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Introduction to Applied Data Analysis and Machine Learning
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<span class="caption-text">
Supervised Learning
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1. Elements of Probability Theory and Statistical Data Analysis
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<a class="reference internal" href="chapter3.html">
2. Getting started, our first data and Machine Learning encounters
</a>
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<a class="reference internal" href="chapter4.html">
3. Linear Regression and more Advanced Regression Analysis
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4. Logistic Regression
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5. Neural networks, from the simple perceptron to deep learning
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<a class="reference internal" href="chapter7.html">
6. Support Vector Machines, overarching aims
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<a class="current reference internal" href="#">
7. Dimensionality Reduction
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8. Convolutional Neural Networks
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9. Recurrent Neural Networks
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10. Solving ODEs with Deep Learning
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11. Data Analysis and Machine Learning:
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12. Elements of Bayesian theory and Bayesian Neural Networks
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<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#reducing-the-number-of-degrees-of-freedom-overarching-view">
7.1. Reducing the number of degrees of freedom, overarching view
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#preprocessing-our-data">
7.2. Preprocessing our data
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#more-preprocessing">
7.3. More preprocessing
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#simple-preprocessing-examples-franke-function-and-regression">
7.4. Simple preprocessing examples, Franke function and regression
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#simple-preprocessing-examples-breast-cancer-data-and-classification-support-vector-machines">
7.5. Simple preprocessing examples, breast cancer data and classification, Support Vector Machines
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#more-on-cancer-data-now-with-logistic-regression">
7.6. More on Cancer Data, now with Logistic Regression
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#why-should-we-think-of-reducing-the-dimensionality">
7.7. Why should we think of reducing the dimensionality
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#basic-ideas-of-the-principal-component-analysis-pca">
7.8. Basic ideas of the Principal Component Analysis (PCA)
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#introducing-the-covariance-and-correlation-functions">
7.9. Introducing the Covariance and Correlation functions
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#correlation-function-and-design-feature-matrix">
7.10. Correlation Function and Design/Feature Matrix
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#covariance-matrix-examples">
7.11. Covariance Matrix Examples
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#correlation-matrix">
7.12. Correlation Matrix
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#correlation-matrix-with-pandas">
7.13. Correlation Matrix with Pandas
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#correlation-matrix-with-pandas-and-the-franke-function">
7.14. Correlation Matrix with Pandas and the Franke function
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#rewriting-the-covariance-and-or-correlation-matrix">
7.15. Rewriting the Covariance and/or Correlation Matrix
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#towards-the-pca-theorem">
7.16. Towards the PCA theorem
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#the-algorithm-before-theorem">
7.17. The Algorithm before theorem
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#writing-our-own-pca-code">
7.18. Writing our own PCA code
</a>
<ul class="nav section-nav flex-column">
<li class="toc-h3 nav-item toc-entry">
<a class="reference internal nav-link" href="#compute-the-sample-mean-and-center-the-data">
7.18.1. Compute the sample mean and center the data
</a>
</li>
<li class="toc-h3 nav-item toc-entry">
<a class="reference internal nav-link" href="#compute-the-sample-covariance">
7.18.2. Compute the sample covariance
</a>
</li>
<li class="toc-h3 nav-item toc-entry">
<a class="reference internal nav-link" href="#diagonalize-the-sample-covariance-matrix-to-obtain-the-principal-components">
7.18.3. Diagonalize the sample covariance matrix to obtain the principal components
</a>
</li>
</ul>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#classical-pca-theorem">
7.19. Classical PCA Theorem
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#proof-of-the-pca-theorem">
7.20. Proof of the PCA Theorem
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#pca-proof-continued">
7.21. PCA Proof continued
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#the-final-step">
7.22. The final step
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#geometric-interpretation-and-link-with-singular-value-decomposition">
7.23. Geometric Interpretation and link with Singular Value Decomposition
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#principal-component-analysis">
7.24. Principal Component Analysis
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#pca-and-scikit-learn">
7.25. PCA and scikit-learn
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#back-to-the-cancer-data">
7.26. Back to the Cancer Data
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#more-on-the-pca">
7.27. More on the PCA
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#incremental-pca">
7.28. Incremental PCA
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#randomized-pca">
7.29. Randomized PCA
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#kernel-pca">
7.30. Kernel PCA
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#lle">
7.31. LLE
</a>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#other-techniques">
7.32. Other techniques
</a>
</li>
</ul>
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<div class="section" id="dimensionality-reduction">
<h1><span class="section-number">7. </span>Dimensionality Reduction<a class="headerlink" href="#dimensionality-reduction" title="Permalink to this headline"></a></h1>
<div class="section" id="reducing-the-number-of-degrees-of-freedom-overarching-view">
<h2><span class="section-number">7.1. </span>Reducing the number of degrees of freedom, overarching view<a class="headerlink" href="#reducing-the-number-of-degrees-of-freedom-overarching-view" title="Permalink to this headline"></a></h2>
<p>Many Machine Learning problems involve thousands or even millions of
features for each training instance. Not only does this make training
extremely slow, it can also make it much harder to find a good
solution, as we will see. This problem is often referred to as the
curse of dimensionality. Fortunately, in real-world problems, it is
often possible to reduce the number of features considerably, turning
an intractable problem into a tractable one.</p>
<p>Here we will discuss some of the most popular dimensionality reduction
techniques: the principal component analysis (PCA), Kernel PCA, and
Locally Linear Embedding (LLE). Furthermore, we will start by looking
at some simple preprocessing of the data which allow us to rescale the
data.</p>
<p>Principal component analysis and its various variants deal with the
problem of fitting a low-dimensional <a class="reference external" href="https://en.wikipedia.org/wiki/Affine_space">affine
subspace</a> to a set of of
data points in a high-dimensional space. With its family of methods it
is one of the most used tools in data modeling, compression and
visualization.</p>
</div>
<div class="section" id="preprocessing-our-data">
<h2><span class="section-number">7.2. </span>Preprocessing our data<a class="headerlink" href="#preprocessing-our-data" title="Permalink to this headline"></a></h2>
<p>Before we proceed however, we will discuss how to preprocess our
data. Till now and in connection with our previous examples we have
not met so many cases where we are too sensitive to the scaling of our
data. Normally the data may need a rescaling and/or may be sensitive
to extreme values. Scaling the data renders our inputs much more
suitable for the algorithms we want to employ.</p>
<p><strong>Scikit-Learn</strong> has several functions which allow us to rescale the
data, normally resulting in much better results in terms of various
accuracy scores. The <strong>StandardScaler</strong> function in <strong>Scikit-Learn</strong>
ensures that for each feature/predictor we study the mean value is
zero and the variance is one (every column in the design/feature
matrix). This scaling has the drawback that it does not ensure that
we have a particular maximum or minimum in our data set. Another
function included in <strong>Scikit-Learn</strong> is the <strong>MinMaxScaler</strong> which
ensures that all features are exactly between <span class="math notranslate nohighlight">\(0\)</span> and <span class="math notranslate nohighlight">\(1\)</span>. The</p>
</div>
<div class="section" id="more-preprocessing">
<h2><span class="section-number">7.3. </span>More preprocessing<a class="headerlink" href="#more-preprocessing" title="Permalink to this headline"></a></h2>
<p>The <strong>Normalizer</strong> scales each data
point such that the feature vector has a euclidean length of one. In other words, it
projects a data point on the circle (or sphere in the case of higher dimensions) with a
radius of 1. This means every data point is scaled by a different number (by the
inverse of it’s length).
This normalization is often used when only the direction (or angle) of the data matters,
not the length of the feature vector.</p>
<p>The <strong>RobustScaler</strong> works similarly to the StandardScaler in that it
ensures statistical properties for each feature that guarantee that
they are on the same scale. However, the RobustScaler uses the median
and quartiles, instead of mean and variance. This makes the
RobustScaler ignore data points that are very different from the rest
(like measurement errors). These odd data points are also called
outliers, and might often lead to trouble for other scaling
techniques.</p>
</div>
<div class="section" id="simple-preprocessing-examples-franke-function-and-regression">
<h2><span class="section-number">7.4. </span>Simple preprocessing examples, Franke function and regression<a class="headerlink" href="#simple-preprocessing-examples-franke-function-and-regression" title="Permalink to this headline"></a></h2>
<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
<span class="c1"># Common imports</span>
<span class="kn">import</span> <span class="nn">os</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">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">sklearn.linear_model</span> <span class="k">as</span> <span class="nn">skl</span>
<span class="kn">from</span> <span class="nn">sklearn.metrics</span> <span class="kn">import</span> <span class="n">mean_squared_error</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.preprocessing</span> <span class="kn">import</span> <span class="n">MinMaxScaler</span><span class="p">,</span> <span class="n">StandardScaler</span><span class="p">,</span> <span class="n">Normalizer</span>
<span class="kn">from</span> <span class="nn">sklearn.svm</span> <span class="kn">import</span> <span class="n">SVR</span>
<span class="c1"># Where to save the figures and data files</span>
<span class="n">PROJECT_ROOT_DIR</span> <span class="o">=</span> <span class="s2">&quot;Results&quot;</span>
<span class="n">FIGURE_ID</span> <span class="o">=</span> <span class="s2">&quot;Results/FigureFiles&quot;</span>
<span class="n">DATA_ID</span> <span class="o">=</span> <span class="s2">&quot;DataFiles/&quot;</span>
<span class="k">if</span> <span class="ow">not</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">exists</span><span class="p">(</span><span class="n">PROJECT_ROOT_DIR</span><span class="p">):</span>
<span class="n">os</span><span class="o">.</span><span class="n">mkdir</span><span class="p">(</span><span class="n">PROJECT_ROOT_DIR</span><span class="p">)</span>
<span class="k">if</span> <span class="ow">not</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">exists</span><span class="p">(</span><span class="n">FIGURE_ID</span><span class="p">):</span>
<span class="n">os</span><span class="o">.</span><span class="n">makedirs</span><span class="p">(</span><span class="n">FIGURE_ID</span><span class="p">)</span>
