster.HDBSCAN.html#sklearn.cluster.HDBSCAN" title="sklearn.cluster.HDBSCAN" class="sphx-glr-backref-module-sklearn-cluster sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">HDBSCAN</span></a>
<span class="kn">from</span> <span class="nn">sklearn.datasets</span> <span class="kn">import</span> <a href="../../modules/generated/sklearn.datasets.make_blobs.html#sklearn.datasets.make_blobs" title="sklearn.datasets.make_blobs" class="sphx-glr-backref-module-sklearn-datasets sphx-glr-backref-type-py-function"><span class="n">make_blobs</span></a>
<span class="k">def</span> <span class="nf">plot</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">labels</span><span class="p">,</span> <span class="n">probabilities</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">parameters</span><span class="o">=</span><span class="kc">None</span><span class="p">,</span> <span class="n">ground_truth</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span> <span class="n">ax</span><span class="o">=</span><span class="kc">None</span><span class="p">):</span>
<span class="k">if</span> <span class="n">ax</span> <span class="ow">is</span> <span class="kc">None</span><span class="p">:</span>
<span class="n">_</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.subplots.html#matplotlib.pyplot.subplots" title="matplotlib.pyplot.subplots" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">subplots</span></a><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span> <span class="mi">4</span><span class="p">))</span>
<span class="n">labels</span> <span class="o">=</span> <span class="n">labels</span> <span class="k">if</span> <span class="n">labels</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span> <span class="k">else</span> <a href="https://numpy.org/doc/stable/reference/generated/numpy.ones.html#numpy.ones" title="numpy.ones" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">ones</span></a><span class="p">(</span><span class="n">X</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
<span class="n">probabilities</span> <span class="o">=</span> <span class="n">probabilities</span> <span class="k">if</span> <span class="n">probabilities</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span> <span class="k">else</span> <a href="https://numpy.org/doc/stable/reference/generated/numpy.ones.html#numpy.ones" title="numpy.ones" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">ones</span></a><span class="p">(</span><span class="n">X</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
<span class="c1"># Black removed and is used for noise instead.</span>
<span class="n">unique_labels</span> <span class="o">=</span> <span class="nb">set</span><span class="p">(</span><span class="n">labels</span><span class="p">)</span>
<span class="n">colors</span> <span class="o">=</span> <span class="p">[</span><span class="n">plt</span><span class="o">.</span><span class="n">cm</span><span class="o">.</span><span class="n">Spectral</span><span class="p">(</span><span class="n">each</span><span class="p">)</span> <span class="k">for</span> <span class="n">each</span> <span class="ow">in</span> <a href="https://numpy.org/doc/stable/reference/generated/numpy.linspace.html#numpy.linspace" title="numpy.linspace" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">linspace</span></a><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">unique_labels</span><span class="p">))]</span>
<span class="c1"># The probability of a point belonging to its labeled cluster determines</span>
<span class="c1"># the size of its marker</span>
<span class="n">proba_map</span> <span class="o">=</span> <span class="p">{</span><span class="n">idx</span><span class="p">:</span> <span class="n">probabilities</span><span class="p">[</span><span class="n">idx</span><span class="p">]</span> <span class="k">for</span> <span class="n">idx</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">labels</span><span class="p">))}</span>
<span class="k">for</span> <span class="n">k</span><span class="p">,</span> <span class="n">col</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">unique_labels</span><span class="p">,</span> <span class="n">colors</span><span class="p">):</span>
<span class="k">if</span> <span class="n">k</span> <span class="o">==</span> <span class="o">-</span><span class="mi">1</span><span class="p">:</span>
<span class="c1"># Black used for noise.</span>
<span class="n">col</span> <span class="o">=</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</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="n">class_index</span> <span class="o">=</span> <a href="https://numpy.org/doc/stable/reference/generated/numpy.where.html#numpy.where" title="numpy.where" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">where</span></a><span class="p">(</span><span class="n">labels</span> <span class="o">==</span> <span class="n">k</span><span class="p">)[</span><span class="mi">0</span><span class="p">]</span>
<span class="k">for</span> <span class="n">ci</span> <span class="ow">in</span> <span class="n">class_index</span><span class="p">:</span>
<span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span>
<span class="n">X</span><span class="p">[</span><span class="n">ci</span><span class="p">,</span> <span class="mi">0</span><span class="p">],</span>
<span class="n">X</span><span class="p">[</span><span class="n">ci</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span>
<span class="s2">"x"</span> <span class="k">if</span> <span class="n">k</span> <span class="o">==</span> <span class="o">-</span><span class="mi">1</span> <span class="k">else</span> <span class="s2">"o"</span><span class="p">,</span>
