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<meta property="og:description" content="sklearn.neighbors provides functionality for unsupervised and supervised neighbors-based learning methods. Unsupervised nearest neighbors is the foundation of many other learning methods, notably m..." />
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<meta name="description" content="sklearn.neighbors provides functionality for unsupervised and supervised neighbors-based learning methods. Unsupervised nearest neighbors is the foundation of many other learning methods, notably m..." />
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<section id="nearest-neighbors">
<span id="neighbors"></span><h1><span class="section-number">1.6. </span>Nearest Neighbors<a class="headerlink" href="#nearest-neighbors" title="Link to this heading">#</a></h1>
<p><a class="reference internal" href="../api/sklearn.neighbors.html#module-sklearn.neighbors" title="sklearn.neighbors"><code class="xref py py-mod docutils literal notranslate"><span class="pre">sklearn.neighbors</span></code></a> provides functionality for unsupervised and
supervised neighbors-based learning methods. Unsupervised nearest neighbors
is the foundation of many other learning methods,
notably manifold learning and spectral clustering. Supervised neighbors-based
learning comes in two flavors: <a class="reference internal" href="#classification">classification</a> for data with
discrete labels, and <a class="reference internal" href="#regression">regression</a> for data with continuous labels.</p>
<p>The principle behind nearest neighbor methods is to find a predefined number
of training samples closest in distance to the new point, and
predict the label from these. The number of samples can be a user-defined
constant (k-nearest neighbor learning), or vary based
on the local density of points (radius-based neighbor learning).
The distance can, in general, be any metric measure: standard Euclidean
distance is the most common choice.
Neighbors-based methods are known as <em>non-generalizing</em> machine
learning methods, since they simply “remember” all of its training data
(possibly transformed into a fast indexing structure such as a
<a class="reference internal" href="#ball-tree"><span class="std std-ref">Ball Tree</span></a> or <a class="reference internal" href="#kd-tree"><span class="std std-ref">KD Tree</span></a>).</p>
<p>Despite its simplicity, nearest neighbors has been successful in a
large number of classification and regression problems, including
handwritten digits and satellite image scenes. Being a non-parametric method,
it is often successful in classification situations where the decision
boundary is very irregular.</p>
<p>The classes in <a class="reference internal" href="../api/sklearn.neighbors.html#module-sklearn.neighbors" title="sklearn.neighbors"><code class="xref py py-mod docutils literal notranslate"><span class="pre">sklearn.neighbors</span></code></a> can handle either NumPy arrays or
<code class="docutils literal notranslate"><span class="pre">scipy.sparse</span></code> matrices as input. For dense matrices, a large number of
possible distance metrics are supported. For sparse matrices, arbitrary
Minkowski metrics are supported for searches.</p>
<p>There are many learning routines which rely on nearest neighbors at their
core. One example is <a class="reference internal" href="density.html#kernel-density"><span class="std std-ref">kernel density estimation</span></a>,
discussed in the <a class="reference internal" href="density.html#density-estimation"><span class="std std-ref">density estimation</span></a> section.</p>
<section id="unsupervised-nearest-neighbors">
<span id="unsupervised-neighbors"></span><h2><span class="section-number">1.6.1. </span>Unsupervised Nearest Neighbors<a class="headerlink" href="#unsupervised-nearest-neighbors" title="Link to this heading">#</a></h2>
<p><a class="reference internal" href="generated/sklearn.neighbors.NearestNeighbors.html#sklearn.neighbors.NearestNeighbors" title="sklearn.neighbors.NearestNeighbors"><code class="xref py py-class docutils literal notranslate"><span class="pre">NearestNeighbors</span></code></a> implements unsupervised nearest neighbors learning.
It acts as a uniform interface to three different nearest neighbors
algorithms: <a class="reference internal" href="generated/sklearn.neighbors.BallTree.html#sklearn.neighbors.BallTree" title="sklearn.neighbors.BallTree"><code class="xref py py-class docutils literal notranslate"><span class="pre">BallTree</span></code></a>, <a class="reference internal" href="generated/sklearn.neighbors.KDTree.html#sklearn.neighbors.KDTree" title="sklearn.neighbors.KDTree"><code class="xref py py-class docutils literal notranslate"><span class="pre">KDTree</span></code></a>, and a
brute-force algorithm based on routines in <a class="reference internal" href="../api/sklearn.metrics.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>.
