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<li><a class="reference internal" href="#">2.3. Clustering</a><ul>
<li><a class="reference internal" href="#overview-of-clustering-methods">2.3.1. Overview of clustering methods</a></li>
<li><a class="reference internal" href="#k-means">2.3.2. K-means</a><ul>
<li><a class="reference internal" href="#low-level-parallelism">2.3.2.1. Low-level parallelism</a></li>
<li><a class="reference internal" href="#mini-batch-k-means">2.3.2.2. Mini Batch K-Means</a></li>
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<li><a class="reference internal" href="#mean-shift">2.3.4. Mean Shift</a></li>
<li><a class="reference internal" href="#spectral-clustering">2.3.5. Spectral clustering</a><ul>
<li><a class="reference internal" href="#different-label-assignment-strategies">2.3.5.1. Different label assignment strategies</a></li>
<li><a class="reference internal" href="#spectral-clustering-graphs">2.3.5.2. Spectral Clustering Graphs</a></li>
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<li><a class="reference internal" href="#different-linkage-type-ward-complete-average-and-single-linkage">2.3.6.1. Different linkage type: Ward, complete, average, and single linkage</a></li>
<li><a class="reference internal" href="#visualization-of-cluster-hierarchy">2.3.6.2. Visualization of cluster hierarchy</a></li>
<li><a class="reference internal" href="#adding-connectivity-constraints">2.3.6.3. Adding connectivity constraints</a></li>
<li><a class="reference internal" href="#varying-the-metric">2.3.6.4. Varying the metric</a></li>
<li><a class="reference internal" href="#bisecting-k-means">2.3.6.5. Bisecting K-Means</a></li>
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<li><a class="reference internal" href="#id11">2.3.8.2. Hierarchical Clustering</a></li>
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<li><a class="reference internal" href="#optics">2.3.9. OPTICS</a></li>
<li><a class="reference internal" href="#birch">2.3.10. BIRCH</a></li>
<li><a class="reference internal" href="#clustering-performance-evaluation">2.3.11. Clustering performance evaluation</a><ul>
<li><a class="reference internal" href="#rand-index">2.3.11.1. Rand index</a></li>
<li><a class="reference internal" href="#mutual-information-based-scores">2.3.11.2. Mutual Information based scores</a></li>
<li><a class="reference internal" href="#homogeneity-completeness-and-v-measure">2.3.11.3. Homogeneity, completeness and V-measure</a></li>
<li><a class="reference internal" href="#fowlkes-mallows-scores">2.3.11.4. Fowlkes-Mallows scores</a></li>
<li><a class="reference internal" href="#silhouette-coefficient">2.3.11.5. Silhouette Coefficient</a></li>
<li><a class="reference internal" href="#calinski-harabasz-index">2.3.11.6. Calinski-Harabasz Index</a></li>
<li><a class="reference internal" href="#davies-bouldin-index">2.3.11.7. Davies-Bouldin Index</a></li>
<li><a class="reference internal" href="#contingency-matrix">2.3.11.8. Contingency Matrix</a></li>
<li><a class="reference internal" href="#pair-confusion-matrix">2.3.11.9. Pair Confusion Matrix</a></li>
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<section id="clustering">
<span id="id1"></span><h1><span class="section-number">2.3. </span>Clustering<a class="headerlink" href="#clustering" title="Link to this heading">¶</a></h1>
<p><a class="reference external" href="https://en.wikipedia.org/wiki/Cluster_analysis">Clustering</a> of
unlabeled data can be performed with the module <a class="reference internal" href="classes.html#module-sklearn.cluster" title="sklearn.cluster"><code class="xref py py-mod docutils literal notranslate"><span class="pre">sklearn.cluster</span></code></a>.</p>
<p>Each clustering algorithm comes in two variants: a class, that implements
the <code class="docutils literal notranslate"><span class="pre">fit</span></code> method to learn the clusters on train data, and a function,
that, given train data, returns an array of integer labels corresponding
to the different clusters. For the class, the labels over the training
data can be found in the <code class="docutils literal notranslate"><span class="pre">labels_</span></code> attribute.</p>
<aside class="topic">
<p class="topic-title">Input data</p>
<p>One important thing to note is that the algorithms implemented in
this module can take different kinds of matrix as input. All the
methods accept standard data matrices of shape <code class="docutils literal notranslate"><span class="pre">(n_samples,</span> <span class="pre">n_features)</span></code>.
These can be obtained from the classes in the <a class="reference internal" href="classes.html#module-sklearn.feature_extraction" title="sklearn.feature_extraction"><code class="xref py py-mod docutils literal notranslate"><span class="pre">sklearn.feature_extraction</span></code></a>
module. For <a class="reference internal" href="generated/sklearn.cluster.AffinityPropagation.html#sklearn.cluster.AffinityPropagation" title="sklearn.cluster.AffinityPropagation"><code class="xref py py-class docutils literal notranslate"><span class="pre">AffinityPropagation</span></code></a>, <a class="reference internal" href="generated/sklearn.cluster.SpectralClustering.html#sklearn.cluster.SpectralClustering" title="sklearn.cluster.SpectralClustering"><code class="xref py py-class docutils literal notranslate"><span class="pre">SpectralClustering</span></code></a>
and <a class="reference internal" href="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> one can also input similarity matrices of shape
<code class="docutils literal notranslate"><span class="pre">(n_samples,</span> <span class="pre">n_samples)</span></code>. These can be obtained from the functions
in the <a class="reference internal" href="classes.html#module-sklearn.metrics.pairwise" title="sklearn.metrics.pairwise"><code class="xref py py-mod docutils literal notranslate"><span class="pre">sklearn.metrics.pairwise</span></code></a> module.</p>
</aside>
<section id="overview-of-clustering-methods">
<h2><span class="section-number">2.3.1. </span>Overview of clustering methods<a class="headerlink" href="#overview-of-clustering-methods" title="Link to this heading">¶</a></h2>
<figure class="align-center" id="id24">
<a class="reference external image-reference" href="../auto_examples/cluster/plot_cluster_comparison.html"><img alt="../_images/sphx_glr_plot_cluster_comparison_001.png" src="../_images/sphx_glr_plot_cluster_comparison_001.png" style="width: 1050.0px; height: 650.0px;" />
</a>
<figcaption>
<p><span class="caption-text">A comparison of the clustering algorithms in scikit-learn</span><a class="headerlink" href="#id24" title="Link to this image">¶</a></p>
</figcaption>
</figure>
<table class="docutils align-default">
<colgroup>
<col style="width: 15.1%" />
<col style="width: 16.1%" />
<col style="width: 20.4%" />
<col style="width: 26.9%" />
<col style="width: 21.5%" />
</colgroup>
<thead>
<tr class="row-odd"><th class="head"><p>Method name</p></th>
<th class="head"><p>Parameters</p></th>
<th class="head"><p>Scalability</p></th>
<th class="head"><p>Usecase</p></th>
<th class="head"><p>Geometry (metric used)</p></th>
</tr>
</thead>
<tbody>
<tr class="row-even"><td><p><a class="reference internal" href="#k-means"><span class="std std-ref">K-Means</span></a></p></td>
<td><p>number of clusters</p></td>
<td><p>Very large <code class="docutils literal notranslate"><span class="pre">n_samples</span></code>, medium <code class="docutils literal notranslate"><span class="pre">n_clusters</span></code> with
<a class="reference internal" href="#mini-batch-kmeans"><span class="std std-ref">MiniBatch code</span></a></p></td>
<td><p>General-purpose, even cluster size, flat geometry,
not too many clusters, inductive</p></td>
<td><p>Distances between points</p></td>
</tr>
<tr class="row-odd"><td><p><a class="reference internal" href="#affinity-propagation"><span class="std std-ref">Affinity propagation</span></a></p></td>
<td><p>damping, sample preference</p></td>
<td><p>Not scalable with n_samples</p></td>
<td><p>Many clusters, uneven cluster size, non-flat geometry, inductive</p></td>
<td><p>Graph distance (e.g. nearest-neighbor graph)</p></td>
</tr>
<tr class="row-even"><td><p><a class="reference internal" href="#mean-shift"><span class="std std-ref">Mean-shift</span></a></p></td>
<td><p>bandwidth</p></td>
<td><p>Not scalable with <code class="docutils literal notranslate"><span class="pre">n_samples</span></code></p></td>
<td><p>Many clusters, uneven cluster size, non-flat geometry, inductive</p></td>
<td><p>Distances between points</p></td>
</tr>
<tr class="row-odd"><td><p><a class="reference internal" href="#spectral-clustering"><span class="std std-ref">Spectral clustering</span></a></p></td>
<td><p>number of clusters</p></td>
<td><p>Medium <code class="docutils literal notranslate"><span class="pre">n_samples</span></code>, small <code class="docutils literal notranslate"><span class="pre">n_clusters</span></code></p></td>
<td><p>Few clusters, even cluster size, non-flat geometry, transductive</p></td>
