testing local website build
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KarlHenrik
@@ -28,7 +28,7 @@
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<link rel="preload" as="font" type="font/woff2" crossorigin href="_static/vendor/fontawesome/6.5.2/webfonts/fa-brands-400.woff2" />
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<link rel="preload" as="font" type="font/woff2" crossorigin href="_static/vendor/fontawesome/6.5.2/webfonts/fa-regular-400.woff2" />
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<link rel="stylesheet" type="text/css" href="_static/pygments.css?v=fa44fd50" />
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<link rel="stylesheet" type="text/css" href="_static/pygments.css?v=03e43079" />
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<link rel="stylesheet" type="text/css" href="_static/styles/sphinx-book-theme.css?v=eba8b062" />
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<link rel="stylesheet" type="text/css" href="_static/togglebutton.css?v=13237357" />
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<link rel="stylesheet" type="text/css" href="_static/copybutton.css?v=76b2166b" />
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@@ -183,7 +183,7 @@
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</ul>
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<p aria-level="2" class="caption" role="heading"><span class="caption-text">About the course</span></p>
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<ul class="nav bd-sidenav">
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<li class="toctree-l1"><a class="reference internal" href="schedule.html">Teaching schedule with links to material</a></li>
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<li class="toctree-l1"><a class="reference internal" href="schedule.html">Course setting</a></li>
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<li class="toctree-l1"><a class="reference internal" href="teachers.html">Teachers and Grading</a></li>
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<li class="toctree-l1"><a class="reference internal" href="textbooks.html">Textbooks</a></li>
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@@ -271,37 +271,6 @@
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<div class="dropdown dropdown-launch-buttons">
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<button class="btn dropdown-toggle" type="button" data-bs-toggle="dropdown" aria-expanded="false" aria-label="Launch interactive content">
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<i class="fas fa-rocket"></i>
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</button>
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<ul class="dropdown-menu">
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<li><a href="https://mybinder.org/v2/git/https%3A//compphysics.github.io/MachineLearning/doc/LectureNotes/_build/html/index.html/master?urlpath=tree/chapter8.ipynb" target="_blank"
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class="btn btn-sm dropdown-item"
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title="Launch on Binder"
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data-bs-placement="left" data-bs-toggle="tooltip"
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>
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<span class="btn__icon-container">
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<img alt="Binder logo" src="_static/images/logo_binder.svg">
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</span>
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<span class="btn__text-container">Binder</span>
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</a>
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</li>
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</ul>
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</div>
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<div class="dropdown dropdown-download-buttons">
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<button class="btn dropdown-toggle" type="button" data-bs-toggle="dropdown" aria-expanded="false" aria-label="Download this page">
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<i class="fas fa-download"></i>
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@@ -572,24 +541,16 @@ function <strong>np.mean(x)</strong>. We can also extract the eigenvalues of the
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covariance matrix through the <strong>np.linalg.eig()</strong> function.</p>
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<div class="cell docutils container">
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<div class="cell_input docutils container">
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Importing various packages</span>
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<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
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<span class="n">n</span> <span class="o">=</span> <span class="mi">100</span>
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<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">normal</span><span class="p">(</span><span class="n">size</span><span class="o">=</span><span class="n">n</span><span class="p">)</span>
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<span class="nb">print</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>
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<span class="n">y</span> <span class="o">=</span> <span class="mi">4</span><span class="o">+</span><span class="mi">3</span><span class="o">*</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">normal</span><span class="p">(</span><span class="n">size</span><span class="o">=</span><span class="n">n</span><span class="p">)</span>
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<span class="nb">print</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">y</span><span class="p">))</span>
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<span class="n">W</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">vstack</span><span class="p">((</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">))</span>
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<span class="n">C</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">cov</span><span class="p">(</span><span class="n">W</span><span class="p">)</span>
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<span class="nb">print</span><span class="p">(</span><span class="n">C</span><span class="p">)</span>
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</pre></div>
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</div>
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</div>
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<div class="cell_output docutils container">
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>0.022443845445495113
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4.000760768324305
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[[ 1.00052738 3.04596527]
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[ 3.04596527 10.40546987]]
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<div class="highlight-none notranslate"><div class="highlight"><pre><span></span># Importing various packages
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import numpy as np
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n = 100
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x = np.random.normal(size=n)
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print(np.mean(x))
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y = 4+3*x+np.random.normal(size=n)
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print(np.mean(y))
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W = np.vstack((x, y))
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C = np.cov(W)
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print(C)
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</pre></div>
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</div>
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</div>
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@@ -604,35 +565,27 @@ code which sets up the correlations matrix for the previous example in
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a more brute force way. Here we scale the mean values for each column of the design matrix, calculate the relevant mean values and variances and then finally set up the <span class="math notranslate nohighlight">\(2\times 2\)</span> correlation matrix (since we have only two vectors).</p>
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<div class="cell docutils container">
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<div class="cell_input docutils container">
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
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<span class="n">n</span> <span class="o">=</span> <span class="mi">100</span>
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<span class="c1"># define two vectors </span>
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<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="n">size</span><span class="o">=</span><span class="n">n</span><span class="p">)</span>
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<span class="n">y</span> <span class="o">=</span> <span class="mi">4</span><span class="o">+</span><span class="mi">3</span><span class="o">*</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">normal</span><span class="p">(</span><span class="n">size</span><span class="o">=</span><span class="n">n</span><span class="p">)</span>
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<span class="c1">#scaling the x and y vectors </span>
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<span class="n">x</span> <span class="o">=</span> <span class="n">x</span> <span class="o">-</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
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<span class="n">y</span> <span class="o">=</span> <span class="n">y</span> <span class="o">-</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">y</span><span class="p">)</span>
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<span class="n">variance_x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">x</span><span class="nd">@x</span><span class="p">)</span><span class="o">/</span><span class="n">n</span>
