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08.sklearn_stock_market.slides.html
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08.sklearn_stock_market.slides.html
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<div class=" highlight hl-ipython2"><pre><span></span><span class="c1"># Author: Gael Varoquaux gael.varoquaux@normalesup.org</span>
<span class="c1"># License: BSD 3 clause</span>
<span class="o">%</span><span class="k">matplotlib</span> inline
<span class="kn">import</span> <span class="nn">datetime</span>
<span class="kn">import</span> <span class="nn">numpy</span> <span class="kn">as</span> <span class="nn">np</span>
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="kn">as</span> <span class="nn">plt</span>
<span class="k">try</span><span class="p">:</span>
<span class="kn">from</span> <span class="nn">matplotlib.finance</span> <span class="kn">import</span> <span class="n">quotes_historical_yahoo</span>
<span class="k">except</span> <span class="ne">ImportError</span><span class="p">:</span>
<span class="kn">from</span> <span class="nn">matplotlib.finance</span> <span class="kn">import</span> <span class="n">quotes_historical_yahoo_ochl</span> <span class="k">as</span> <span class="n">quotes_historical_yahoo</span>
<span class="kn">from</span> <span class="nn">matplotlib.collections</span> <span class="kn">import</span> <span class="n">LineCollection</span>
<span class="kn">from</span> <span class="nn">sklearn</span> <span class="kn">import</span> <span class="n">cluster</span><span class="p">,</span> <span class="n">covariance</span><span class="p">,</span> <span class="n">manifold</span>
</pre></div>
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<div class=" highlight hl-ipython2"><pre><span></span><span class="c1"># Retrieve the data from Internet</span>
<span class="c1"># Choose a time period reasonnably calm (not too long ago so that we get</span>
<span class="c1"># high-tech firms, and before the 2008 crash)</span>
<span class="n">d1</span> <span class="o">=</span> <span class="n">datetime</span><span class="o">.</span><span class="n">datetime</span><span class="p">(</span><span class="mi">2003</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="n">d2</span> <span class="o">=</span> <span class="n">datetime</span><span class="o">.</span><span class="n">datetime</span><span class="p">(</span><span class="mi">2008</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="c1"># kraft symbol has now changed from KFT to MDLZ in yahoo</span>
<span class="n">symbol_dict</span> <span class="o">=</span> <span class="p">{</span>
<span class="s1">'TOT'</span><span class="p">:</span> <span class="s1">'Total'</span><span class="p">,</span>
<span class="s1">'XOM'</span><span class="p">:</span> <span class="s1">'Exxon'</span><span class="p">,</span>
<span class="s1">'CVX'</span><span class="p">:</span> <span class="s1">'Chevron'</span><span class="p">,</span>
<span class="s1">'COP'</span><span class="p">:</span> <span class="s1">'ConocoPhillips'</span><span class="p">,</span>
<span class="s1">'VLO'</span><span class="p">:</span> <span class="s1">'Valero Energy'</span><span class="p">,</span>
<span class="s1">'MSFT'</span><span class="p">:</span> <span class="s1">'Microsoft'</span><span class="p">,</span>
<span class="s1">'IBM'</span><span class="p">:</span> <span class="s1">'IBM'</span><span class="p">,</span>
<span class="s1">'TWX'</span><span class="p">:</span> <span class="s1">'Time Warner'</span><span class="p">,</span>
<span class="s1">'CMCSA'</span><span class="p">:</span> <span class="s1">'Comcast'</span><span class="p">,</span>
<span class="s1">'CVC'</span><span class="p">:</span> <span class="s1">'Cablevision'</span><span class="p">,</span>
<span class="s1">'YHOO'</span><span class="p">:</span> <span class="s1">'Yahoo'</span><span class="p">,</span>
<span class="s1">'DELL'</span><span class="p">:</span> <span class="s1">'Dell'</span><span class="p">,</span>
<span class="s1">'HPQ'</span><span class="p">:</span> <span class="s1">'HP'</span><span class="p">,</span>
<span class="s1">'AMZN'</span><span class="p">:</span> <span class="s1">'Amazon'</span><span class="p">,</span>
<span class="s1">'TM'</span><span class="p">:</span> <span class="s1">'Toyota'</span><span class="p">,</span>
<span class="s1">'CAJ'</span><span class="p">:</span> <span class="s1">'Canon'</span><span class="p">,</span>
<span class="s1">'MTU'</span><span class="p">:</span> <span class="s1">'Mitsubishi'</span><span class="p">,</span>
<span class="s1">'SNE'</span><span class="p">:</span> <span class="s1">'Sony'</span><span class="p">,</span>
<span class="s1">'F'</span><span class="p">:</span> <span class="s1">'Ford'</span><span class="p">,</span>