<td><p>Graph distance (e.g. nearest-neighbor graph)</p></td>
</tr>
<tr class="row-even"><td><p><a class="reference internal" href="#hierarchical-clustering"><span class="std std-ref">Ward hierarchical clustering</span></a></p></td>
<td><p>number of clusters or distance threshold</p></td>
<td><p>Large <code class="docutils literal notranslate"><span class="pre">n_samples</span></code> and <code class="docutils literal notranslate"><span class="pre">n_clusters</span></code></p></td>
<td><p>Many clusters, possibly connectivity constraints, transductive</p></td>
<td><p>Distances between points</p></td>
</tr>
<tr class="row-odd"><td><p><a class="reference internal" href="#hierarchical-clustering"><span class="std std-ref">Agglomerative clustering</span></a></p></td>
<td><p>number of clusters or distance threshold, linkage type, distance</p></td>
<td><p>Large <code class="docutils literal notranslate"><span class="pre">n_samples</span></code> and <code class="docutils literal notranslate"><span class="pre">n_clusters</span></code></p></td>
<td><p>Many clusters, possibly connectivity constraints, non Euclidean
distances, transductive</p></td>
<td><p>Any pairwise distance</p></td>
</tr>
<tr class="row-even"><td><p><a class="reference internal" href="#dbscan"><span class="std std-ref">DBSCAN</span></a></p></td>
<td><p>neighborhood size</p></td>
<td><p>Very large <code class="docutils literal notranslate"><span class="pre">n_samples</span></code>, medium <code class="docutils literal notranslate"><span class="pre">n_clusters</span></code></p></td>
<td><p>Non-flat geometry, uneven cluster sizes, outlier removal,
transductive</p></td>
<td><p>Distances between nearest points</p></td>
</tr>
<tr class="row-odd"><td><p><a class="reference internal" href="#hdbscan"><span class="std std-ref">HDBSCAN</span></a></p></td>
<td><p>minimum cluster membership, minimum point neighbors</p></td>
<td><p>large <code class="docutils literal notranslate"><span class="pre">n_samples</span></code>, medium <code class="docutils literal notranslate"><span class="pre">n_clusters</span></code></p></td>
<td><p>Non-flat geometry, uneven cluster sizes, outlier removal,
transductive, hierarchical, variable cluster density</p></td>
<td><p>Distances between nearest points</p></td>
</tr>
<tr class="row-even"><td><p><a class="reference internal" href="#optics"><span class="std std-ref">OPTICS</span></a></p></td>
<td><p>minimum cluster membership</p></td>
<td><p>Very large <code class="docutils literal notranslate"><span class="pre">n_samples</span></code>, large <code class="docutils literal notranslate"><span class="pre">n_clusters</span></code></p></td>
<td><p>Non-flat geometry, uneven cluster sizes, variable cluster density,
outlier removal, transductive</p></td>
<td><p>Distances between points</p></td>
</tr>
<tr class="row-odd"><td><p><a class="reference internal" href="mixture.html#mixture"><span class="std std-ref">Gaussian mixtures</span></a></p></td>
<td><p>many</p></td>
<td><p>Not scalable</p></td>
<td><p>Flat geometry, good for density estimation, inductive</p></td>
<td><p>Mahalanobis distances to centers</p></td>
</tr>
<tr class="row-even"><td><p><a class="reference internal" href="#birch"><span class="std std-ref">BIRCH</span></a></p></td>
<td><p>branching factor, threshold, optional global clusterer.</p></td>
<td><p>Large <code class="docutils literal notranslate"><span class="pre">n_clusters</span></code> and <code class="docutils literal notranslate"><span class="pre">n_samples</span></code></p></td>
<td><p>Large dataset, outlier removal, data reduction, inductive</p></td>
<td><p>Euclidean distance between points</p></td>
</tr>
<tr class="row-odd"><td><p><a class="reference internal" href="#bisect-k-means"><span class="std std-ref">Bisecting K-Means</span></a></p></td>
<td><p>number of clusters</p></td>
<td><p>Very large <code class="docutils literal notranslate"><span class="pre">n_samples</span></code>, medium <code class="docutils literal notranslate"><span class="pre">n_clusters</span></code></p></td>
<td><p>General-purpose, even cluster size, flat geometry,
no empty clusters, inductive, hierarchical</p></td>
<td><p>Distances between points</p></td>
</tr>
</tbody>
</table>
<p>Non-flat geometry clustering is useful when the clusters have a specific
shape, i.e. a non-flat manifold, and the standard euclidean distance is
not the right metric. This case arises in the two top rows of the figure
above.</p>
<p>Gaussian mixture models, useful for clustering, are described in
<a class="reference internal" href="mixture.html#mixture"><span class="std std-ref">another chapter of the documentation</span></a> dedicated to
mixture models. KMeans can be seen as a special case of Gaussian mixture
model with equal covariance per component.</p>
<p><a class="reference internal" href="../glossary.html#term-transductive"><span class="xref std std-term">Transductive</span></a> clustering methods (in contrast to
<a class="reference internal" href="../glossary.html#term-inductive"><span class="xref std std-term">inductive</span></a> clustering methods) are not designed to be applied to new,
unseen data.</p>
</section>
<section id="k-means">
<span id="id2"></span><h2><span class="section-number">2.3.2. </span>K-means<a class="headerlink" href="#k-means" title="Link to this heading">¶</a></h2>
<p>The <a class="reference internal" href="generated/sklearn.cluster.KMeans.html#sklearn.cluster.KMeans" title="sklearn.cluster.KMeans"><code class="xref py py-class docutils literal notranslate"><span class="pre">KMeans</span></code></a> algorithm clusters data by trying to separate samples in n
groups of equal variance, minimizing a criterion known as the <em>inertia</em> or
within-cluster sum-of-squares (see below). This algorithm requires the number
of clusters to be specified. It scales well to large numbers of samples and has
been used across a large range of application areas in many different fields.</p>
<p>The k-means algorithm divides a set of <span class="math notranslate nohighlight">\(N\)</span> samples <span class="math notranslate nohighlight">\(X\)</span> into
<span class="math notranslate nohighlight">\(K\)</span> disjoint clusters <span class="math notranslate nohighlight">\(C\)</span>, each described by the mean <span class="math notranslate nohighlight">\(\mu_j\)</span>
of the samples in the cluster. The means are commonly called the cluster
“centroids”; note that they are not, in general, points from <span class="math notranslate nohighlight">\(X\)</span>,
although they live in the same space.</p>
<p>The K-means algorithm aims to choose centroids that minimise the <strong>inertia</strong>,
or <strong>within-cluster sum-of-squares criterion</strong>:</p>
<div class="math notranslate nohighlight">
\[\sum_{i=0}^{n}\min_{\mu_j \in C}(||x_i - \mu_j||^2)\]</div>
<p>Inertia can be recognized as a measure of how internally coherent clusters are.
It suffers from various drawbacks:</p>
<ul class="simple">
<li><p>Inertia makes the assumption that clusters are convex and isotropic,
which is not always the case. It responds poorly to elongated clusters,
or manifolds with irregular shapes.</p></li>
<li><p>Inertia is not a normalized metric: we just know that lower values are
better and zero is optimal. But in very high-dimensional spaces, Euclidean
distances tend to become inflated
(this is an instance of the so-called “curse of dimensionality”).
Running a dimensionality reduction algorithm such as <a class="reference internal" href="decomposition.html#pca"><span class="std std-ref">Principal component analysis (PCA)</span></a> prior to
k-means clustering can alleviate this problem and speed up the
computations.</p></li>
</ul>
<a class="reference external image-reference" href="../auto_examples/cluster/plot_kmeans_assumptions.html"><img alt="../_images/sphx_glr_plot_kmeans_assumptions_002.png" class="align-center" src="../_images/sphx_glr_plot_kmeans_assumptions_002.png" style="width: 600.0px; height: 600.0px;" />
</a>
<p>For more detailed descriptions of the issues shown above and how to address them,
refer to the examples <a class="reference internal" href="../auto_examples/cluster/plot_kmeans_assumptions.html#sphx-glr-auto-examples-cluster-plot-kmeans-assumptions-py"><span class="std std-ref">Demonstration of k-means assumptions</span></a>
and <a class="reference internal" href="../auto_examples/cluster/plot_kmeans_silhouette_analysis.html#sphx-glr-auto-examples-cluster-plot-kmeans-silhouette-analysis-py"><span class="std std-ref">Selecting the number of clusters with silhouette analysis on KMeans clustering</span></a>.</p>
<p>K-means is often referred to as Lloyd’s algorithm. In basic terms, the
algorithm has three steps. The first step chooses the initial centroids, with
the most basic method being to choose <span class="math notranslate nohighlight">\(k\)</span> samples from the dataset
<span class="math notranslate nohighlight">\(X\)</span>. After initialization, K-means consists of looping between the
two other steps. The first step assigns each sample to its nearest centroid.