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<span class="n">variance_y</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">y</span><span class="nd">@y</span><span class="p">)</span><span class="o">/</span><span class="n">n</span>
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<span class="nb">print</span><span class="p">(</span><span class="n">variance_x</span><span class="p">)</span>
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<span class="nb">print</span><span class="p">(</span><span class="n">variance_y</span><span class="p">)</span>
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<span class="n">cov_xy</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">x</span><span class="nd">@y</span><span class="p">)</span><span class="o">/</span><span class="n">n</span>
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<span class="n">cov_xx</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">x</span><span class="nd">@x</span><span class="p">)</span><span class="o">/</span><span class="n">n</span>
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<span class="n">cov_yy</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">y</span><span class="nd">@y</span><span class="p">)</span><span class="o">/</span><span class="n">n</span>
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<span class="n">C</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="mi">2</span><span class="p">,</span><span class="mi">2</span><span class="p">))</span>
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<span class="n">C</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">]</span><span class="o">=</span> <span class="n">cov_xx</span><span class="o">/</span><span class="n">variance_x</span>
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<span class="n">C</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="o">=</span> <span class="n">cov_yy</span><span class="o">/</span><span class="n">variance_y</span>
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<span class="n">C</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="o">=</span> <span class="n">cov_xy</span><span class="o">/</span><span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">variance_y</span><span class="o">*</span><span class="n">variance_x</span><span class="p">)</span>
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<span class="n">C</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">]</span><span class="o">=</span> <span class="n">C</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">]</span>
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<span class="nb">print</span><span class="p">(</span><span class="n">C</span><span class="p">)</span>
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</pre></div>
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</div>
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</div>
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<div class="cell_output docutils container">
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>0.08808936438772381
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1.8955512810801536
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[[1. 0.66003687]
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[0.66003687 1. ]]
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<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>import numpy as np
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n = 100
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# define two vectors
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x = np.random.random(size=n)
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y = 4+3*x+np.random.normal(size=n)
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#scaling the x and y vectors
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x = x - np.mean(x)
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y = y - np.mean(y)
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variance_x = np.sum(x@x)/n
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variance_y = np.sum(y@y)/n
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print(variance_x)
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print(variance_y)
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cov_xy = np.sum(x@y)/n
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cov_xx = np.sum(x@x)/n
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cov_yy = np.sum(y@y)/n
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C = np.zeros((2,2))
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C[0,0]= cov_xx/variance_x
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C[1,1]= cov_yy/variance_y
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C[0,1]= cov_xy/np.sqrt(variance_y*variance_x)
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C[1,0]= C[0,1]
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print(C)
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</pre></div>
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</div>
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</div>
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@@ -644,47 +597,19 @@ this matrix we easily see that it is a positive definite matrix.</p>
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<p>We whow here how we can set up the correlation matrix using <strong>pandas</strong>, as done in this simple code</p>
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<div class="cell docutils container">
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<div class="cell_input docutils container">
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
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<span class="kn">import</span> <span class="nn">pandas</span> <span class="k">as</span> <span class="nn">pd</span>
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<span class="n">n</span> <span class="o">=</span> <span class="mi">10</span>
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<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">normal</span><span class="p">(</span><span class="n">size</span><span class="o">=</span><span class="n">n</span><span class="p">)</span>
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<span class="n">x</span> <span class="o">=</span> <span class="n">x</span> <span class="o">-</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
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<span class="n">y</span> <span class="o">=</span> <span class="mi">4</span><span class="o">+</span><span class="mi">3</span><span class="o">*</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">normal</span><span class="p">(</span><span class="n">size</span><span class="o">=</span><span class="n">n</span><span class="p">)</span>
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<span class="n">y</span> <span class="o">=</span> <span class="n">y</span> <span class="o">-</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">y</span><span class="p">)</span>
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<span class="n">X</span> <span class="o">=</span> <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">vstack</span><span class="p">((</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)))</span><span class="o">.</span><span class="n">T</span>
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<span class="nb">print</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
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<span class="n">Xpd</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
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<span class="nb">print</span><span class="p">(</span><span class="n">Xpd</span><span class="p">)</span>
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<span class="n">correlation_matrix</span> <span class="o">=</span> <span class="n">Xpd</span><span class="o">.</span><span class="n">corr</span><span class="p">()</span>
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<span class="nb">print</span><span class="p">(</span><span class="n">correlation_matrix</span><span class="p">)</span>
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</pre></div>
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</div>
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</div>
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<div class="cell_output docutils container">
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>[[-0.72540078 -2.98990574]
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[ 0.96045962 2.58951054]
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[ 0.77755709 1.47011271]
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[-0.60907896 -0.58087521]
|
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[-0.27034896 -0.53030696]
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[-0.22425043 -0.07533845]
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[-2.15767447 -6.91457183]
|
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[ 0.95668475 3.51161154]
|
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[ 0.25664022 0.84460603]
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[ 1.03541191 2.67515735]]
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0 1
|
||||
0 -0.725401 -2.989906
|
||||
1 0.960460 2.589511
|
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2 0.777557 1.470113
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3 -0.609079 -0.580875
|
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4 -0.270349 -0.530307
|
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5 -0.224250 -0.075338
|
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6 -2.157674 -6.914572
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7 0.956685 3.511612
|
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8 0.256640 0.844606
|
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9 1.035412 2.675157
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0 1
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0 1.000000 0.974962
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1 0.974962 1.000000
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<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>import numpy as np
|
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import pandas as pd
|
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n = 10
