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Morten Hjorth-Jensen
@@ -7,8 +7,8 @@
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Decision Trees, Ensemble Methods and Boosting
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@@ -204,7 +210,7 @@ const thebe_selector_output = ".output, .cell_output"
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<h1>Basic ideas of the Principal Component Analysis (PCA)</h1>
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<li class="toc-h2 nav-item toc-entry">
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<a class="reference internal nav-link" href="#introducing-the-covariance-and-correlation-functions">
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11.1. Introducing the Covariance and Correlation functions
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<li class="toc-h2 nav-item toc-entry">
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<a class="reference internal nav-link" href="#correlation-matrix">
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11.2. Correlation Matrix
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</a>
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</li>
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<li class="toc-h2 nav-item toc-entry">
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<a class="reference internal nav-link" href="#towards-the-pca-theorem">
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11.3. Towards the PCA theorem
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</a>
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<ul class="nav section-nav flex-column">
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<li class="toc-h3 nav-item toc-entry">
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<a class="reference internal nav-link" href="#the-algorithm-before-theorem">
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11.3.1. The Algorithm before theorem
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</a>
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<li class="toc-h3 nav-item toc-entry">
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<a class="reference internal nav-link" href="#writing-our-own-pca-code">
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11.3.2. Writing our own PCA code
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</a>
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</li>
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<li class="toc-h3 nav-item toc-entry">
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<a class="reference internal nav-link" href="#diagonalize-the-sample-covariance-matrix-to-obtain-the-principal-components">
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11.3.3. Diagonalize the sample covariance matrix to obtain the principal components
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</a>
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</li>
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</ul>
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</li>
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<li class="toc-h2 nav-item toc-entry">
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<a class="reference internal nav-link" href="#classical-pca-theorem">
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11.4. Classical PCA Theorem
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</a>
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</li>
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<li class="toc-h2 nav-item toc-entry">
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<a class="reference internal nav-link" href="#geometric-interpretation-and-link-with-singular-value-decomposition">
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11.5. Geometric Interpretation and link with Singular Value Decomposition
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</a>
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</li>
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<li class="toc-h2 nav-item toc-entry">
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<a class="reference internal nav-link" href="#pca-and-scikit-learn">
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11.6. PCA and scikit-learn
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</a>
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</li>
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<li class="toc-h2 nav-item toc-entry">
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<a class="reference internal nav-link" href="#back-to-the-cancer-data">
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11.7. Back to the Cancer Data
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</a>
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<ul class="nav section-nav flex-column">
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<li class="toc-h3 nav-item toc-entry">
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<a class="reference internal nav-link" href="#incremental-pca">
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11.7.1. Incremental PCA
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</a>
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</li>
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<li class="toc-h3 nav-item toc-entry">
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<a class="reference internal nav-link" href="#randomized-pca">
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11.7.2. Randomized PCA
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</a>
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</li>
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<li class="toc-h3 nav-item toc-entry">
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<a class="reference internal nav-link" href="#kernel-pca">
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11.7.3. Kernel PCA
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</a>
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</li>
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</ul>
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</li>
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<li class="toc-h2 nav-item toc-entry">
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<a class="reference internal nav-link" href="#other-techniques">
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11.8. Other techniques
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</a>
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</ul>
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</nav>
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</div>
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</div>