The choice of neighbors search algorithm is controlled through the keyword
<code class="docutils literal notranslate"><span class="pre">'algorithm'</span></code>, which must be one of
<code class="docutils literal notranslate"><span class="pre">['auto',</span> <span class="pre">'ball_tree',</span> <span class="pre">'kd_tree',</span> <span class="pre">'brute']</span></code>. When the default value
<code class="docutils literal notranslate"><span class="pre">'auto'</span></code> is passed, the algorithm attempts to determine the best approach
from the training data. For a discussion of the strengths and weaknesses
of each option, see <a class="reference internal" href="#nearest-neighbor-algorithms">Nearest Neighbor Algorithms</a>.</p>
<div class="admonition warning">
<p class="admonition-title">Warning</p>
<p>Regarding the Nearest Neighbors algorithms, if two
neighbors <span class="math notranslate nohighlight">\(k+1\)</span> and <span class="math notranslate nohighlight">\(k\)</span> have identical distances
but different labels, the result will depend on the ordering of the
training data.</p>
</div>
<section id="finding-the-nearest-neighbors">
<h3><span class="section-number">1.6.1.1. </span>Finding the Nearest Neighbors<a class="headerlink" href="#finding-the-nearest-neighbors" title="Link to this heading">#</a></h3>
<p>For the simple task of finding the nearest neighbors between two sets of
data, the unsupervised algorithms within <a class="reference internal" href="../api/sklearn.neighbors.html#module-sklearn.neighbors" title="sklearn.neighbors"><code class="xref py py-mod docutils literal notranslate"><span class="pre">sklearn.neighbors</span></code></a> can be
used:</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span><span class="w"> </span><span class="nn">sklearn.neighbors</span><span class="w"> </span><span class="kn">import</span> <span class="n">NearestNeighbors</span>
<span class="gp">>>> </span><span class="kn">import</span><span class="w"> </span><span class="nn">numpy</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">np</span>
<span class="gp">>>> </span><span class="n">X</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</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="o">-</span><span class="mi">2</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="o">-</span><span class="mi">3</span><span class="p">,</span> <span class="o">-</span><span class="mi">2</span><span class="p">],</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="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">]])</span>
<span class="gp">>>> </span><span class="n">nbrs</span> <span class="o">=</span> <span class="n">NearestNeighbors</span><span class="p">(</span><span class="n">n_neighbors</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">algorithm</span><span class="o">=</span><span class="s1">'ball_tree'</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="gp">>>> </span><span class="n">distances</span><span class="p">,</span> <span class="n">indices</span> <span class="o">=</span> <span class="n">nbrs</span><span class="o">.</span><span class="n">kneighbors</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">indices</span>
<span class="go">array([[0, 1],</span>
<span class="go"> [1, 0],</span>
<span class="go"> [2, 1],</span>
<span class="go"> [3, 4],</span>
<span class="go"> [4, 3],</span>
<span class="go"> [5, 4]]...)</span>
<span class="gp">>>> </span><span class="n">distances</span>
<span class="go">array([[0. , 1. ],</span>
<span class="go"> [0. , 1. ],</span>
<span class="go"> [0. , 1.41421356],</span>
<span class="go"> [0. , 1. ],</span>
<span class="go"> [0. , 1. ],</span>
<span class="go"> [0. , 1.41421356]])</span>
</pre></div>
</div>
<p>Because the query set matches the training set, the nearest neighbor of each
point is the point itself, at a distance of zero.</p>
<p>It is also possible to efficiently produce a sparse graph showing the
connections between neighboring points:</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">nbrs</span><span class="o">.</span><span class="n">kneighbors_graph</span><span class="p">(</span><span class="n">X</span><span class="p">)</span><span class="o">.</span><span class="n">toarray</span><span class="p">()</span>
<span class="go">array([[1., 1., 0., 0., 0., 0.],</span>
<span class="go"> [1., 1., 0., 0., 0., 0.],</span>
<span class="go"> [0., 1., 1., 0., 0., 0.],</span>
<span class="go"> [0., 0., 0., 1., 1., 0.],</span>
