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<!DOCTYPE html>
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<title>6. Two-point Correlation Functions — astroML 1.0 documentation</title>
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1. Introduction
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2. Installation of astroML
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3. Data Sets
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4. Unsupervised Learning: Density Estimation
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5. Unsupervised Learning: Clustering
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6. Two-point Correlation Functions
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7. Unsupervised Learning: Dimensionality Reduction
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8. Supervised Learning: Regression
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9. Supervised Learning: Classification
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10. Time Series Analysis
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11. Class reference
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<li class="toctree-l2">
<a class="reference internal" href="../modules/generated/astroML.plotting.hist.html">
11.1.1.1. astroML.plotting.hist
</a>
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<li class="toctree-l2">
<a class="reference internal" href="../modules/generated/astroML.plotting.scatter_contour.html">
11.1.1.2. astroML.plotting.scatter_contour
</a>
</li>
<li class="toctree-l2">
<a class="reference internal" href="../modules/generated/astroML.plotting.MultiAxes.html">
11.1.2.1. astroML.plotting.MultiAxes
</a>
</li>
<li class="toctree-l2">
<a class="reference internal" href="../modules/generated/astroML.density_estimation.histogram.html">
11.2.1.1. astroML.density_estimation.histogram
</a>
</li>
<li class="toctree-l2">
<a class="reference internal" href="../modules/generated/astroML.density_estimation.bayesian_blocks.html">
11.2.1.2. astroML.density_estimation.bayesian_blocks
</a>
</li>
<li class="toctree-l2">
<a class="reference internal" href="../modules/generated/astroML.density_estimation.knuth_bin_width.html">
11.2.1.3. astroML.density_estimation.knuth_bin_width
</a>
</li>
<li class="toctree-l2">
<a class="reference internal" href="../modules/generated/astroML.density_estimation.scotts_bin_width.html">
11.2.1.4. astroML.density_estimation.scotts_bin_width
</a>
</li>
<li class="toctree-l2">
<a class="reference internal" href="../modules/generated/astroML.density_estimation.freedman_bin_width.html">
11.2.1.5. astroML.density_estimation.freedman_bin_width
</a>
</li>
<li class="toctree-l2">
<a class="reference internal" href="../modules/generated/astroML.density_estimation.XDGMM.html">
11.2.2.1. astroML.density_estimation.XDGMM
</a>
</li>
<li class="toctree-l2">
<a class="reference internal" href="../modules/generated/astroML.density_estimation.KNeighborsDensity.html">
11.2.2.2. astroML.density_estimation.KNeighborsDensity
</a>
</li>
<li class="toctree-l2">
<a class="reference internal" href="../modules/generated/astroML.density_estimation.EmpiricalDistribution.html">
11.2.2.3. astroML.density_estimation.EmpiricalDistribution
</a>
</li>
<li class="toctree-l2">
<a class="reference internal" href="../modules/generated/astroML.density_estimation.FunctionDistribution.html">
11.2.2.4. astroML.density_estimation.FunctionDistribution
</a>
</li>
<li class="toctree-l2">
<a class="reference internal" href="../modules/generated/astroML.linear_model.LinearRegression.html">
11.3.1.1. astroML.linear_model.LinearRegression
</a>
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<a class="reference internal" href="../modules/generated/astroML.linear_model.PolynomialRegression.html">
11.3.1.2. astroML.linear_model.PolynomialRegression
</a>
</li>
<li class="toctree-l2">
<a class="reference internal" href="../modules/generated/astroML.linear_model.BasisFunctionRegression.html">
11.3.1.3. astroML.linear_model.BasisFunctionRegression
</a>
</li>
<li class="toctree-l2">
<a class="reference internal" href="../modules/generated/astroML.linear_model.NadarayaWatson.html">
11.3.1.4. astroML.linear_model.NadarayaWatson
</a>
</li>
<li class="toctree-l2">
<a class="reference internal" href="../modules/generated/astroML.linear_model.TLS_logL.html">
11.3.2.1. astroML.linear_model.TLS_logL
</a>
</li>
<li class="toctree-l2">
<a class="reference internal" href="../modules/generated/astroML.datasets.fetch_sdss_spectrum.html">
11.4.2. astroML.datasets.fetch_sdss_spectrum
</a>
</li>
<li class="toctree-l2">
<a class="reference internal" href="../modules/generated/astroML.datasets.fetch_sdss_corrected_spectra.html">
11.4.3. astroML.datasets.fetch_sdss_corrected_spectra
</a>
</li>
<li class="toctree-l2">
<a class="reference internal" href="../modules/generated/astroML.datasets.fetch_sdss_S82standards.html">
11.4.4. astroML.datasets.fetch_sdss_S82standards
</a>
</li>
<li class="toctree-l2">
<a class="reference internal" href="../modules/generated/astroML.datasets.fetch_dr7_quasar.html">
11.4.5. astroML.datasets.fetch_dr7_quasar
</a>
</li>
<li class="toctree-l2">
<a class="reference internal" href="../modules/generated/astroML.datasets.fetch_moving_objects.html">