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</div>
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<div>
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<div class="tex2jax_ignore mathjax_ignore section" id="basic-ideas-of-the-principal-component-analysis-pca">
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@@ -544,10 +642,10 @@ covariance matrix through the <strong>np.linalg.eig()</strong> function.</p>
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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>0.03382304823545749
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4.021250026482402
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[[0.9252772 2.69061276]
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[2.69061276 8.90540529]]
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>0.046785461905835435
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4.240670854503034
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[[0.94986593 2.88137798]
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[2.88137798 9.93586895]]
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</pre></div>
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</div>
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</div>
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@@ -587,10 +685,10 @@ a more brute force way. Here we scale the mean values for each column of the des
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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>0.08328216846752691
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2.094472507965532
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[[1. 0.67697934]
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[0.67697934 1. ]]
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>0.08271198519070039
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1.7306310662842432
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[[1. 0.58084359]
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[0.58084359 1. ]]
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</pre></div>
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</div>
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</div>
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@@ -619,30 +717,30 @@ this matrix we easily see that it is a positive definite matrix.</p>
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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>[[ 2.38295101 5.51289697]
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[ 0.51803019 2.61399851]
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[-1.09849763 -1.02255619]
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[-0.54016188 -0.53794784]
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[ 0.28634473 1.32721178]
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[-1.66619972 -6.88017651]
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[ 1.52811546 4.29512284]
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[ 0.427017 2.43493232]
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[-0.80842254 -4.37964744]
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[-1.02917662 -3.36383443]]
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<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>[[-0.50488131 -2.2493023 ]
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[-0.26115367 -1.92631966]
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[-1.43556723 -2.99992698]
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[ 0.64528459 2.57643113]
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[-0.55273102 -2.09964817]
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[ 0.31681097 1.26466619]
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[-0.04673082 -0.56607416]
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[ 1.53394148 5.38629412]
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[ 0.15092012 -0.22014758]
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[ 0.15410688 0.83402742]]
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0 1
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0 2.382951 5.512897
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1 0.518030 2.613999
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2 -1.098498 -1.022556
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3 -0.540162 -0.537948
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4 0.286345 1.327212
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5 -1.666200 -6.880177
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6 1.528115 4.295123
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7 0.427017 2.434932
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8 -0.808423 -4.379647
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9 -1.029177 -3.363834
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0 1
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0 1.000000 0.930583
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1 0.930583 1.000000
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0 -0.504881 -2.249302
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1 -0.261154 -1.926320
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2 -1.435567 -2.999927
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3 0.645285 2.576431
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4 -0.552731 -2.099648
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5 0.316811 1.264666
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6 -0.046731 -0.566074
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7 1.533941 5.386294
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8 0.150920 -0.220148
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9 0.154107 0.834027
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0 1
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0 1.00000 0.95302
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1 0.95302 1.00000
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</pre></div>
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</div>
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</div>
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@@ -699,37 +797,37 @@ this matrix we easily see that it is a positive definite matrix.</p>
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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 1 2 3 4 5 6 7 \
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0 0.0 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000
|
||||
1 0.0 0.086074 0.080593 0.088526 0.083276 0.078269 0.081046 0.076604
|