<span class="s1">'HMC'</span><span class="p">:</span> <span class="s1">'Honda'</span><span class="p">,</span>
<span class="s1">'NAV'</span><span class="p">:</span> <span class="s1">'Navistar'</span><span class="p">,</span>
<span class="s1">'NOC'</span><span class="p">:</span> <span class="s1">'Northrop Grumman'</span><span class="p">,</span>
<span class="s1">'BA'</span><span class="p">:</span> <span class="s1">'Boeing'</span><span class="p">,</span>
<span class="s1">'KO'</span><span class="p">:</span> <span class="s1">'Coca Cola'</span><span class="p">,</span>
<span class="s1">'MMM'</span><span class="p">:</span> <span class="s1">'3M'</span><span class="p">,</span>
<span class="s1">'MCD'</span><span class="p">:</span> <span class="s1">'Mc Donalds'</span><span class="p">,</span>
<span class="s1">'PEP'</span><span class="p">:</span> <span class="s1">'Pepsi'</span><span class="p">,</span>
<span class="s1">'MDLZ'</span><span class="p">:</span> <span class="s1">'Kraft Foods'</span><span class="p">,</span>
<span class="s1">'K'</span><span class="p">:</span> <span class="s1">'Kellogg'</span><span class="p">,</span>
<span class="s1">'UN'</span><span class="p">:</span> <span class="s1">'Unilever'</span><span class="p">,</span>
<span class="s1">'MAR'</span><span class="p">:</span> <span class="s1">'Marriott'</span><span class="p">,</span>
<span class="s1">'PG'</span><span class="p">:</span> <span class="s1">'Procter Gamble'</span><span class="p">,</span>
<span class="s1">'CL'</span><span class="p">:</span> <span class="s1">'Colgate-Palmolive'</span><span class="p">,</span>
<span class="s1">'GE'</span><span class="p">:</span> <span class="s1">'General Electrics'</span><span class="p">,</span>
<span class="s1">'WFC'</span><span class="p">:</span> <span class="s1">'Wells Fargo'</span><span class="p">,</span>
<span class="s1">'JPM'</span><span class="p">:</span> <span class="s1">'JPMorgan Chase'</span><span class="p">,</span>
<span class="s1">'AIG'</span><span class="p">:</span> <span class="s1">'AIG'</span><span class="p">,</span>
<span class="s1">'AXP'</span><span class="p">:</span> <span class="s1">'American express'</span><span class="p">,</span>
<span class="s1">'BAC'</span><span class="p">:</span> <span class="s1">'Bank of America'</span><span class="p">,</span>
<span class="s1">'GS'</span><span class="p">:</span> <span class="s1">'Goldman Sachs'</span><span class="p">,</span>
<span class="s1">'AAPL'</span><span class="p">:</span> <span class="s1">'Apple'</span><span class="p">,</span>
<span class="s1">'SAP'</span><span class="p">:</span> <span class="s1">'SAP'</span><span class="p">,</span>
<span class="s1">'CSCO'</span><span class="p">:</span> <span class="s1">'Cisco'</span><span class="p">,</span>
<span class="s1">'TXN'</span><span class="p">:</span> <span class="s1">'Texas instruments'</span><span class="p">,</span>
<span class="s1">'XRX'</span><span class="p">:</span> <span class="s1">'Xerox'</span><span class="p">,</span>
<span class="s1">'LMT'</span><span class="p">:</span> <span class="s1">'Lookheed Martin'</span><span class="p">,</span>
<span class="s1">'WMT'</span><span class="p">:</span> <span class="s1">'Wal-Mart'</span><span class="p">,</span>
<span class="s1">'WBA'</span><span class="p">:</span> <span class="s1">'Walgreen'</span><span class="p">,</span>
<span class="s1">'HD'</span><span class="p">:</span> <span class="s1">'Home Depot'</span><span class="p">,</span>
<span class="s1">'GSK'</span><span class="p">:</span> <span class="s1">'GlaxoSmithKline'</span><span class="p">,</span>
<span class="s1">'PFE'</span><span class="p">:</span> <span class="s1">'Pfizer'</span><span class="p">,</span>
<span class="s1">'SNY'</span><span class="p">:</span> <span class="s1">'Sanofi-Aventis'</span><span class="p">,</span>
<span class="s1">'NVS'</span><span class="p">:</span> <span class="s1">'Novartis'</span><span class="p">,</span>
<span class="s1">'KMB'</span><span class="p">:</span> <span class="s1">'Kimberly-Clark'</span><span class="p">,</span>
<span class="s1">'R'</span><span class="p">:</span> <span class="s1">'Ryder'</span><span class="p">,</span>
<span class="s1">'GD'</span><span class="p">:</span> <span class="s1">'General Dynamics'</span><span class="p">,</span>
<span class="s1">'RTN'</span><span class="p">:</span> <span class="s1">'Raytheon'</span><span class="p">,</span>
<span class="s1">'CVS'</span><span class="p">:</span> <span class="s1">'CVS'</span><span class="p">,</span>
<span class="s1">'CAT'</span><span class="p">:</span> <span class="s1">'Caterpillar'</span><span class="p">,</span>
<span class="s1">'DD'</span><span class="p">:</span> <span class="s1">'DuPont de Nemours'</span><span class="p">}</span>