<span class="n">markerfacecolor</span><span class="o">=</span><span class="nb">tuple</span><span class="p">(</span><span class="n">col</span><span class="p">),</span>
<span class="n">markeredgecolor</span><span class="o">=</span><span class="s2">"k"</span><span class="p">,</span>
<span class="n">markersize</span><span class="o">=</span><span class="mi">4</span> <span class="k">if</span> <span class="n">k</span> <span class="o">==</span> <span class="o">-</span><span class="mi">1</span> <span class="k">else</span> <span class="mi">1</span> <span class="o">+</span> <span class="mi">5</span> <span class="o">*</span> <span class="n">proba_map</span><span class="p">[</span><span class="n">ci</span><span class="p">],</span>
<span class="p">)</span>
<span class="n">n_clusters_</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="nb">set</span><span class="p">(</span><span class="n">labels</span><span class="p">))</span> <span class="o">-</span> <span class="p">(</span><span class="mi">1</span> <span class="k">if</span> <span class="o">-</span><span class="mi">1</span> <span class="ow">in</span> <span class="n">labels</span> <span class="k">else</span> <span class="mi">0</span><span class="p">)</span>
<span class="n">preamble</span> <span class="o">=</span> <span class="s2">"True"</span> <span class="k">if</span> <span class="n">ground_truth</span> <span class="k">else</span> <span class="s2">"Estimated"</span>
<span class="n">title</span> <span class="o">=</span> <span class="sa">f</span><span class="s2">"</span><span class="si">{</span><span class="n">preamble</span><span class="si">}</span><span class="s2"> number of clusters: </span><span class="si">{</span><span class="n">n_clusters_</span><span class="si">}</span><span class="s2">"</span>
<span class="k">if</span> <span class="n">parameters</span> <span class="ow">is</span> <span class="ow">not</span> <span class="kc">None</span><span class="p">:</span>
<span class="n">parameters_str</span> <span class="o">=</span> <span class="s2">", "</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="sa">f</span><span class="s2">"</span><span class="si">{</span><span class="n">k</span><span class="si">}</span><span class="s2">=</span><span class="si">{</span><span class="n">v</span><span class="si">}</span><span class="s2">"</span> <span class="k">for</span> <span class="n">k</span><span class="p">,</span> <span class="n">v</span> <span class="ow">in</span> <span class="n">parameters</span><span class="o">.</span><span class="n">items</span><span class="p">())</span>
<span class="n">title</span> <span class="o">+=</span> <span class="sa">f</span><span class="s2">" | </span><span class="si">{</span><span class="n">parameters_str</span><span class="si">}</span><span class="s2">"</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="n">title</span><span class="p">)</span>
<a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.tight_layout.html#matplotlib.pyplot.tight_layout" title="matplotlib.pyplot.tight_layout" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">tight_layout</span></a><span class="p">()</span>
</pre></div>
</div>
<section id="generate-sample-data">
<h2>Generate sample data<a class="headerlink" href="#generate-sample-data" title="Link to this heading">¶</a></h2>
<p>One of the greatest advantages of HDBSCAN over DBSCAN is its out-of-the-box
robustness. It’s especially remarkable on heterogeneous mixtures of data.
Like DBSCAN, it can model arbitrary shapes and distributions, however unlike
DBSCAN it does not require specification of an arbitrary and sensitive
<code class="docutils literal notranslate"><span class="pre">eps</span></code> hyperparameter.</p>
<p>For example, below we generate a dataset from a mixture of three bi-dimensional
and isotropic Gaussian distributions.</p>
<div class="highlight-Python notranslate"><div class="highlight"><pre><span></span><span class="n">centers</span> <span class="o">=</span> <span class="p">[[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mf">1.5</span><span class="p">,</span> <span class="o">-</span><span class="mf">1.5</span><span class="p">]]</span>
<span class="n">X</span><span class="p">,</span> <span class="n">labels_true</span> <span class="o">=</span> <a href="../../modules/generated/sklearn.datasets.make_blobs.html#sklearn.datasets.make_blobs" title="sklearn.datasets.make_blobs" class="sphx-glr-backref-module-sklearn-datasets sphx-glr-backref-type-py-function"><span class="n">make_blobs</span></a><span class="p">(</span>
<span class="n">n_samples</span><span class="o">=</span><span class="mi">750</span><span class="p">,</span> <span class="n">centers</span><span class="o">=</span><span class="n">centers</span><span class="p">,</span> <span class="n">cluster_std</span><span class="o">=</span><span class="p">[</span><span class="mf">0.4</span><span class="p">,</span> <span class="mf">0.1</span><span class="p">,</span> <span class="mf">0.75</span><span class="p">],</span> <span class="n">random_state</span><span class="o">=</span><span class="mi">0</span>
<span class="p">)</span>
<span class="n">plot</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">labels</span><span class="o">=</span><span class="n">labels_true</span><span class="p">,</span> <span class="n">ground_truth</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
</pre></div>
</div>
<img src="../../_images/sphx_glr_plot_hdbscan_001.png" srcset="../../_images/sphx_glr_plot_hdbscan_001.png" alt="True number of clusters: 3" class = "sphx-glr-single-img"/></section>
<section id="scale-invariance">
<h2>Scale Invariance<a class="headerlink" href="#scale-invariance" title="Link to this heading">¶</a></h2>
<p>It’s worth remembering that, while DBSCAN provides a default value for <code class="docutils literal notranslate"><span class="pre">eps</span></code>