The second step creates new centroids by taking the mean value of all of the
samples assigned to each previous centroid. The difference between the old
and the new centroids are computed and the algorithm repeats these last two
steps until this value is less than a threshold. In other words, it repeats
until the centroids do not move significantly.</p>
<a class="reference external image-reference" href="../auto_examples/cluster/plot_kmeans_digits.html"><img alt="../_images/sphx_glr_plot_kmeans_digits_001.png" class="align-right" src="../_images/sphx_glr_plot_kmeans_digits_001.png" style="width: 224.0px; height: 168.0px;" />
</a>
<p>K-means is equivalent to the expectation-maximization algorithm
with a small, all-equal, diagonal covariance matrix.</p>
<p>The algorithm can also be understood through the concept of <a class="reference external" href="https://en.wikipedia.org/wiki/Voronoi_diagram">Voronoi diagrams</a>. First the Voronoi diagram of
the points is calculated using the current centroids. Each segment in the
Voronoi diagram becomes a separate cluster. Secondly, the centroids are updated
to the mean of each segment. The algorithm then repeats this until a stopping
criterion is fulfilled. Usually, the algorithm stops when the relative decrease
in the objective function between iterations is less than the given tolerance
value. This is not the case in this implementation: iteration stops when
centroids move less than the tolerance.</p>
<p>Given enough time, K-means will always converge, however this may be to a local
minimum. This is highly dependent on the initialization of the centroids.
As a result, the computation is often done several times, with different
initializations of the centroids. One method to help address this issue is the
k-means++ initialization scheme, which has been implemented in scikit-learn
(use the <code class="docutils literal notranslate"><span class="pre">init='k-means++'</span></code> parameter). This initializes the centroids to be
(generally) distant from each other, leading to probably better results than
random initialization, as shown in the reference. For a detailed example of
comaparing different initialization schemes, refer to
<a class="reference internal" href="../auto_examples/cluster/plot_kmeans_digits.html#sphx-glr-auto-examples-cluster-plot-kmeans-digits-py"><span class="std std-ref">A demo of K-Means clustering on the handwritten digits data</span></a>.</p>
<p>K-means++ can also be called independently to select seeds for other
clustering algorithms, see <a class="reference internal" href="generated/sklearn.cluster.kmeans_plusplus.html#sklearn.cluster.kmeans_plusplus" title="sklearn.cluster.kmeans_plusplus"><code class="xref py py-func docutils literal notranslate"><span class="pre">sklearn.cluster.kmeans_plusplus</span></code></a> for details
and example usage.</p>
<p>The algorithm supports sample weights, which can be given by a parameter
<code class="docutils literal notranslate"><span class="pre">sample_weight</span></code>. This allows to assign more weight to some samples when
computing cluster centers and values of inertia. For example, assigning a
weight of 2 to a sample is equivalent to adding a duplicate of that sample
to the dataset <span class="math notranslate nohighlight">\(X\)</span>.</p>
<p>K-means can be used for vector quantization. This is achieved using the
<code class="docutils literal notranslate"><span class="pre">transform</span></code> method of a trained model of <a class="reference internal" href="generated/sklearn.cluster.KMeans.html#sklearn.cluster.KMeans" title="sklearn.cluster.KMeans"><code class="xref py py-class docutils literal notranslate"><span class="pre">KMeans</span></code></a>. For an example of
performing vector quantization on an image refer to
<a class="reference internal" href="../auto_examples/cluster/plot_color_quantization.html#sphx-glr-auto-examples-cluster-plot-color-quantization-py"><span class="std std-ref">Color Quantization using K-Means</span></a>.</p>
<aside class="topic">
<p class="topic-title">Examples:</p>
<ul class="simple">
<li><p><a class="reference internal" href="../auto_examples/cluster/plot_cluster_iris.html#sphx-glr-auto-examples-cluster-plot-cluster-iris-py"><span class="std std-ref">K-means Clustering</span></a>: Example usage of
<a class="reference internal" href="generated/sklearn.cluster.KMeans.html#sklearn.cluster.KMeans" title="sklearn.cluster.KMeans"><code class="xref py py-class docutils literal notranslate"><span class="pre">KMeans</span></code></a> using the iris dataset</p></li>
<li><p><a class="reference internal" href="../auto_examples/text/plot_document_clustering.html#sphx-glr-auto-examples-text-plot-document-clustering-py"><span class="std std-ref">Clustering text documents using k-means</span></a>: Document clustering
using <a class="reference internal" href="generated/sklearn.cluster.KMeans.html#sklearn.cluster.KMeans" title="sklearn.cluster.KMeans"><code class="xref py py-class docutils literal notranslate"><span class="pre">KMeans</span></code></a> and <a class="reference internal" href="generated/sklearn.cluster.MiniBatchKMeans.html#sklearn.cluster.MiniBatchKMeans" title="sklearn.cluster.MiniBatchKMeans"><code class="xref py py-class docutils literal notranslate"><span class="pre">MiniBatchKMeans</span></code></a> based on sparse data</p></li>
</ul>
</aside>
<section id="low-level-parallelism">
<h3><span class="section-number">2.3.2.1. </span>Low-level parallelism<a class="headerlink" href="#low-level-parallelism" title="Link to this heading">¶</a></h3>
<p><a class="reference internal" href="generated/sklearn.cluster.KMeans.html#sklearn.cluster.KMeans" title="sklearn.cluster.KMeans"><code class="xref py py-class docutils literal notranslate"><span class="pre">KMeans</span></code></a> benefits from OpenMP based parallelism through Cython. Small
chunks of data (256 samples) are processed in parallel, which in addition
yields a low memory footprint. For more details on how to control the number of
threads, please refer to our <a class="reference internal" href="../computing/parallelism.html#parallelism"><span class="std std-ref">Parallelism</span></a> notes.</p>
<aside class="topic">
<p class="topic-title">Examples:</p>
<ul class="simple">
<li><p><a class="reference internal" href="../auto_examples/cluster/plot_kmeans_assumptions.html#sphx-glr-auto-examples-cluster-plot-kmeans-assumptions-py"><span class="std std-ref">Demonstration of k-means assumptions</span></a>: Demonstrating
when k-means performs intuitively and when it does not</p></li>
<li><p><a class="reference internal" href="../auto_examples/cluster/plot_kmeans_digits.html#sphx-glr-auto-examples-cluster-plot-kmeans-digits-py"><span class="std std-ref">A demo of K-Means clustering on the handwritten digits data</span></a>: Clustering
handwritten digits</p></li>
</ul>
</aside>
<p><details id="summary-anchor">
<summary class="btn btn-light">
<strong>References</strong>
<span class="tooltiptext">Click for more details</span>
<a class="headerlink" href="#summary-anchor" title="Permalink to this heading">¶</a>
</summary>
<div class="card"></p>
<ul class="simple">
<li><p><a class="reference external" href="http://ilpubs.stanford.edu:8090/778/1/2006-13.pdf">“k-means++: The advantages of careful seeding”</a> Arthur, David, and
Sergei Vassilvitskii, <em>Proceedings of the eighteenth annual ACM-SIAM symposium
on Discrete algorithms</em>, Society for Industrial and Applied Mathematics (2007)</p></li>
</ul>
<p></div>
</details></p>
</section>
<section id="mini-batch-k-means">
<span id="mini-batch-kmeans"></span><h3><span class="section-number">2.3.2.2. </span>Mini Batch K-Means<a class="headerlink" href="#mini-batch-k-means" title="Link to this heading">¶</a></h3>
<p>The <a class="reference internal" href="generated/sklearn.cluster.MiniBatchKMeans.html#sklearn.cluster.MiniBatchKMeans" title="sklearn.cluster.MiniBatchKMeans"><code class="xref py py-class docutils literal notranslate"><span class="pre">MiniBatchKMeans</span></code></a> is a variant of the <a class="reference internal" href="generated/sklearn.cluster.KMeans.html#sklearn.cluster.KMeans" title="sklearn.cluster.KMeans"><code class="xref py py-class docutils literal notranslate"><span class="pre">KMeans</span></code></a> algorithm
which uses mini-batches to reduce the computation time, while still attempting
to optimise the same objective function. Mini-batches are subsets of the input
data, randomly sampled in each training iteration. These mini-batches
drastically reduce the amount of computation required to converge to a local
solution. In contrast to other algorithms that reduce the convergence time of
k-means, mini-batch k-means produces results that are generally only slightly
worse than the standard algorithm.</p>
<p>The algorithm iterates between two major steps, similar to vanilla k-means.