<span class="go"> [0., 0., 0., 1., 1., 0.],</span>
<span class="go"> [0., 0., 0., 0., 1., 1.]])</span>
</pre></div>
</div>
<p>The dataset is structured such that points nearby in index order are nearby
in parameter space, leading to an approximately block-diagonal matrix of
K-nearest neighbors. Such a sparse graph is useful in a variety of
circumstances which make use of spatial relationships between points for
unsupervised learning: in particular, see <a class="reference internal" href="generated/sklearn.manifold.Isomap.html#sklearn.manifold.Isomap" title="sklearn.manifold.Isomap"><code class="xref py py-class docutils literal notranslate"><span class="pre">Isomap</span></code></a>,
<a class="reference internal" href="generated/sklearn.manifold.LocallyLinearEmbedding.html#sklearn.manifold.LocallyLinearEmbedding" title="sklearn.manifold.LocallyLinearEmbedding"><code class="xref py py-class docutils literal notranslate"><span class="pre">LocallyLinearEmbedding</span></code></a>, and
<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>.</p>
</section>
<section id="kdtree-and-balltree-classes">
<h3><span class="section-number">1.6.1.2. </span>KDTree and BallTree Classes<a class="headerlink" href="#kdtree-and-balltree-classes" title="Link to this heading">#</a></h3>
<p>Alternatively, one can use the <a class="reference internal" href="generated/sklearn.neighbors.KDTree.html#sklearn.neighbors.KDTree" title="sklearn.neighbors.KDTree"><code class="xref py py-class docutils literal notranslate"><span class="pre">KDTree</span></code></a> or <a class="reference internal" href="generated/sklearn.neighbors.BallTree.html#sklearn.neighbors.BallTree" title="sklearn.neighbors.BallTree"><code class="xref py py-class docutils literal notranslate"><span class="pre">BallTree</span></code></a> classes
directly to find nearest neighbors. This is the functionality wrapped by
the <a class="reference internal" href="generated/sklearn.neighbors.NearestNeighbors.html#sklearn.neighbors.NearestNeighbors" title="sklearn.neighbors.NearestNeighbors"><code class="xref py py-class docutils literal notranslate"><span class="pre">NearestNeighbors</span></code></a> class used above. The Ball Tree and KD Tree
have the same interface; we’ll show an example of using the KD Tree here:</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span><span class="w"> </span><span class="nn">sklearn.neighbors</span><span class="w"> </span><span class="kn">import</span> <span class="n">KDTree</span>
<span class="gp">>>> </span><span class="kn">import</span><span class="w"> </span><span class="nn">numpy</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">np</span>
<span class="gp">>>> </span><span class="n">X</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</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="o">-</span><span class="mi">2</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="o">-</span><span class="mi">3</span><span class="p">,</span> <span class="o">-</span><span class="mi">2</span><span class="p">],</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="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">]])</span>
<span class="gp">>>> </span><span class="n">kdt</span> <span class="o">=</span> <span class="n">KDTree</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">leaf_size</span><span class="o">=</span><span class="mi">30</span><span class="p">,</span> <span class="n">metric</span><span class="o">=</span><span class="s1">'euclidean'</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">kdt</span><span class="o">.</span><span class="n">query</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">k</span><span class="o">=</span><span class="mi">2</span><span class="p">,</span> <span class="n">return_distance</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
<span class="go">array([[0, 1],</span>
<span class="go"> [1, 0],</span>
<span class="go"> [2, 1],</span>
<span class="go"> [3, 4],</span>
<span class="go"> [4, 3],</span>
<span class="go"> [5, 4]]...)</span>
</pre></div>
</div>
<p>Refer to the <a class="reference internal" href="generated/sklearn.neighbors.KDTree.html#sklearn.neighbors.KDTree" title="sklearn.neighbors.KDTree"><code class="xref py py-class docutils literal notranslate"><span class="pre">KDTree</span></code></a> and <a class="reference internal" href="generated/sklearn.neighbors.BallTree.html#sklearn.neighbors.BallTree" title="sklearn.neighbors.BallTree"><code class="xref py py-class docutils literal notranslate"><span class="pre">BallTree</span></code></a> class documentation