parameter, it hardly has a proper default value and must be tuned for the
specific dataset at use.</p>
<p>As a simple demonstration, consider the clustering for a <code class="docutils literal notranslate"><span class="pre">eps</span></code> value tuned
for one dataset, and clustering obtained with the same value but applied to
rescaled versions of the dataset.</p>
<div class="highlight-Python notranslate"><div class="highlight"><pre><span></span><span class="n">fig</span><span class="p">,</span> <span class="n">axes</span> <span class="o">=</span> <a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.subplots.html#matplotlib.pyplot.subplots" title="matplotlib.pyplot.subplots" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">subplots</span></a><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span> <span class="mi">12</span><span class="p">))</span>
<span class="n">dbs</span> <span class="o">=</span> <a href="../../modules/generated/sklearn.cluster.DBSCAN.html#sklearn.cluster.DBSCAN" title="sklearn.cluster.DBSCAN" class="sphx-glr-backref-module-sklearn-cluster sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">DBSCAN</span></a><span class="p">(</span><span class="n">eps</span><span class="o">=</span><span class="mf">0.3</span><span class="p">)</span>
<span class="k">for</span> <span class="n">idx</span><span class="p">,</span> <span class="n">scale</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">,</span> <span class="mi">3</span><span class="p">]):</span>
<span class="n">dbs</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X</span> <span class="o">*</span> <span class="n">scale</span><span class="p">)</span>
<span class="n">plot</span><span class="p">(</span><span class="n">X</span> <span class="o">*</span> <span class="n">scale</span><span class="p">,</span> <span class="n">dbs</span><span class="o">.</span><span class="n">labels_</span><span class="p">,</span> <span class="n">parameters</span><span class="o">=</span><span class="p">{</span><span class="s2">"scale"</span><span class="p">:</span> <span class="n">scale</span><span class="p">,</span> <span class="s2">"eps"</span><span class="p">:</span> <span class="mf">0.3</span><span class="p">},</span> <span class="n">ax</span><span class="o">=</span><span class="n">axes</span><span class="p">[</span><span class="n">idx</span><span class="p">])</span>
</pre></div>
</div>
<img src="../../_images/sphx_glr_plot_hdbscan_002.png" srcset="../../_images/sphx_glr_plot_hdbscan_002.png" alt="Estimated number of clusters: 3 | scale=1, eps=0.3, Estimated number of clusters: 1 | scale=0.5, eps=0.3, Estimated number of clusters: 11 | scale=3, eps=0.3" class = "sphx-glr-single-img"/><p>Indeed, in order to maintain the same results we would have to scale <code class="docutils literal notranslate"><span class="pre">eps</span></code> by
the same factor.</p>
<div class="highlight-Python notranslate"><div class="highlight"><pre><span></span><span class="n">fig</span><span class="p">,</span> <span class="n">axis</span> <span class="o">=</span> <a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.subplots.html#matplotlib.pyplot.subplots" title="matplotlib.pyplot.subplots" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">subplots</span></a><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">12</span><span class="p">,</span> <span class="mi">5</span><span class="p">))</span>
<span class="n">dbs</span> <span class="o">=</span> <a href="../../modules/generated/sklearn.cluster.DBSCAN.html#sklearn.cluster.DBSCAN" title="sklearn.cluster.DBSCAN" class="sphx-glr-backref-module-sklearn-cluster sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">DBSCAN</span></a><span class="p">(</span><span class="n">eps</span><span class="o">=</span><span class="mf">0.9</span><span class="p">)</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="mi">3</span> <span class="o">*</span> <span class="n">X</span><span class="p">)</span>
<span class="n">plot</span><span class="p">(</span><span class="mi">3</span> <span class="o">*</span> <span class="n">X</span><span class="p">,</span> <span class="n">dbs</span><span class="o">.</span><span class="n">labels_</span><span class="p">,</span> <span class="n">parameters</span><span class="o">=</span><span class="p">{</span><span class="s2">"scale"</span><span class="p">:</span> <span class="mi">3</span><span class="p">,</span> <span class="s2">"eps"</span><span class="p">:</span> <span class="mf">0.9</span><span class="p">},</span> <span class="n">ax</span><span class="o">=</span><span class="n">axis</span><span class="p">)</span>
</pre></div>
</div>
<img src="../../_images/sphx_glr_plot_hdbscan_003.png" srcset="../../_images/sphx_glr_plot_hdbscan_003.png" alt="Estimated number of clusters: 3 | scale=3, eps=0.9" class = "sphx-glr-single-img"/><p>While standardizing data (e.g. using
<a class="reference internal" href="../../modules/generated/sklearn.preprocessing.StandardScaler.html#sklearn.preprocessing.StandardScaler" title="sklearn.preprocessing.StandardScaler"><code class="xref py py-class docutils literal notranslate"><span class="pre">sklearn.preprocessing.StandardScaler</span></code></a>) helps mitigate this problem,
great care must be taken to select the appropriate value for <code class="docutils literal notranslate"><span class="pre">eps</span></code>.</p>
<p>HDBSCAN is much more robust in this sense: HDBSCAN can be seen as
clustering over all possible values of <code class="docutils literal notranslate"><span class="pre">eps</span></code> and extracting the best
clusters from all possible clusters (see <a class="reference internal" href="../../modules/clustering.html#hdbscan"><span class="std std-ref">User Guide</span></a>).