In the first step, <span class="math notranslate nohighlight">\(b\)</span> samples are drawn randomly from the dataset, to form
a mini-batch. These are then assigned to the nearest centroid. In the second
step, the centroids are updated. In contrast to k-means, this is done on a
per-sample basis. For each sample in the mini-batch, the assigned centroid
is updated by taking the streaming average of the sample and all previous
samples assigned to that centroid. This has the effect of decreasing the
rate of change for a centroid over time. These steps are performed until
convergence or a predetermined number of iterations is reached.</p>
<p><a class="reference internal" href="generated/sklearn.cluster.MiniBatchKMeans.html#sklearn.cluster.MiniBatchKMeans" title="sklearn.cluster.MiniBatchKMeans"><code class="xref py py-class docutils literal notranslate"><span class="pre">MiniBatchKMeans</span></code></a> converges faster than <a class="reference internal" href="generated/sklearn.cluster.KMeans.html#sklearn.cluster.KMeans" title="sklearn.cluster.KMeans"><code class="xref py py-class docutils literal notranslate"><span class="pre">KMeans</span></code></a>, but the quality
of the results is reduced. In practice this difference in quality can be quite
small, as shown in the example and cited reference.</p>
<figure class="align-center">
<a class="reference external image-reference" href="../auto_examples/cluster/plot_mini_batch_kmeans.html"><img alt="../_images/sphx_glr_plot_mini_batch_kmeans_001.png" src="../_images/sphx_glr_plot_mini_batch_kmeans_001.png" style="width: 800.0px; height: 300.0px;" />
</a>
</figure>
<aside class="topic">
<p class="topic-title">Examples:</p>
<ul class="simple">
<li><p><a class="reference internal" href="../auto_examples/cluster/plot_mini_batch_kmeans.html#sphx-glr-auto-examples-cluster-plot-mini-batch-kmeans-py"><span class="std std-ref">Comparison of the K-Means and MiniBatchKMeans clustering algorithms</span></a>: Comparison of
<a class="reference internal" href="generated/sklearn.cluster.KMeans.html#sklearn.cluster.KMeans" title="sklearn.cluster.KMeans"><code class="xref py py-class docutils literal notranslate"><span class="pre">KMeans</span></code></a> and <a class="reference internal" href="generated/sklearn.cluster.MiniBatchKMeans.html#sklearn.cluster.MiniBatchKMeans" title="sklearn.cluster.MiniBatchKMeans"><code class="xref py py-class docutils literal notranslate"><span class="pre">MiniBatchKMeans</span></code></a></p></li>
<li><p><a class="reference internal" href="../auto_examples/text/plot_document_clustering.html#sphx-glr-auto-examples-text-plot-document-clustering-py"><span class="std std-ref">Clustering text documents using k-means</span></a>: Document clustering
using <a class="reference internal" href="generated/sklearn.cluster.KMeans.html#sklearn.cluster.KMeans" title="sklearn.cluster.KMeans"><code class="xref py py-class docutils literal notranslate"><span class="pre">KMeans</span></code></a> and <a class="reference internal" href="generated/sklearn.cluster.MiniBatchKMeans.html#sklearn.cluster.MiniBatchKMeans" title="sklearn.cluster.MiniBatchKMeans"><code class="xref py py-class docutils literal notranslate"><span class="pre">MiniBatchKMeans</span></code></a> based on sparse data</p></li>
</ul>
</aside>
<p><details id="summary-anchor">
<summary class="btn btn-light">
<strong>References</strong>
<span class="tooltiptext">Click for more details</span>
<a class="headerlink" href="#summary-anchor" title="Permalink to this heading">¶</a>
</summary>
<div class="card"></p>
<ul class="simple">
<li><p><a class="reference external" href="https://www.eecs.tufts.edu/~dsculley/papers/fastkmeans.pdf">“Web Scale K-Means clustering”</a>
D. Sculley, <em>Proceedings of the 19th international conference on World
wide web</em> (2010)</p></li>
</ul>
<p></div>
</details></p>
</section>
</section>
<section id="affinity-propagation">
<span id="id3"></span><h2><span class="section-number">2.3.3. </span>Affinity Propagation<a class="headerlink" href="#affinity-propagation" title="Link to this heading">¶</a></h2>
<p><a class="reference internal" href="generated/sklearn.cluster.AffinityPropagation.html#sklearn.cluster.AffinityPropagation" title="sklearn.cluster.AffinityPropagation"><code class="xref py py-class docutils literal notranslate"><span class="pre">AffinityPropagation</span></code></a> creates clusters by sending messages between
pairs of samples until convergence. A dataset is then described using a small
number of exemplars, which are identified as those most representative of other
samples. The messages sent between pairs represent the suitability for one
sample to be the exemplar of the other, which is updated in response to the
values from other pairs. This updating happens iteratively until convergence,
at which point the final exemplars are chosen, and hence the final clustering
is given.</p>
<figure class="align-center">
<a class="reference external image-reference" href="../auto_examples/cluster/plot_affinity_propagation.html"><img alt="../_images/sphx_glr_plot_affinity_propagation_001.png" src="../_images/sphx_glr_plot_affinity_propagation_001.png" style="width: 320.0px; height: 240.0px;" />
</a>
</figure>
<p>Affinity Propagation can be interesting as it chooses the number of
clusters based on the data provided. For this purpose, the two important
parameters are the <em>preference</em>, which controls how many exemplars are
used, and the <em>damping factor</em> which damps the responsibility and
availability messages to avoid numerical oscillations when updating these
messages.</p>
<p>The main drawback of Affinity Propagation is its complexity. The
algorithm has a time complexity of the order <span class="math notranslate nohighlight">\(O(N^2 T)\)</span>, where <span class="math notranslate nohighlight">\(N\)</span>
is the number of samples and <span class="math notranslate nohighlight">\(T\)</span> is the number of iterations until
convergence. Further, the memory complexity is of the order
<span class="math notranslate nohighlight">\(O(N^2)\)</span> if a dense similarity matrix is used, but reducible if a
sparse similarity matrix is used. This makes Affinity Propagation most
appropriate for small to medium sized datasets.</p>
<p><details id="summary-anchor">
<summary class="btn btn-light">
<strong>Algorithm description</strong>
<span class="tooltiptext">Click for more details</span>
<a class="headerlink" href="#summary-anchor" title="Permalink to this heading">¶</a>
</summary>
<div class="card"></p>
<p>The messages sent between points belong to one of two categories. The first is
the responsibility <span class="math notranslate nohighlight">\(r(i, k)\)</span>, which is the accumulated evidence that
sample <span class="math notranslate nohighlight">\(k\)</span> should be the exemplar for sample <span class="math notranslate nohighlight">\(i\)</span>. The second is the
availability <span class="math notranslate nohighlight">\(a(i, k)\)</span> which is the accumulated evidence that sample
<span class="math notranslate nohighlight">\(i\)</span> should choose sample <span class="math notranslate nohighlight">\(k\)</span> to be its exemplar, and considers the
values for all other samples that <span class="math notranslate nohighlight">\(k\)</span> should be an exemplar. In this way,
exemplars are chosen by samples if they are (1) similar enough to many samples
and (2) chosen by many samples to be representative of themselves.</p>
<p>More formally, the responsibility of a sample <span class="math notranslate nohighlight">\(k\)</span> to be the exemplar of
sample <span class="math notranslate nohighlight">\(i\)</span> is given by:</p>
<div class="math notranslate nohighlight">
\[r(i, k) \leftarrow s(i, k) - max [ a(i, k') + s(i, k') \forall k' \neq k ]\]</div>
<p>Where <span class="math notranslate nohighlight">\(s(i, k)\)</span> is the similarity between samples <span class="math notranslate nohighlight">\(i\)</span> and <span class="math notranslate nohighlight">\(k\)</span>.