|
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x = np.random.normal(size=n)
|
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x = x - np.mean(x)
|
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y = 4+3*x+np.random.normal(size=n)
|
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y = y - np.mean(y)
|
||||
X = (np.vstack((x, y))).T
|
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print(X)
|
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Xpd = pd.DataFrame(X)
|
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print(Xpd)
|
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correlation_matrix = Xpd.corr()
|
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print(correlation_matrix)
|
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</pre></div>
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</div>
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</div>
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@@ -692,86 +617,49 @@ this matrix we easily see that it is a positive definite matrix.</p>
|
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<p>We expand this model to the Franke function discussed above.</p>
|
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<div class="cell docutils container">
|
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<div class="cell_input docutils container">
|
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Common imports</span>
|
||||
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
|
||||
<span class="kn">import</span> <span class="nn">pandas</span> <span class="k">as</span> <span class="nn">pd</span>
|
||||
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span># Common imports
|
||||
import numpy as np
|
||||
import pandas as pd
|
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|
||||
|
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<span class="k">def</span> <span class="nf">FrankeFunction</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">y</span><span class="p">):</span>
|
||||
<span class="n">term1</span> <span class="o">=</span> <span class="mf">0.75</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="p">(</span><span class="mf">0.25</span><span class="o">*</span><span class="p">(</span><span class="mi">9</span><span class="o">*</span><span class="n">x</span><span class="o">-</span><span class="mi">2</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span> <span class="o">-</span> <span class="mf">0.25</span><span class="o">*</span><span class="p">((</span><span class="mi">9</span><span class="o">*</span><span class="n">y</span><span class="o">-</span><span class="mi">2</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">))</span>
|
||||
<span class="n">term2</span> <span class="o">=</span> <span class="mf">0.75</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="p">((</span><span class="mi">9</span><span class="o">*</span><span class="n">x</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">/</span><span class="mf">49.0</span> <span class="o">-</span> <span class="mf">0.1</span><span class="o">*</span><span class="p">(</span><span class="mi">9</span><span class="o">*</span><span class="n">y</span><span class="o">+</span><span class="mi">1</span><span class="p">))</span>
|
||||
<span class="n">term3</span> <span class="o">=</span> <span class="mf">0.5</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="p">(</span><span class="mi">9</span><span class="o">*</span><span class="n">x</span><span class="o">-</span><span class="mi">7</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="o">/</span><span class="mf">4.0</span> <span class="o">-</span> <span class="mf">0.25</span><span class="o">*</span><span class="p">((</span><span class="mi">9</span><span class="o">*</span><span class="n">y</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="n">term4</span> <span class="o">=</span> <span class="o">-</span><span class="mf">0.2</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="p">(</span><span class="mi">9</span><span class="o">*</span><span class="n">x</span><span class="o">-</span><span class="mi">4</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span> <span class="o">-</span> <span class="p">(</span><span class="mi">9</span><span class="o">*</span><span class="n">y</span><span class="o">-</span><span class="mi">7</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span>
|
||||
<span class="k">return</span> <span class="n">term1</span> <span class="o">+</span> <span class="n">term2</span> <span class="o">+</span> <span class="n">term3</span> <span class="o">+</span> <span class="n">term4</span>
|
||||
def FrankeFunction(x,y):
|
||||
term1 = 0.75*np.exp(-(0.25*(9*x-2)**2) - 0.25*((9*y-2)**2))
|
||||
term2 = 0.75*np.exp(-((9*x+1)**2)/49.0 - 0.1*(9*y+1))
|
||||
term3 = 0.5*np.exp(-(9*x-7)**2/4.0 - 0.25*((9*y-3)**2))
|
||||
term4 = -0.2*np.exp(-(9*x-4)**2 - (9*y-7)**2)
|
||||
return term1 + term2 + term3 + term4
|
||||
|
||||
|
||||
<span class="k">def</span> <span class="nf">create_X</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">n</span> <span class="p">):</span>
|
||||
<span class="k">if</span> <span class="nb">len</span><span class="p">(</span><span class="n">x</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span> <span class="o">></span> <span class="mi">1</span><span class="p">:</span>
|
||||
<span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">ravel</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
|
||||
<span class="n">y</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">ravel</span><span class="p">(</span><span class="n">y</span><span class="p">)</span>
|
||||
def create_X(x, y, n ):
|
||||
if len(x.shape) > 1:
|
||||
x = np.ravel(x)
|
||||
y = np.ravel(y)
|
||||
|
||||
<span class="n">N</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
|
||||
<span class="n">l</span> <span class="o">=</span> <span class="nb">int</span><span class="p">((</span><span class="n">n</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">n</span><span class="o">+</span><span class="mi">2</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span> <span class="c1"># Number of elements in beta</span>
|
||||
<span class="n">X</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">ones</span><span class="p">((</span><span class="n">N</span><span class="p">,</span><span class="n">l</span><span class="p">))</span>
|
||||
N = len(x)
|
||||
l = int((n+1)*(n+2)/2) # Number of elements in beta
|
||||
X = np.ones((N,l))
|
||||
|
||||
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span><span class="n">n</span><span class="o">+</span><span class="mi">1</span><span class="p">):</span>
|
||||
<span class="n">q</span> <span class="o">=</span> <span class="nb">int</span><span class="p">((</span><span class="n">i</span><span class="p">)</span><span class="o">*</span><span class="p">(</span><span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span>
|
||||
<span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">):</span>
|
||||
<span class="n">X</span><span class="p">[:,</span><span class="n">q</span><span class="o">+</span><span class="n">k</span><span class="p">]</span> <span class="o">=</span> <span class="p">(</span><span class="n">x</span><span class="o">**</span><span class="p">(</span><span class="n">i</span><span class="o">-</span><span class="n">k</span><span class="p">))</span><span class="o">*</span><span class="p">(</span><span class="n">y</span><span class="o">**</span><span class="n">k</span><span class="p">)</span>
|
||||
for i in range(1,n+1):
|
||||
q = int((i)*(i+1)/2)
|
||||
for k in range(i+1):
|
||||
X[:,q+k] = (x**(i-k))*(y**k)
|
||||
|
||||
<span class="k">return</span> <span class="n">X</span>
|
||||
return X
|
||||
|
||||
|
||||
<span class="c1"># Making meshgrid of datapoints and compute Franke's function</span>
|
||||
<span class="n">n</span> <span class="o">=</span> <span class="mi">4</span>
|
||||
<span class="n">N</span> <span class="o">=</span> <span class="mi">100</span>
|
||||
<span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">sort</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">uniform</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">N</span><span class="p">))</span>
|
||||
<span class="n">y</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">sort</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">uniform</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">N</span><span class="p">))</span>
|
||||
<span class="n">z</span> <span class="o">=</span> <span class="n">FrankeFunction</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">)</span>
|
||||
<span class="n">X</span> <span class="o">=</span> <span class="n">create_X</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="n">n</span><span class="o">=</span><span class="n">n</span><span class="p">)</span>
|
||||
# Making meshgrid of datapoints and compute Franke's function
|
||||
n = 4
|
||||
N = 100
|
||||
x = np.sort(np.random.uniform(0, 1, N))
|
||||
y = np.sort(np.random.uniform(0, 1, N))
|
||||
z = FrankeFunction(x, y)
|
||||
X = create_X(x, y, n=n)
|
||||
|
||||
<span class="n">Xpd</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
|
||||
<span class="c1"># subtract the mean values and set up the covariance matrix</span>
|
||||
<span class="n">Xpd</span> <span class="o">=</span> <span class="n">Xpd</span> <span class="o">-</span> <span class="n">Xpd</span><span class="o">.</span><span class="n">mean</span><span class="p">()</span>
|
||||
<span class="n">covariance_matrix</span> <span class="o">=</span> <span class="n">Xpd</span><span class="o">.</span><span class="n">cov</span><span class="p">()</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="n">covariance_matrix</span><span class="p">)</span>
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell_output docutils container">
|
||||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span> 0 1 2 3 4 5 6 7 \
|
||||
0 0.0 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000
|
||||
1 0.0 0.087010 0.083890 0.089681 0.085024 0.080592 0.082783 0.078206
|
||||
2 0.0 0.083890 0.081567 0.087183 0.083033 0.079041 0.080807 0.076604
|
||||
3 0.0 0.089681 0.087183 0.098391 0.093630 0.089065 0.094407 0.089439
|
||||
4 0.0 0.085024 0.083033 0.093630 0.089365 0.085248 0.090084 0.085550
|
||||
5 0.0 0.080592 0.079041 0.089065 0.085248 0.081539 0.085918 0.081782
|
||||