for more information on the options available for nearest neighbors searches,
including specification of query strategies, distance metrics, etc. For a list
of valid metrics use <code class="docutils literal notranslate"><span class="pre">KDTree.valid_metrics</span></code> and <code class="docutils literal notranslate"><span class="pre">BallTree.valid_metrics</span></code>:</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">from</span><span class="w"> </span><span class="nn">sklearn.neighbors</span><span class="w"> </span><span class="kn">import</span> <span class="n">KDTree</span><span class="p">,</span> <span class="n">BallTree</span>
<span class="gp">>>> </span><span class="n">KDTree</span><span class="o">.</span><span class="n">valid_metrics</span>
<span class="go">['euclidean', 'l2', 'minkowski', 'p', 'manhattan', 'cityblock', 'l1', 'chebyshev', 'infinity']</span>
<span class="gp">>>> </span><span class="n">BallTree</span><span class="o">.</span><span class="n">valid_metrics</span>
<span class="go">['euclidean', 'l2', 'minkowski', 'p', 'manhattan', 'cityblock', 'l1', 'chebyshev', 'infinity', 'seuclidean', 'mahalanobis', 'hamming', 'canberra', 'braycurtis', 'jaccard', 'dice', 'rogerstanimoto', 'russellrao', 'sokalmichener', 'sokalsneath', 'haversine', 'pyfunc']</span>
</pre></div>
</div>
</section>
</section>
<section id="nearest-neighbors-classification">
<span id="classification"></span><h2><span class="section-number">1.6.2. </span>Nearest Neighbors Classification<a class="headerlink" href="#nearest-neighbors-classification" title="Link to this heading">#</a></h2>
<p>Neighbors-based classification is a type of <em>instance-based learning</em> or
<em>non-generalizing learning</em>: it does not attempt to construct a general
internal model, but simply stores instances of the training data.
Classification is computed from a simple majority vote of the nearest
neighbors of each point: a query point is assigned the data class which
has the most representatives within the nearest neighbors of the point.</p>
<p>scikit-learn implements two different nearest neighbors classifiers:
<a class="reference internal" href="generated/sklearn.neighbors.KNeighborsClassifier.html#sklearn.neighbors.KNeighborsClassifier" title="sklearn.neighbors.KNeighborsClassifier"><code class="xref py py-class docutils literal notranslate"><span class="pre">KNeighborsClassifier</span></code></a> implements learning based on the <span class="math notranslate nohighlight">\(k\)</span>
nearest neighbors of each query point, where <span class="math notranslate nohighlight">\(k\)</span> is an integer value
specified by the user. <a class="reference internal" href="generated/sklearn.neighbors.RadiusNeighborsClassifier.html#sklearn.neighbors.RadiusNeighborsClassifier" title="sklearn.neighbors.RadiusNeighborsClassifier"><code class="xref py py-class docutils literal notranslate"><span class="pre">RadiusNeighborsClassifier</span></code></a> implements learning
based on the number of neighbors within a fixed radius <span class="math notranslate nohighlight">\(r\)</span> of each
training point, where <span class="math notranslate nohighlight">\(r\)</span> is a floating-point value specified by
the user.</p>
<p>The <span class="math notranslate nohighlight">\(k\)</span>-neighbors classification in <a class="reference internal" href="generated/sklearn.neighbors.KNeighborsClassifier.html#sklearn.neighbors.KNeighborsClassifier" title="sklearn.neighbors.KNeighborsClassifier"><code class="xref py py-class docutils literal notranslate"><span class="pre">KNeighborsClassifier</span></code></a>
is the most commonly used technique. The optimal choice of the value <span class="math notranslate nohighlight">\(k\)</span>
is highly data-dependent: in general a larger <span class="math notranslate nohighlight">\(k\)</span> suppresses the effects
of noise, but makes the classification boundaries less distinct.</p>
<p>In cases where the data is not uniformly sampled, radius-based neighbors
classification in <a class="reference internal" href="generated/sklearn.neighbors.RadiusNeighborsClassifier.html#sklearn.neighbors.RadiusNeighborsClassifier" title="sklearn.neighbors.RadiusNeighborsClassifier"><code class="xref py py-class docutils literal notranslate"><span class="pre">RadiusNeighborsClassifier</span></code></a> can be a better choice.