11.4.6. astroML.datasets.fetch_moving_objects
</a>
</li>
<li class="toctree-l2">
<a class="reference internal" href="../modules/generated/astroML.datasets.fetch_sdss_galaxy_colors.html">
11.4.7. astroML.datasets.fetch_sdss_galaxy_colors
</a>
</li>
<li class="toctree-l2">
<a class="reference internal" href="../modules/generated/astroML.datasets.fetch_nasa_atlas.html">
11.4.8. astroML.datasets.fetch_nasa_atlas
</a>
</li>
<li class="toctree-l2">
<a class="reference internal" href="../modules/generated/astroML.datasets.fetch_sdss_sspp.html">
11.4.9. astroML.datasets.fetch_sdss_sspp
</a>
</li>
<li class="toctree-l2">
<a class="reference internal" href="../modules/generated/astroML.datasets.fetch_sdss_specgals.html">
11.4.10. astroML.datasets.fetch_sdss_specgals
</a>
</li>
<li class="toctree-l2">
<a class="reference internal" href="../modules/generated/astroML.datasets.fetch_great_wall.html">
11.4.11. astroML.datasets.fetch_great_wall
</a>
</li>
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<a class="reference internal" href="../modules/generated/astroML.datasets.fetch_imaging_sample.html">
11.4.12. astroML.datasets.fetch_imaging_sample
</a>
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<a class="reference internal" href="../modules/generated/astroML.datasets.fetch_wmap_temperatures.html">
11.4.13. astroML.datasets.fetch_wmap_temperatures
</a>
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<a class="reference internal" href="../modules/generated/astroML.datasets.fetch_rrlyrae_mags.html">
11.4.14. astroML.datasets.fetch_rrlyrae_mags
</a>
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<a class="reference internal" href="../modules/generated/astroML.datasets.fetch_rrlyrae_combined.html">
11.4.15. astroML.datasets.fetch_rrlyrae_combined
</a>
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<a class="reference internal" href="../modules/generated/astroML.datasets.fetch_LINEAR_sample.html">
11.4.16. astroML.datasets.fetch_LINEAR_sample
</a>
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<a class="reference internal" href="../modules/generated/astroML.datasets.fetch_LINEAR_geneva.html">
11.4.17. astroML.datasets.fetch_LINEAR_geneva
</a>
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11.4.18. astroML.datasets.fetch_LIGO_bigdog
</a>
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11.4.19. astroML.datasets.fetch_LIGO_large
</a>
</li>
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<a class="reference internal" href="../modules/generated/astroML.datasets.fetch_hogg2010test.html">
11.4.20. astroML.datasets.fetch_hogg2010test
</a>
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11.4.21. astroML.datasets.fetch_rrlyrae_templates
</a>
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11.4.22. astroML.datasets.fetch_sdss_filter
</a>
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11.4.23. astroML.datasets.fetch_vega_spectrum
</a>
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<a class="reference internal" href="../modules/generated/astroML.datasets.generate_mu_z.html">
11.4.24. astroML.datasets.generate_mu_z
</a>
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11.5.1.1. astroML.time_series.lomb_scargle
</a>
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11.5.1.2. astroML.time_series.lomb_scargle_bootstrap
</a>
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<a class="reference internal" href="../modules/generated/astroML.time_series.multiterm_periodogram.html">
11.5.1.3. astroML.time_series.multiterm_periodogram
</a>
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<a class="reference internal" href="../modules/generated/astroML.time_series.search_frequencies.html">
11.5.1.4. astroML.time_series.search_frequencies
</a>
</li>
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<a class="reference internal" href="../modules/generated/astroML.time_series.MultiTermFit.html">
11.5.1.5. astroML.time_series.MultiTermFit
</a>
</li>
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<a class="reference internal" href="../modules/generated/astroML.time_series.ACF_scargle.html">
11.5.2.1. astroML.time_series.ACF_scargle
</a>
</li>
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<a class="reference internal" href="../modules/generated/astroML.time_series.ACF_EK.html">
11.5.2.2. astroML.time_series.ACF_EK
</a>
</li>
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<a class="reference internal" href="../modules/generated/astroML.time_series.generate_power_law.html">
11.5.2.3. astroML.time_series.generate_power_law
</a>
</li>
<li class="toctree-l2">
<a class="reference internal" href="../modules/generated/astroML.time_series.generate_damped_RW.html">
11.5.2.4. astroML.time_series.generate_damped_RW
</a>
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11.6.1. astroML.stats.binned_statistic
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11.6.4. astroML.stats.sigmaG
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11.6.6. astroML.stats.mean_sigma
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11.6.7. astroML.stats.fit_bivariate_normal
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11.6.8. astroML.stats.bivariate_normal
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11.6.9. astroML.stats.trunc_exp