The availability of sample <span class="math notranslate nohighlight">\(k\)</span> to be the exemplar of sample <span class="math notranslate nohighlight">\(i\)</span> is
given by:</p>
<div class="math notranslate nohighlight">
\[a(i, k) \leftarrow min [0, r(k, k) + \sum_{i'~s.t.~i' \notin \{i, k\}}{r(i',
k)}]\]</div>
<p>To begin with, all values for <span class="math notranslate nohighlight">\(r\)</span> and <span class="math notranslate nohighlight">\(a\)</span> are set to zero, and the
calculation of each iterates until convergence. As discussed above, in order to
avoid numerical oscillations when updating the messages, the damping factor
<span class="math notranslate nohighlight">\(\lambda\)</span> is introduced to iteration process:</p>
<div class="math notranslate nohighlight">
\[r_{t+1}(i, k) = \lambda\cdot r_{t}(i, k) + (1-\lambda)\cdot r_{t+1}(i, k)\]</div>
<div class="math notranslate nohighlight">
\[a_{t+1}(i, k) = \lambda\cdot a_{t}(i, k) + (1-\lambda)\cdot a_{t+1}(i, k)\]</div>
<p>where <span class="math notranslate nohighlight">\(t\)</span> indicates the iteration times.</p>
<p></div>
</details></p>
<aside class="topic">
<p class="topic-title">Examples:</p>
<ul class="simple">
<li><p><a class="reference internal" href="../auto_examples/cluster/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>: Affinity
Propagation on a synthetic 2D datasets with 3 classes.</p></li>
<li><p><a class="reference internal" href="../auto_examples/applications/plot_stock_market.html#sphx-glr-auto-examples-applications-plot-stock-market-py"><span class="std std-ref">Visualizing the stock market structure</span></a> Affinity
Propagation on Financial time series to find groups of companies</p></li>
</ul>
</aside>
</section>
<section id="mean-shift">
<span id="id4"></span><h2><span class="section-number">2.3.4. </span>Mean Shift<a class="headerlink" href="#mean-shift" title="Link to this heading">¶</a></h2>
<p><a class="reference internal" href="generated/sklearn.cluster.MeanShift.html#sklearn.cluster.MeanShift" title="sklearn.cluster.MeanShift"><code class="xref py py-class docutils literal notranslate"><span class="pre">MeanShift</span></code></a> clustering aims to discover <em>blobs</em> in a smooth density of
samples. It is a centroid based algorithm, which works by updating candidates
for centroids to be the mean of the points within a given region. These
candidates are then filtered in a post-processing stage to eliminate
near-duplicates to form the final set of centroids.</p>
<p><details id="summary-anchor">
<summary class="btn btn-light">
<strong>Mathematical details</strong>
<span class="tooltiptext">Click for more details</span>
<a class="headerlink" href="#summary-anchor" title="Permalink to this heading">¶</a>
</summary>
<div class="card"></p>
<p>The position of centroid candidates is iteratively adjusted using a technique
called hill climbing, which finds local maxima of the estimated probability
density. Given a candidate centroid <span class="math notranslate nohighlight">\(x\)</span> for iteration <span class="math notranslate nohighlight">\(t\)</span>, the
candidate is updated according to the following equation:</p>
<div class="math notranslate nohighlight">
\[x^{t+1} = x^t + m(x^t)\]</div>
<p>Where <span class="math notranslate nohighlight">\(m\)</span> is the <em>mean shift</em> vector that is computed for each centroid
that points towards a region of the maximum increase in the density of points.
To compute <span class="math notranslate nohighlight">\(m\)</span> we define <span class="math notranslate nohighlight">\(N(x)\)</span> as the neighborhood of samples
within a given distance around <span class="math notranslate nohighlight">\(x\)</span>. Then <span class="math notranslate nohighlight">\(m\)</span> is computed using the
following equation, effectively updating a centroid to be the mean of the
samples within its neighborhood:</p>
<div class="math notranslate nohighlight">
\[m(x) = \frac{1}{|N(x)|} \sum_{x_j \in N(x)}x_j - x\]</div>
<p>In general, the equation for <span class="math notranslate nohighlight">\(m\)</span> depends on a kernel used for density
estimation. The generic formula is:</p>
<div class="math notranslate nohighlight">
\[m(x) = \frac{\sum_{x_j \in N(x)}K(x_j - x)x_j}{\sum_{x_j \in N(x)}K(x_j -
x)} - x\]</div>
<p>In our implementation, <span class="math notranslate nohighlight">\(K(x)\)</span> is equal to 1 if <span class="math notranslate nohighlight">\(x\)</span> is small enough
and is equal to 0 otherwise. Effectively <span class="math notranslate nohighlight">\(K(y - x)\)</span> indicates whether
<span class="math notranslate nohighlight">\(y\)</span> is in the neighborhood of <span class="math notranslate nohighlight">\(x\)</span>.</p>
<p></div>
</details></p>
<p>The algorithm automatically sets the number of clusters, instead of relying on a
parameter <code class="docutils literal notranslate"><span class="pre">bandwidth</span></code>, which dictates the size of the region to search through.
This parameter can be set manually, but can be estimated using the provided
<code class="docutils literal notranslate"><span class="pre">estimate_bandwidth</span></code> function, which is called if the bandwidth is not set.</p>
<p>The algorithm is not highly scalable, as it requires multiple nearest neighbor
searches during the execution of the algorithm. The algorithm is guaranteed to
converge, however the algorithm will stop iterating when the change in centroids
is small.</p>
<p>Labelling a new sample is performed by finding the nearest centroid for a
given sample.</p>
<figure class="align-center">
<a class="reference external image-reference" href="../auto_examples/cluster/plot_mean_shift.html"><img alt="../_images/sphx_glr_plot_mean_shift_001.png" src="../_images/sphx_glr_plot_mean_shift_001.png" style="width: 320.0px; height: 240.0px;" />
</a>
</figure>
<aside class="topic">
<p class="topic-title">Examples:</p>
<ul class="simple">
<li><p><a class="reference internal" href="../auto_examples/cluster/plot_mean_shift.html#sphx-glr-auto-examples-cluster-plot-mean-shift-py"><span class="std std-ref">A demo of the mean-shift clustering algorithm</span></a>: Mean Shift
clustering on a synthetic 2D datasets with 3 classes.</p></li>
</ul>
</aside>
<p><details id="summary-anchor">
<summary class="btn btn-light">
<strong>References</strong>
<span class="tooltiptext">Click for more details</span>
<a class="headerlink" href="#summary-anchor" title="Permalink to this heading">¶</a>
</summary>
<div class="card"></p>
<ul class="simple">
<li><p><a class="reference external" href="https://doi.org/10.1109/34.1000236">“Mean shift: A robust approach toward feature space analysis”</a> D. Comaniciu and P. Meer, <em>IEEE Transactions on Pattern
Analysis and Machine Intelligence</em> (2002)</p></li>
</ul>
<p></div>
</details></p>
</section>
<section id="spectral-clustering">
<span id="id5"></span><h2><span class="section-number">2.3.5. </span>Spectral clustering<a class="headerlink" href="#spectral-clustering" title="Link to this heading">¶</a></h2>
<p><a class="reference internal" href="generated/sklearn.cluster.SpectralClustering.html#sklearn.cluster.SpectralClustering" title="sklearn.cluster.SpectralClustering"><code class="xref py py-class docutils literal notranslate"><span class="pre">SpectralClustering</span></code></a> performs a low-dimension embedding of the
affinity matrix between samples, followed by clustering, e.g., by KMeans,
of the components of the eigenvectors in the low dimensional space.
It is especially computationally efficient if the affinity matrix is sparse
and the <code class="docutils literal notranslate"><span class="pre">amg</span></code> solver is used for the eigenvalue problem (Note, the <code class="docutils literal notranslate"><span class="pre">amg</span></code> solver
requires that the <a class="reference external" href="https://github.com/pyamg/pyamg">pyamg</a> module is installed.)</p>
<p>The present version of SpectralClustering requires the number of clusters
to be specified in advance. It works well for a small number of clusters,
but is not advised for many clusters.</p>
<p>For two clusters, SpectralClustering solves a convex relaxation of the
<a class="reference external" href="https://people.eecs.berkeley.edu/~malik/papers/SM-ncut.pdf">normalized cuts</a>
problem on the similarity graph: cutting the graph in two so that the weight of
the edges cut is small compared to the weights of the edges inside each
cluster. This criteria is especially interesting when working on images, where
graph vertices are pixels, and weights of the edges of the similarity graph are
computed using a function of a gradient of the image.</p>
<p class="centered">
<strong><a class="reference external" href="../auto_examples/cluster/plot_segmentation_toy.html"><img alt="noisy_img" src="../_images/sphx_glr_plot_segmentation_toy_001.png" style="width: 500.0px; height: 250.0px;" /></a> <a class="reference external" href="../auto_examples/cluster/plot_segmentation_toy.html"><img alt="segmented_img" src="../_images/sphx_glr_plot_segmentation_toy_002.png" style="width: 500.0px; height: 250.0px;" /></a></strong></p><div class="admonition warning">
<p class="admonition-title">Warning</p>
<p>Transforming distance to well-behaved similarities</p>
<p>Note that if the values of your similarity matrix are not well
distributed, e.g. with negative values or with a distance matrix
rather than a similarity, the spectral problem will be singular and
the problem not solvable. In which case it is advised to apply a
transformation to the entries of the matrix. For instance, in the
case of a signed distance matrix, is common to apply a heat kernel:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">similarity</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">beta</span> <span class="o">*</span> <span class="n">distance</span> <span class="o">/</span> <span class="n">distance</span><span class="o">.</span><span class="n">std</span><span class="p">())</span>
</pre></div>
</div>
<p>See the examples for such an application.</p>
</div>
<aside class="topic">
<p class="topic-title">Examples:</p>
<ul class="simple">
<li><p><a class="reference internal" href="../auto_examples/cluster/plot_segmentation_toy.html#sphx-glr-auto-examples-cluster-plot-segmentation-toy-py"><span class="std std-ref">Spectral clustering for image segmentation</span></a>: Segmenting
objects from a noisy background using spectral clustering.</p></li>
<li><p><a class="reference internal" href="../auto_examples/cluster/plot_coin_segmentation.html#sphx-glr-auto-examples-cluster-plot-coin-segmentation-py"><span class="std std-ref">Segmenting the picture of greek coins in regions</span></a>: Spectral
clustering to split the image of coins in regions.</p></li>
</ul>
</aside>
<section id="different-label-assignment-strategies">
<h3><span class="section-number">2.3.5.1. </span>Different label assignment strategies<a class="headerlink" href="#different-label-assignment-strategies" title="Link to this heading">¶</a></h3>
<p>Different label assignment strategies can be used, corresponding to the
<code class="docutils literal notranslate"><span class="pre">assign_labels</span></code> parameter of <a class="reference internal" href="generated/sklearn.cluster.SpectralClustering.html#sklearn.cluster.SpectralClustering" title="sklearn.cluster.SpectralClustering"><code class="xref py py-class docutils literal notranslate"><span class="pre">SpectralClustering</span></code></a>.