The user specifies a fixed radius <span class="math notranslate nohighlight">\(r\)</span>, such that points in sparser
neighborhoods use fewer nearest neighbors for the classification. For
high-dimensional parameter spaces, this method becomes less effective due
to the so-called “curse of dimensionality”.</p>
<p>The basic nearest neighbors classification uses uniform weights: that is, the
value assigned to a query point is computed from a simple majority vote of
the nearest neighbors. Under some circumstances, it is better to weight the
neighbors such that nearer neighbors contribute more to the fit. This can
be accomplished through the <code class="docutils literal notranslate"><span class="pre">weights</span></code> keyword. The default value,
<code class="docutils literal notranslate"><span class="pre">weights</span> <span class="pre">=</span> <span class="pre">'uniform'</span></code>, assigns uniform weights to each neighbor.
<code class="docutils literal notranslate"><span class="pre">weights</span> <span class="pre">=</span> <span class="pre">'distance'</span></code> assigns weights proportional to the inverse of the
distance from the query point. Alternatively, a user-defined function of the
distance can be supplied to compute the weights.</p>
<p class="centered">
<strong><a class="reference external" href="../auto_examples/neighbors/plot_classification.html"><img alt="classification_1" src="../_images/sphx_glr_plot_classification_001.png" style="width: 900.0px; height: 375.0px;" /></a></strong></p><p class="rubric">Examples</p>
<ul class="simple">
<li><p><a class="reference internal" href="../auto_examples/neighbors/plot_classification.html#sphx-glr-auto-examples-neighbors-plot-classification-py"><span class="std std-ref">Nearest Neighbors Classification</span></a>: an example of
classification using nearest neighbors.</p></li>
</ul>
</section>
<section id="nearest-neighbors-regression">
<span id="regression"></span><h2><span class="section-number">1.6.3. </span>Nearest Neighbors Regression<a class="headerlink" href="#nearest-neighbors-regression" title="Link to this heading">#</a></h2>
<p>Neighbors-based regression can be used in cases where the data labels are
continuous rather than discrete variables. The label assigned to a query
point is computed based on the mean of the labels of its nearest neighbors.</p>
<p>scikit-learn implements two different neighbors regressors:
<a class="reference internal" href="generated/sklearn.neighbors.KNeighborsRegressor.html#sklearn.neighbors.KNeighborsRegressor" title="sklearn.neighbors.KNeighborsRegressor"><code class="xref py py-class docutils literal notranslate"><span class="pre">KNeighborsRegressor</span></code></a> implements learning based on the <span class="math notranslate nohighlight">\(k\)</span>
nearest neighbors of each query point, where <span class="math notranslate nohighlight">\(k\)</span> is an integer
value specified by the user. <a class="reference internal" href="generated/sklearn.neighbors.RadiusNeighborsRegressor.html#sklearn.neighbors.RadiusNeighborsRegressor" title="sklearn.neighbors.RadiusNeighborsRegressor"><code class="xref py py-class docutils literal notranslate"><span class="pre">RadiusNeighborsRegressor</span></code></a> implements
learning based on the neighbors within a fixed radius <span class="math notranslate nohighlight">\(r\)</span> of the
query point, where <span class="math notranslate nohighlight">\(r\)</span> is a floating-point value specified by the
user.</p>
<p>The basic nearest neighbors regression uses uniform weights: that is,
each point in the local neighborhood contributes uniformly to the
classification of a query point. Under some circumstances, it can be
advantageous to weight points such that nearby points contribute more
to the regression than faraway points. This can be accomplished through
the <code class="docutils literal notranslate"><span class="pre">weights</span></code> keyword. The default value, <code class="docutils literal notranslate"><span class="pre">weights</span> <span class="pre">=</span> <span class="pre">'uniform'</span></code>,
assigns equal weights to all points. <code class="docutils literal notranslate"><span class="pre">weights</span> <span class="pre">=</span> <span class="pre">'distance'</span></code> assigns
weights proportional to the inverse of the distance from the query point.