<span class="k">if</span> <span class="ow">not</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">exists</span><span class="p">(</span><span class="n">DATA_ID</span><span class="p">):</span>
<span class="n">os</span><span class="o">.</span><span class="n">makedirs</span><span class="p">(</span><span class="n">DATA_ID</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">image_path</span><span class="p">(</span><span class="n">fig_id</span><span class="p">):</span>
<span class="k">return</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">FIGURE_ID</span><span class="p">,</span> <span class="n">fig_id</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">data_path</span><span class="p">(</span><span class="n">dat_id</span><span class="p">):</span>
<span class="k">return</span> <span class="n">os</span><span class="o">.</span><span class="n">path</span><span class="o">.</span><span class="n">join</span><span class="p">(</span><span class="n">DATA_ID</span><span class="p">,</span> <span class="n">dat_id</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">save_fig</span><span class="p">(</span><span class="n">fig_id</span><span class="p">):</span>
<span class="n">plt</span><span class="o">.</span><span class="n">savefig</span><span class="p">(</span><span class="n">image_path</span><span class="p">(</span><span class="n">fig_id</span><span class="p">)</span> <span class="o">+</span> <span class="s2">&quot;.png&quot;</span><span class="p">,</span> <span class="nb">format</span><span class="o">=</span><span class="s1">&#39;png&#39;</span><span class="p">)</span>
<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>
<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">&gt;</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>
<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>
<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>
<span class="k">return</span> <span class="n">X</span>
<span class="c1"># Making meshgrid of datapoints and compute Franke&#39;s function</span>
<span class="n">n</span> <span class="o">=</span> <span class="mi">5</span>
<span class="n">N</span> <span class="o">=</span> <span class="mi">1000</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>
<span class="c1"># split in training and test data</span>
<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">X</span><span class="p">,</span><span class="n">z</span><span class="p">,</span><span class="n">test_size</span><span class="o">=</span><span class="mf">0.2</span><span class="p">)</span>
<span class="n">svm</span> <span class="o">=</span> <span class="n">SVR</span><span class="p">(</span><span class="n">gamma</span><span class="o">=</span><span class="s1">&#39;auto&#39;</span><span class="p">,</span><span class="n">C</span><span class="o">=</span><span class="mf">10.0</span><span class="p">)</span>
<span class="n">svm</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="c1"># The mean squared error and R2 score</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;MSE before scaling: </span><span class="si">{:.2f}</span><span class="s2">&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">mean_squared_error</span><span class="p">(</span><span class="n">svm</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X_test</span><span class="p">),</span> <span class="n">y_test</span><span class="p">)))</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;R2 score before scaling </span><span class="si">{:.2f}</span><span class="s2">&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">svm</span><span class="o">.</span><span class="n">score</span><span class="p">(</span><span class="n">X_test</span><span class="p">,</span><span class="n">y_test</span><span class="p">)))</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="nb">print</span><span class="p">(</span><span class="s2">&quot;Feature min values before scaling:</span><span class="se">\n</span><span class="s2"> </span><span class="si">{}</span><span class="s2">&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">X_train</span><span class="o">.</span><span class="n">min</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="nb">print</span><span class="p">(</span><span class="s2">&quot;Feature max values before scaling:</span><span class="se">\n</span><span class="s2"> </span><span class="si">{}</span><span class="s2">&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">X_train</span><span class="o">.</span><span class="n">max</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="nb">print</span><span class="p">(</span><span class="s2">&quot;Feature min values after scaling:</span><span class="se">\n</span><span class="s2"> </span><span class="si">{}</span><span class="s2">&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">X_train_scaled</span><span class="o">.</span><span class="n">min</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="nb">print</span><span class="p">(</span><span class="s2">&quot;Feature max values after scaling:</span><span class="se">\n</span><span class="s2"> </span><span class="si">{}</span><span class="s2">&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">X_train_scaled</span><span class="o">.</span><span class="n">max</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">svm</span> <span class="o">=</span> <span class="n">SVR</span><span class="p">(</span><span class="n">gamma</span><span class="o">=</span><span class="s1">&#39;auto&#39;</span><span class="p">,</span><span class="n">C</span><span class="o">=</span><span class="mf">10.0</span><span class="p">)</span>
<span class="n">svm</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">&quot;MSE after scaling: </span><span class="si">{:.2f}</span><span class="s2">&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">mean_squared_error</span><span class="p">(</span><span class="n">svm</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X_test_scaled</span><span class="p">),</span> <span class="n">y_test</span><span class="p">)))</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">&quot;R2 score for scaled data: </span><span class="si">{:.2f}</span><span class="s2">&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">svm</span><span class="o">.</span><span class="n">score</span><span class="p">(</span><span class="n">X_test_scaled</span><span class="p">,</span><span class="n">y_test</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>MSE before scaling: 0.01
R2 score before scaling 0.94
Feature min values before scaling:
[1.00000000e+00 3.86972479e-03 3.66528972e-03 1.49747700e-05
1.41836625e-05 1.34343487e-05 5.79482386e-08 5.48868704e-08
5.19872323e-08 4.92407803e-08 2.24243736e-10 2.12397083e-10
2.01176282e-10 1.90548268e-10 1.80481726e-10 8.67761543e-13
8.21918258e-13 7.78496845e-13 7.37369358e-13 6.98414609e-13
6.61517814e-13]
Feature max values before scaling:
[1. 0.99791107 0.99418827 0.99582651 0.99211148 0.98841032
0.9937463 0.99003903 0.9863456 0.98266595 0.99167043 0.98797091
0.9842852 0.98061323 0.97695496 0.9895989 0.98590711 0.98222909
0.9785648 0.97491417 0.97127716]
Feature min values after scaling:
[ 0. -1.66745992 -1.72823447 -1.09228638 -1.1114811 -1.1313648
-0.86506245 -0.87590574 -0.88698187 -0.89829265 -0.73687981 -0.74440021
-0.75205856 -0.75985475 -0.76778823 -0.65190491 -0.65755533 -0.66330693
-0.66916042 -0.67511627 -0.68117476]
Feature max values after scaling:
[0. 1.76330702 1.70952132 2.29506297 2.25330559 2.21033632
2.73427311 2.69911557 2.66307432 2.6261167 3.11835914 3.08796041
3.05688927 3.02512212 2.99263434 3.46490261 3.4379033 3.41038453
3.38232999 3.35372259 3.32454443]
MSE after scaling: 0.00
R2 score for scaled data: 0.97
</pre></div>
</div>
</div>
</div>
</div>
<div class="section" id="simple-preprocessing-examples-breast-cancer-data-and-classification-support-vector-machines">
<h2><span class="section-number">7.5. </span>Simple preprocessing examples, breast cancer data and classification, Support Vector Machines<a class="headerlink" href="#simple-preprocessing-examples-breast-cancer-data-and-classification-support-vector-machines" title="Permalink to this headline"></a></h2>
<p>We show here how we can use a simple regression case on the breast
cancer data using support vector machines (SVM) as algorithm for
classification.</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.svm</span> <span class="kn">import</span> <span class="n">SVC</span>
<span class="n">cancer</span> <span class="o">=</span> <span class="n">load_breast_cancer</span><span class="p">()</span>
<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>
<span class="nb">print</span><span class="p">(</span><span class="n">X_train</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">X_test</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
<span class="n">svm</span> <span class="o">=</span> <span class="n">SVC</span><span class="p">(</span><span class="n">C</span><span class="o">=</span><span class="mi">100</span><span class="p">)</span>
<span class="n">svm</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">&quot;Test set accuracy: </span><span class="si">{:.2f}</span><span class="s2">&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">svm</span><span class="o">.</span><span class="n">score</span><span class="p">(</span><span class="n">X_test</span><span class="p">,</span><span class="n">y_test</span><span class="p">)))</span>
<span class="kn">from</span> <span class="nn">sklearn.preprocessing</span> <span class="kn">import</span> <span class="n">MinMaxScaler</span><span class="p">,</span> <span class="n">StandardScaler</span>
<span class="n">scaler</span> <span class="o">=</span> <span class="n">MinMaxScaler</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="nb">print</span><span class="p">(</span><span class="s2">&quot;Feature min values before scaling:</span><span class="se">\n</span><span class="s2"> </span><span class="si">{}</span><span class="s2">&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">X_train</span><span class="o">.</span><span class="n">min</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="nb">print</span><span class="p">(</span><span class="s2">&quot;Feature max values before scaling:</span><span class="se">\n</span><span class="s2"> </span><span class="si">{}</span><span class="s2">&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">X_train</span><span class="o">.</span><span class="n">max</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="nb">print</span><span class="p">(</span><span class="s2">&quot;Feature min values before scaling:</span><span class="se">\n</span><span class="s2"> </span><span class="si">{}</span><span class="s2">&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">X_train_scaled</span><span class="o">.</span><span class="n">min</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="nb">print</span><span class="p">(</span><span class="s2">&quot;Feature max values before scaling:</span><span class="se">\n</span><span class="s2"> </span><span class="si">{}</span><span class="s2">&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">X_train_scaled</span><span class="o">.</span><span class="n">max</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">svm</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">&quot;Test set accuracy scaled data with Min-Max scaling: </span><span class="si">{:.2f}</span><span class="s2">&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">svm</span><span class="o">.</span><span class="n">score</span><span class="p">(</span><span class="n">X_test_scaled</span><span class="p">,</span><span class="n">y_test</span><span class="p">)))</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="n">svm</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">&quot;Test set accuracy scaled data with Standar Scaler: </span><span class="si">{:.2f}</span><span class="s2">&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">svm</span><span class="o">.</span><span class="n">score</span><span class="p">(</span><span class="n">X_test_scaled</span><span class="p">,</span><span class="n">y_test</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>(426, 30)
(143, 30)
Test set accuracy: 0.94
Feature min values before scaling:
[6.981e+00 9.710e+00 4.379e+01 1.435e+02 5.263e-02 1.938e-02 0.000e+00
0.000e+00 1.060e-01 4.996e-02 1.115e-01 3.628e-01 7.570e-01 7.228e+00
1.713e-03 2.252e-03 0.000e+00 0.000e+00 7.882e-03 8.948e-04 7.930e+00
1.202e+01 5.041e+01 1.852e+02 7.117e-02 2.729e-02 0.000e+00 0.000e+00
1.565e-01 5.504e-02]
Feature max values before scaling:
[2.811e+01 3.381e+01 1.885e+02 2.501e+03 1.447e-01 3.114e-01 4.268e-01
2.012e-01 3.040e-01 9.744e-02 2.873e+00 4.885e+00 2.198e+01 5.422e+02
2.333e-02 1.064e-01 3.960e-01 5.279e-02 6.146e-02 2.984e-02 3.604e+01
4.954e+01 2.512e+02 4.254e+03 2.226e-01 1.058e+00 1.252e+00 2.903e-01
6.638e-01 2.075e-01]
Feature min values before scaling:
[0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.
0. 0. 0. 0. 0. 0.]
Feature max values before scaling:
[1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1. 1.
1. 1. 1. 1. 1. 1.]