<span class="n">symbols</span><span class="p">,</span> <span class="n">names</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="n">symbol_dict</span><span class="o">.</span><span class="n">items</span><span class="p">()))</span><span class="o">.</span><span class="n">T</span>
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<div class=" highlight hl-ipython2"><pre><span></span><span class="n">quotes</span> <span class="o">=</span> <span class="p">[</span><span class="n">quotes_historical_yahoo</span><span class="p">(</span><span class="n">symbol</span><span class="p">,</span> <span class="n">d1</span><span class="p">,</span> <span class="n">d2</span><span class="p">,</span> <span class="n">asobject</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
<span class="k">for</span> <span class="n">symbol</span> <span class="ow">in</span> <span class="n">symbols</span><span class="p">]</span>
<span class="n">stock_open</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="n">q</span><span class="o">.</span><span class="n">open</span> <span class="k">for</span> <span class="n">q</span> <span class="ow">in</span> <span class="n">quotes</span><span class="p">])</span><span class="o">.</span><span class="n">astype</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">float</span><span class="p">)</span>
<span class="n">stock_close</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="n">q</span><span class="o">.</span><span class="n">close</span> <span class="k">for</span> <span class="n">q</span> <span class="ow">in</span> <span class="n">quotes</span><span class="p">])</span><span class="o">.</span><span class="n">astype</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">float</span><span class="p">)</span>
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<pre>/Users/chengjun/anaconda/lib/python2.7/site-packages/matplotlib/finance.py:485: MatplotlibDeprecationWarning: This function has been deprecated in 1.4 in favor of `quotes_historical_yahoo_ochl`, which maintains the original argument order, or `quotes_historical_yahoo_ohlc`, which uses the open-high-low-close order. This function will be removed in 1.5
mplDeprecation)
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<div class=" highlight hl-ipython2"><pre><span></span><span class="kn">import</span> <span class="nn">pickle</span>
<span class="n">output</span> <span class="o">=</span> <span class="nb">open</span><span class="p">(</span><span class="s1">'/Users/chengjun/GitHub/cjc2016/data/stock_open.pkl'</span><span class="p">,</span> <span class="s1">'wb'</span><span class="p">)</span>
<span class="c1"># Pickle dictionary using protocol 0.</span>
<span class="n">pickle</span><span class="o">.</span><span class="n">dump</span><span class="p">(</span><span class="n">stock_open</span><span class="p">,</span> <span class="n">output</span><span class="p">)</span>
<span class="n">output</span><span class="o">.</span><span class="n">close</span><span class="p">()</span>
<span class="n">output</span> <span class="o">=</span> <span class="nb">open</span><span class="p">(</span><span class="s1">'/Users/chengjun/GitHub/cjc2016/data/stock_close.pkl'</span><span class="p">,</span> <span class="s1">'wb'</span><span class="p">)</span>
<span class="c1"># Pickle dictionary using protocol 0.</span>
<span class="n">pickle</span><span class="o">.</span><span class="n">dump</span><span class="p">(</span><span class="n">stock_close</span><span class="p">,</span> <span class="n">output</span><span class="p">)</span>
<span class="n">output</span><span class="o">.</span><span class="n">close</span><span class="p">()</span>
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<h1 id="Load-Pickle-Data">Load Pickle Data<a class="anchor-link" href="#Load-Pickle-Data">¶</a></h1>
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<div class=" highlight hl-ipython2"><pre><span></span><span class="kn">import</span> <span class="nn">pickle</span>
<span class="n">data</span> <span class="o">=</span> <span class="nb">open</span><span class="p">(</span><span class="s1">'/Users/chengjun/GitHub/cjc2016/data/stock_open.pkl'</span><span class="p">,</span> <span class="s1">'r'</span><span class="p">)</span>
<span class="c1"># Pickle dictionary using protocol 0.</span>
<span class="n">stock_open</span> <span class="o">=</span> <span class="n">pickle</span><span class="o">.</span><span class="n">load</span><span class="p">(</span><span class="n">data</span><span class="p">)</span>
<span class="n">data</span><span class="o">.</span><span class="n">close</span><span class="p">()</span>
<span class="n">data</span> <span class="o">=</span> <span class="nb">open</span><span class="p">(</span><span class="s1">'/Users/chengjun/GitHub/cjc2016/data/stock_close.pkl'</span><span class="p">,</span> <span class="s1">'r'</span><span class="p">)</span>