Alternatively, a user-defined function of the distance can be supplied,
which will be used to compute the weights.</p>
<figure class="align-center">
<a class="reference external image-reference" href="../auto_examples/neighbors/plot_regression.html"><img alt="../_images/sphx_glr_plot_regression_001.png" src="../_images/sphx_glr_plot_regression_001.png" style="width: 480.0px; height: 360.0px;" />
</a>
</figure>
<p>The use of multi-output nearest neighbors for regression is demonstrated in
<a class="reference internal" href="../auto_examples/miscellaneous/plot_multioutput_face_completion.html#sphx-glr-auto-examples-miscellaneous-plot-multioutput-face-completion-py"><span class="std std-ref">Face completion with a multi-output estimators</span></a>. In this example, the inputs
X are the pixels of the upper half of faces and the outputs Y are the pixels of
the lower half of those faces.</p>
<figure class="align-center">
<a class="reference external image-reference" href="../auto_examples/miscellaneous/plot_multioutput_face_completion.html"><img alt="../_images/sphx_glr_plot_multioutput_face_completion_001.png" src="../_images/sphx_glr_plot_multioutput_face_completion_001.png" style="width: 750.0px; height: 847.5px;" />
</a>
</figure>
<p class="rubric">Examples</p>
<ul class="simple">
<li><p><a class="reference internal" href="../auto_examples/neighbors/plot_regression.html#sphx-glr-auto-examples-neighbors-plot-regression-py"><span class="std std-ref">Nearest Neighbors regression</span></a>: an example of regression
using nearest neighbors.</p></li>
<li><p><a class="reference internal" href="../auto_examples/miscellaneous/plot_multioutput_face_completion.html#sphx-glr-auto-examples-miscellaneous-plot-multioutput-face-completion-py"><span class="std std-ref">Face completion with a multi-output estimators</span></a>:
an example of multi-output regression using nearest neighbors.</p></li>
</ul>
</section>
<section id="nearest-neighbor-algorithms">
<h2><span class="section-number">1.6.4. </span>Nearest Neighbor Algorithms<a class="headerlink" href="#nearest-neighbor-algorithms" title="Link to this heading">#</a></h2>
<section id="brute-force">
<span id="id1"></span><h3><span class="section-number">1.6.4.1. </span>Brute Force<a class="headerlink" href="#brute-force" title="Link to this heading">#</a></h3>
<p>Fast computation of nearest neighbors is an active area of research in
machine learning. The most naive neighbor search implementation involves
the brute-force computation of distances between all pairs of points in the
dataset: for <span class="math notranslate nohighlight">\(N\)</span> samples in <span class="math notranslate nohighlight">\(D\)</span> dimensions, this approach scales
as <span class="math notranslate nohighlight">\(O[D N^2]\)</span>. Efficient brute-force neighbors searches can be very
competitive for small data samples.
However, as the number of samples <span class="math notranslate nohighlight">\(N\)</span> grows, the brute-force
approach quickly becomes infeasible. In the classes within
<a class="reference internal" href="../api/sklearn.neighbors.html#module-sklearn.neighbors" title="sklearn.neighbors"><code class="xref py py-mod docutils literal notranslate"><span class="pre">sklearn.neighbors</span></code></a>, brute-force neighbors searches are specified
using the keyword <code class="docutils literal notranslate"><span class="pre">algorithm</span> <span class="pre">=</span> <span class="pre">'brute'</span></code>, and are computed using the
routines available in <a class="reference internal" href="../api/sklearn.metrics.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>.</p>
</section>
<section id="k-d-tree">
<span id="kd-tree"></span><h3><span class="section-number">1.6.4.2. </span>K-D Tree<a class="headerlink" href="#k-d-tree" title="Link to this heading">#</a></h3>
<p>To address the computational inefficiencies of the brute-force approach, a
variety of tree-based data structures have been invented. In general, these
structures attempt to reduce the required number of distance calculations
by efficiently encoding aggregate distance information for the sample.