Test set accuracy scaled data with Min-Max scaling: 0.97
Test set accuracy scaled data with Standar Scaler: 0.96
</pre></div>
</div>
</div>
</div>
</div>
<div class="section" id="more-on-cancer-data-now-with-logistic-regression">
<h2><span class="section-number">7.6. </span>More on Cancer Data, now with Logistic Regression<a class="headerlink" href="#more-on-cancer-data-now-with-logistic-regression" title="Permalink to this headline"></a></h2>
<div class="cell 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">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>
<span class="c1"># Set up training data</span>
<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>
<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">&quot;Test set accuracy: </span><span class="si">{:.2f}</span><span class="s2">&quot;</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_test</span><span class="p">,</span><span class="n">y_test</span><span class="p">)))</span>
<span class="c1"># Scale 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="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">&quot;Test set accuracy scaled data: </span><span class="si">{:.2f}</span><span class="s2">&quot;</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_test_scaled</span><span class="p">,</span><span class="n">y_test</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>Test set accuracy: 0.95
Test set accuracy scaled data: 0.96
</pre></div>
</div>
<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>/Users/hjensen/opt/anaconda3/lib/python3.8/site-packages/sklearn/linear_model/_logistic.py:762: 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(
</pre></div>
</div>
</div>
</div>
</div>
<div class="section" id="why-should-we-think-of-reducing-the-dimensionality">
<h2><span class="section-number">7.7. </span>Why should we think of reducing the dimensionality<a class="headerlink" href="#why-should-we-think-of-reducing-the-dimensionality" title="Permalink to this headline"></a></h2>
<p>In addition to the plot of the features, we study now also the covariance (and the correlation matrix).
We use also <strong>Pandas</strong> to compute the correlation matrix.</p>
<div class="cell 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">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>
<span class="kn">import</span> <span class="nn">pandas</span> <span class="k">as</span> <span class="nn">pd</span>
<span class="c1"># Making a data frame</span>
<span class="n">cancerpd</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">cancer</span><span class="o">.</span><span class="n">data</span><span class="p">,</span> <span class="n">columns</span><span class="o">=</span><span class="n">cancer</span><span class="o">.</span><span class="n">feature_names</span><span class="p">)</span>
<span class="n">fig</span><span class="p">,</span> <span class="n">axes</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">subplots</span><span class="p">(</span><span class="mi">15</span><span class="p">,</span><span class="mi">2</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">20</span><span class="p">))</span>
<span class="n">malignant</span> <span class="o">=</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="o">==</span> <span class="mi">0</span><span class="p">]</span>
<span class="n">benign</span> <span class="o">=</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="o">==</span> <span class="mi">1</span><span class="p">]</span>
<span class="n">ax</span> <span class="o">=</span> <span class="n">axes</span><span class="o">.</span><span class="n">ravel</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">30</span><span class="p">):</span>
<span class="n">_</span><span class="p">,</span> <span class="n">bins</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">histogram</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">i</span><span class="p">],</span> <span class="n">bins</span> <span class="o">=</span><span class="mi">50</span><span class="p">)</span>
<span class="n">ax</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">hist</span><span class="p">(</span><span class="n">malignant</span><span class="p">[:,</span><span class="n">i</span><span class="p">],</span> <span class="n">bins</span> <span class="o">=</span> <span class="n">bins</span><span class="p">,</span> <span class="n">alpha</span> <span class="o">=</span> <span class="mf">0.5</span><span class="p">)</span>
<span class="n">ax</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">hist</span><span class="p">(</span><span class="n">benign</span><span class="p">[:,</span><span class="n">i</span><span class="p">],</span> <span class="n">bins</span> <span class="o">=</span> <span class="n">bins</span><span class="p">,</span> <span class="n">alpha</span> <span class="o">=</span> <span class="mf">0.5</span><span class="p">)</span>
<span class="n">ax</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">set_title</span><span class="p">(</span><span class="n">cancer</span><span class="o">.</span><span class="n">feature_names</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
<span class="n">ax</span><span class="p">[</span><span class="n">i</span><span class="p">]</span><span class="o">.</span><span class="n">set_yticks</span><span class="p">(())</span>
<span class="n">ax</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s2">&quot;Feature magnitude&quot;</span><span class="p">)</span>
<span class="n">ax</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s2">&quot;Frequency&quot;</span><span class="p">)</span>
<span class="n">ax</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">legend</span><span class="p">([</span><span class="s2">&quot;Malignant&quot;</span><span class="p">,</span> <span class="s2">&quot;Benign&quot;</span><span class="p">],</span> <span class="n">loc</span> <span class="o">=</span><span class="s2">&quot;best&quot;</span><span class="p">)</span>
<span class="n">fig</span><span class="o">.</span><span class="n">tight_layout</span><span class="p">()</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
<span class="kn">import</span> <span class="nn">seaborn</span> <span class="k">as</span> <span class="nn">sns</span>
<span class="n">correlation_matrix</span> <span class="o">=</span> <span class="n">cancerpd</span><span class="o">.</span><span class="n">corr</span><span class="p">()</span><span class="o">.</span><span class="n">round</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
<span class="c1"># use the heatmap function from seaborn to plot the correlation matrix</span>
<span class="c1"># annot = True to print the values inside the square</span>
<span class="n">sns</span><span class="o">.</span><span class="n">heatmap</span><span class="p">(</span><span class="n">data</span><span class="o">=</span><span class="n">correlation_matrix</span><span class="p">,</span> <span class="n">annot</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
<span class="c1">#print eigvalues of correlation matrix</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">correlation_matrix</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">EigValues</span><span class="p">)</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<img alt="_images/chapter8_7_0.png" src="_images/chapter8_7_0.png" />
<img alt="_images/chapter8_7_1.png" src="_images/chapter8_7_1.png" />
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>[ 1.33026664e+01 5.69238112e+00 2.83259341e+00 1.99126621e+00
1.69095941e+00 1.19614372e+00 7.13037918e-01 5.77804918e-01
5.14549644e-01 4.30579019e-01 3.58977964e-01 -1.91269450e-01
2.86195387e-01 -1.59310961e-01 -1.38483045e-01 2.44786221e-01
-8.79318674e-02 1.91811329e-01 1.70896266e-01 -7.17683028e-02
1.39860289e-01 1.18721828e-01 -4.83194721e-02 8.92983892e-02
6.81402316e-02 6.01968016e-02 3.31819744e-02 1.56432255e-02
-1.61028757e-02 -6.50565973e-03]
</pre></div>
</div>
</div>
</div>
<p>In the above example we note two things. In the first plot we display
the overlap of benign and malignant tumors as functions of the various
features in the Wisconsing breast cancer data set. We see that for
some of the features we can distinguish clearly the benign and
malignant cases while for other features we cannot. This can point to
us which features may be of greater interest when we wish to classify
a benign or not benign tumour.</p>
<p>In the second figure we have computed the so-called correlation
matrix, which in our case with thirty features becomes a <span class="math notranslate nohighlight">\(30\times 30\)</span>
matrix.</p>
<p>We constructed this matrix using <strong>pandas</strong> via the statements</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">cancerpd</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">cancer</span><span class="o">.</span><span class="n">data</span><span class="p">,</span> <span class="n">columns</span><span class="o">=</span><span class="n">cancer</span><span class="o">.</span><span class="n">feature_names</span><span class="p">)</span>
</pre></div>
</div>
</div>
</div>
<p>and then</p>
<div class="cell docutils container">
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<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">correlation_matrix</span> <span class="o">=</span> <span class="n">cancerpd</span><span class="o">.</span><span class="n">corr</span><span class="p">()</span><span class="o">.</span><span class="n">round</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
</pre></div>
</div>
</div>
</div>
<p>Diagonalizing this matrix we can in turn say something about which
features are of relevance and which are not. But before we proceed we
need to define covariance and correlation matrices. This leads us to
the classical Principal Component Analysis (PCA) theorem with
applications.</p>
</div>
<div class="section" id="basic-ideas-of-the-principal-component-analysis-pca">
<h2><span class="section-number">7.8. </span>Basic ideas of the Principal Component Analysis (PCA)<a class="headerlink" href="#basic-ideas-of-the-principal-component-analysis-pca" title="Permalink to this headline"></a></h2>
<p>The principal component analysis deals with the problem of fitting a
low-dimensional affine subspace <span class="math notranslate nohighlight">\(S\)</span> of dimension <span class="math notranslate nohighlight">\(d\)</span> much smaller than
the totaldimension <span class="math notranslate nohighlight">\(D\)</span> of the problem at hand (our data
set). Mathematically it can be formulated as a statistical problem or
a geometric problem. In our discussion of the theorem for the
classical PCA, we will stay with a statistical approach. This is also
what set the scene historically which for the PCA.</p>
<p>We have a data set defined by a design/feature matrix <span class="math notranslate nohighlight">\(\boldsymbol{X}\)</span> (see below for its definition)</p>
<ul class="simple">
<li><p>Each data point is determined by <span class="math notranslate nohighlight">\(p\)</span> extrinsic (measurement) variables</p></li>
<li><p>We may want to ask the following question: Are there fewer intrinsic variables (say <span class="math notranslate nohighlight">\(d &lt;&lt; p\)</span>) that still approximately describe the data?</p></li>
<li><p>If so, these intrinsic variables may tell us something important and finding these intrinsic variables is what dimension reduction methods do.</p></li>
</ul>
</div>
<div class="section" id="introducing-the-covariance-and-correlation-functions">
<h2><span class="section-number">7.9. </span>Introducing the Covariance and Correlation functions<a class="headerlink" href="#introducing-the-covariance-and-correlation-functions" title="Permalink to this headline"></a></h2>
<p>Before we discuss the PCA theorem, we need to remind ourselves about
the definition of the covariance and the correlation function. These are quantities</p>
<p>Suppose we have defined two vectors
<span class="math notranslate nohighlight">\(\hat{x}\)</span> and <span class="math notranslate nohighlight">\(\hat{y}\)</span> with <span class="math notranslate nohighlight">\(n\)</span> elements each. The covariance matrix <span class="math notranslate nohighlight">\(\boldsymbol{C}\)</span> is defined as</p>
<div class="math notranslate nohighlight">
\[\begin{split}
\boldsymbol{C}[\boldsymbol{x},\boldsymbol{y}] = \begin{bmatrix} \mathrm{cov}[\boldsymbol{x},\boldsymbol{x}] &amp; \mathrm{cov}[\boldsymbol{x},\boldsymbol{y}] \\
\mathrm{cov}[\boldsymbol{y},\boldsymbol{x}] &amp; \mathrm{cov}[\boldsymbol{y},\boldsymbol{y}] \\
\end{bmatrix},
\end{split}\]</div>
<p>where for example</p>
<div class="math notranslate nohighlight">
\[
\mathrm{cov}[\boldsymbol{x},\boldsymbol{y}] =\frac{1}{n} \sum_{i=0}^{n-1}(x_i- \overline{x})(y_i- \overline{y}).
\]</div>
<p>With this definition and recalling that the variance is defined as</p>
<div class="math notranslate nohighlight">
\[
\mathrm{var}[\boldsymbol{x}]=\frac{1}{n} \sum_{i=0}^{n-1}(x_i- \overline{x})^2,
\]</div>
<p>we can rewrite the covariance matrix as</p>
<div class="math notranslate nohighlight">
\[\begin{split}
\boldsymbol{C}[\boldsymbol{x},\boldsymbol{y}] = \begin{bmatrix} \mathrm{var}[\boldsymbol{x}] &amp; \mathrm{cov}[\boldsymbol{x},\boldsymbol{y}] \\
\mathrm{cov}[\boldsymbol{x},\boldsymbol{y}] &amp; \mathrm{var}[\boldsymbol{y}] \\
\end{bmatrix}.
