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2 changes: 1 addition & 1 deletion README.md
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The software package is released under the BSD 2-Clause (FreeBSD) License.

Copyright (c) by
Marion Neumann, Daniel Marthaler, Shan Huang & Kristian Kersting, 30/09/2013
Marion Neumann, Daniel Marthaler, Shan Huang & Kristian Kersting, 18/02/2014
================================================================================

pyGPs is a library containing code for Gaussian Process (GP) Regression and Classification.
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# Sphinx build info version 1
# This file hashes the configuration used when building these files. When it is not found, a full rebuild will be done.
config: ed39426012cff07da796141960224737
tags: 645f666f9bcd5a90fca523b33c5a78b7
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<script type="text/javascript">
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</div>
<div class="footer">
&copy; Copyright 2013, Marion Neumann, Shan Huang, Daniel Marthaler, Kristian Kersting.
Created using <a href="http://sphinx-doc.org/">Sphinx</a> 1.2b3.
Created using <a href="http://sphinx.pocoo.org/">Sphinx</a> 1.1.3.
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</div>
<div class="footer">
&copy; Copyright 2013, Marion Neumann, Shan Huang, Daniel Marthaler, Kristian Kersting.
Created using <a href="http://sphinx-doc.org/">Sphinx</a> 1.2b3.
Created using <a href="http://sphinx.pocoo.org/">Sphinx</a> 1.1.3.
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</div>
<div class="footer">
&copy; Copyright 2013, Marion Neumann, Shan Huang, Daniel Marthaler, Kristian Kersting.
Created using <a href="http://sphinx-doc.org/">Sphinx</a> 1.2b3.
Created using <a href="http://sphinx.pocoo.org/">Sphinx</a> 1.1.3.
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</div>
<div class="footer">
&copy; Copyright 2013, Marion Neumann, Shan Huang, Daniel Marthaler, Kristian Kersting.
Created using <a href="http://sphinx-doc.org/">Sphinx</a> 1.2b3.
Created using <a href="http://sphinx.pocoo.org/">Sphinx</a> 1.1.3.
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</div>
<div class="footer">
&copy; Copyright 2013, Marion Neumann, Shan Huang, Daniel Marthaler, Kristian Kersting.
Created using <a href="http://sphinx-doc.org/">Sphinx</a> 1.2b3.
Created using <a href="http://sphinx.pocoo.org/">Sphinx</a> 1.1.3.
</div>
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</div>
<div class="footer">
&copy; Copyright 2013, Marion Neumann, Shan Huang, Daniel Marthaler, Kristian Kersting.
Created using <a href="http://sphinx-doc.org/">Sphinx</a> 1.2b3.
Created using <a href="http://sphinx.pocoo.org/">Sphinx</a> 1.1.3.
</div>
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</pre></div>
</div>
<p>The left figure below shows the result of fixed inducing points, and the right figure shows the result for equidistant <img class="math" src="_images/math/9ad99798ec4c38e165cf517cb9e02b1c9e824103.png" alt="u"/>.</p>
<a class="reference internal image-reference" href="_images/d3_2.png"><img alt="_images/d3_2.png" src="_images/d3_2.png" style="width: 45%;" /></a>
<a class="reference internal image-reference" href="_images/d3_3.png"><img alt="_images/d3_3.png" src="_images/d3_3.png" style="width: 45%;" /></a>
<img alt="_images/d3_2.png" src="_images/d3_2.png" style="width: 45%;" />
<img alt="_images/d3_3.png" src="_images/d3_3.png" style="width: 45%;" />
<p><strong>[Theory]</strong>
Note that the predictive variance is
overestimated outside the support of the inducing inputs. In a multivariate example where densely sampled inducing inputs are infeasible, one can
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</div>
<div class="footer">
&copy; Copyright 2013, Marion Neumann, Shan Huang, Daniel Marthaler, Kristian Kersting.
Created using <a href="http://sphinx-doc.org/">Sphinx</a> 1.2b3.
Created using <a href="http://sphinx.pocoo.org/">Sphinx</a> 1.1.3.
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</div>
<div class="footer">
&copy; Copyright 2013, Marion Neumann, Shan Huang, Daniel Marthaler, Kristian Kersting.
Created using <a href="http://sphinx-doc.org/">Sphinx</a> 1.2b3.
Created using <a href="http://sphinx.pocoo.org/">Sphinx</a> 1.1.3.
</div>
</body>
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</div>
<div class="footer">
&copy; Copyright 2013, Marion Neumann, Shan Huang, Daniel Marthaler, Kristian Kersting.
Created using <a href="http://sphinx-doc.org/">Sphinx</a> 1.2b3.
Created using <a href="http://sphinx.pocoo.org/">Sphinx</a> 1.1.3.
</div>
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<tbody valign="top">
<tr class="field-odd field"><th class="field-name">Parameters:</th><td class="field-body"><strong>D</strong> &#8211; dimension of training data. Set if you want default alpha, which is 0.5 for each dimension.</td>
</tr>
<tr class="field-even field"><th class="field-name">Alpha_list:</th><td class="field-body">scalar alpha for each dimension</td>
<tr class="field-even field"><th class="field-name">Alpha_list :</th><td class="field-body">scalar alpha for each dimension</td>
</tr>
</tbody>
</table>
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</div>
<div class="footer">
&copy; Copyright 2013, Marion Neumann, Shan Huang, Daniel Marthaler, Kristian Kersting.