One immediate advantage is that HDBSCAN is scale-invariant.</p>
<div class="highlight-Python notranslate"><div class="highlight"><pre><span></span><span class="n">fig</span><span class="p">,</span> <span class="n">axes</span> <span class="o">=</span> <a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.subplots.html#matplotlib.pyplot.subplots" title="matplotlib.pyplot.subplots" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">subplots</span></a><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span> <span class="mi">12</span><span class="p">))</span>
<span class="n">hdb</span> <span class="o">=</span> <a href="../../modules/generated/sklearn.cluster.HDBSCAN.html#sklearn.cluster.HDBSCAN" title="sklearn.cluster.HDBSCAN" class="sphx-glr-backref-module-sklearn-cluster sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">HDBSCAN</span></a><span class="p">()</span>
<span class="k">for</span> <span class="n">idx</span><span class="p">,</span> <span class="n">scale</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">([</span><span class="mi">1</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">,</span> <span class="mi">3</span><span class="p">]):</span>
<span class="n">hdb</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X</span> <span class="o">*</span> <span class="n">scale</span><span class="p">)</span>
<span class="n">plot</span><span class="p">(</span>
<span class="n">X</span> <span class="o">*</span> <span class="n">scale</span><span class="p">,</span>
<span class="n">hdb</span><span class="o">.</span><span class="n">labels_</span><span class="p">,</span>
<span class="n">hdb</span><span class="o">.</span><span class="n">probabilities_</span><span class="p">,</span>
<span class="n">ax</span><span class="o">=</span><span class="n">axes</span><span class="p">[</span><span class="n">idx</span><span class="p">],</span>
<span class="n">parameters</span><span class="o">=</span><span class="p">{</span><span class="s2">"scale"</span><span class="p">:</span> <span class="n">scale</span><span class="p">},</span>
<span class="p">)</span>
</pre></div>
</div>
<img src="../../_images/sphx_glr_plot_hdbscan_004.png" srcset="../../_images/sphx_glr_plot_hdbscan_004.png" alt="Estimated number of clusters: 3 | scale=1, Estimated number of clusters: 3 | scale=0.5, Estimated number of clusters: 3 | scale=3" class = "sphx-glr-single-img"/></section>
<section id="multi-scale-clustering">
<h2>Multi-Scale Clustering<a class="headerlink" href="#multi-scale-clustering" title="Link to this heading">¶</a></h2>
<p>HDBSCAN is much more than scale invariant though – it is capable of
multi-scale clustering, which accounts for clusters with varying density.