6 0.0 0.082783 0.080807 0.094407 0.090084 0.085918 0.092964 0.088281
|
||||
7 0.0 0.078206 0.076604 0.089439 0.085550 0.081782 0.088281 0.084004
|
||||
8 0.0 0.073941 0.072667 0.084787 0.081289 0.077882 0.083881 0.079972
|
||||
9 0.0 0.069966 0.068980 0.080432 0.077288 0.074206 0.079747 0.076174
|
||||
10 0.0 0.075203 0.073623 0.088019 0.084189 0.080482 0.088274 0.084012
|
||||
11 0.0 0.071049 0.069761 0.083356 0.079897 0.076533 0.083777 0.079875
|
||||
12 0.0 0.067191 0.066160 0.079008 0.075885 0.072831 0.079572 0.075997
|
||||
13 0.0 0.063604 0.062799 0.074953 0.072133 0.069360 0.075638 0.072360
|
||||
14 0.0 0.060267 0.059662 0.071167 0.068621 0.066104 0.071955 0.068949
|
||||
|
||||
8 9 10 11 12 13 14
|
||||
0 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000
|
||||
1 0.073941 0.069966 0.075203 0.071049 0.067191 0.063604 0.060267
|
||||
2 0.072667 0.068980 0.073623 0.069761 0.066160 0.062799 0.059662
|
||||
3 0.084787 0.080432 0.088019 0.083356 0.079008 0.074953 0.071167
|
||||
4 0.081289 0.077288 0.084189 0.079897 0.075885 0.072133 0.068621
|
||||
5 0.077882 0.074206 0.080482 0.076533 0.072831 0.069360 0.066104
|
||||
6 0.083881 0.079747 0.088274 0.083777 0.079572 0.075638 0.071955
|
||||
7 0.079972 0.076174 0.084012 0.079875 0.075997 0.072360 0.068949
|
||||
8 0.076277 0.072786 0.079993 0.076185 0.072607 0.069244 0.066082
|
||||
9 0.072786 0.069575 0.076206 0.072699 0.069396 0.066284 0.063352
|
||||
10 0.079993 0.076206 0.084959 0.080794 0.076889 0.073225 0.069785
|
||||
11 0.076185 0.072699 0.080794 0.076957 0.073349 0.069958 0.066767
|
||||
12 0.072607 0.069396 0.076889 0.073349 0.070015 0.066874 0.063912
|
||||
13 0.069244 0.066284 0.073225 0.069958 0.066874 0.063961 0.061211
|
||||
14 0.066082 0.063352 0.069785 0.066767 0.063912 0.061211 0.058654
|
||||
Xpd = pd.DataFrame(X)
|
||||
# subtract the mean values and set up the covariance matrix
|
||||
Xpd = Xpd - Xpd.mean()
|
||||
covariance_matrix = Xpd.cov()
|
||||
print(covariance_matrix)
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
@@ -895,16 +783,16 @@ this distribution, and store them in the <span class="math notranslate nohighlig
|
||||
Note that the function <strong>multivariate</strong> returns also the covariance discussed above and that it is defined by dividing by <span class="math notranslate nohighlight">\(n-1\)</span> instead of <span class="math notranslate nohighlight">\(n\)</span>.</p>
|
||||
<div class="cell docutils container">
|
||||
<div class="cell_input docutils container">
|
||||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="o">%</span><span class="k">matplotlib</span> inline
|
||||
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>%matplotlib inline
|
||||
|
||||
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
|
||||
<span class="kn">import</span> <span class="nn">pandas</span> <span class="k">as</span> <span class="nn">pd</span>
|
||||
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
|
||||
<span class="kn">from</span> <span class="nn">IPython.display</span> <span class="kn">import</span> <span class="n">display</span>
|
||||
<span class="n">n</span> <span class="o">=</span> <span class="mi">10000</span>
|
||||
<span class="n">mean</span> <span class="o">=</span> <span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
|
||||
<span class="n">cov</span> <span class="o">=</span> <span class="p">[[</span><span class="mi">4</span><span class="p">,</span> <span class="mi">2</span><span class="p">],</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">]]</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">multivariate_normal</span><span class="p">(</span><span class="n">mean</span><span class="p">,</span> <span class="n">cov</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span>
|
||||
import numpy as np
|
||||
import pandas as pd
|
||||
import matplotlib.pyplot as plt
|
||||
from IPython.display import display
|
||||
n = 10000
|
||||
mean = (-1, 2)
|
||||
cov = [[4, 2], [2, 2]]
|
||||
X = np.random.multivariate_normal(mean, cov, n)
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
@@ -926,11 +814,11 @@ centered at the origin!
|
||||
The following code elements perform these operations using <strong>pandas</strong> or using our own functionality for doing so. The latter, using <strong>numpy</strong> is rather simple through the <strong>mean()</strong> function.</p>
|
||||
<div class="cell docutils container">
|
||||
<div class="cell_input docutils container">
|
||||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">df</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
|
||||
<span class="c1"># Pandas does the centering for us</span>
|
||||
<span class="n">df</span> <span class="o">=</span> <span class="n">df</span> <span class="o">-</span><span class="n">df</span><span class="o">.</span><span class="n">mean</span><span class="p">()</span>
|
||||
<span class="c1"># we center it ourselves</span>
|
||||
<span class="n">X_centered</span> <span class="o">=</span> <span class="n">X</span> <span class="o">-</span> <span class="n">X</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
|
||||
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>df = pd.DataFrame(X)
|
||||
# Pandas does the centering for us
|
||||
df = df -df.mean()
|
||||
# we center it ourselves
|
||||
X_centered = X - X.mean(axis=0)
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
@@ -953,17 +841,8 @@ specific case.</p>
|
||||
We can write our own code or simply use either the functionaly of <strong>numpy</strong> or that of <strong>pandas</strong>, as follows</p>
|
||||
<div class="cell docutils container">
|
||||
<div class="cell_input docutils container">
|
||||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="nb">print</span><span class="p">(</span><span class="n">df</span><span class="o">.</span><span class="n">cov</span><span class="p">())</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">cov</span><span class="p">(</span><span class="n">X_centered</span><span class="o">.</span><span class="n">T</span><span class="p">))</span>
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell_output docutils container">
|
||||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span> 0 1
|
||||
0 3.935482 1.945984
|
||||
1 1.945984 1.962204
|
||||
[[3.93548177 1.94598372]
|
||||
[1.94598372 1.96220435]]
|
||||
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>print(df.cov())
|
||||
print(np.cov(X_centered.T))
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
@@ -972,30 +851,22 @@ We can write our own code or simply use either the functionaly of <strong>numpy<
|
||||
Our own code here is not very elegant and asks for obvious improvements. It is tailored to this specific <span class="math notranslate nohighlight">\(2\times 2\)</span> covariance matrix.</p>
|
||||
<div class="cell docutils container">
|
||||
<div class="cell_input docutils container">
|
||||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># extract the relevant columns from the centered design matrix of dim n x 2</span>
|
||||
<span class="n">x</span> <span class="o">=</span> <span class="n">X_centered</span><span class="p">[:,</span><span class="mi">0</span><span class="p">]</span>
|
||||
<span class="n">y</span> <span class="o">=</span> <span class="n">X_centered</span><span class="p">[:,</span><span class="mi">1</span><span class="p">]</span>
|
||||
<span class="n">Cov</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="mi">2</span><span class="p">,</span><span class="mi">2</span><span class="p">))</span>
|
||||
<span class="n">Cov</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="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">x</span><span class="o">.</span><span class="n">T</span><span class="nd">@y</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">n</span><span class="o">-</span><span class="mf">1.0</span><span class="p">)</span>
|
||||
<span class="n">Cov</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">x</span><span class="o">.</span><span class="n">T</span><span class="nd">@x</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">n</span><span class="o">-</span><span class="mf">1.0</span><span class="p">)</span>
|
||||
<span class="n">Cov</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="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">y</span><span class="o">.</span><span class="n">T</span><span class="nd">@y</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">n</span><span class="o">-</span><span class="mf">1.0</span><span class="p">)</span>
|
||||
<span class="n">Cov</span><span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">0</span><span class="p">]</span><span class="o">=</span> <span class="n">Cov</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="nb">print</span><span class="p">(</span><span class="s2">"Centered covariance using own code"</span><span class="p">)</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="n">Cov</span><span class="p">)</span>
|
||||
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">,</span> <span class="s1">'x'</span><span class="p">)</span>
|
||||
<span class="n">plt</span><span class="o">.</span><span class="n">axis</span><span class="p">(</span><span class="s1">'equal'</span><span class="p">)</span>
|
||||
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
|
||||
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span># extract the relevant columns from the centered design matrix of dim n x 2
|
||||
x = X_centered[:,0]
|
||||
y = X_centered[:,1]
|
||||
Cov = np.zeros((2,2))
|
||||
Cov[0,1] = np.sum(x.T@y)/(n-1.0)
|
||||
Cov[0,0] = np.sum(x.T@x)/(n-1.0)
|
||||
Cov[1,1] = np.sum(y.T@y)/(n-1.0)
|
||||
Cov[1,0]= Cov[0,1]
|
||||
print("Centered covariance using own code")
|
||||
print(Cov)
|
||||
plt.plot(x, y, 'x')
|
||||
plt.axis('equal')
|
||||
plt.show()
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell_output docutils container">
|
||||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Centered covariance using own code
|
||||
[[3.93548177 1.94598372]
|
||||
[1.94598372 1.96220435]]
|
||||
</pre></div>
|
||||
</div>
|
||||
<img alt="_images/628da35a18d31f48e5e25bd7c4c9039bb751daa0301a94c33fa0bb162c1c6f54.png" src="_images/628da35a18d31f48e5e25bd7c4c9039bb751daa0301a94c33fa0bb162c1c6f54.png" />
|
||||
</div>
|
||||
</div>
|
||||
<p>Depending on the number of points <span class="math notranslate nohighlight">\(n\)</span>, we will get results that are close to the covariance values defined above.