<code class="docutils literal notranslate"><span class="pre">"kmeans"</span></code> strategy can match finer details, but can be unstable.
In particular, unless you control the <code class="docutils literal notranslate"><span class="pre">random_state</span></code>, it may not be
reproducible from run-to-run, as it depends on random initialization.
The alternative <code class="docutils literal notranslate"><span class="pre">"discretize"</span></code> strategy is 100% reproducible, but tends
to create parcels of fairly even and geometrical shape.
The recently added <code class="docutils literal notranslate"><span class="pre">"cluster_qr"</span></code> option is a deterministic alternative that
tends to create the visually best partitioning on the example application
below.</p>
<table class="docutils align-default">
<thead>
<tr class="row-odd"><th class="head"><p><code class="docutils literal notranslate"><span class="pre">assign_labels="kmeans"</span></code></p></th>
<th class="head"><p><code class="docutils literal notranslate"><span class="pre">assign_labels="discretize"</span></code></p></th>
<th class="head"><p><code class="docutils literal notranslate"><span class="pre">assign_labels="cluster_qr"</span></code></p></th>
</tr>
</thead>
<tbody>
<tr class="row-even"><td><p><a class="reference external" href="../auto_examples/cluster/plot_coin_segmentation.html"><img alt="coin_kmeans" src="../_images/sphx_glr_plot_coin_segmentation_001.png" style="width: 175.0px; height: 175.0px;" /></a></p></td>
<td><p><a class="reference external" href="../auto_examples/cluster/plot_coin_segmentation.html"><img alt="coin_discretize" src="../_images/sphx_glr_plot_coin_segmentation_002.png" style="width: 175.0px; height: 175.0px;" /></a></p></td>
<td><p><a class="reference external" href="../auto_examples/cluster/plot_coin_segmentation.html"><img alt="coin_cluster_qr" src="../_images/sphx_glr_plot_coin_segmentation_003.png" style="width: 175.0px; height: 175.0px;" /></a></p></td>
</tr>
</tbody>
</table>
<p><details id="summary-anchor">
<summary class="btn btn-light">
<strong>References</strong>
<span class="tooltiptext">Click for more details</span>
<a class="headerlink" href="#summary-anchor" title="Permalink to this heading">¶</a>
</summary>
<div class="card"></p>
<ul class="simple">
<li><p><a class="reference external" href="https://people.eecs.berkeley.edu/~jordan/courses/281B-spring04/readings/yu-shi.pdf">“Multiclass spectral clustering”</a>
Stella X. Yu, Jianbo Shi, 2003</p></li>
<li><p><a class="reference external" href="https://doi.org/10.1093/imaiai/iay008">“Simple, direct, and efficient multi-way spectral clustering”</a>
Anil Damle, Victor Minden, Lexing Ying, 2019</p></li>
</ul>
<p></div>
</details></p>
</section>
<section id="spectral-clustering-graphs">
<span id="spectral-clustering-graph"></span><h3><span class="section-number">2.3.5.2. </span>Spectral Clustering Graphs<a class="headerlink" href="#spectral-clustering-graphs" title="Link to this heading">¶</a></h3>
<p>Spectral Clustering can also be used to partition graphs via their spectral
embeddings. In this case, the affinity matrix is the adjacency matrix of the
graph, and SpectralClustering is initialized with <code class="docutils literal notranslate"><span class="pre">affinity='precomputed'</span></code>:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span> <span class="nn">sklearn.cluster</span> <span class="kn">import</span> <span class="n">SpectralClustering</span>
<span class="gp">>>> </span><span class="n">sc</span> <span class="o">=</span> <span class="n">SpectralClustering</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="n">affinity</span><span class="o">=</span><span class="s1">'precomputed'</span><span class="p">,</span> <span class="n">n_init</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span>
<span class="gp">... </span> <span class="n">assign_labels</span><span class="o">=</span><span class="s1">'discretize'</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">sc</span><span class="o">.</span><span class="n">fit_predict</span><span class="p">(</span><span class="n">adjacency_matrix</span><span class="p">)</span>
</pre></div>
</div>
<p><details id="summary-anchor">
<summary class="btn btn-light">
<strong>References</strong>
<span class="tooltiptext">Click for more details</span>
<a class="headerlink" href="#summary-anchor" title="Permalink to this heading">¶</a>
</summary>
<div class="card"></p>
<ul class="simple">
<li><p><a class="reference external" href="https://doi.org/10.1007/s11222-007-9033-z">“A Tutorial on Spectral Clustering”</a> Ulrike
von Luxburg, 2007</p></li>
<li><p><a class="reference external" href="https://doi.org/10.1109/34.868688">“Normalized cuts and image segmentation”</a> Jianbo
Shi, Jitendra Malik, 2000</p></li>
<li><p><a class="reference external" href="https://citeseerx.ist.psu.edu/doc_view/pid/84a86a69315e994cfd1e0c7debb86d62d7bd1f44">“A Random Walks View of Spectral Segmentation”</a>
Marina Meila, Jianbo Shi, 2001</p></li>
<li><p><a class="reference external" href="https://citeseerx.ist.psu.edu/doc_view/pid/796c5d6336fc52aa84db575fb821c78918b65f58">“On Spectral Clustering: Analysis and an algorithm”</a>
Andrew Y. Ng, Michael I. Jordan, Yair Weiss, 2001</p></li>
<li><p><a class="reference external" href="https://arxiv.org/abs/1708.07481">“Preconditioned Spectral Clustering for Stochastic Block Partition
Streaming Graph Challenge”</a> David Zhuzhunashvili, Andrew Knyazev</p></li>
</ul>
<p></div>
</details></p>
</section>
</section>
<section id="hierarchical-clustering">
<span id="id6"></span><h2><span class="section-number">2.3.6. </span>Hierarchical clustering<a class="headerlink" href="#hierarchical-clustering" title="Link to this heading">¶</a></h2>
<p>Hierarchical clustering is a general family of clustering algorithms that
build nested clusters by merging or splitting them successively. This
hierarchy of clusters is represented as a tree (or dendrogram). The root of the
tree is the unique cluster that gathers all the samples, the leaves being the
clusters with only one sample. See the <a class="reference external" href="https://en.wikipedia.org/wiki/Hierarchical_clustering">Wikipedia page</a> for more details.</p>
<p>The <a class="reference internal" href="generated/sklearn.cluster.AgglomerativeClustering.html#sklearn.cluster.AgglomerativeClustering" title="sklearn.cluster.AgglomerativeClustering"><code class="xref py py-class docutils literal notranslate"><span class="pre">AgglomerativeClustering</span></code></a> object performs a hierarchical clustering
using a bottom up approach: each observation starts in its own cluster, and
clusters are successively merged together. The linkage criteria determines the
metric used for the merge strategy:</p>
<ul class="simple">
<li><p><strong>Ward</strong> minimizes the sum of squared differences within all clusters. It is a
variance-minimizing approach and in this sense is similar to the k-means
objective function but tackled with an agglomerative hierarchical
approach.</p></li>
<li><p><strong>Maximum</strong> or <strong>complete linkage</strong> minimizes the maximum distance between
observations of pairs of clusters.</p></li>
<li><p><strong>Average linkage</strong> minimizes the average of the distances between all
observations of pairs of clusters.</p></li>
<li><p><strong>Single linkage</strong> minimizes the distance between the closest
observations of pairs of clusters.</p></li>
</ul>
<p><a class="reference internal" href="generated/sklearn.cluster.AgglomerativeClustering.html#sklearn.cluster.AgglomerativeClustering" title="sklearn.cluster.AgglomerativeClustering"><code class="xref py py-class docutils literal notranslate"><span class="pre">AgglomerativeClustering</span></code></a> can also scale to large number of samples
when it is used jointly with a connectivity matrix, but is computationally
expensive when no connectivity constraints are added between samples: it
considers at each step all the possible merges.</p>
<aside class="topic">
<p class="topic-title"><a class="reference internal" href="generated/sklearn.cluster.FeatureAgglomeration.html#sklearn.cluster.FeatureAgglomeration" title="sklearn.cluster.FeatureAgglomeration"><code class="xref py py-class docutils literal notranslate"><span class="pre">FeatureAgglomeration</span></code></a></p>
<p>The <a class="reference internal" href="generated/sklearn.cluster.FeatureAgglomeration.html#sklearn.cluster.FeatureAgglomeration" title="sklearn.cluster.FeatureAgglomeration"><code class="xref py py-class docutils literal notranslate"><span class="pre">FeatureAgglomeration</span></code></a> uses agglomerative clustering to
group together features that look very similar, thus decreasing the
number of features. It is a dimensionality reduction tool, see
<a class="reference internal" href="unsupervised_reduction.html#data-reduction"><span class="std std-ref">Unsupervised dimensionality reduction</span></a>.</p>
</aside>
<section id="different-linkage-type-ward-complete-average-and-single-linkage">
<h3><span class="section-number">2.3.6.1. </span>Different linkage type: Ward, complete, average, and single linkage<a class="headerlink" href="#different-linkage-type-ward-complete-average-and-single-linkage" title="Link to this heading">¶</a></h3>
<p><a class="reference internal" href="generated/sklearn.cluster.AgglomerativeClustering.html#sklearn.cluster.AgglomerativeClustering" title="sklearn.cluster.AgglomerativeClustering"><code class="xref py py-class docutils literal notranslate"><span class="pre">AgglomerativeClustering</span></code></a> supports Ward, single, average, and complete
linkage strategies.</p>
<a class="reference external image-reference" href="../auto_examples/cluster/plot_linkage_comparison.html"><img alt="../_images/sphx_glr_plot_linkage_comparison_001.png" src="../_images/sphx_glr_plot_linkage_comparison_001.png" style="width: 589.1px; height: 623.5px;" />
</a>
<p>Agglomerative cluster has a “rich get richer” behavior that leads to
uneven cluster sizes. In this regard, single linkage is the worst
strategy, and Ward gives the most regular sizes. However, the affinity
(or distance used in clustering) cannot be varied with Ward, thus for non
Euclidean metrics, average linkage is a good alternative. Single linkage,
while not robust to noisy data, can be computed very efficiently and can
therefore be useful to provide hierarchical clustering of larger datasets.