Traditional DBSCAN assumes that any potential clusters are homogeneous in
density. HDBSCAN is free from such constraints. To demonstrate this we
consider the following dataset</p>
<div class="highlight-Python notranslate"><div class="highlight"><pre><span></span><span class="n">centers</span> <span class="o">=</span> <span class="p">[[</span><span class="o">-</span><span class="mf">0.85</span><span class="p">,</span> <span class="o">-</span><span class="mf">0.85</span><span class="p">],</span> <span class="p">[</span><span class="o">-</span><span class="mf">0.85</span><span class="p">,</span> <span class="mf">0.85</span><span class="p">],</span> <span class="p">[</span><span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">],</span> <span class="p">[</span><span class="mi">3</span><span class="p">,</span> <span class="o">-</span><span class="mi">3</span><span class="p">]]</span>
<span class="n">X</span><span class="p">,</span> <span class="n">labels_true</span> <span class="o">=</span> <a href="../../modules/generated/sklearn.datasets.make_blobs.html#sklearn.datasets.make_blobs" title="sklearn.datasets.make_blobs" class="sphx-glr-backref-module-sklearn-datasets sphx-glr-backref-type-py-function"><span class="n">make_blobs</span></a><span class="p">(</span>
<span class="n">n_samples</span><span class="o">=</span><span class="mi">750</span><span class="p">,</span> <span class="n">centers</span><span class="o">=</span><span class="n">centers</span><span class="p">,</span> <span class="n">cluster_std</span><span class="o">=</span><span class="p">[</span><span class="mf">0.2</span><span class="p">,</span> <span class="mf">0.35</span><span class="p">,</span> <span class="mf">1.35</span><span class="p">,</span> <span class="mf">1.35</span><span class="p">],</span> <span class="n">random_state</span><span class="o">=</span><span class="mi">0</span>
<span class="p">)</span>
<span class="n">plot</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">labels</span><span class="o">=</span><span class="n">labels_true</span><span class="p">,</span> <span class="n">ground_truth</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
</pre></div>
</div>
<img src="../../_images/sphx_glr_plot_hdbscan_005.png" srcset="../../_images/sphx_glr_plot_hdbscan_005.png" alt="True number of clusters: 4" class = "sphx-glr-single-img"/><p>This dataset is more difficult for DBSCAN due to the varying densities and
spatial separation:</p>
<ul class="simple">
<li><p>If <code class="docutils literal notranslate"><span class="pre">eps</span></code> is too large then we risk falsely clustering the two dense
clusters as one since their mutual reachability will extend
clusters.</p></li>
<li><p>If <code class="docutils literal notranslate"><span class="pre">eps</span></code> is too small, then we risk fragmenting the sparser clusters
into many false clusters.</p></li>
</ul>
<p>Not to mention this requires manually tuning choices of <code class="docutils literal notranslate"><span class="pre">eps</span></code> until we
find a tradeoff that we are comfortable with.</p>
<div class="highlight-Python notranslate"><div class="highlight"><pre><span></span><span class="n">fig</span><span class="p">,</span> <span class="n">axes</span> <span class="o">=</span> <a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.subplots.html#matplotlib.pyplot.subplots" title="matplotlib.pyplot.subplots" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">subplots</span></a><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span> <span class="mi">8</span><span class="p">))</span>
<span class="n">params</span> <span class="o">=</span> <span class="p">{</span><span class="s2">"eps"</span><span class="p">:</span> <span class="mf">0.7</span><span class="p">}</span>
<span class="n">dbs</span> <span class="o">=</span> <a href="../../modules/generated/sklearn.cluster.DBSCAN.html#sklearn.cluster.DBSCAN" title="sklearn.cluster.DBSCAN" class="sphx-glr-backref-module-sklearn-cluster sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">DBSCAN</span></a><span class="p">(</span><span class="o">**</span><span class="n">params</span><span class="p">)</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">plot</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">dbs</span><span class="o">.</span><span class="n">labels_</span><span class="p">,</span> <span class="n">parameters</span><span class="o">=</span><span class="n">params</span><span class="p">,</span> <span class="n">ax</span><span class="o">=</span><span class="n">axes</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
<span class="n">params</span> <span class="o">=</span> <span class="p">{</span><span class="s2">"eps"</span><span class="p">:</span> <span class="mf">0.3</span><span class="p">}</span>
<span class="n">dbs</span> <span class="o">=</span> <a href="../../modules/generated/sklearn.cluster.DBSCAN.html#sklearn.cluster.DBSCAN" title="sklearn.cluster.DBSCAN" class="sphx-glr-backref-module-sklearn-cluster sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">DBSCAN</span></a><span class="p">(</span><span class="o">**</span><span class="n">params</span><span class="p">)</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">plot</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">dbs</span><span class="o">.</span><span class="n">labels_</span><span class="p">,</span> <span class="n">parameters</span><span class="o">=</span><span class="n">params</span><span class="p">,</span> <span class="n">ax</span><span class="o">=</span><span class="n">axes</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span>
</pre></div>
</div>
<img src="../../_images/sphx_glr_plot_hdbscan_006.png" srcset="../../_images/sphx_glr_plot_hdbscan_006.png" alt="Estimated number of clusters: 3 | eps=0.7, Estimated number of clusters: 14 | eps=0.3" class = "sphx-glr-single-img"/><p>To properly cluster the two dense clusters, we would need a smaller value of
epsilon, however at <code class="docutils literal notranslate"><span class="pre">eps=0.3</span></code> we are already fragmenting the sparse clusters,
which would only become more severe as we decrease epsilon. Indeed it seems
that DBSCAN is incapable of simultaneously separating the two dense clusters
while preventing the sparse clusters from fragmenting. Let’s compare with
HDBSCAN.</p>
<div class="highlight-Python notranslate"><div class="highlight"><pre><span></span><span class="n">hdb</span> <span class="o">=</span> <a href="../../modules/generated/sklearn.cluster.HDBSCAN.html#sklearn.cluster.HDBSCAN" title="sklearn.cluster.HDBSCAN" class="sphx-glr-backref-module-sklearn-cluster sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">HDBSCAN</span></a><span class="p">()</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">plot</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">hdb</span><span class="o">.</span><span class="n">labels_</span><span class="p">,</span> <span class="n">hdb</span><span class="o">.</span><span class="n">probabilities_</span><span class="p">)</span>
</pre></div>
</div>
<img src="../../_images/sphx_glr_plot_hdbscan_007.png" srcset="../../_images/sphx_glr_plot_hdbscan_007.png" alt="Estimated number of clusters: 4" class = "sphx-glr-single-img"/><p>HDBSCAN is able to adapt to the multi-scale structure of the dataset without
requiring parameter tuning. While any sufficiently interesting dataset will
require tuning, this case demonstrates that HDBSCAN can yield qualitatively
better classes of clusterings without users’ intervention which are
inaccessible via DBSCAN.</p>
</section>
<section id="hyperparameter-robustness">
<h2>Hyperparameter Robustness<a class="headerlink" href="#hyperparameter-robustness" title="Link to this heading">¶</a></h2>
<p>Ultimately tuning will be an important step in any real world application, so
let’s take a look at some of the most important hyperparameters for HDBSCAN.