|
||||
The plot shows how the data are clustered around a line with slope close to one. Is this expected? Try to change the covariance and the mean values. For example, try to make the variance of the first element much larger than that of the second diagonal element. Try also to shrink the covariance (the non-diagonal elements) and see how the data points are distributed.</p>
|
||||
@@ -1025,42 +896,27 @@ analysis. Feel free to extend upon this in order to address the above
|
||||
questions.</p>
|
||||
<div class="cell docutils container">
|
||||
<div class="cell_input docutils container">
|
||||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># diagonalize and obtain eigenvalues, not necessarily sorted</span>
|
||||
<span class="n">EigValues</span><span class="p">,</span> <span class="n">EigVectors</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">eig</span><span class="p">(</span><span class="n">Cov</span><span class="p">)</span>
|
||||
<span class="c1"># sort eigenvectors and eigenvalues</span>
|
||||
<span class="c1">#permute = EigValues.argsort()</span>
|
||||
<span class="c1">#EigValues = EigValues[permute]</span>
|
||||
<span class="c1">#EigVectors = EigVectors[:,permute]</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="s2">"Eigenvalues of Covariance matrix"</span><span class="p">)</span>
|
||||
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">2</span><span class="p">):</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="n">EigValues</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
|
||||
<span class="n">FirstEigvector</span> <span class="o">=</span> <span class="n">EigVectors</span><span class="p">[:,</span><span class="mi">0</span><span class="p">]</span>
|
||||
<span class="n">SecondEigvector</span> <span class="o">=</span> <span class="n">EigVectors</span><span class="p">[:,</span><span class="mi">1</span><span class="p">]</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="s2">"First eigenvector"</span><span class="p">)</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="n">FirstEigvector</span><span class="p">)</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="s2">"Second eigenvector"</span><span class="p">)</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="n">SecondEigvector</span><span class="p">)</span>
|
||||
<span class="c1">#thereafter we do a PCA with Scikit-learn</span>
|
||||
<span class="kn">from</span> <span class="nn">sklearn.decomposition</span> <span class="kn">import</span> <span class="n">PCA</span>
|
||||
<span class="n">pca</span> <span class="o">=</span> <span class="n">PCA</span><span class="p">(</span><span class="n">n_components</span> <span class="o">=</span> <span class="mi">2</span><span class="p">)</span>
|
||||
<span class="n">X2Dsl</span> <span class="o">=</span> <span class="n">pca</span><span class="o">.</span><span class="n">fit_transform</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="s2">"Eigenvector of largest eigenvalue"</span><span class="p">)</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="n">pca</span><span class="o">.</span><span class="n">components_</span><span class="o">.</span><span class="n">T</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">])</span>
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell_output docutils container">
|
||||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Eigenvalues of Covariance matrix
|
||||
5.130656204495118
|
||||
0.7670299149882235
|
||||
First eigenvector
|
||||
[0.85211808 0.52334957]
|
||||
Second eigenvector
|
||||
[-0.52334957 0.85211808]
|
||||
</pre></div>
|
||||
</div>
|
||||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Eigenvector of largest eigenvalue
|
||||
[0.85211808 0.52334957]
|
||||
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span># diagonalize and obtain eigenvalues, not necessarily sorted
|
||||
EigValues, EigVectors = np.linalg.eig(Cov)
|
||||
# sort eigenvectors and eigenvalues
|
||||
#permute = EigValues.argsort()
|
||||
#EigValues = EigValues[permute]
|
||||
#EigVectors = EigVectors[:,permute]
|
||||
print("Eigenvalues of Covariance matrix")
|
||||
for i in range(2):
|
||||
print(EigValues[i])
|
||||
FirstEigvector = EigVectors[:,0]
|
||||
SecondEigvector = EigVectors[:,1]
|
||||
print("First eigenvector")
|
||||
print(FirstEigvector)
|
||||
print("Second eigenvector")
|
||||
print(SecondEigvector)
|
||||
#thereafter we do a PCA with Scikit-learn
|
||||
from sklearn.decomposition import PCA
|
||||
pca = PCA(n_components = 2)
|
||||
X2Dsl = pca.fit_transform(X)
|
||||
print("Eigenvector of largest eigenvalue")
|
||||
print(pca.components_.T[:, 0])
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
@@ -1137,163 +993,30 @@ First it identifies the hyperplane that lies closest to the data, and then it pr
|
||||
training set, then extracts the first two principal components. First we center the data using either <strong>pandas</strong> or our own code</p>
|
||||
<div class="cell docutils container">
|
||||
<div class="cell_input docutils container">
|
||||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
|
||||
<span class="kn">import</span> <span class="nn">pandas</span> <span class="k">as</span> <span class="nn">pd</span>
|
||||
<span class="kn">from</span> <span class="nn">IPython.display</span> <span class="kn">import</span> <span class="n">display</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">100</span><span class="p">)</span>
|
||||
<span class="c1"># setting up a 10 x 5 vanilla matrix </span>
|
||||
<span class="n">rows</span> <span class="o">=</span> <span class="mi">10</span>
|
||||
<span class="n">cols</span> <span class="o">=</span> <span class="mi">5</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">randn</span><span class="p">(</span><span class="n">rows</span><span class="p">,</span><span class="n">cols</span><span class="p">)</span>
|
||||
<span class="n">df</span> <span class="o">=</span> <span class="n">pd</span><span class="o">.</span><span class="n">DataFrame</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
|
||||
<span class="c1"># Pandas does the centering for us</span>
|
||||
<span class="n">df</span> <span class="o">=</span> <span class="n">df</span> <span class="o">-</span><span class="n">df</span><span class="o">.</span><span class="n">mean</span><span class="p">()</span>
|
||||
<span class="n">display</span><span class="p">(</span><span class="n">df</span><span class="p">)</span>
|
||||
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>import numpy as np
|
||||
import pandas as pd
|
||||
from IPython.display import display
|
||||
np.random.seed(100)
|
||||
# setting up a 10 x 5 vanilla matrix
|
||||
rows = 10
|
||||
cols = 5
|
||||
X = np.random.randn(rows,cols)