Created using <a href="http://sphinx-doc.org/">Sphinx</a> 1.2b3.
Created using <a href="http://sphinx.pocoo.org/">Sphinx</a> 1.1.3.
</div>
</body>
</html>
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</div>
<div class="footer">
&copy; Copyright 2013, Marion Neumann, Shan Huang, Daniel Marthaler, Kristian Kersting.
Created using <a href="http://sphinx-doc.org/">Sphinx</a> 1.2b3.
Created using <a href="http://sphinx.pocoo.org/">Sphinx</a> 1.1.3.
</div>
</body>
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</div>
<div class="footer">
&copy; Copyright 2013, Marion Neumann, Shan Huang, Daniel Marthaler, Kristian Kersting.
Created using <a href="http://sphinx-doc.org/">Sphinx</a> 1.2b3.
Created using <a href="http://sphinx.pocoo.org/">Sphinx</a> 1.1.3.
</div>
</body>
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Each digit of <img class="math" src="_images/math/f3c19812b1b3f3cc95f5f55e7fbd1399eee6004a.png" alt="16*16"/> pixels is flattened into a <img class="math" src="_images/math/abd678361b0f40ec2723283c0f2bab3df5cadc68.png" alt="256"/> dimension vector.
For the simplicity of demo, we only selected digits <img class="math" src="_images/math/dce34f4dfb2406144304ad0d6106c5382ddd1446.png" alt="1"/> s and <img class="math" src="_images/math/41c544263a265ff15498ee45f7392c5f86c6d151.png" alt="2"/> s such that we have a binary classification problem where digit <img class="math" src="_images/math/dce34f4dfb2406144304ad0d6106c5382ddd1446.png" alt="1"/> for class +1 and digit <img class="math" src="_images/math/41c544263a265ff15498ee45f7392c5f86c6d151.png" alt="2"/> for class -1. We also reduced the dataset into <img class="math" src="_images/math/e59e2c6e83eb78cca610a5fd4070ae01c8d4ae60.png" alt="100"/> samples per digit, where the original dataset consist of thousands of samples for each digit.</p>
<p>Here are samples for two digits for <img class="math" src="_images/math/dce34f4dfb2406144304ad0d6106c5382ddd1446.png" alt="1"/></p>
<a class="reference internal image-reference" href="_images/digit1_1.png"><img alt="_images/digit1_1.png" src="_images/digit1_1.png" style="width: 30%;" /></a>
<a class="reference internal image-reference" href="_images/digit1_2.png"><img alt="_images/digit1_2.png" src="_images/digit1_2.png" style="width: 30%;" /></a>
<img alt="_images/digit1_1.png" src="_images/digit1_1.png" style="width: 30%;" />
<img alt="_images/digit1_2.png" src="_images/digit1_2.png" style="width: 30%;" />
<p>and samples for two digits for <img class="math" src="_images/math/41c544263a265ff15498ee45f7392c5f86c6d151.png" alt="2"/></p>
<a class="reference internal image-reference" href="_images/digit2_1.png"><img alt="_images/digit2_1.png" src="_images/digit2_1.png" style="width: 30%;" /></a>
<a class="reference internal image-reference" href="_images/digit2_2.png"><img alt="_images/digit2_2.png" src="_images/digit2_2.png" style="width: 30%;" /></a>
<img alt="_images/digit2_1.png" src="_images/digit2_1.png" style="width: 30%;" />
<img alt="_images/digit2_2.png" src="_images/digit2_2.png" style="width: 30%;" />
</div>
<div class="section" id="form-a-nearest-neighbour-graph">
<h2>Form a nearest neighbour graph<a class="headerlink" href="#form-a-nearest-neighbour-graph" title="Permalink to this headline"></a></h2>
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<p>The left image shows the digit that using diffusion kernel will predict the wrong result (should be <img class="math" src="_images/math/41c544263a265ff15498ee45f7392c5f86c6d151.png" alt="2"/>),
but rbf kernel does the job fine. The right image shows the digit that rbf kernel predicts the wrong class, diffusion kernel on the other hand, predicts correctly due to graph information! (should be <img class="math" src="_images/math/dce34f4dfb2406144304ad0d6106c5382ddd1446.png" alt="1"/>).</p>
<p>Interestingly, using a composite kernel with diffusion kernel on graph and an rbf kernel together. All test cases including the following are predicted correctly.</p>
<a class="reference internal image-reference" href="_images/digitDiffwrong.png"><img alt="_images/digitDiffwrong.png" src="_images/digitDiffwrong.png" style="width: 50%;" /></a>
<a class="reference internal image-reference" href="_images/digitRBFwrong.png"><img alt="_images/digitRBFwrong.png" src="_images/digitRBFwrong.png" style="width: 50%;" /></a>
<img alt="_images/digitDiffwrong.png" src="_images/digitDiffwrong.png" style="width: 50%;" />
<img alt="_images/digitRBFwrong.png" src="_images/digitRBFwrong.png" style="width: 50%;" />
<table class="docutils footnote" frame="void" id="id3" rules="none">
<colgroup><col class="label" /><col /></colgroup>
<tbody valign="top">
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</div>
<div class="footer">
&copy; Copyright 2013, Marion Neumann, Shan Huang, Daniel Marthaler, Kristian Kersting.
Created using <a href="http://sphinx-doc.org/">Sphinx</a> 1.2b3.
Created using <a href="http://sphinx.pocoo.org/">Sphinx</a> 1.1.3.
</div>
</body>
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