The basic idea is that if point <span class="math notranslate nohighlight">\(A\)</span> is very distant from point
<span class="math notranslate nohighlight">\(B\)</span>, and point <span class="math notranslate nohighlight">\(B\)</span> is very close to point <span class="math notranslate nohighlight">\(C\)</span>,
then we know that points <span class="math notranslate nohighlight">\(A\)</span> and <span class="math notranslate nohighlight">\(C\)</span>
are very distant, <em>without having to explicitly calculate their distance</em>.
In this way, the computational cost of a nearest neighbors search can be
reduced to <span class="math notranslate nohighlight">\(O[D N \log(N)]\)</span> or better. This is a significant
improvement over brute-force for large <span class="math notranslate nohighlight">\(N\)</span>.</p>
<p>An early approach to taking advantage of this aggregate information was
the <em>KD tree</em> data structure (short for <em>K-dimensional tree</em>), which
generalizes two-dimensional <em>Quad-trees</em> and 3-dimensional <em>Oct-trees</em>
to an arbitrary number of dimensions. The KD tree is a binary tree
structure which recursively partitions the parameter space along the data
axes, dividing it into nested orthotropic regions into which data points
are filed. The construction of a KD tree is very fast: because partitioning
is performed only along the data axes, no <span class="math notranslate nohighlight">\(D\)</span>-dimensional distances
need to be computed. Once constructed, the nearest neighbor of a query
point can be determined with only <span class="math notranslate nohighlight">\(O[\log(N)]\)</span> distance computations.
Though the KD tree approach is very fast for low-dimensional (<span class="math notranslate nohighlight">\(D < 20\)</span>)
neighbors searches, it becomes inefficient as <span class="math notranslate nohighlight">\(D\)</span> grows very large:
this is one manifestation of the so-called “curse of dimensionality”.
In scikit-learn, KD tree neighbors searches are specified using the
keyword <code class="docutils literal notranslate"><span class="pre">algorithm</span> <span class="pre">=</span> <span class="pre">'kd_tree'</span></code>, and are computed using the class
<a class="reference internal" href="generated/sklearn.neighbors.KDTree.html#sklearn.neighbors.KDTree" title="sklearn.neighbors.KDTree"><code class="xref py py-class docutils literal notranslate"><span class="pre">KDTree</span></code></a>.</p>
<details class="sd-sphinx-override sd-dropdown sd-card sd-mb-3" id="references">
<summary class="sd-summary-title sd-card-header">
<span class="sd-summary-text">References<a class="headerlink" href="#references" title="Link to this dropdown">#</a></span><span class="sd-summary-state-marker sd-summary-chevron-right"><svg version="1.1" width="1.5em" height="1.5em" class="sd-octicon sd-octicon-chevron-right" viewBox="0 0 24 24" aria-hidden="true"><path d="M8.72 18.78a.75.75 0 0 1 0-1.06L14.44 12 8.72 6.28a.751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018l6.25 6.25a.75.75 0 0 1 0 1.06l-6.25 6.25a.75.75 0 0 1-1.06 0Z"></path></svg></span></summary><div class="sd-summary-content sd-card-body docutils">
<ul class="simple">
<li><p class="sd-card-text"><a class="reference external" href="https://dl.acm.org/citation.cfm?doid=361002.361007">“Multidimensional binary search trees used for associative searching”</a>,
Bentley, J.L., Communications of the ACM (1975)</p></li>
</ul>
</div>
</details></section>
<section id="ball-tree">
<span id="id2"></span><h3><span class="section-number">1.6.4.3. </span>Ball Tree<a class="headerlink" href="#ball-tree" title="Link to this heading">#</a></h3>
<p>To address the inefficiencies of KD Trees in higher dimensions, the <em>ball tree</em>
data structure was developed. Where KD trees partition data along
Cartesian axes, ball trees partition data in a series of nesting
hyper-spheres. This makes tree construction more costly than that of the
KD tree, but results in a data structure which can be very efficient on
highly structured data, even in very high dimensions.</p>
<p>A ball tree recursively divides the data into
nodes defined by a centroid <span class="math notranslate nohighlight">\(C\)</span> and radius <span class="math notranslate nohighlight">\(r\)</span>, such that each
point in the node lies within the hyper-sphere defined by <span class="math notranslate nohighlight">\(r\)</span> and
<span class="math notranslate nohighlight">\(C\)</span>. The number of candidate points for a neighbor search
is reduced through use of the <em>triangle inequality</em>:</p>
<div class="math notranslate nohighlight">
\[|x+y| \leq |x| + |y|\]</div>
<p>With this setup, a single distance calculation between a test point and
the centroid is sufficient to determine a lower and upper bound on the
distance to all points within the node.