<span class="c1"># Pickle dictionary using protocol 0.</span>
<span class="n">stock_close</span> <span class="o">=</span> <span class="n">pickle</span><span class="o">.</span><span class="n">load</span><span class="p">(</span><span class="n">data</span><span class="p">)</span>
<span class="n">data</span><span class="o">.</span><span class="n">close</span><span class="p">()</span>
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<div class=" highlight hl-ipython2"><pre><span></span><span class="c1"># The daily variations of the quotes are what carry most information</span>
<span class="n">variation</span> <span class="o">=</span> <span class="n">stock_close</span> <span class="o">-</span> <span class="n">stock_open</span>
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<div class=" highlight hl-ipython2"><pre><span></span><span class="n">variation</span>
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<pre>array([[ 0.12936386, -0.03172999, 0.20258755, ..., -0.14865278,
0.07157136, -0.44045379],
[ 0.63269122, 0.0799928 , 0.99630258, ..., -1.11184795,
-0.75570651, 1.01629786],
[ 0.34160551, 0.01423356, 0.20638595, ..., -0.79103983,
-0.43507428, -0.57986219],
...,
[ 0.91294843, -0.13416206, 0.29449672, ..., -0.62319818,
-0.33666256, -0.18624163],
[ 0.00920966, -0.05986028, -0.16116061, ..., -0.30019468,
-0.02633234, -0.08426518],
[ 0.06352973, 0.01270637, -0.01694387, ..., -0.19319241,
-0.05201374, -0.63159692]])</pre>
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<h1 id="GraphLassoCV">GraphLassoCV<a class="anchor-link" href="#GraphLassoCV">¶</a></h1><blockquote><p>Sparse inverse covariance w/ cross-validated choice of the l1 penalty</p>
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<div class=" highlight hl-ipython2"><pre><span></span><span class="c1"># Learn a graphical structure from the correlations</span>
<span class="n">edge_model</span> <span class="o">=</span> <span class="n">covariance</span><span class="o">.</span><span class="n">GraphLassoCV</span><span class="p">()</span>
<span class="c1"># standardize the time series: using correlations rather than covariance</span>
<span class="c1"># is more efficient for structure recovery</span>
<span class="n">X</span> <span class="o">=</span> <span class="n">variation</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span><span class="o">.</span><span class="n">T</span>
<span class="n">X</span> <span class="o">/=</span> <span class="n">X</span><span class="o">.</span><span class="n">std</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="n">edge_model</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
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<pre>GraphLassoCV(alphas=4, assume_centered=False, cv=None, enet_tol=0.0001,
max_iter=100, mode='cd', n_jobs=1, n_refinements=4, tol=0.0001,
verbose=False)</pre>
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<h1 id="Cluster-using-affinity-propagation">Cluster using affinity propagation<a class="anchor-link" href="#Cluster-using-affinity-propagation">¶</a></h1>
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<div class=" highlight hl-ipython2"><pre><span></span><span class="n">_</span><span class="p">,</span> <span class="n">labels</span> <span class="o">=</span> <span class="n">cluster</span><span class="o">.</span><span class="n">affinity_propagation</span><span class="p">(</span><span class="n">edge_model</span><span class="o">.</span><span class="n">covariance_</span><span class="p">)</span>
<span class="n">n_labels</span> <span class="o">=</span> <span class="n">labels</span><span class="o">.</span><span class="n">max</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="n">n_labels</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
<span class="k">print</span><span class="p">(</span><span class="s1">'Cluster </span><span class="si">%i</span><span class="s1">: </span><span class="si">%s</span><span class="s1">'</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="s1">', '</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">names</span><span class="p">[</span><span class="n">labels</span> <span class="o">==</span> <span class="n">i</span><span class="p">])))</span>
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<pre>Cluster 1: Pepsi, Coca Cola, Kellogg
Cluster 2: Apple, Amazon, Yahoo
Cluster 3: GlaxoSmithKline, Novartis, Sanofi-Aventis
Cluster 4: Comcast, Time Warner, Cablevision
Cluster 5: ConocoPhillips, Chevron, Total, Valero Energy, Exxon
Cluster 6: CVS, Walgreen
Cluster 7: Navistar, Sony, Marriott, Caterpillar, Canon, Toyota, Honda, Mitsubishi, Xerox, Unilever
Cluster 8: Kimberly-Clark, Colgate-Palmolive, Procter Gamble