Single linkage can also perform well on non-globular data.</p>
<aside class="topic">
<p class="topic-title">Examples:</p>
<ul class="simple">
<li><p><a class="reference internal" href="../auto_examples/cluster/plot_digits_linkage.html#sphx-glr-auto-examples-cluster-plot-digits-linkage-py"><span class="std std-ref">Various Agglomerative Clustering on a 2D embedding of digits</span></a>: exploration of
the different linkage strategies in a real dataset.</p></li>
<li><p><a class="reference internal" href="../auto_examples/cluster/plot_linkage_comparison.html#sphx-glr-auto-examples-cluster-plot-linkage-comparison-py"><span class="std std-ref">Comparing different hierarchical linkage methods on toy datasets</span></a>: exploration of
the different linkage strategies in toy datasets.</p></li>
</ul>
</aside>
</section>
<section id="visualization-of-cluster-hierarchy">
<h3><span class="section-number">2.3.6.2. </span>Visualization of cluster hierarchy<a class="headerlink" href="#visualization-of-cluster-hierarchy" title="Link to this heading">¶</a></h3>
<p>It’s possible to visualize the tree representing the hierarchical merging of clusters
as a dendrogram. Visual inspection can often be useful for understanding the structure
of the data, though more so in the case of small sample sizes.</p>
<a class="reference external image-reference" href="../auto_examples/cluster/plot_agglomerative_dendrogram.html"><img alt="../_images/sphx_glr_plot_agglomerative_dendrogram_001.png" src="../_images/sphx_glr_plot_agglomerative_dendrogram_001.png" style="width: 268.8px; height: 201.6px;" />
</a>
<aside class="topic">
<p class="topic-title">Examples:</p>
<ul class="simple">
<li><p><a class="reference internal" href="../auto_examples/cluster/plot_agglomerative_dendrogram.html#sphx-glr-auto-examples-cluster-plot-agglomerative-dendrogram-py"><span class="std std-ref">Plot Hierarchical Clustering Dendrogram</span></a></p></li>
</ul>
</aside>
</section>
<section id="adding-connectivity-constraints">
<h3><span class="section-number">2.3.6.3. </span>Adding connectivity constraints<a class="headerlink" href="#adding-connectivity-constraints" title="Link to this heading">¶</a></h3>
<p>An interesting aspect of <a class="reference internal" href="generated/sklearn.cluster.AgglomerativeClustering.html#sklearn.cluster.AgglomerativeClustering" title="sklearn.cluster.AgglomerativeClustering"><code class="xref py py-class docutils literal notranslate"><span class="pre">AgglomerativeClustering</span></code></a> is that
connectivity constraints can be added to this algorithm (only adjacent
clusters can be merged together), through a connectivity matrix that defines
for each sample the neighboring samples following a given structure of the
data. For instance, in the swiss-roll example below, the connectivity
constraints forbid the merging of points that are not adjacent on the swiss
roll, and thus avoid forming clusters that extend across overlapping folds of
the roll.</p>
<p class="centered">
<strong><a class="reference external" href="../auto_examples/cluster/plot_ward_structured_vs_unstructured.html"><img alt="unstructured" src="../_images/sphx_glr_plot_ward_structured_vs_unstructured_001.png" style="width: 313.6px; height: 235.2px;" /></a> <a class="reference external" href="../auto_examples/cluster/plot_ward_structured_vs_unstructured.html"><img alt="structured" src="../_images/sphx_glr_plot_ward_structured_vs_unstructured_002.png" style="width: 313.6px; height: 235.2px;" /></a></strong></p><p>These constraint are useful to impose a certain local structure, but they
also make the algorithm faster, especially when the number of the samples
is high.</p>
<p>The connectivity constraints are imposed via an connectivity matrix: a
scipy sparse matrix that has elements only at the intersection of a row
and a column with indices of the dataset that should be connected. This
matrix can be constructed from a-priori information: for instance, you
may wish to cluster web pages by only merging pages with a link pointing
from one to another. It can also be learned from the data, for instance
using <a class="reference internal" href="generated/sklearn.neighbors.kneighbors_graph.html#sklearn.neighbors.kneighbors_graph" title="sklearn.neighbors.kneighbors_graph"><code class="xref py py-func docutils literal notranslate"><span class="pre">sklearn.neighbors.kneighbors_graph</span></code></a> to restrict
merging to nearest neighbors as in <a class="reference internal" href="../auto_examples/cluster/plot_agglomerative_clustering.html#sphx-glr-auto-examples-cluster-plot-agglomerative-clustering-py"><span class="std std-ref">this example</span></a>, or
using <a class="reference internal" href="generated/sklearn.feature_extraction.image.grid_to_graph.html#sklearn.feature_extraction.image.grid_to_graph" title="sklearn.feature_extraction.image.grid_to_graph"><code class="xref py py-func docutils literal notranslate"><span class="pre">sklearn.feature_extraction.image.grid_to_graph</span></code></a> to
enable only merging of neighboring pixels on an image, as in the
<a class="reference internal" href="../auto_examples/cluster/plot_coin_ward_segmentation.html#sphx-glr-auto-examples-cluster-plot-coin-ward-segmentation-py"><span class="std std-ref">coin</span></a> example.</p>
<div class="admonition warning">
<p class="admonition-title">Warning</p>
<p><strong>Connectivity constraints with single, average and complete linkage</strong></p>
<p>Connectivity constraints and single, complete or average linkage can enhance
the ‘rich getting richer’ aspect of agglomerative clustering,
particularly so if they are built with
<a class="reference internal" href="generated/sklearn.neighbors.kneighbors_graph.html#sklearn.neighbors.kneighbors_graph" title="sklearn.neighbors.kneighbors_graph"><code class="xref py py-func docutils literal notranslate"><span class="pre">sklearn.neighbors.kneighbors_graph</span></code></a>. In the limit of a small
number of clusters, they tend to give a few macroscopically occupied
clusters and almost empty ones. (see the discussion in
<a class="reference internal" href="../auto_examples/cluster/plot_agglomerative_clustering.html#sphx-glr-auto-examples-cluster-plot-agglomerative-clustering-py"><span class="std std-ref">Agglomerative clustering with and without structure</span></a>).