While HDBSCAN is free from the <code class="docutils literal notranslate"><span class="pre">eps</span></code> parameter of DBSCAN, it does still have
some hyperparameters like <code class="docutils literal notranslate"><span class="pre">min_cluster_size</span></code> and <code class="docutils literal notranslate"><span class="pre">min_samples</span></code> which tune its
results regarding density. We will however see that HDBSCAN is relatively robust
to various real world examples thanks to those parameters whose clear meaning
helps tuning them.</p>
<section id="min-cluster-size">
<h3><code class="docutils literal notranslate"><span class="pre">min_cluster_size</span></code><a class="headerlink" href="#min-cluster-size" title="Link to this heading">¶</a></h3>
<p><code class="docutils literal notranslate"><span class="pre">min_cluster_size</span></code> is the minimum number of samples in a group for that
group to be considered a cluster.</p>
<p>Clusters smaller than the ones of this size will be left as noise.
The default value is 5. This parameter is generally tuned to
larger values as needed. Smaller values will likely to lead to results with
fewer points labeled as noise. However values which too small will lead to
false sub-clusters being picked up and preferred. Larger values tend to be
more robust with respect to noisy datasets, e.g. high-variance clusters with
significant overlap.</p>
<div class="highlight-Python notranslate"><div class="highlight"><pre><span></span><span class="n">PARAM</span> <span class="o">=</span> <span class="p">({</span><span class="s2">"min_cluster_size"</span><span class="p">:</span> <span class="mi">5</span><span class="p">},</span> <span class="p">{</span><span class="s2">"min_cluster_size"</span><span class="p">:</span> <span class="mi">3</span><span class="p">},</span> <span class="p">{</span><span class="s2">"min_cluster_size"</span><span class="p">:</span> <span class="mi">25</span><span class="p">})</span>
<span class="n">fig</span><span class="p">,</span> <span class="n">axes</span> <span class="o">=</span> <a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.subplots.html#matplotlib.pyplot.subplots" title="matplotlib.pyplot.subplots" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">subplots</span></a><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span> <span class="mi">12</span><span class="p">))</span>
<span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">param</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">PARAM</span><span class="p">):</span>
<span class="n">hdb</span> <span class="o">=</span> <a href="../../modules/generated/sklearn.cluster.HDBSCAN.html#sklearn.cluster.HDBSCAN" title="sklearn.cluster.HDBSCAN" class="sphx-glr-backref-module-sklearn-cluster sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">HDBSCAN</span></a><span class="p">(</span><span class="o">**</span><span class="n">param</span><span class="p">)</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">labels</span> <span class="o">=</span> <span class="n">hdb</span><span class="o">.</span><span class="n">labels_</span>
<span class="n">plot</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">labels</span><span class="p">,</span> <span class="n">hdb</span><span class="o">.</span><span class="n">probabilities_</span><span class="p">,</span> <span class="n">param</span><span class="p">,</span> <span class="n">ax</span><span class="o">=</span><span class="n">axes</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
</pre></div>
</div>
<img src="../../_images/sphx_glr_plot_hdbscan_008.png" srcset="../../_images/sphx_glr_plot_hdbscan_008.png" alt="Estimated number of clusters: 4 | min_cluster_size=5, Estimated number of clusters: 91 | min_cluster_size=3, Estimated number of clusters: 4 | min_cluster_size=25" class = "sphx-glr-single-img"/></section>
<section id="min-samples">
<h3><code class="docutils literal notranslate"><span class="pre">min_samples</span></code><a class="headerlink" href="#min-samples" title="Link to this heading">¶</a></h3>
<p><code class="docutils literal notranslate"><span class="pre">min_samples</span></code> is the number of samples in a neighborhood for a point to
be considered as a core point, including the point itself.
<code class="docutils literal notranslate"><span class="pre">min_samples</span></code> defaults to <code class="docutils literal notranslate"><span class="pre">min_cluster_size</span></code>.