Because of the spherical geometry of the ball tree nodes, it can out-perform
a <em>KD-tree</em> in high dimensions, though the actual performance is highly
dependent on the structure of the training data.
In scikit-learn, ball-tree-based
neighbors searches are specified using the keyword <code class="docutils literal notranslate"><span class="pre">algorithm</span> <span class="pre">=</span> <span class="pre">'ball_tree'</span></code>,
and are computed using the class <a class="reference internal" href="generated/sklearn.neighbors.BallTree.html#sklearn.neighbors.BallTree" title="sklearn.neighbors.BallTree"><code class="xref py py-class docutils literal notranslate"><span class="pre">BallTree</span></code></a>.
Alternatively, the user can work with the <a class="reference internal" href="generated/sklearn.neighbors.BallTree.html#sklearn.neighbors.BallTree" title="sklearn.neighbors.BallTree"><code class="xref py py-class docutils literal notranslate"><span class="pre">BallTree</span></code></a> class directly.</p>
<details class="sd-sphinx-override sd-dropdown sd-card sd-mb-3" id="references-2">
<summary class="sd-summary-title sd-card-header">
<span class="sd-summary-text">References<a class="headerlink" href="#references-2" title="Link to this dropdown">#</a></span><span class="sd-summary-state-marker sd-summary-chevron-right"><svg version="1.1" width="1.5em" height="1.5em" class="sd-octicon sd-octicon-chevron-right" viewBox="0 0 24 24" aria-hidden="true"><path d="M8.72 18.78a.75.75 0 0 1 0-1.06L14.44 12 8.72 6.28a.751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018l6.25 6.25a.75.75 0 0 1 0 1.06l-6.25 6.25a.75.75 0 0 1-1.06 0Z"></path></svg></span></summary><div class="sd-summary-content sd-card-body docutils">
<ul class="simple">
<li><p class="sd-card-text"><a class="reference external" href="https://citeseerx.ist.psu.edu/doc_view/pid/17ac002939f8e950ffb32ec4dc8e86bdd8cb5ff1">“Five Balltree Construction Algorithms”</a>,
Omohundro, S.M., International Computer Science Institute
Technical Report (1989)</p></li>
</ul>
</div>
</details><details class="sd-sphinx-override sd-dropdown sd-card sd-mb-3" id="choice-of-nearest-neighbors-algorithm">
<summary class="sd-summary-title sd-card-header">
<span class="sd-summary-text">Choice of Nearest Neighbors Algorithm<a class="headerlink" href="#choice-of-nearest-neighbors-algorithm" title="Link to this dropdown">#</a></span><span class="sd-summary-state-marker sd-summary-chevron-right"><svg version="1.1" width="1.5em" height="1.5em" class="sd-octicon sd-octicon-chevron-right" viewBox="0 0 24 24" aria-hidden="true"><path d="M8.72 18.78a.75.75 0 0 1 0-1.06L14.44 12 8.72 6.28a.751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018l6.25 6.25a.75.75 0 0 1 0 1.06l-6.25 6.25a.75.75 0 0 1-1.06 0Z"></path></svg></span></summary><div class="sd-summary-content sd-card-body docutils">
<p class="sd-card-text">The optimal algorithm for a given dataset is a complicated choice, and
depends on a number of factors:</p>
<ul>
<li><p class="sd-card-text">number of samples <span class="math notranslate nohighlight">\(N\)</span> (i.e. <code class="docutils literal notranslate"><span class="pre">n_samples</span></code>) and dimensionality
<span class="math notranslate nohighlight">\(D\)</span> (i.e. <code class="docutils literal notranslate"><span class="pre">n_features</span></code>).</p>