\end{split}\]</div>
<p>The covariance takes values between zero and infinity and may thus
lead to problems with loss of numerical precision for particularly
large values. It is common to scale the covariance matrix by
introducing instead the correlation matrix defined via the so-called
correlation function</p>
<div class="math notranslate nohighlight">
\[
\mathrm{corr}[\boldsymbol{x},\boldsymbol{y}]=\frac{\mathrm{cov}[\boldsymbol{x},\boldsymbol{y}]}{\sqrt{\mathrm{var}[\boldsymbol{x}] \mathrm{var}[\boldsymbol{y}]}}.
\]</div>
<p>The correlation function is then given by values <span class="math notranslate nohighlight">\(\mathrm{corr}[\boldsymbol{x},\boldsymbol{y}]
\in [-1,1]\)</span>. This avoids eventual problems with too large values. We
can then define the correlation matrix for the two vectors <span class="math notranslate nohighlight">\(\boldsymbol{x}\)</span>
and <span class="math notranslate nohighlight">\(\boldsymbol{y}\)</span> as</p>
<div class="math notranslate nohighlight">
\[\begin{split}
\boldsymbol{K}[\boldsymbol{x},\boldsymbol{y}] = \begin{bmatrix} 1 &amp; \mathrm{corr}[\boldsymbol{x},\boldsymbol{y}] \\
\mathrm{corr}[\boldsymbol{y},\boldsymbol{x}] &amp; 1 \\
\end{bmatrix},
\end{split}\]</div>
<p>In the above example this is the function we constructed using <strong>pandas</strong>.</p>
</div>
<div class="section" id="correlation-function-and-design-feature-matrix">
<h2><span class="section-number">7.10. </span>Correlation Function and Design/Feature Matrix<a class="headerlink" href="#correlation-function-and-design-feature-matrix" title="Permalink to this headline"></a></h2>
<p>In our derivation of the various regression algorithms like <strong>Ordinary Least Squares</strong> or <strong>Ridge regression</strong>
we defined the design/feature matrix <span class="math notranslate nohighlight">\(\boldsymbol{X}\)</span> as</p>
<div class="math notranslate nohighlight">
\[\begin{split}
\boldsymbol{X}=\begin{bmatrix}
x_{0,0} &amp; x_{0,1} &amp; x_{0,2}&amp; \dots &amp; \dots x_{0,p-1}\\
x_{1,0} &amp; x_{1,1} &amp; x_{1,2}&amp; \dots &amp; \dots x_{1,p-1}\\
x_{2,0} &amp; x_{2,1} &amp; x_{2,2}&amp; \dots &amp; \dots x_{2,p-1}\\
\dots &amp; \dots &amp; \dots &amp; \dots \dots &amp; \dots \\
x_{n-2,0} &amp; x_{n-2,1} &amp; x_{n-2,2}&amp; \dots &amp; \dots x_{n-2,p-1}\\
x_{n-1,0} &amp; x_{n-1,1} &amp; x_{n-1,2}&amp; \dots &amp; \dots x_{n-1,p-1}\\
\end{bmatrix},
\end{split}\]</div>
<p>with <span class="math notranslate nohighlight">\(\boldsymbol{X}\in {\mathbb{R}}^{n\times p}\)</span>, with the predictors/features <span class="math notranslate nohighlight">\(p\)</span> refering to the column numbers and the
entries <span class="math notranslate nohighlight">\(n\)</span> being the row elements.
We can rewrite the design/feature matrix in terms of its column vectors as</p>
<div class="math notranslate nohighlight">
\[
\boldsymbol{X}=\begin{bmatrix} \boldsymbol{x}_0 &amp; \boldsymbol{x}_1 &amp; \boldsymbol{x}_2 &amp; \dots &amp; \dots &amp; \boldsymbol{x}_{p-1}\end{bmatrix},
\]</div>
<p>with a given vector</p>
<div class="math notranslate nohighlight">
\[
\boldsymbol{x}_i^T = \begin{bmatrix}x_{0,i} &amp; x_{1,i} &amp; x_{2,i}&amp; \dots &amp; \dots x_{n-1,i}\end{bmatrix}.
\]</div>
<p>With these definitions, we can now rewrite our <span class="math notranslate nohighlight">\(2\times 2\)</span>
correaltion/covariance matrix in terms of a moe general design/feature
matrix <span class="math notranslate nohighlight">\(\boldsymbol{X}\in {\mathbb{R}}^{n\times p}\)</span>. This leads to a <span class="math notranslate nohighlight">\(p\times p\)</span>
covariance matrix for the vectors <span class="math notranslate nohighlight">\(\boldsymbol{x}_i\)</span> with <span class="math notranslate nohighlight">\(i=0,1,\dots,p-1\)</span></p>
<div class="math notranslate nohighlight">
\[\begin{split}
\boldsymbol{C}[\boldsymbol{x}] = \begin{bmatrix}
\mathrm{var}[\boldsymbol{x}_0] &amp; \mathrm{cov}[\boldsymbol{x}_0,\boldsymbol{x}_1] &amp; \mathrm{cov}[\boldsymbol{x}_0,\boldsymbol{x}_2] &amp; \dots &amp; \dots &amp; \mathrm{cov}[\boldsymbol{x}_0,\boldsymbol{x}_{p-1}]\\
\mathrm{cov}[\boldsymbol{x}_1,\boldsymbol{x}_0] &amp; \mathrm{var}[\boldsymbol{x}_1] &amp; \mathrm{cov}[\boldsymbol{x}_1,\boldsymbol{x}_2] &amp; \dots &amp; \dots &amp; \mathrm{cov}[\boldsymbol{x}_1,\boldsymbol{x}_{p-1}]\\
\mathrm{cov}[\boldsymbol{x}_2,\boldsymbol{x}_0] &amp; \mathrm{cov}[\boldsymbol{x}_2,\boldsymbol{x}_1] &amp; \mathrm{var}[\boldsymbol{x}_2] &amp; \dots &amp; \dots &amp; \mathrm{cov}[\boldsymbol{x}_2,\boldsymbol{x}_{p-1}]\\
\dots &amp; \dots &amp; \dots &amp; \dots &amp; \dots &amp; \dots \\
\dots &amp; \dots &amp; \dots &amp; \dots &amp; \dots &amp; \dots \\
\mathrm{cov}[\boldsymbol{x}_{p-1},\boldsymbol{x}_0] &amp; \mathrm{cov}[\boldsymbol{x}_{p-1},\boldsymbol{x}_1] &amp; \mathrm{cov}[\boldsymbol{x}_{p-1},\boldsymbol{x}_{2}] &amp; \dots &amp; \dots &amp; \mathrm{var}[\boldsymbol{x}_{p-1}]\\
\end{bmatrix},
\end{split}\]</div>
<p>and the correlation matrix</p>
<div class="math notranslate nohighlight">
\[\begin{split}
\boldsymbol{K}[\boldsymbol{x}] = \begin{bmatrix}
1 &amp; \mathrm{corr}[\boldsymbol{x}_0,\boldsymbol{x}_1] &amp; \mathrm{corr}[\boldsymbol{x}_0,\boldsymbol{x}_2] &amp; \dots &amp; \dots &amp; \mathrm{corr}[\boldsymbol{x}_0,\boldsymbol{x}_{p-1}]\\
\mathrm{corr}[\boldsymbol{x}_1,\boldsymbol{x}_0] &amp; 1 &amp; \mathrm{corr}[\boldsymbol{x}_1,\boldsymbol{x}_2] &amp; \dots &amp; \dots &amp; \mathrm{corr}[\boldsymbol{x}_1,\boldsymbol{x}_{p-1}]\\
\mathrm{corr}[\boldsymbol{x}_2,\boldsymbol{x}_0] &amp; \mathrm{corr}[\boldsymbol{x}_2,\boldsymbol{x}_1] &amp; 1 &amp; \dots &amp; \dots &amp; \mathrm{corr}[\boldsymbol{x}_2,\boldsymbol{x}_{p-1}]\\
\dots &amp; \dots &amp; \dots &amp; \dots &amp; \dots &amp; \dots \\
\dots &amp; \dots &amp; \dots &amp; \dots &amp; \dots &amp; \dots \\
\mathrm{corr}[\boldsymbol{x}_{p-1},\boldsymbol{x}_0] &amp; \mathrm{corr}[\boldsymbol{x}_{p-1},\boldsymbol{x}_1] &amp; \mathrm{corr}[\boldsymbol{x}_{p-1},\boldsymbol{x}_{2}] &amp; \dots &amp; \dots &amp; 1\\
\end{bmatrix},
\end{split}\]</div>
</div>
<div class="section" id="covariance-matrix-examples">
<h2><span class="section-number">7.11. </span>Covariance Matrix Examples<a class="headerlink" href="#covariance-matrix-examples" title="Permalink to this headline"></a></h2>
<p>The Numpy function <strong>np.cov</strong> calculates the covariance elements using
the factor <span class="math notranslate nohighlight">\(1/(n-1)\)</span> instead of <span class="math notranslate nohighlight">\(1/n\)</span> since it assumes we do not have
the exact mean values. The following simple function uses the
<strong>np.vstack</strong> function which takes each vector of dimension <span class="math notranslate nohighlight">\(1\times n\)</span>
and produces a <span class="math notranslate nohighlight">\(2\times n\)</span> matrix <span class="math notranslate nohighlight">\(\boldsymbol{W}\)</span></p>
<div class="math notranslate nohighlight">
\[\begin{split}
\boldsymbol{W} = \begin{bmatrix} x_0 &amp; y_0 \\
x_1 &amp; y_1 \\
x_2 &amp; y_2\\
\dots &amp; \dots \\
x_{n-2} &amp; y_{n-2}\\
x_{n-1} &amp; y_{n-1} &amp;
\end{bmatrix},
\end{split}\]</div>
<p>which in turn is converted into into the <span class="math notranslate nohighlight">\(2\times 2\)</span> covariance matrix
<span class="math notranslate nohighlight">\(\boldsymbol{C}\)</span> via the Numpy function <strong>np.cov()</strong>. We note that we can also calculate
the mean value of each set of samples <span class="math notranslate nohighlight">\(\boldsymbol{x}\)</span> etc using the Numpy
function <strong>np.mean(x)</strong>. We can also extract the eigenvalues of the
covariance matrix through the <strong>np.linalg.eig()</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="c1"># Importing various packages</span>
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</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">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>
<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>
<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>
<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>
<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>
<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>
<span class="nb">print</span><span class="p">(</span><span class="n">C</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>0.11455861677725063
4.414311328416008
[[1.04460586 2.98508511]
[2.98508511 9.60711152]]
</pre></div>
</div>
</div>
</div>
</div>
<div class="section" id="correlation-matrix">
<h2><span class="section-number">7.12. </span>Correlation Matrix<a class="headerlink" href="#correlation-matrix" title="Permalink to this headline"></a></h2>
<p>The previous example can be converted into the correlation matrix by
simply scaling the matrix elements with the variances. We should also
subtract the mean values for each column. This leads to the following
code which sets up the correlations matrix for the previous example in
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>
<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="n">n</span> <span class="o">=</span> <span class="mi">100</span>
<span class="c1"># define two vectors </span>
<span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">random</span><span class="p">(</span><span class="n">size</span><span class="o">=</span><span class="n">n</span><span class="p">)</span>