|
||||
df = pd.DataFrame(X)
|
||||
# Pandas does the centering for us
|
||||
df = df -df.mean()
|
||||
display(df)
|
||||
|
||||
<span class="c1"># we center it ourselves</span>
|
||||
<span class="n">X_centered</span> <span class="o">=</span> <span class="n">X</span> <span class="o">-</span> <span class="n">X</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">axis</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
|
||||
<span class="c1"># Then check the difference between pandas and our own set up</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="n">X_centered</span><span class="o">-</span><span class="n">df</span><span class="p">)</span>
|
||||
<span class="c1">#Now we do an SVD</span>
|
||||
<span class="n">U</span><span class="p">,</span> <span class="n">s</span><span class="p">,</span> <span class="n">V</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">svd</span><span class="p">(</span><span class="n">X_centered</span><span class="p">)</span>
|
||||
<span class="n">c1</span> <span class="o">=</span> <span class="n">V</span><span class="o">.</span><span class="n">T</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">]</span>
|
||||
<span class="n">c2</span> <span class="o">=</span> <span class="n">V</span><span class="o">.</span><span class="n">T</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">]</span>
|
||||
<span class="n">W2</span> <span class="o">=</span> <span class="n">V</span><span class="o">.</span><span class="n">T</span><span class="p">[:,</span> <span class="p">:</span><span class="mi">2</span><span class="p">]</span>
|
||||
<span class="n">X2D</span> <span class="o">=</span> <span class="n">X_centered</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">W2</span><span class="p">)</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="n">X2D</span><span class="p">)</span>
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell_output docutils container">
|
||||
<div class="output text_html"><div>
|
||||
<style scoped>
|
||||
.dataframe tbody tr th:only-of-type {
|
||||
vertical-align: middle;
|
||||
}
|
||||
|
||||
.dataframe tbody tr th {
|
||||
vertical-align: top;
|
||||
}
|
||||
|
||||
.dataframe thead th {
|
||||
text-align: right;
|
||||
}
|
||||
</style>
|
||||
<table border="1" class="dataframe">
|
||||
<thead>
|
||||
<tr style="text-align: right;">
|
||||
<th></th>
|
||||
<th>0</th>
|
||||
<th>1</th>
|
||||
<th>2</th>
|
||||
<th>3</th>
|
||||
<th>4</th>
|
||||
</tr>
|
||||
</thead>
|
||||
<tbody>
|
||||
<tr>
|
||||
<th>0</th>
|
||||
<td>-1.574465</td>
|
||||
<td>0.259153</td>
|
||||
<td>1.197370</td>
|
||||
<td>0.147400</td>
|
||||
<td>0.649382</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<th>1</th>
|
||||
<td>0.689519</td>
|
||||
<td>0.137652</td>
|
||||
<td>-1.025709</td>
|
||||
<td>0.210340</td>
|
||||
<td>-0.076938</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<th>2</th>
|
||||
<td>-0.282727</td>
|
||||
<td>0.351636</td>
|
||||
<td>-0.539261</td>
|
||||
<td>1.216683</td>
|
||||
<td>0.340782</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<th>3</th>
|
||||
<td>0.070889</td>
|
||||
<td>-0.614808</td>
|
||||
<td>1.074067</td>
|
||||
<td>-0.038300</td>
|
||||
<td>-1.450257</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<th>4</th>
|
||||
<td>1.794282</td>
|
||||
<td>1.458078</td>
|
||||
<td>-0.207545</td>
|
||||
<td>-0.442600</td>
|
||||
<td>-0.147420</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<th>5</th>
|
||||
<td>1.112383</td>
|
||||
<td>0.647473</td>
|
||||
<td>1.405890</td>
|
||||
<td>0.073598</td>
|
||||
<td>-0.276263</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<th>6</th>
|
||||
<td>0.397700</td>
|
||||
<td>-1.526744</td>
|
||||
<td>-0.712018</td>
|
||||
<td>1.216290</td>
|
||||
<td>0.418506</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<th>7</th>
|
||||
<td>-0.280647</td>
|
||||
<td>1.106095</td>
|
||||
<td>-1.646283</td>
|
||||
<td>-0.956563</td>
|
||||
<td>-1.564374</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<th>8</th>
|
||||
<td>-0.369139</td>
|
||||
<td>-0.751699</td>
|
||||
<td>0.051649</td>
|
||||
<td>-0.213103</td>
|
||||
<td>0.967809</td>
|
||||
</tr>
|
||||
<tr>
|
||||
<th>9</th>
|
||||
<td>-1.557795</td>
|
||||
<td>-1.066837</td>
|
||||
<td>0.401842</td>
|
||||
<td>-1.213743</td>
|
||||
<td>1.138775</td>
|
||||
</tr>
|
||||
</tbody>
|
||||
</table>
|
||||
</div></div><div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span> 0 1 2 3 4
|
||||
0 0.0 0.0 0.0 0.0 0.0
|
||||
1 0.0 0.0 0.0 0.0 0.0
|
||||
2 0.0 0.0 0.0 0.0 0.0
|
||||
3 0.0 0.0 0.0 0.0 0.0
|
||||
4 0.0 0.0 0.0 0.0 0.0
|
||||
5 0.0 0.0 0.0 0.0 0.0
|
||||
6 0.0 0.0 0.0 0.0 0.0
|
||||
7 0.0 0.0 0.0 0.0 0.0
|
||||
8 0.0 0.0 0.0 0.0 0.0
|
||||
9 0.0 0.0 0.0 0.0 0.0
|
||||
[[-1.5378811 0.94639099]
|
||||
[ 0.86145244 -0.89288636]
|
||||
[-0.00445655 -0.81633628]
|
||||
[ 0.07145103 1.00433417]
|
||||
[ 2.03707133 0.48476997]
|
||||
[ 0.72174172 1.4557763 ]
|
||||
[-0.55854694 -1.60673226]
|
||||
[ 1.6999536 -0.43766686]
|
||||
[-1.10405456 -0.31718909]
|
||||
[-2.18673098 0.17953942]]
|
||||
# we center it ourselves
|
||||
X_centered = X - X.mean(axis=0)
|
||||
# Then check the difference between pandas and our own set up
|
||||
print(X_centered-df)
|
||||
#Now we do an SVD
|
||||
U, s, V = np.linalg.svd(X_centered)
|
||||
c1 = V.T[:, 0]
|
||||
c2 = V.T[:, 1]
|
||||
W2 = V.T[:, :2]
|
||||
X2D = X_centered.dot(W2)
|
||||
print(X2D)
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
@@ -1306,8 +1029,8 @@ down to <span class="math notranslate nohighlight">\(d\)</span> dimensions by pr
|
||||
Selecting this hyperplane ensures that the projection will preserve as much variance as possible.</p>
|
||||
<div class="cell docutils container">
|
||||
<div class="cell_input docutils container">
|
||||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">W2</span> <span class="o">=</span> <span class="n">V</span><span class="o">.</span><span class="n">T</span><span class="p">[:,</span> <span class="p">:</span><span class="mi">2</span><span class="p">]</span>
|
||||
<span class="n">X2D</span> <span class="o">=</span> <span class="n">X_centered</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">W2</span><span class="p">)</span>
|
||||
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>W2 = V.T[:, :2]
|
||||
X2D = X_centered.dot(W2)
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
@@ -1320,25 +1043,11 @@ following code applies PCA to reduce the dimensionality of the dataset down to t
|
||||
that it automatically takes care of centering the data):</p>
|
||||
<div class="cell docutils container">
|
||||
<div class="cell_input docutils container">
|
||||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1">#thereafter we do a PCA with Scikit-learn</span>