Single linkage is the most brittle linkage option with regard to this issue.</p>
</div>
<a class="reference external image-reference" href="../auto_examples/cluster/plot_agglomerative_clustering.html"><img alt="../_images/sphx_glr_plot_agglomerative_clustering_001.png" src="../_images/sphx_glr_plot_agglomerative_clustering_001.png" style="width: 380.0px; height: 152.0px;" />
</a>
<a class="reference external image-reference" href="../auto_examples/cluster/plot_agglomerative_clustering.html"><img alt="../_images/sphx_glr_plot_agglomerative_clustering_002.png" src="../_images/sphx_glr_plot_agglomerative_clustering_002.png" style="width: 380.0px; height: 152.0px;" />
</a>
<a class="reference external image-reference" href="../auto_examples/cluster/plot_agglomerative_clustering.html"><img alt="../_images/sphx_glr_plot_agglomerative_clustering_003.png" src="../_images/sphx_glr_plot_agglomerative_clustering_003.png" style="width: 380.0px; height: 152.0px;" />
</a>
<a class="reference external image-reference" href="../auto_examples/cluster/plot_agglomerative_clustering.html"><img alt="../_images/sphx_glr_plot_agglomerative_clustering_004.png" src="../_images/sphx_glr_plot_agglomerative_clustering_004.png" style="width: 380.0px; height: 152.0px;" />
</a>
<aside class="topic">
<p class="topic-title">Examples:</p>
<ul class="simple">
<li><p><a class="reference internal" href="../auto_examples/cluster/plot_coin_ward_segmentation.html#sphx-glr-auto-examples-cluster-plot-coin-ward-segmentation-py"><span class="std std-ref">A demo of structured Ward hierarchical clustering on an image of coins</span></a>: Ward
clustering to split the image of coins in regions.</p></li>
<li><p><a class="reference internal" href="../auto_examples/cluster/plot_ward_structured_vs_unstructured.html#sphx-glr-auto-examples-cluster-plot-ward-structured-vs-unstructured-py"><span class="std std-ref">Hierarchical clustering: structured vs unstructured ward</span></a>: Example
of Ward algorithm on a swiss-roll, comparison of structured approaches
versus unstructured approaches.</p></li>
<li><p><a class="reference internal" href="../auto_examples/cluster/plot_feature_agglomeration_vs_univariate_selection.html#sphx-glr-auto-examples-cluster-plot-feature-agglomeration-vs-univariate-selection-py"><span class="std std-ref">Feature agglomeration vs. univariate selection</span></a>: Example
of dimensionality reduction with feature agglomeration based on Ward
hierarchical clustering.</p></li>
<li><p><a class="reference internal" href="../auto_examples/cluster/plot_agglomerative_clustering.html#sphx-glr-auto-examples-cluster-plot-agglomerative-clustering-py"><span class="std std-ref">Agglomerative clustering with and without structure</span></a></p></li>
</ul>
</aside>
</section>
<section id="varying-the-metric">
<h3><span class="section-number">2.3.6.4. </span>Varying the metric<a class="headerlink" href="#varying-the-metric" title="Link to this heading">¶</a></h3>
<p>Single, average and complete linkage can be used with a variety of distances (or
affinities), in particular Euclidean distance (<em>l2</em>), Manhattan distance
(or Cityblock, or <em>l1</em>), cosine distance, or any precomputed affinity
matrix.</p>
<ul class="simple">
<li><p><em>l1</em> distance is often good for sparse features, or sparse noise: i.e.
many of the features are zero, as in text mining using occurrences of
rare words.</p></li>
<li><p><em>cosine</em> distance is interesting because it is invariant to global
scalings of the signal.</p></li>
</ul>
<p>The guidelines for choosing a metric is to use one that maximizes the
distance between samples in different classes, and minimizes that within
each class.</p>
<a class="reference external image-reference" href="../auto_examples/cluster/plot_agglomerative_clustering_metrics.html"><img alt="../_images/sphx_glr_plot_agglomerative_clustering_metrics_005.png" src="../_images/sphx_glr_plot_agglomerative_clustering_metrics_005.png" style="width: 204.8px; height: 153.6px;" />
</a>
<a class="reference external image-reference" href="../auto_examples/cluster/plot_agglomerative_clustering_metrics.html"><img alt="../_images/sphx_glr_plot_agglomerative_clustering_metrics_006.png" src="../_images/sphx_glr_plot_agglomerative_clustering_metrics_006.png" style="width: 204.8px; height: 153.6px;" />
</a>
<a class="reference external image-reference" href="../auto_examples/cluster/plot_agglomerative_clustering_metrics.html"><img alt="../_images/sphx_glr_plot_agglomerative_clustering_metrics_007.png" src="../_images/sphx_glr_plot_agglomerative_clustering_metrics_007.png" style="width: 204.8px; height: 153.6px;" />
</a>
<aside class="topic">
<p class="topic-title">Examples:</p>
<ul class="simple">
<li><p><a class="reference internal" href="../auto_examples/cluster/plot_agglomerative_clustering_metrics.html#sphx-glr-auto-examples-cluster-plot-agglomerative-clustering-metrics-py"><span class="std std-ref">Agglomerative clustering with different metrics</span></a></p></li>
</ul>
</aside>
</section>
<section id="bisecting-k-means">
<h3><span class="section-number">2.3.6.5. </span>Bisecting K-Means<a class="headerlink" href="#bisecting-k-means" title="Link to this heading">¶</a></h3>
<p id="bisect-k-means">The <a class="reference internal" href="generated/sklearn.cluster.BisectingKMeans.html#sklearn.cluster.BisectingKMeans" title="sklearn.cluster.BisectingKMeans"><code class="xref py py-class docutils literal notranslate"><span class="pre">BisectingKMeans</span></code></a> is an iterative variant of <a class="reference internal" href="generated/sklearn.cluster.KMeans.html#sklearn.cluster.KMeans" title="sklearn.cluster.KMeans"><code class="xref py py-class docutils literal notranslate"><span class="pre">KMeans</span></code></a>, using
divisive hierarchical clustering. Instead of creating all centroids at once, centroids
are picked progressively based on a previous clustering: a cluster is split into two
new clusters repeatedly until the target number of clusters is reached.</p>
<p><a class="reference internal" href="generated/sklearn.cluster.BisectingKMeans.html#sklearn.cluster.BisectingKMeans" title="sklearn.cluster.BisectingKMeans"><code class="xref py py-class docutils literal notranslate"><span class="pre">BisectingKMeans</span></code></a> is more efficient than <a class="reference internal" href="generated/sklearn.cluster.KMeans.html#sklearn.cluster.KMeans" title="sklearn.cluster.KMeans"><code class="xref py py-class docutils literal notranslate"><span class="pre">KMeans</span></code></a> when the number of
clusters is large since it only works on a subset of the data at each bisection
while <a class="reference internal" href="generated/sklearn.cluster.KMeans.html#sklearn.cluster.KMeans" title="sklearn.cluster.KMeans"><code class="xref py py-class docutils literal notranslate"><span class="pre">KMeans</span></code></a> always works on the entire dataset.</p>
<p>Although <a class="reference internal" href="generated/sklearn.cluster.BisectingKMeans.html#sklearn.cluster.BisectingKMeans" title="sklearn.cluster.BisectingKMeans"><code class="xref py py-class docutils literal notranslate"><span class="pre">BisectingKMeans</span></code></a> can’t benefit from the advantages of the <code class="docutils literal notranslate"><span class="pre">"k-means++"</span></code>
initialization by design, it will still produce comparable results than
<code class="docutils literal notranslate"><span class="pre">KMeans(init="k-means++")</span></code> in terms of inertia at cheaper computational costs, and will
likely produce better results than <code class="docutils literal notranslate"><span class="pre">KMeans</span></code> with a random initialization.</p>
<p>This variant is more efficient to agglomerative clustering if the number of clusters is
small compared to the number of data points.</p>
<p>This variant also does not produce empty clusters.</p>
<dl class="simple">
<dt>There exist two strategies for selecting the cluster to split:</dt><dd><ul class="simple">
<li><p><code class="docutils literal notranslate"><span class="pre">bisecting_strategy="largest_cluster"</span></code> selects the cluster having the most points</p></li>
<li><p><code class="docutils literal notranslate"><span class="pre">bisecting_strategy="biggest_inertia"</span></code> selects the cluster with biggest inertia
(cluster with biggest Sum of Squared Errors within)</p></li>
</ul>
</dd>
</dl>
<p>Picking by largest amount of data points in most cases produces result as
accurate as picking by inertia and is faster (especially for larger amount of data
points, where calculating error may be costly).</p>
<p>Picking by largest amount of data points will also likely produce clusters of similar
sizes while <code class="docutils literal notranslate"><span class="pre">KMeans</span></code> is known to produce clusters of different sizes.</p>
<p>Difference between Bisecting K-Means and regular K-Means can be seen on example
<a class="reference internal" href="../auto_examples/cluster/plot_bisect_kmeans.html#sphx-glr-auto-examples-cluster-plot-bisect-kmeans-py"><span class="std std-ref">Bisecting K-Means and Regular K-Means Performance Comparison</span></a>.
While the regular K-Means algorithm tends to create non-related clusters,
clusters from Bisecting K-Means are well ordered and create quite a visible hierarchy.</p>
<p><details id="summary-anchor">
<summary class="btn btn-light">
<strong>References</strong>