<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>
<span class="c1">#scaling the x and y vectors </span>
<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>
<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>
<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>
<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>
<span class="nb">print</span><span class="p">(</span><span class="n">variance_x</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">variance_y</span><span class="p">)</span>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<span class="nb">print</span><span class="p">(</span><span class="n">C</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>0.07650519724946082
1.5077307406836722
[[1. 0.63582252]
[0.63582252 1. ]]
</pre></div>
</div>
</div>
</div>
<p>We see that the matrix elements along the diagonal are one as they
should be and that the matrix is symmetric. Furthermore, diagonalizing
this matrix we easily see that it is a positive definite matrix.</p>
<p>The above procedure with <strong>numpy</strong> can be made more compact if we use <strong>pandas</strong>.</p>
</div>
<div class="section" id="correlation-matrix-with-pandas">
<h2><span class="section-number">7.13. </span>Correlation Matrix with Pandas<a class="headerlink" href="#correlation-matrix-with-pandas" title="Permalink to this headline"></a></h2>
<p>We whow here how we can set up the correlation matrix using <strong>pandas</strong>, as done in this simple 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="n">n</span> <span class="o">=</span> <span class="mi">10</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>
<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>
<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>
<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>
<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>
<span class="nb">print</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
<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="nb">print</span><span class="p">(</span><span class="n">Xpd</span><span class="p">)</span>
<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>
<span class="nb">print</span><span class="p">(</span><span class="n">correlation_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>[[-0.21054401 -1.67179046]
[-0.27924545 -1.98111498]
[-1.24089109 -3.51965915]
[ 2.17042981 6.77708985]
[-0.36489938 -0.63821317]
[-1.61222872 -5.67668711]
[-0.67184873 -0.62798419]
[ 1.21860807 4.36876959]
[ 0.0351748 -0.93789716]
[ 0.95544469 3.90748678]]
0 1
0 -0.210544 -1.671790
1 -0.279245 -1.981115
2 -1.240891 -3.519659
3 2.170430 6.777090
4 -0.364899 -0.638213
5 -1.612229 -5.676687
6 -0.671849 -0.627984
7 1.218608 4.368770
8 0.035175 -0.937897
9 0.955445 3.907487
0 1
0 1.000000 0.972249
1 0.972249 1.000000
</pre></div>
</div>
</div>
</div>
<p>We expand this model to the Franke function discussed above.</p>
</div>
<div class="section" id="correlation-matrix-with-pandas-and-the-franke-function">
<h2><span class="section-number">7.14. </span>Correlation Matrix with Pandas and the Franke function<a class="headerlink" href="#correlation-matrix-with-pandas-and-the-franke-function" title="Permalink to this headline"></a></h2>
<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"># 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>
<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>
<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">&gt;</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>
<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>
<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>
<span class="k">return</span> <span class="n">X</span>
<span class="c1"># Making meshgrid of datapoints and compute Franke&#39;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>
<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 0.062851 0.073533 0.065769 0.068538 0.069947 0.059538 0.060320
2 0.0 0.073533 0.087625 0.078561 0.082559 0.084744 0.071705 0.072983
3 0.0 0.065769 0.078561 0.072769 0.076696 0.079022 0.068407 0.069803
4 0.0 0.068538 0.082559 0.076696 0.081211 0.083977 0.072511 0.074206
5 0.0 0.069947 0.084744 0.079022 0.083977 0.087102 0.075127 0.077070
6 0.0 0.059538 0.071705 0.068407 0.072511 0.075127 0.066156 0.067782
7 0.0 0.060320 0.072983 0.069803 0.074206 0.077070 0.067782 0.069583
8 0.0 0.060735 0.073737 0.070715 0.075353 0.078424 0.068928 0.070878
9 0.0 0.060967 0.074217 0.071359 0.076188 0.079434 0.069792 0.071870
10 0.0 0.052371 0.063245 0.061909 0.065807 0.068404 0.061247 0.062897
11 0.0 0.052613 0.063714 0.062491 0.066554 0.069297 0.062003 0.063759
12 0.0 0.052756 0.064027 0.062920 0.067118 0.069987 0.062592 0.064441
13 0.0 0.052853 0.064260 0.063261 0.067574 0.070552 0.063079 0.065009
14 0.0 0.052933 0.064453 0.063553 0.067966 0.071040 0.063502 0.065504
8 9 10 11 12 13 14
0 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000
1 0.060735 0.060967 0.052371 0.052613 0.052756 0.052853 0.052933
2 0.073737 0.074217 0.063245 0.063714 0.064027 0.064260 0.064453
3 0.070715 0.071359 0.061909 0.062491 0.062920 0.063261 0.063553
4 0.075353 0.076188 0.065807 0.066554 0.067118 0.067574 0.067966
5 0.078424 0.079434 0.068404 0.069297 0.069987 0.070552 0.071040
6 0.068928 0.069792 0.061247 0.062003 0.062592 0.063079 0.063502
7 0.070878 0.071870 0.062897 0.063759 0.064441 0.065009 0.065504
8 0.072303 0.073408 0.064109 0.065065 0.065830 0.066473 0.067035
9 0.073408 0.074615 0.065054 0.066094 0.066934 0.067645 0.068268
10 0.064109 0.065054 0.057780 0.058594 0.059248 0.059799 0.060281
11 0.065065 0.066094 0.058594 0.059478 0.060193 0.060800 0.061333
12 0.065830 0.066934 0.059248 0.060193 0.060965 0.061622 0.062202
13 0.066473 0.067645 0.059799 0.060800 0.061622 0.062326 0.062949
14 0.067035 0.068268 0.060281 0.061333 0.062202 0.062949 0.063612
</pre></div>
</div>
</div>
</div>
<p>We note here that the covariance is zero for the first rows and
columns since all matrix elements in the design matrix were set to one
(we are fitting the function in terms of a polynomial of degree <span class="math notranslate nohighlight">\(n\)</span>).</p>
<p>This means that the variance for these elements will be zero and will
cause problems when we set up the correlation matrix. We can simply
drop these elements and construct a correlation
matrix without these elements.</p>
</div>
<div class="section" id="rewriting-the-covariance-and-or-correlation-matrix">
<h2><span class="section-number">7.15. </span>Rewriting the Covariance and/or Correlation Matrix<a class="headerlink" href="#rewriting-the-covariance-and-or-correlation-matrix" title="Permalink to this headline"></a></h2>
<p>We can rewrite the covariance matrix in a more compact form in terms of the design/feature matrix <span class="math notranslate nohighlight">\(\boldsymbol{X}\)</span> as</p>
<div class="math notranslate nohighlight">
\[
\boldsymbol{C}[\boldsymbol{x}] = \frac{1}{n}\boldsymbol{X}\boldsymbol{X}^T= \mathbb{E}[\boldsymbol{X}\boldsymbol{X}^T].
\]</div>
<p>To see this let us simply look at a design matrix <span class="math notranslate nohighlight">\(\boldsymbol{X}\in {\mathbb{R}}^{2\times 2}\)</span></p>
<div class="math notranslate nohighlight">
\[\begin{split}
\boldsymbol{X}=\begin{bmatrix}
x_{00} &amp; x_{01}\\
x_{10} &amp; x_{11}\\
\end{bmatrix}=\begin{bmatrix}
\boldsymbol{x}_{0} &amp; \boldsymbol{x}_{1}\\
\end{bmatrix}.
\end{split}\]</div>
<p>If we then compute the expectation value</p>
<div class="math notranslate nohighlight">
\[\begin{split}
\mathbb{E}[\boldsymbol{X}\boldsymbol{X}^T] = \frac{1}{n}\boldsymbol{X}\boldsymbol{X}^T=\begin{bmatrix}
x_{00}^2+x_{01}^2 &amp; x_{00}x_{10}+x_{01}x_{11}\\
x_{10}x_{00}+x_{11}x_{01} &amp; x_{10}^2+x_{11}^2\\
\end{bmatrix},
\end{split}\]</div>
<p>which is just</p>
<div class="math notranslate nohighlight">
\[\begin{split}
\boldsymbol{C}[\boldsymbol{x}_0,\boldsymbol{x}_1] = \boldsymbol{C}[\boldsymbol{x}]=\begin{bmatrix} \mathrm{var}[\boldsymbol{x}_0] &amp; \mathrm{cov}[\boldsymbol{x}_0,\boldsymbol{x}_1] \\
\mathrm{cov}[\boldsymbol{x}_1,\boldsymbol{x}_0] &amp; \mathrm{var}[\boldsymbol{x}_1] \\
\end{bmatrix},
\end{split}\]</div>
<p>where we wrote $<span class="math notranslate nohighlight">\(\boldsymbol{C}[\boldsymbol{x}_0,\boldsymbol{x}_1] = \boldsymbol{C}[\boldsymbol{x}]\)</span><span class="math notranslate nohighlight">\( to indicate that this the covariance of the vectors \)</span>\boldsymbol{x}<span class="math notranslate nohighlight">\( of the design/feature matrix \)</span>\boldsymbol{X}$.</p>
<p>It is easy to generalize this to a matrix <span class="math notranslate nohighlight">\(\boldsymbol{X}\in {\mathbb{R}}^{n\times p}\)</span>.</p>
</div>
<div class="section" id="towards-the-pca-theorem">
<h2><span class="section-number">7.16. </span>Towards the PCA theorem<a class="headerlink" href="#towards-the-pca-theorem" title="Permalink to this headline"></a></h2>
<p>We have that the covariance matrix (the correlation matrix involves a simple rescaling) is given as</p>
<div class="math notranslate nohighlight">
\[
\boldsymbol{C}[\boldsymbol{x}] = \frac{1}{n}\boldsymbol{X}\boldsymbol{X}^T= \mathbb{E}[\boldsymbol{X}\boldsymbol{X}^T].
\]</div>
<p>Let us now assume that we can perform a series of orthogonal transformations where we employ some orthogonal matrices <span class="math notranslate nohighlight">\(\boldsymbol{S}\)</span>.