|
||||
<span class="kn">from</span> <span class="nn">sklearn.decomposition</span> <span class="kn">import</span> <span class="n">PCA</span>
|
||||
<span class="n">pca</span> <span class="o">=</span> <span class="n">PCA</span><span class="p">(</span><span class="n">n_components</span> <span class="o">=</span> <span class="mi">2</span><span class="p">)</span>
|
||||
<span class="n">X2D</span> <span class="o">=</span> <span class="n">pca</span><span class="o">.</span><span class="n">fit_transform</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="n">X2D</span><span class="p">)</span>
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell_output docutils container">
|
||||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>[[-1.5378811 0.94639099]
|
||||
[ 0.86145244 -0.89288636]
|
||||
[-0.00445655 -0.81633628]
|
||||
[ 0.07145103 1.00433417]
|
||||
[ 2.03707133 0.48476997]
|
||||
[ 0.72174172 1.4557763 ]
|
||||
[-0.55854694 -1.60673226]
|
||||
[ 1.6999536 -0.43766686]
|
||||
[-1.10405456 -0.31718909]
|
||||
[-2.18673098 0.17953942]]
|
||||
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>#thereafter we do a PCA with Scikit-learn
|
||||
from sklearn.decomposition import PCA
|
||||
pca = PCA(n_components = 2)
|
||||
X2D = pca.fit_transform(X)
|
||||
print(X2D)
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
@@ -1348,12 +1057,7 @@ components variable (note that it contains the PCs as horizontal vectors, so, fo
|
||||
principal component is equal to</p>
|
||||
<div class="cell docutils container">
|
||||
<div class="cell_input docutils container">
|
||||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">pca</span><span class="o">.</span><span class="n">components_</span><span class="o">.</span><span class="n">T</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">]</span>
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell_output docutils container">
|
||||
<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>array([ 0.62373464, 0.5303329 , -0.317367 , -0.01873344, -0.47815203])
|
||||
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>pca.components_.T[:, 0]
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
@@ -1368,51 +1072,34 @@ variance that lies along the axis of each principal component.</p>
|
||||
Here we compute performance scores on the training data using logistic regression.</p>
|
||||
<div class="cell docutils container">
|
||||
<div class="cell_input docutils container">
|
||||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
|
||||
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
|
||||
<span class="kn">from</span> <span class="nn">sklearn.model_selection</span> <span class="kn">import</span> <span class="n">train_test_split</span>
|
||||
<span class="kn">from</span> <span class="nn">sklearn.datasets</span> <span class="kn">import</span> <span class="n">load_breast_cancer</span>
|
||||
<span class="kn">from</span> <span class="nn">sklearn.linear_model</span> <span class="kn">import</span> <span class="n">LogisticRegression</span>
|
||||
<span class="n">cancer</span> <span class="o">=</span> <span class="n">load_breast_cancer</span><span class="p">()</span>
|
||||
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>import matplotlib.pyplot as plt
|
||||
import numpy as np
|
||||
from sklearn.model_selection import train_test_split
|
||||
from sklearn.datasets import load_breast_cancer
|
||||
from sklearn.linear_model import LogisticRegression
|
||||
cancer = load_breast_cancer()
|
||||
|
||||
<span class="n">X_train</span><span class="p">,</span> <span class="n">X_test</span><span class="p">,</span> <span class="n">y_train</span><span class="p">,</span> <span class="n">y_test</span> <span class="o">=</span> <span class="n">train_test_split</span><span class="p">(</span><span class="n">cancer</span><span class="o">.</span><span class="n">data</span><span class="p">,</span><span class="n">cancer</span><span class="o">.</span><span class="n">target</span><span class="p">,</span><span class="n">random_state</span><span class="o">=</span><span class="mi">0</span><span class="p">)</span>
|
||||
X_train, X_test, y_train, y_test = train_test_split(cancer.data,cancer.target,random_state=0)
|
||||
|
||||
<span class="n">logreg</span> <span class="o">=</span> <span class="n">LogisticRegression</span><span class="p">()</span>
|
||||
<span class="n">logreg</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span> <span class="n">y_train</span><span class="p">)</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="s2">"Train set accuracy from Logistic Regression: </span><span class="si">{:.2f}</span><span class="s2">"</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">logreg</span><span class="o">.</span><span class="n">score</span><span class="p">(</span><span class="n">X_train</span><span class="p">,</span><span class="n">y_train</span><span class="p">)))</span>
|
||||
<span class="c1"># We scale the data</span>
|
||||
<span class="kn">from</span> <span class="nn">sklearn.preprocessing</span> <span class="kn">import</span> <span class="n">StandardScaler</span>
|
||||
<span class="n">scaler</span> <span class="o">=</span> <span class="n">StandardScaler</span><span class="p">()</span>
|
||||
<span class="n">scaler</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train</span><span class="p">)</span>
|
||||
<span class="n">X_train_scaled</span> <span class="o">=</span> <span class="n">scaler</span><span class="o">.</span><span class="n">transform</span><span class="p">(</span><span class="n">X_train</span><span class="p">)</span>
|
||||
<span class="n">X_test_scaled</span> <span class="o">=</span> <span class="n">scaler</span><span class="o">.</span><span class="n">transform</span><span class="p">(</span><span class="n">X_test</span><span class="p">)</span>
|
||||
<span class="c1"># Then perform again a log reg fit</span>
|
||||
<span class="n">logreg</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train_scaled</span><span class="p">,</span> <span class="n">y_train</span><span class="p">)</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="s2">"Train set accuracy scaled data: </span><span class="si">{:.2f}</span><span class="s2">"</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">logreg</span><span class="o">.</span><span class="n">score</span><span class="p">(</span><span class="n">X_train_scaled</span><span class="p">,</span><span class="n">y_train</span><span class="p">)))</span>
|
||||
<span class="c1">#thereafter we do a PCA with Scikit-learn</span>
|
||||
<span class="kn">from</span> <span class="nn">sklearn.decomposition</span> <span class="kn">import</span> <span class="n">PCA</span>
|
||||