Cluster 9: American express, Ryder, Goldman Sachs, Wal-Mart, General Electrics, Pfizer, Wells Fargo, DuPont de Nemours, Bank of America, AIG, Home Depot, Ford, JPMorgan Chase, Mc Donalds
Cluster 10: Microsoft, SAP, 3M, IBM, Texas instruments, HP, Dell, Cisco
Cluster 11: Raytheon, Boeing, Lookheed Martin, General Dynamics, Northrop Grumman
Cluster 12: Kraft Foods
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<div class=" highlight hl-ipython2"><pre><span></span><span class="c1"># Find a low-dimension embedding for visualization: find the best position of</span>
<span class="c1"># the nodes (the stocks) on a 2D plane</span>
<span class="c1"># We use a dense eigen_solver to achieve reproducibility (arpack is</span>
<span class="c1"># initiated with random vectors that we don't control). In addition, we</span>
<span class="c1"># use a large number of neighbors to capture the large-scale structure.</span>
<span class="n">node_position_model</span> <span class="o">=</span> <span class="n">manifold</span><span class="o">.</span><span class="n">LocallyLinearEmbedding</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">eigen_solver</span><span class="o">=</span><span class="s1">'dense'</span><span class="p">,</span> <span class="n">n_neighbors</span><span class="o">=</span><span class="mi">6</span><span class="p">)</span>
<span class="n">embedding</span> <span class="o">=</span> <span class="n">node_position_model</span><span class="o">.</span><span class="n">fit_transform</span><span class="p">(</span><span class="n">X</span><span class="o">.</span><span class="n">T</span><span class="p">)</span><span class="o">.</span><span class="n">T</span>
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<div class=" highlight hl-ipython2"><pre><span></span><span class="c1"># Display a graph of the partial correlations</span>
<span class="n">partial_correlations</span> <span class="o">=</span> <span class="n">edge_model</span><span class="o">.</span><span class="n">precision_</span><span class="o">.</span><span class="n">copy</span><span class="p">()</span>
<span class="n">d</span> <span class="o">=</span> <span class="mi">1</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">np</span><span class="o">.</span><span class="n">diag</span><span class="p">(</span><span class="n">partial_correlations</span><span class="p">))</span>
<span class="n">partial_correlations</span> <span class="o">*=</span> <span class="n">d</span>
<span class="n">partial_correlations</span> <span class="o">*=</span> <span class="n">d</span><span class="p">[:,</span> <span class="n">np</span><span class="o">.</span><span class="n">newaxis</span><span class="p">]</span>
<span class="n">non_zero</span> <span class="o">=</span> <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">triu</span><span class="p">(</span><span class="n">partial_correlations</span><span class="p">,</span> <span class="n">k</span><span class="o">=</span><span class="mi">1</span><span class="p">))</span> <span class="o">></span> <span class="mf">0.02</span><span class="p">)</span>
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<div class=" highlight hl-ipython2"><pre><span></span><span class="c1"># Plot the nodes using the coordinates of our embedding</span>
<span class="n">plt</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">embedding</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">embedding</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">s</span><span class="o">=</span><span class="mi">100</span> <span class="o">*</span> <span class="n">d</span> <span class="o">**</span> <span class="mi">2</span><span class="p">,</span> <span class="n">c</span><span class="o">=</span><span class="n">labels</span><span class="p">,</span>
<span class="n">cmap</span><span class="o">=</span><span class="n">plt</span><span class="o">.</span><span class="n">cm</span><span class="o">.</span><span class="n">spectral</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
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<div class=" highlight hl-ipython2"><pre><span></span><span class="c1"># Visualization</span>
<span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">facecolor</span><span class="o">=</span><span class="s1">'w'</span><span class="p">,</span> <span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span> <span class="mi">8</span><span class="p">))</span>
<span class="n">plt</span><span class="o">.</span><span class="n">clf</span><span class="p">()</span>
<span class="n">ax</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">axes</span><span class="p">([</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">,</span> <span class="mf">1.</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">'off'</span><span class="p">)</span>