These matrices are defined as <span class="math notranslate nohighlight">\(\boldsymbol{S}\in {\mathbb{R}}^{p\times p}\)</span> and obey the orthogonality requirements <span class="math notranslate nohighlight">\(\boldsymbol{S}\boldsymbol{S}^T=\boldsymbol{S}^T\boldsymbol{S}=\boldsymbol{I}\)</span>. The matrix can be written out in terms of the column vectors <span class="math notranslate nohighlight">\(\boldsymbol{s}_i\)</span> as <span class="math notranslate nohighlight">\(\boldsymbol{S}=[\boldsymbol{s}_0,\boldsymbol{s}_1,\dots,\boldsymbol{s}_{p-1}]\)</span> and <span class="math notranslate nohighlight">\(\boldsymbol{s}_i \in {\mathbb{R}}^{p}\)</span>.</p>
<p>Assume also that there is a transformation <span class="math notranslate nohighlight">\(\boldsymbol{S}\boldsymbol{C}[\boldsymbol{x}]\boldsymbol{S}^T=\boldsymbol{C}[\boldsymbol{y}]\)</span> such that the new matrix <span class="math notranslate nohighlight">\(\boldsymbol{C}[\boldsymbol{y}]\)</span> is diagonal with elements <span class="math notranslate nohighlight">\([\lambda_0,\lambda_1,\lambda_2,\dots,\lambda_{p-1}]\)</span>.</p>
<p>That is we have</p>
<div class="math notranslate nohighlight">
\[
\boldsymbol{C}[\boldsymbol{y}] = \mathbb{E}[\boldsymbol{S}\boldsymbol{X}\boldsymbol{X}^T\boldsymbol{S}^T]=\boldsymbol{S}\boldsymbol{C}[\boldsymbol{x}]\boldsymbol{S}^T,
\]</div>
<p>since the matrix <span class="math notranslate nohighlight">\(\boldsymbol{S}\)</span> is not a data dependent matrix. Multiplying with <span class="math notranslate nohighlight">\(\boldsymbol{S}^T\)</span> from the left we have</p>
<div class="math notranslate nohighlight">
\[
\boldsymbol{S}^T\boldsymbol{C}[\boldsymbol{y}] = \boldsymbol{C}[\boldsymbol{x}]\boldsymbol{S}^T,
\]</div>
<p>and since <span class="math notranslate nohighlight">\(\boldsymbol{C}[\boldsymbol{y}]\)</span> is diagonal we have for a given eigenvalue <span class="math notranslate nohighlight">\(i\)</span> of the covariance matrix that</p>
<div class="math notranslate nohighlight">
\[
\boldsymbol{S}^T_i\lambda_i = \boldsymbol{C}[\boldsymbol{x}]\boldsymbol{S}^T_i.
\]</div>
<p>In the derivation of the PCA theorem we will assume that the eigenvalues are ordered in descending order, that is
<span class="math notranslate nohighlight">\(\lambda_0 &gt; \lambda_1 &gt; \dots &gt; \lambda_{p-1}\)</span>.</p>
<p>The eigenvalues tell us then how much we need to stretch the
corresponding eigenvectors. Dimensions with large eigenvalues have
thus large variations (large variance) and define therefore useful
dimensions. The data points are more spread out in the direction of
these eigenvectors. Smaller eigenvalues mean on the other hand that
the corresponding eigenvectors are shrunk accordingly and the data
points are tightly bunched together and there is not much variation in
these specific directions. Hopefully then we could leave it out
dimensions where the eigenvalues are very small. If <span class="math notranslate nohighlight">\(p\)</span> is very large,
we could then aim at reducing <span class="math notranslate nohighlight">\(p\)</span> to <span class="math notranslate nohighlight">\(l &lt;&lt; p\)</span> and handle only <span class="math notranslate nohighlight">\(l\)</span>
features/predictors.</p>
</div>
<div class="section" id="the-algorithm-before-theorem">
<h2><span class="section-number">7.17. </span>The Algorithm before theorem<a class="headerlink" href="#the-algorithm-before-theorem" title="Permalink to this headline"></a></h2>
<p>Here’s how we would proceed in setting up the algorithm for the PCA, see also discussion below here.</p>
<ul class="simple">
<li><p>Set up the datapoints for the design/feature matrix <span class="math notranslate nohighlight">\(\boldsymbol{X}\)</span> with <span class="math notranslate nohighlight">\(\boldsymbol{X}\in {\mathbb{R}}^{n\times p}\)</span>, with the predictors/features <span class="math notranslate nohighlight">\(p\)</span> referring to the column numbers and the entries <span class="math notranslate nohighlight">\(n\)</span> being the row elements.</p></li>
</ul>
<div class="math notranslate nohighlight">
\[\begin{split}
\boldsymbol{X}=\begin{bmatrix}
x_{0,0} &amp; x_{0,1} &amp; x_{0,2}&amp; \dots &amp; \dots x_{0,p-1}\\
x_{1,0} &amp; x_{1,1} &amp; x_{1,2}&amp; \dots &amp; \dots x_{1,p-1}\\
x_{2,0} &amp; x_{2,1} &amp; x_{2,2}&amp; \dots &amp; \dots x_{2,p-1}\\
\dots &amp; \dots &amp; \dots &amp; \dots \dots &amp; \dots \\
x_{n-2,0} &amp; x_{n-2,1} &amp; x_{n-2,2}&amp; \dots &amp; \dots x_{n-2,p-1}\\
x_{n-1,0} &amp; x_{n-1,1} &amp; x_{n-1,2}&amp; \dots &amp; \dots x_{n-1,p-1}\\
\end{bmatrix},
\end{split}\]</div>
<ul class="simple">
<li><p>Center the data by subtracting the mean value for each column. This leads to a new matrix <span class="math notranslate nohighlight">\(\boldsymbol{X}\rightarrow \overline{\boldsymbol{X}}\)</span>.</p></li>
<li><p>Compute then the covariance/correlation matrix <span class="math notranslate nohighlight">\(\mathbb{E}[\overline{\boldsymbol{X}}\overline{\boldsymbol{X}}^T]\)</span>.</p></li>
<li><p>Find the eigenpairs of <span class="math notranslate nohighlight">\(\boldsymbol{C}\)</span> with eigenvalues <span class="math notranslate nohighlight">\([\lambda_0,\lambda_1,\dots,\lambda_{p-1}]\)</span> and eigenvectors <span class="math notranslate nohighlight">\([\boldsymbol{s}_0,\boldsymbol{s}_1,\dots,\boldsymbol{s}_{p-1}]\)</span>.</p></li>
<li><p>Order the eigenvalue (and the eigenvectors accordingly) in order of decreasing eigenvalues.</p></li>
<li><p>Keep only those <span class="math notranslate nohighlight">\(l\)</span> eigenvalues larger than a selected threshold value, discarding thus <span class="math notranslate nohighlight">\(p-l\)</span> features since we expect small variations in the data here.</p></li>
</ul>
</div>
<div class="section" id="writing-our-own-pca-code">
<h2><span class="section-number">7.18. </span>Writing our own PCA code<a class="headerlink" href="#writing-our-own-pca-code" title="Permalink to this headline"></a></h2>
<p>We will use a simple example first with two-dimensional data
drawn from a multivariate normal distribution with the following mean and covariance matrix:</p>
<div class="math notranslate nohighlight">
\[\begin{split}
\mu = (-1,2) \qquad \Sigma = \begin{bmatrix} 4 &amp; 2 \\
2 &amp; 2
\end{bmatrix}
\end{split}\]</div>
<p>Note that the mean refers to each column of data.
We will generate <span class="math notranslate nohighlight">\(n = 1000\)</span> points <span class="math notranslate nohighlight">\(X = \{ x_1, \ldots, x_N \}\)</span> from
this distribution, and store them in the <span class="math notranslate nohighlight">\(1000 \times 2\)</span> matrix <span class="math notranslate nohighlight">\(\boldsymbol{X}\)</span>.</p>
<p>The following Python code aids in setting up the data and writing out the design matrix.
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="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>
</pre></div>
</div>
</div>
</div>
<p>Now we are going to implement the PCA algorithm. We will break it down into various substeps.</p>
<div class="section" id="compute-the-sample-mean-and-center-the-data">
<h3><span class="section-number">7.18.1. </span>Compute the sample mean and center the data<a class="headerlink" href="#compute-the-sample-mean-and-center-the-data" title="Permalink to this headline"></a></h3>
<p>The first step of PCA is to compute the sample mean of the data and use it to center the data. Recall that the sample mean is</p>
<div class="math notranslate nohighlight">
\[
\mu_n = \frac{1}{n} \sum_{i=1}^n x_i
\]</div>
<p>and the mean-centered data <span class="math notranslate nohighlight">\(\bar{X} = \{ \bar{x}_1, \ldots, \bar{x}_n \}\)</span> takes the form</p>
<div class="math notranslate nohighlight">
\[
\bar{x}_i = x_i - \mu_n.
\]</div>
<p>When you are done with these steps, print out <span class="math notranslate nohighlight">\(\mu_n\)</span> to verify it is
close to <span class="math notranslate nohighlight">\(\mu\)</span> and plot your mean centered data to verify it is
centered at the origin! Compare your code with the functionality from <strong>Scikit-Learn</strong> discussed above.
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>
</pre></div>
</div>
</div>
</div>
<p>Alternatively, we could use the functions we discussed
earlier for scaling the data set. That is, we could have used the
<strong>StandardScaler</strong> function in <strong>Scikit-Learn</strong>, a function which ensures
that for each feature/predictor we study the mean value is zero and
the variance is one (every column in the design/feature matrix). You
would then not get the same results, since we divide by the
variance. The diagonal covariance matrix elements will then be one,
while the non-diagonal ones need to be divided by <span class="math notranslate nohighlight">\(2\sqrt{2}\)</span> for our
specific case.</p>
</div>
<div class="section" id="compute-the-sample-covariance">
<h3><span class="section-number">7.18.2. </span>Compute the sample covariance<a class="headerlink" href="#compute-the-sample-covariance" title="Permalink to this headline"></a></h3>
<p>Now we are going to use the mean centered data to compute the sample covariance of the data by using the following equation</p>
<div class="math notranslate nohighlight">
\[
\Sigma_n = \frac{1}{n-1} \sum_{i=1}^n \bar{x}_i^T \bar{x}_i = \frac{1}{n-1} \sum_{i=1}^n (x_i - \mu_n)^T (x_i - \mu_n)
\]</div>
<p>where the data points <span class="math notranslate nohighlight">\(x_i \in \mathbb{R}^p\)</span> (here in this example <span class="math notranslate nohighlight">\(p = 2\)</span>) are column vectors and <span class="math notranslate nohighlight">\(x^T\)</span> is the transpose of <span class="math notranslate nohighlight">\(x\)</span>.
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 3.963873 1.979586
1 1.979586 1.976796
[[3.96387325 1.97958566]
[1.97958566 1.97679551]]
</pre></div>
</div>
</div>
</div>
<p>Note that the way we define the covariance matrix here has a factor <span class="math notranslate nohighlight">\(n-1\)</span> instead of <span class="math notranslate nohighlight">\(n\)</span>. This is included in the <strong>cov()</strong> function by <strong>numpy</strong> and <strong>pandas</strong>.