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2 0.0 0.080593 0.076130 0.082896 0.078254 0.073810 0.076216 0.072216
|
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3 0.0 0.088526 0.082896 0.096472 0.091051 0.085879 0.091696 0.086956
|
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4 0.0 0.083276 0.078254 0.091051 0.086112 0.081388 0.086890 0.082536
|
||||
5 0.0 0.078269 0.073810 0.085879 0.081388 0.077083 0.082295 0.078299
|
||||
6 0.0 0.081046 0.076216 0.091696 0.086890 0.082295 0.089501 0.085163
|
||||
7 0.0 0.076604 0.072216 0.086956 0.082536 0.078299 0.085163 0.081144
|
||||
8 0.0 0.072442 0.068461 0.082503 0.078436 0.074530 0.081072 0.077349
|
||||
9 0.0 0.068538 0.064931 0.078314 0.074573 0.070972 0.077211 0.073761
|
||||
10 0.0 0.073062 0.069085 0.084862 0.080738 0.076780 0.084484 0.080647
|
||||
11 0.0 0.069275 0.065641 0.080706 0.076895 0.073230 0.080582 0.077013
|
||||
12 0.0 0.065730 0.062411 0.076804 0.073280 0.069884 0.076905 0.073583
|
||||
13 0.0 0.062409 0.059378 0.073136 0.069877 0.066729 0.073436 0.070343
|
||||
14 0.0 0.059294 0.056528 0.069685 0.066670 0.063752 0.070163 0.067282
|
||||
1 0.0 0.088104 0.081216 0.088760 0.086997 0.085010 0.080343 0.079306
|
||||
2 0.0 0.081216 0.075612 0.080764 0.079618 0.078300 0.072530 0.071897
|
||||
3 0.0 0.088760 0.080764 0.094925 0.092249 0.089310 0.089233 0.087615
|
||||
4 0.0 0.086997 0.079618 0.092249 0.089982 0.087470 0.086257 0.084927
|
||||
5 0.0 0.085010 0.078300 0.089310 0.087470 0.085410 0.083021 0.081989
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||||
6 0.0 0.080343 0.072530 0.089233 0.086257 0.083021 0.086109 0.084269
|
||||
7 0.0 0.079306 0.071897 0.087615 0.084927 0.081989 0.084269 0.082642
|
||||
8 0.0 0.078323 0.071329 0.086021 0.083629 0.080998 0.082431 0.081022
|
||||
9 0.0 0.077372 0.070810 0.084426 0.082339 0.080026 0.080571 0.079383
|
||||
10 0.0 0.071946 0.064637 0.081990 0.079001 0.075770 0.080642 0.078768
|
||||
11 0.0 0.071044 0.064041 0.080699 0.077930 0.074922 0.079218 0.077511
|
||||
12 0.0 0.070222 0.063524 0.079476 0.076927 0.074144 0.077846 0.076306
|
||||
13 0.0 0.069475 0.063084 0.078314 0.075987 0.073433 0.076518 0.075145
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14 0.0 0.068799 0.062719 0.077203 0.075102 0.072783 0.075222 0.074019
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8 9 10 11 12 13 14
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||||
0 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000
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1 0.072442 0.068538 0.073062 0.069275 0.065730 0.062409 0.059294
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2 0.068461 0.064931 0.069085 0.065641 0.062411 0.059378 0.056528
|
||||
3 0.082503 0.078314 0.084862 0.080706 0.076804 0.073136 0.069685
|
||||
4 0.078436 0.074573 0.080738 0.076895 0.073280 0.069877 0.066670
|
||||
5 0.074530 0.070972 0.076780 0.073230 0.069884 0.066729 0.063752
|
||||
6 0.081072 0.077211 0.084484 0.080582 0.076905 0.073436 0.070163
|
||||
7 0.077349 0.073761 0.080647 0.077013 0.073583 0.070343 0.067282
|
||||
8 0.073827 0.070492 0.077015 0.073629 0.070429 0.067402 0.064538
|
||||
9 0.070492 0.067392 0.073575 0.070419 0.067432 0.064604 0.061923
|
||||
10 0.077015 0.073575 0.080984 0.077452 0.074111 0.070950 0.067956
|
||||
11 0.073629 0.070419 0.077452 0.074149 0.071022 0.068059 0.065249
|
||||
12 0.070429 0.067432 0.074111 0.071022 0.068093 0.065314 0.062676
|
||||
13 0.067402 0.064604 0.070950 0.068059 0.065314 0.062706 0.060228
|
||||
14 0.064538 0.061923 0.067956 0.065249 0.062676 0.060228 0.057899
|
||||
1 0.078323 0.077372 0.071946 0.071044 0.070222 0.069475 0.068799
|
||||
2 0.071329 0.070810 0.064637 0.064041 0.063524 0.063084 0.062719
|
||||
3 0.086021 0.084426 0.081990 0.080699 0.079476 0.078314 0.077203
|
||||
4 0.083629 0.082339 0.079001 0.077930 0.076927 0.075987 0.075102
|
||||
5 0.080998 0.080026 0.075770 0.074922 0.074144 0.073433 0.072783
|
||||
6 0.082431 0.080571 0.080642 0.079218 0.077846 0.076518 0.075222
|
||||
7 0.081022 0.079383 0.078768 0.077511 0.076306 0.075145 0.074019
|
||||
8 0.079622 0.078211 0.076886 0.075797 0.074760 0.073768 0.072813
|
||||
9 0.078211 0.077033 0.074969 0.074050 0.073183 0.072364 0.071585
|
||||
10 0.076886 0.074969 0.076632 0.075201 0.073811 0.072452 0.071115
|
||||
11 0.075797 0.074050 0.075201 0.073904 0.072647 0.071421 0.070216
|
||||
12 0.074760 0.073183 0.073811 0.072647 0.071523 0.070428 0.069355
|
||||
13 0.073768 0.072364 0.072452 0.071421 0.070428 0.069465 0.068525
|
||||
14 0.072813 0.071585 0.071115 0.070216 0.069355 0.068525 0.067719
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
@@ -918,10 +1016,10 @@ We can write our own code or simply use either the functionaly of <strong>numpy<
|
||||
</div>
|
||||
<div class="cell_output docutils container">
|
||||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span> 0 1
|
||||
0 4.038506 2.027865
|
||||
1 2.027865 2.037559
|
||||
[[4.03850557 2.02786488]
|
||||
[2.02786488 2.03755944]]
|
||||
0 3.967536 1.983164
|
||||
1 1.983164 2.000755
|
||||
[[3.9675364 1.98316352]
|
||||
[1.98316352 2.00075534]]
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
@@ -948,8 +1046,8 @@ Our own code here is not very elegant and asks for obvious improvements. It is t
|
||||
</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
|
||||
[[4.03850557 2.02786488]
|
||||
[2.02786488 2.03755944]]
|
||||
[[3.9675364 1.98316352]
|
||||
[1.98316352 2.00075534]]
|
||||
</pre></div>
|
||||
</div>
|
||||
<img alt="_images/chapter8_65_1.png" src="_images/chapter8_65_1.png" />
|
||||
@@ -1009,16 +1107,16 @@ questions.</p>
|
||||
</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
|
||||
5.299267190588216
|
||||
0.7767978193240488
|
||||
5.197738983259782
|
||||
0.7705527590466072
|
||||
First eigenvector
|
||||
[0.84924834 0.52799362]
|
||||
[0.84977962 0.52713812]
|
||||
Second eigenvector
|
||||
[-0.52799362 0.84924834]
|
||||
[-0.52713812 0.84977962]
|
||||
</pre></div>
|
||||
</div>
|
||||
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Eigenvector of largest eigenvalue
|
||||
[0.84924834 0.52799362]
|
||||
[-0.84977962 -0.52713812]
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
@@ -1242,16 +1340,16 @@ training set, then extracts the first two principal components. First we center
|
||||
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]]
|
||||
[[-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]]
|
||||
</pre></div>
|
||||
</div>
|
||||
</div>
|
||||
@@ -1475,54 +1573,42 @@ For example, the following code uses Scikit-Learn’s KernelPCA class to perform
|
||||
|
||||
</div>
|
||||
|
||||
|
||||
|
||||
|
||||
|
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|
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