Similarly to <code class="docutils literal notranslate"><span class="pre">min_cluster_size</span></code>, larger values for <code class="docutils literal notranslate"><span class="pre">min_samples</span></code> increase
the model’s robustness to noise, but risks ignoring or discarding
potentially valid but small clusters.
<code class="docutils literal notranslate"><span class="pre">min_samples</span></code> better be tuned after finding a good value for <code class="docutils literal notranslate"><span class="pre">min_cluster_size</span></code>.</p>
<div class="highlight-Python notranslate"><div class="highlight"><pre><span></span><span class="n">PARAM</span> <span class="o">=</span> <span class="p">(</span>
<span class="p">{</span><span class="s2">"min_cluster_size"</span><span class="p">:</span> <span class="mi">20</span><span class="p">,</span> <span class="s2">"min_samples"</span><span class="p">:</span> <span class="mi">5</span><span class="p">},</span>
<span class="p">{</span><span class="s2">"min_cluster_size"</span><span class="p">:</span> <span class="mi">20</span><span class="p">,</span> <span class="s2">"min_samples"</span><span class="p">:</span> <span class="mi">3</span><span class="p">},</span>
<span class="p">{</span><span class="s2">"min_cluster_size"</span><span class="p">:</span> <span class="mi">20</span><span class="p">,</span> <span class="s2">"min_samples"</span><span class="p">:</span> <span class="mi">25</span><span class="p">},</span>
<span class="p">)</span>
<span class="n">fig</span><span class="p">,</span> <span class="n">axes</span> <span class="o">=</span> <a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.subplots.html#matplotlib.pyplot.subplots" title="matplotlib.pyplot.subplots" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">subplots</span></a><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span> <span class="mi">12</span><span class="p">))</span>
<span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">param</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">PARAM</span><span class="p">):</span>
<span class="n">hdb</span> <span class="o">=</span> <a href="../../modules/generated/sklearn.cluster.HDBSCAN.html#sklearn.cluster.HDBSCAN" title="sklearn.cluster.HDBSCAN" class="sphx-glr-backref-module-sklearn-cluster sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">HDBSCAN</span></a><span class="p">(</span><span class="o">**</span><span class="n">param</span><span class="p">)</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">labels</span> <span class="o">=</span> <span class="n">hdb</span><span class="o">.</span><span class="n">labels_</span>
<span class="n">plot</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">labels</span><span class="p">,</span> <span class="n">hdb</span><span class="o">.</span><span class="n">probabilities_</span><span class="p">,</span> <span class="n">param</span><span class="p">,</span> <span class="n">ax</span><span class="o">=</span><span class="n">axes</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
</pre></div>
</div>
<img src="../../_images/sphx_glr_plot_hdbscan_009.png" srcset="../../_images/sphx_glr_plot_hdbscan_009.png" alt="Estimated number of clusters: 4 | min_cluster_size=20, min_samples=5, Estimated number of clusters: 4 | min_cluster_size=20, min_samples=3, Estimated number of clusters: 4 | min_cluster_size=20, min_samples=25" class = "sphx-glr-single-img"/></section>
<section id="dbscan-clustering">
<h3><code class="docutils literal notranslate"><span class="pre">dbscan_clustering</span></code><a class="headerlink" href="#dbscan-clustering" title="Link to this heading">¶</a></h3>
<p>During <code class="docutils literal notranslate"><span class="pre">fit</span></code>, <code class="docutils literal notranslate"><span class="pre">HDBSCAN</span></code> builds a single-linkage tree which encodes the
clustering of all points across all values of <a class="reference internal" href="../../modules/generated/sklearn.cluster.DBSCAN.html#sklearn.cluster.DBSCAN" title="sklearn.cluster.DBSCAN"><code class="xref py py-class docutils literal notranslate"><span class="pre">DBSCAN</span></code></a>’s
<code class="docutils literal notranslate"><span class="pre">eps</span></code> parameter.