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">&quot;Centered covariance using own code&quot;</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">&#39;x&#39;</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">&#39;equal&#39;</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</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>Centered covariance using own code
[[3.96387325 1.97958566]
[1.97958566 1.97679551]]
</pre></div>
</div>
<img alt="_images/chapter8_77_1.png" src="_images/chapter8_77_1.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?</p>
</div>
<div class="section" id="diagonalize-the-sample-covariance-matrix-to-obtain-the-principal-components">
<h3><span class="section-number">7.18.3. </span>Diagonalize the sample covariance matrix to obtain the principal components<a class="headerlink" href="#diagonalize-the-sample-covariance-matrix-to-obtain-the-principal-components" title="Permalink to this headline"></a></h3>
<p>Now we are ready to solve for the principal components! To do so we
diagonalize the sample covariance matrix <span class="math notranslate nohighlight">\(\Sigma\)</span>. We can use the
function <strong>np.linalg.eig</strong> to do so. It will return the eigenvalues and
eigenvectors of <span class="math notranslate nohighlight">\(\Sigma\)</span>. Once we have these we can perform the
following tasks:</p>
<ul class="simple">
<li><p>We compute the percentage of the total variance captured by the first principal component</p></li>
<li><p>We plot the mean centered data and lines along the first and second principal components</p></li>
<li><p>Then we project the mean centered data onto the first and second principal components, and plot the projected data.</p></li>
<li><p>Finally, we approximate the data as</p></li>
</ul>
<div class="math notranslate nohighlight">
\[
x_i \approx \tilde{x}_i = \mu_n + \langle x_i, v_0 \rangle v_0
\]</div>
<p>where <span class="math notranslate nohighlight">\(v_0\)</span> is the first principal component.</p>
<p>Collecting all these steps we can write our own PCA function and
compare this with the functionality included in <strong>Scikit-Learn</strong>.</p>
<p>The code here outlines some of the elements we could include in the
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">&quot;Eigenvalues of Covariance matrix&quot;</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">&quot;First eigenvector&quot;</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">&quot;Second eigenvector&quot;</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">&quot;Eigenvector of largest eigenvalue&quot;</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
5.185256249406441
0.7554125101643117
First eigenvector
[0.85104821 0.52508756]
Second eigenvector
[-0.52508756 0.85104821]
Eigenvector of largest eigenvalue
[-0.85104821 -0.52508756]
</pre></div>
</div>
</div>
</div>
<p>This code does not contain all the above elements, but it shows how we can use <strong>Scikit-Learn</strong> to extract the eigenvector which corresponds to the largest eigenvalue. Try to address the questions we pose before the above code. Try also to change the values of the covariance matrix by making one of the diagonal elements much larger than the other. What do you observe then?</p>
</div>
</div>
<div class="section" id="classical-pca-theorem">
<h2><span class="section-number">7.19. </span>Classical PCA Theorem<a class="headerlink" href="#classical-pca-theorem" title="Permalink to this headline"></a></h2>
<p>We assume now that we have a design matrix <span class="math notranslate nohighlight">\(\boldsymbol{X}\)</span> which has been
centered as discussed above. For the sake of simplicity we skip the
overline symbol. The matrix is defined in terms of the various column
vectors <span class="math notranslate nohighlight">\([\boldsymbol{x}_0,\boldsymbol{x}_1,\dots, \boldsymbol{x}_{p-1}]\)</span> each with dimension
<span class="math notranslate nohighlight">\(\boldsymbol{x}\in {\mathbb{R}}^{n}\)</span>.</p>
<p>We assume also that we have an orthogonal transformation <span class="math notranslate nohighlight">\(\boldsymbol{W}\in {\mathbb{R}}^{p\times p}\)</span>. We define the reconstruction error (which is similar to the mean squared error we have seen before) as</p>
<div class="math notranslate nohighlight">
\[
J(\boldsymbol{W},\boldsymbol{Z}) = \frac{1}{n}\sum_i (\boldsymbol{x}_i - \overline{\boldsymbol{x}}_i)^2,
\]</div>
<p>with <span class="math notranslate nohighlight">\(\overline{\boldsymbol{x}}_i = \boldsymbol{W}\boldsymbol{z}_i\)</span>, where <span class="math notranslate nohighlight">\(\boldsymbol{z}_i\)</span> is a row vector with dimension <span class="math notranslate nohighlight">\({\mathbb{R}}^{n}\)</span> of the matrix
<span class="math notranslate nohighlight">\(\boldsymbol{Z}\in{\mathbb{R}}^{p\times n}\)</span>. When doing PCA we want to reduce this dimensionality.</p>
<p>The PCA theorem states that minimizing the above reconstruction error
corresponds to setting <span class="math notranslate nohighlight">\(\boldsymbol{W}=\boldsymbol{S}\)</span>, the orthogonal matrix which
diagonalizes the empirical covariance(correlation) matrix. The optimal
low-dimensional encoding of the data is then given by a set of vectors
<span class="math notranslate nohighlight">\(\boldsymbol{z}_i\)</span> with at most <span class="math notranslate nohighlight">\(l\)</span> vectors, with <span class="math notranslate nohighlight">\(l &lt;&lt; p\)</span>, defined by the
orthogonal projection of the data onto the columns spanned by the
eigenvectors of the covariance(correlations matrix).</p>
<p>The proof which follows will be updated by mid January 2020.</p>
</div>
<div class="section" id="proof-of-the-pca-theorem">
<h2><span class="section-number">7.20. </span>Proof of the PCA Theorem<a class="headerlink" href="#proof-of-the-pca-theorem" title="Permalink to this headline"></a></h2>
<p>To show the PCA theorem let us start with the assumption that there is one vector <span class="math notranslate nohighlight">\(\boldsymbol{w}_0\)</span> which corresponds to a solution which minimized the reconstruction error <span class="math notranslate nohighlight">\(J\)</span>. This is an orthogonal vector. It means that we now approximate the reconstruction error in terms of <span class="math notranslate nohighlight">\(\boldsymbol{w}_0\)</span> and <span class="math notranslate nohighlight">\(\boldsymbol{z}_0\)</span> as</p>
<div class="math notranslate nohighlight">
\[
J(\boldsymbol{w}_0,\boldsymbol{z}_0)= \frac{1}{n}\sum_i (\boldsymbol{x}_i - z_{i0}\boldsymbol{w}_0)^2=\frac{1}{n}\sum_i (\boldsymbol{x}_i^T\boldsymbol{x}_i - 2z_{i0}\boldsymbol{w}_0^T\boldsymbol{x}_i+z_{i0}^2\boldsymbol{w}_0^T\boldsymbol{w}_0),
\]</div>
<p>which we can rewrite due to the orthogonality of <span class="math notranslate nohighlight">\(\boldsymbol{w}_i\)</span> as</p>
<div class="math notranslate nohighlight">
\[
J(\boldsymbol{w}_0,\boldsymbol{z}_0)=\frac{1}{n}\sum_i (\boldsymbol{x}_i^T\boldsymbol{x}_i - 2z_{i0}\boldsymbol{w}_0^T\boldsymbol{x}_i+z_{i0}^2).
\]</div>
<p>Minimizing <span class="math notranslate nohighlight">\(J\)</span> with respect to the unknown parameters <span class="math notranslate nohighlight">\(z_{0i}\)</span> we obtain that</p>
<div class="math notranslate nohighlight">
\[
z_{i0}=\boldsymbol{w}_0^T\boldsymbol{x}_i,
\]</div>
<p>where the vectors on the rhs are known.</p>
</div>
<div class="section" id="pca-proof-continued">
<h2><span class="section-number">7.21. </span>PCA Proof continued<a class="headerlink" href="#pca-proof-continued" title="Permalink to this headline"></a></h2>
<p>We have now found the unknown parameters <span class="math notranslate nohighlight">\(z_{i0}\)</span>. These correspond to the projected coordinates and we can write</p>
<div class="math notranslate nohighlight">
\[
J(\boldsymbol{w}_0)= \frac{1}{p}\sum_i (\boldsymbol{x}_i^T\boldsymbol{x}_i - z_{i0}^2)=\mathrm{const}-\frac{1}{n}\sum_i z_{i0}^2.
\]</div>
<p>We can show that the variance of the projected coordinates defined by <span class="math notranslate nohighlight">\(\boldsymbol{w}_0^T\boldsymbol{x}_i\)</span> are given by</p>
<div class="math notranslate nohighlight">
\[
\mathrm{var}[\boldsymbol{w}_0^T\boldsymbol{x}_i] = \frac{1}{n}\sum_i z_{i0}^2,
\]</div>
<p>since the expectation value of</p>
<div class="math notranslate nohighlight">
\[
\mathbb{E}[\boldsymbol{w}_0^T\boldsymbol{x}_i] = \mathbb{E}[z_{i0}]= \boldsymbol{w}_0^T\mathbb{E}[\boldsymbol{x}_i]=0,
\]</div>
<p>where we have used the fact that our data are centered.</p>
<p>Recalling our definition of the covariance as</p>
<div class="math notranslate nohighlight">
\[
\boldsymbol{C}[\boldsymbol{x}] = \frac{1}{n}\boldsymbol{X}\boldsymbol{X}^T=\mathbb{E}[\boldsymbol{X}\boldsymbol{X}^T],
\]</div>
<p>we have thus that</p>
<div class="math notranslate nohighlight">
\[
\mathrm{var}[\boldsymbol{w}_0^T\boldsymbol{x}_i] = \frac{1}{n}\sum_i z_{i0}^2=\boldsymbol{w}_0^T\boldsymbol{C}[\boldsymbol{x}]\boldsymbol{w}_0.
\]</div>
<p>We are almost there, we have obtained a relation between minimizing
the reconstruction error and the variance and the covariance
matrix. Minimizing the error is equivalent to maximizing the variance
of the projected data.</p>
</div>
<div class="section" id="the-final-step">
<h2><span class="section-number">7.22. </span>The final step<a class="headerlink" href="#the-final-step" title="Permalink to this headline"></a></h2>
<p>We could trivially maximize the variance of the projection (and
thereby minimize the error in the reconstruction function) by letting
the norm-2 of <span class="math notranslate nohighlight">\(\boldsymbol{w}_0\)</span> go to infinity. However, this norm since we
want the matrix <span class="math notranslate nohighlight">\(\boldsymbol{W}\)</span> to be an orthogonal matrix, is constrained by
<span class="math notranslate nohighlight">\(\vert\vert \boldsymbol{w}_0 \vert\vert_2^2=1\)</span>. Imposing this condition via a
Lagrange multiplier we can then in turn maximize</p>
<div class="math notranslate nohighlight">
\[
J(\boldsymbol{w}_0)= \boldsymbol{w}_0^T\boldsymbol{C}[\boldsymbol{x}]\boldsymbol{w}_0+\lambda_0(1-\boldsymbol{w}_0^T\boldsymbol{w}_0).