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11.6.10. astroML.stats.linear
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11.8.1. astroML.correlation.two_point
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11.8.4. astroML.correlation.bootstrap_two_point_angular
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<div class="section" id="two-point-correlation-functions">
<span id="astroml-correlation-functions"></span><h1><span class="section-number">6. </span>Two-point Correlation Functions<a class="headerlink" href="#two-point-correlation-functions" title="Permalink to this headline">¶</a></h1>
<p>The N-point correlation function is a common technique used in astronomy
to extract useful information from multi-dimensional datasets. This is
especially true for spatial distributions of galaxies, because the theoretical
description of the process of galaxy formation directly predicts the
two-point correlation function.</p>
<p>The computation of the two-point correlation is a <em>generalized N-body problem</em>
similar to Nearest Neighbor searches and Kernel Density estimation
(see <a class="reference internal" href="density_estimation.html#astroml-density-estimation"><span class="std std-ref">Unsupervised Learning: Density Estimation</span></a>)</p>
<p>AstroML implements a fast correlation function estimator based on the
scikit-learn <a class="reference external" href="http://scikit-learn.org/stable/modules/neighbors.html#kdtree-and-balltree-classes">BallTree</a> and <a class="reference external" href="http://scikit-learn.org/stable/modules/neighbors.html#kdtree-and-balltree-classes">KDTree</a> data structures.
Here is an example of computing the correlation function (with bootstrap
estimates) over the SDSS spectroscopic galaxies within the redshift range
0.08 < z < 0.12:</p>
<a class="reference external image-reference" href="../book_figures/chapter6/fig_correlation_function.html"><img alt="book_figures/images/chapter6/fig_correlation_function_1.png" class="align-center" src="book_figures/images/chapter6/fig_correlation_function_1.png" /></a>
<p>The correlation function interface is very straightforward. We’ll construct
some random data in two dimensions and compute the two-point correlation here:</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="gp">>>> </span><span class="kn">from</span> <span class="nn">astroML.correlation</span> <span class="kn">import</span> <span class="n">two_point</span>
<span class="gp">>>> </span><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">seed</span><span class="p">(</span><span class="mi">0</span><span class="p">)</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">random</span><span class="o">.</span><span class="n">random</span><span class="p">((</span><span class="mi">5000</span><span class="p">,</span> <span class="mi">2</span><span class="p">))</span>
<span class="gp">>>> </span><span class="n">bins</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">20</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">corr</span> <span class="o">=</span> <span class="n">two_point</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">bins</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">np</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">corr</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">atol</span><span class="o">=</span><span class="mf">0.02</span><span class="p">)</span>
<span class="go">True</span>
</pre></div>
</div>
<p>For uniformly-distributed data, the correlation function is zero (that is,
there is no excess over a uniformly-distributed background). If we wish to
find the error on the result, this can be done via a bootstrap approach:</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="gp">>>> </span><span class="kn">from</span> <span class="nn">astroML.correlation</span> <span class="kn">import</span> <span class="n">bootstrap_two_point</span>
<span class="gp">>>> </span><span class="n">corr</span><span class="p">,</span> <span class="n">dcorr</span> <span class="o">=</span> <span class="n">bootstrap_two_point</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">bins</span><span class="p">,</span> <span class="n">Nbootstrap</span><span class="o">=</span><span class="mi">5</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">np</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">corr</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="n">atol</span><span class="o">=</span><span class="mi">2</span> <span class="o">*</span> <span class="n">dcorr</span><span class="p">)</span>
<span class="go">True</span>
</pre></div>
</div>
<p>These are contrived examples, but they show how easy the computation can be.
For a more realistic example, see the source code associated with the
above figure.</p>
</div>
</div>
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<a class='left-prev' id="prev-link" href="clustering.html" title="previous page"><span class="section-number">5. </span>Unsupervised Learning: Clustering</a>
<a class='right-next' id="next-link" href="dimensionality.html" title="next page"><span class="section-number">7. </span>Unsupervised Learning: Dimensionality Reduction</a>
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