We can thus plot and evaluate these clusterings efficiently without fully
recomputing intermediate values such as core-distances, mutual-reachability,
and the minimum spanning tree. All we need to do is specify the <code class="docutils literal notranslate"><span class="pre">cut_distance</span></code>
(equivalent to <code class="docutils literal notranslate"><span class="pre">eps</span></code>) we want to cluster with.</p>
<div class="highlight-Python notranslate"><div class="highlight"><pre><span></span><span class="n">PARAM</span> <span class="o">=</span> <span class="p">(</span>
<span class="p">{</span><span class="s2">"cut_distance"</span><span class="p">:</span> <span class="mf">0.1</span><span class="p">},</span>
<span class="p">{</span><span class="s2">"cut_distance"</span><span class="p">:</span> <span class="mf">0.5</span><span class="p">},</span>
<span class="p">{</span><span class="s2">"cut_distance"</span><span class="p">:</span> <span class="mf">1.0</span><span class="p">},</span>
<span class="p">)</span>
<span class="n">hdb</span> <span class="o">=</span> <a href="../../modules/generated/sklearn.cluster.HDBSCAN.html#sklearn.cluster.HDBSCAN" title="sklearn.cluster.HDBSCAN" class="sphx-glr-backref-module-sklearn-cluster sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">HDBSCAN</span></a><span class="p">()</span>
<span class="n">hdb</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">fig</span><span class="p">,</span> <span class="n">axes</span> <span class="o">=</span> <a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.subplots.html#matplotlib.pyplot.subplots" title="matplotlib.pyplot.subplots" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">subplots</span></a><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">PARAM</span><span class="p">),</span> <span class="mi">1</span><span class="p">,</span> <span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span> <span class="mi">12</span><span class="p">))</span>
<span class="k">for</span> <span class="n">i</span><span class="p">,</span> <span class="n">param</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">PARAM</span><span class="p">):</span>
<span class="n">labels</span> <span class="o">=</span> <span class="n">hdb</span><span class="o">.</span><span class="n">dbscan_clustering</span><span class="p">(</span><span class="o">**</span><span class="n">param</span><span class="p">)</span>
<span class="n">plot</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">labels</span><span class="p">,</span> <span class="n">hdb</span><span class="o">.</span><span class="n">probabilities_</span><span class="p">,</span> <span class="n">param</span><span class="p">,</span> <span class="n">ax</span><span class="o">=</span><span class="n">axes</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
</pre></div>
</div>
<img src="../../_images/sphx_glr_plot_hdbscan_010.png" srcset="../../_images/sphx_glr_plot_hdbscan_010.png" alt="Estimated number of clusters: 3 | cut_distance=0.1, Estimated number of clusters: 3 | cut_distance=0.5, Estimated number of clusters: 1 | cut_distance=1.0" class = "sphx-glr-single-img"/><p class="sphx-glr-timing"><strong>Total running time of the script:</strong> (0 minutes 15.242 seconds)</p>
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<p class="rubric">Related examples</p>
<div class="sphx-glr-thumbnails"><div class="sphx-glr-thumbcontainer" tooltip="This example shows characteristics of different clustering algorithms on datasets that are "int..."><img alt="" src="../../_images/sphx_glr_plot_cluster_comparison_thumb.png" />
<p><a class="reference internal" href="plot_cluster_comparison.html#sphx-glr-auto-examples-cluster-plot-cluster-comparison-py"><span class="std std-ref">Comparing different clustering algorithms on toy datasets</span></a></p>
<div class="sphx-glr-thumbnail-title">Comparing different clustering algorithms on toy datasets</div>
</div><div class="sphx-glr-thumbcontainer" tooltip="DBSCAN (Density-Based Spatial Clustering of Applications with Noise) finds core samples in regi..."><img alt="" src="../../_images/sphx_glr_plot_dbscan_thumb.png" />
<p><a class="reference internal" href="plot_dbscan.html#sphx-glr-auto-examples-cluster-plot-dbscan-py"><span class="std std-ref">Demo of DBSCAN clustering algorithm</span></a></p>
<div class="sphx-glr-thumbnail-title">Demo of DBSCAN clustering algorithm</div>
</div><div class="sphx-glr-thumbcontainer" tooltip="We are pleased to announce the release of scikit-learn 1.3! Many bug fixes and improvements wer..."><img alt="" src="../../_images/sphx_glr_plot_release_highlights_1_3_0_thumb.png" />
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<div class="sphx-glr-thumbnail-title">Release Highlights for scikit-learn 1.3</div>
</div><div class="sphx-glr-thumbcontainer" tooltip=" The `Johnson-Lindenstrauss lemma`_ states that any high dimensional dataset can be randomly pr..."><img alt="" src="../../_images/sphx_glr_plot_johnson_lindenstrauss_bound_thumb.png" />
<p><a class="reference internal" href="../miscellaneous/plot_johnson_lindenstrauss_bound.html#sphx-glr-auto-examples-miscellaneous-plot-johnson-lindenstrauss-bound-py"><span class="std std-ref">The Johnson-Lindenstrauss bound for embedding with random projections</span></a></p>
<div class="sphx-glr-thumbnail-title">The Johnson-Lindenstrauss bound for embedding with random projections</div>
</div><div class="sphx-glr-thumbcontainer" tooltip="Reference: Brendan J. Frey and Delbert Dueck, "Clustering by Passing Messages Between Data Poin..."><img alt="" src="../../_images/sphx_glr_plot_affinity_propagation_thumb.png" />
<p><a class="reference internal" href="plot_affinity_propagation.html#sphx-glr-auto-examples-cluster-plot-affinity-propagation-py"><span class="std std-ref">Demo of affinity propagation clustering algorithm</span></a></p>
<div class="sphx-glr-thumbnail-title">Demo of affinity propagation clustering algorithm</div>
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