\]</div>
<p>Taking the derivative with respect to <span class="math notranslate nohighlight">\(\boldsymbol{w}_0\)</span> we obtain</p>
<div class="math notranslate nohighlight">
\[
\frac{\partial J(\boldsymbol{w}_0)}{\partial \boldsymbol{w}_0}= 2\boldsymbol{C}[\boldsymbol{x}]\boldsymbol{w}_0-2\lambda_0\boldsymbol{w}_0=0,
\]</div>
<p>meaning that</p>
<div class="math notranslate nohighlight">
\[
\boldsymbol{C}[\boldsymbol{x}]\boldsymbol{w}_0=\lambda_0\boldsymbol{w}_0.
\]</div>
<p><strong>The direction that maximizes the variance (or minimizes the construction error) is an eigenvector of the covariance matrix</strong>! If we left multiply with <span class="math notranslate nohighlight">\(\boldsymbol{w}_0^T\)</span> we have the variance of the projected data is</p>
<div class="math notranslate nohighlight">
\[
\boldsymbol{w}_0^T\boldsymbol{C}[\boldsymbol{x}]\boldsymbol{w}_0=\lambda_0.
\]</div>
<p>If we want to maximize the variance (minimize the construction error)
we simply pick the eigenvector of the covariance matrix with the
largest eigenvalue. This establishes the link between the minimization
of the reconstruction function <span class="math notranslate nohighlight">\(J\)</span> in terms of an orthogonal matrix
and the maximization of the variance and thereby the covariance of our
observations encoded in the design/feature matrix <span class="math notranslate nohighlight">\(\boldsymbol{X}\)</span>.</p>
<p>The proof
for the other eigenvectors <span class="math notranslate nohighlight">\(\boldsymbol{w}_1,\boldsymbol{w}_2,\dots\)</span> can be
established by applying the above arguments and using the fact that
our basis of eigenvectors is orthogonal, see <a class="reference external" href="https://mitpress.mit.edu/books/machine-learning-1">Murphy chapter
12.2</a>. The
discussion in chapter 12.2 of Murphy’s text has also a nice link with
the Singular Value Decomposition theorem. For categorical data, see
chapter 12.4 and discussion therein.</p>
<p>Additional part of the proof for the other eigenvectors will be added by mid January 2020.</p>
</div>
<div class="section" id="geometric-interpretation-and-link-with-singular-value-decomposition">
<h2><span class="section-number">7.23. </span>Geometric Interpretation and link with Singular Value Decomposition<a class="headerlink" href="#geometric-interpretation-and-link-with-singular-value-decomposition" title="Permalink to this headline"></a></h2>
<p>This material will be added by mid January 2020.</p>
</div>
<div class="section" id="principal-component-analysis">
<h2><span class="section-number">7.24. </span>Principal Component Analysis<a class="headerlink" href="#principal-component-analysis" title="Permalink to this headline"></a></h2>
<p>Principal Component Analysis (PCA) is by far the most popular dimensionality reduction algorithm.
First it identifies the hyperplane that lies closest to the data, and then it projects the data onto it.</p>
<p>The following Python code uses NumPy’s <strong>svd()</strong> function to obtain all the principal components of the
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>
<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]]
</pre></div>
</div>
</div>
</div>
<p>PCA assumes that the dataset is centered around the origin. Scikit-Learn’s PCA classes take care of centering
the data for you. However, if you implement PCA yourself (as in the preceding example), or if you use other libraries, don’t
forget to center the data first.</p>
<p>Once you have identified all the principal components, you can reduce the dimensionality of the dataset
down to <span class="math notranslate nohighlight">\(d\)</span> dimensions by projecting it onto the hyperplane defined by the first <span class="math notranslate nohighlight">\(d\)</span> principal components.
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>
</pre></div>
</div>
</div>
</div>
</div>
<div class="section" id="pca-and-scikit-learn">
<h2><span class="section-number">7.25. </span>PCA and scikit-learn<a class="headerlink" href="#pca-and-scikit-learn" title="Permalink to this headline"></a></h2>
<p>Scikit-Learn’s PCA class implements PCA using SVD decomposition just like we did before. The
following code applies PCA to reduce the dimensionality of the dataset down to two dimensions (note
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]]
</pre></div>
</div>
</div>
</div>
<p>After fitting the PCA transformer to the dataset, you can access the principal components using the
components variable (note that it contains the PCs as horizontal vectors, so, for example, the first
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><span class="o">.</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output traceback highlight-ipythontb notranslate"><div class="highlight"><pre><span></span><span class="gt"> File</span><span class="nn"> &quot;&lt;ipython-input-19-17314f270d45&gt;&quot;</span><span class="gt">, line </span><span class="mi">1</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><span class="o">.</span>
<span class="o">^</span>
<span class="ne">SyntaxError</span>: invalid syntax
</pre></div>
</div>
</div>
</div>
<p>Another very useful piece of information is the explained variance ratio of each principal component,
available via the <span class="math notranslate nohighlight">\(explained\_variance\_ratio\)</span> variable. It indicates the proportion of the dataset’s
variance that lies along the axis of each principal component.</p>
</div>
<div class="section" id="back-to-the-cancer-data">
<h2><span class="section-number">7.26. </span>Back to the Cancer Data<a class="headerlink" href="#back-to-the-cancer-data" title="Permalink to this headline"></a></h2>
<p>We can now repeat the above but applied to real data, in this case our breast cancer data.
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>
<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>
<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">&quot;Train set accuracy from Logistic Regression: </span><span class="si">{:.2f}</span><span class="s2">&quot;</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">&quot;Train set accuracy scaled data: </span><span class="si">{:.2f}</span><span class="s2">&quot;</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">&quot;Train set accuracy scaled and PCA data: </span><span class="si">{:.2f}</span><span class="s2">&quot;</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>
<p>We see that our training data after the PCA decomposition has a performance similar to the non-scaled data.</p>
</div>
<div class="section" id="more-on-the-pca">
<h2><span class="section-number">7.27. </span>More on the PCA<a class="headerlink" href="#more-on-the-pca" title="Permalink to this headline"></a></h2>
<p>Instead of arbitrarily choosing the number of dimensions to reduce down to, it is generally preferable to
choose the number of dimensions that add up to a sufficiently large portion of the variance (e.g., 95%).
Unless, of course, you are reducing dimensionality for data visualization — in that case you will
generally want to reduce the dimensionality down to 2 or 3.
The following code computes PCA without reducing dimensionality, then computes the minimum number
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">&gt;=</span> <span class="mf">0.95</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span>
</pre></div>
</div>
</div>
</div>
<p>You could then set <span class="math notranslate nohighlight">\(n\_components=d\)</span> and run PCA again. However, there is a much better option: instead
of specifying the number of principal components you want to preserve, you can set <span class="math notranslate nohighlight">\(n\_components\)</span> to be
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>
</pre></div>
</div>
</div>
</div>
</div>
<div class="section" id="incremental-pca">
<h2><span class="section-number">7.28. </span>Incremental PCA<a class="headerlink" href="#incremental-pca" title="Permalink to this headline"></a></h2>
<p>One problem with the preceding implementation of PCA is that it requires the whole training set to fit in
memory in order for the SVD algorithm to run. Fortunately, Incremental PCA (IPCA) algorithms have
been developed: you can split the training set into mini-batches and feed an IPCA algorithm one minibatch
at a time. This is useful for large training sets, and also to apply PCA online (i.e., on the fly, as new
instances arrive).</p>
</div>
<div class="section" id="randomized-pca">
<h2><span class="section-number">7.29. </span>Randomized PCA<a class="headerlink" href="#randomized-pca" title="Permalink to this headline"></a></h2>
<p>Scikit-Learn offers yet another option to perform PCA, called Randomized PCA. This is a stochastic
algorithm that quickly finds an approximation of the first d principal components. Its computational
complexity is <span class="math notranslate nohighlight">\(O(m \times d^2)+O(d^3)\)</span>, instead of <span class="math notranslate nohighlight">\(O(m \times n^2) + O(n^3)\)</span>, so it is dramatically faster than the
previous algorithms when <span class="math notranslate nohighlight">\(d\)</span> is much smaller than <span class="math notranslate nohighlight">\(n\)</span>.</p>
</div>
<div class="section" id="kernel-pca">
<h2><span class="section-number">7.30. </span>Kernel PCA<a class="headerlink" href="#kernel-pca" title="Permalink to this headline"></a></h2>
<p>The kernel trick is a mathematical technique that implicitly maps instances into a
very high-dimensional space (called the feature space), enabling nonlinear classification and regression
with Support Vector Machines. Recall that a linear decision boundary in the high-dimensional feature
space corresponds to a complex nonlinear decision boundary in the original space.
It turns out that the same trick can be applied to PCA, making it possible to perform complex nonlinear
projections for dimensionality reduction. This is called Kernel PCA (kPCA). It is often good at
preserving clusters of instances after projection, or sometimes even unrolling datasets that lie close to a
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">&quot;rbf&quot;</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>
</pre></div>
</div>
</div>
</div>
</div>
<div class="section" id="lle">
<h2><span class="section-number">7.31. </span>LLE<a class="headerlink" href="#lle" title="Permalink to this headline"></a></h2>
<p>Locally Linear Embedding (LLE) is another very powerful nonlinear dimensionality reduction
(NLDR) technique. It is a Manifold Learning technique that does not rely on projections like the previous
algorithms. In a nutshell, LLE works by first measuring how each training instance linearly relates to its
closest neighbors (c.n.), and then looking for a low-dimensional representation of the training set where
these local relationships are best preserved (more details shortly).</p>
</div>
<div class="section" id="other-techniques">
<h2><span class="section-number">7.32. </span>Other techniques<a class="headerlink" href="#other-techniques" title="Permalink to this headline"></a></h2>
<p>There are many other dimensionality reduction techniques, several of which are available in Scikit-Learn.</p>
<p>Here are some of the most popular:</p>
<ul class="simple">
<li><p><strong>Multidimensional Scaling (MDS)</strong> reduces dimensionality while trying to preserve the distances between the instances.</p></li>
<li><p><strong>Isomap</strong> creates a graph by connecting each instance to its nearest neighbors, then reduces dimensionality while trying to preserve the geodesic distances between the instances.</p></li>
<li><p><strong>t-Distributed Stochastic Neighbor Embedding</strong> (t-SNE) reduces dimensionality while trying to keep similar instances close and dissimilar instances apart. It is mostly used for visualization, in particular to visualize clusters of instances in high-dimensional space (e.g., to visualize the MNIST images in 2D).</p></li>
<li><p>Linear Discriminant Analysis (LDA) is actually a classification algorithm, but during training it learns the most discriminative axes between the classes, and these axes can then be used to define a hyperplane onto which to project the data. The benefit is that the projection will keep classes as far apart as possible, so LDA is a good technique to reduce dimensionality before running another classification algorithm such as a Support Vector Machine (SVM) classifier discussed in the SVM lectures.</p></li>
</ul>
</div>
</div>
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