<span class="n">pca</span> <span class="o">=</span> <span class="n">PCA</span><span class="p">(</span><span class="n">n_components</span> <span class="o">=</span> <span class="mi">2</span><span class="p">)</span>
|
||||
<span class="n">X2D_train</span> <span class="o">=</span> <span class="n">pca</span><span class="o">.</span><span class="n">fit_transform</span><span class="p">(</span><span class="n">X_train_scaled</span><span class="p">)</span>
|
||||
<span class="c1"># and finally compute the log reg fit and the score on the training data </span>
|
||||
<span class="n">logreg</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X2D_train</span><span class="p">,</span><span class="n">y_train</span><span class="p">)</span>
|
||||
<span class="nb">print</span><span class="p">(</span><span class="s2">"Train set accuracy scaled and PCA data: </span><span class="si">{:.2f}</span><span class="s2">"</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">logreg</span><span class="o">.</span><span class="n">score</span><span class="p">(</span><span class="n">X2D_train</span><span class="p">,</span><span class="n">y_train</span><span class="p">)))</span>
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
<div class="cell_output docutils container">
|
||||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Train set accuracy from Logistic Regression: 0.96
|
||||
Train set accuracy scaled data: 0.99
|
||||
Train set accuracy scaled and PCA data: 0.96
|
||||
</pre></div>
|
||||
</div>
|
||||
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/Users/mhjensen/miniforge3/envs/myenv/lib/python3.9/site-packages/sklearn/linear_model/_logistic.py:465: ConvergenceWarning: lbfgs failed to converge (status=1):
|
||||
STOP: TOTAL NO. of ITERATIONS REACHED LIMIT.
|
||||
|
||||
Increase the number of iterations (max_iter) or scale the data as shown in:
|
||||
https://scikit-learn.org/stable/modules/preprocessing.html
|
||||
Please also refer to the documentation for alternative solver options:
|
||||
https://scikit-learn.org/stable/modules/linear_model.html#logistic-regression
|
||||
n_iter_i = _check_optimize_result(
|
||||
logreg = LogisticRegression()
|
||||
logreg.fit(X_train, y_train)
|
||||
print("Train set accuracy from Logistic Regression: {:.2f}".format(logreg.score(X_train,y_train)))
|
||||
# We scale the data
|
||||
from sklearn.preprocessing import StandardScaler
|
||||
scaler = StandardScaler()
|
||||
scaler.fit(X_train)
|
||||
X_train_scaled = scaler.transform(X_train)
|
||||
X_test_scaled = scaler.transform(X_test)
|
||||
# Then perform again a log reg fit
|
||||
logreg.fit(X_train_scaled, y_train)
|
||||
print("Train set accuracy scaled data: {:.2f}".format(logreg.score(X_train_scaled,y_train)))
|
||||
#thereafter we do a PCA with Scikit-learn
|
||||
from sklearn.decomposition import PCA
|
||||
pca = PCA(n_components = 2)
|
||||
X2D_train = pca.fit_transform(X_train_scaled)
|
||||
# and finally compute the log reg fit and the score on the training data
|
||||
logreg.fit(X2D_train,y_train)
|
||||
print("Train set accuracy scaled and PCA data: {:.2f}".format(logreg.score(X2D_train,y_train)))
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
@@ -1426,10 +1113,10 @@ The following code computes PCA without reducing dimensionality, then computes t
|
||||
of dimensions required to preserve 95% of the training set’s variance:</p>
|
||||
<div class="cell docutils container">
|
||||
<div class="cell_input docutils container">
|
||||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">pca</span> <span class="o">=</span> <span class="n">PCA</span><span class="p">()</span>
|
||||
<span class="n">pca</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
|
||||
<span class="n">cumsum</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">cumsum</span><span class="p">(</span><span class="n">pca</span><span class="o">.</span><span class="n">explained_variance_ratio_</span><span class="p">)</span>
|
||||
<span class="n">d</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">argmax</span><span class="p">(</span><span class="n">cumsum</span> <span class="o">>=</span> <span class="mf">0.95</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span>
|
||||
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>pca = PCA()
|
||||
pca.fit(X)
|
||||
cumsum = np.cumsum(pca.explained_variance_ratio_)
|
||||
d = np.argmax(cumsum >= 0.95) + 1
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
@@ -1439,8 +1126,8 @@ of specifying the number of principal components you want to preserve, you can s
|
||||
a float between 0.0 and 1.0, indicating the ratio of variance you wish to preserve:</p>
|
||||
<div class="cell docutils container">
|
||||
<div class="cell_input docutils container">
|
||||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">pca</span> <span class="o">=</span> <span class="n">PCA</span><span class="p">(</span><span class="n">n_components</span><span class="o">=</span><span class="mf">0.95</span><span class="p">)</span>
|
||||
<span class="n">X_reduced</span> <span class="o">=</span> <span class="n">pca</span><span class="o">.</span><span class="n">fit_transform</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
|
||||
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>pca = PCA(n_components=0.95)
|
||||
X_reduced = pca.fit_transform(X)
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
@@ -1473,9 +1160,9 @@ twisted manifold.
|
||||
For example, the following code uses Scikit-Learn’s KernelPCA class to perform kPCA with an</p>
|
||||
<div class="cell docutils container">
|
||||
<div class="cell_input docutils container">
|
||||
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">sklearn.decomposition</span> <span class="kn">import</span> <span class="n">KernelPCA</span>
|
||||
<span class="n">rbf_pca</span> <span class="o">=</span> <span class="n">KernelPCA</span><span class="p">(</span><span class="n">n_components</span> <span class="o">=</span> <span class="mi">2</span><span class="p">,</span> <span class="n">kernel</span><span class="o">=</span><span class="s2">"rbf"</span><span class="p">,</span> <span class="n">gamma</span><span class="o">=</span><span class="mf">0.04</span><span class="p">)</span>
|
||||
<span class="n">X_reduced</span> <span class="o">=</span> <span class="n">rbf_pca</span><span class="o">.</span><span class="n">fit_transform</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
|
||||
<div class="highlight-none notranslate"><div class="highlight"><pre><span></span>from sklearn.decomposition import KernelPCA
|
||||
rbf_pca = KernelPCA(n_components = 2, kernel="rbf", gamma=0.04)
|
||||
X_reduced = rbf_pca.fit_transform(X)
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
|
||||
Reference in New Issue
Block a user