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<!-- navigation toc: --> <li><a href="._DimRed-bs001.html#___sec0" style="font-size: 80%;">Reducing the number of degrees of freedom, overarching view</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs002.html#___sec1" style="font-size: 80%;">Preprocessing our data</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs003.html#___sec2" style="font-size: 80%;">More preprocessing</a></li>
<!-- navigation toc: --> <li><a href="#___sec3" style="font-size: 80%;">Simple preprocessing examples, Franke function and regression</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs005.html#___sec4" style="font-size: 80%;">Simple preprocessing examples, breast cancer data and classification, Support Vector Machines</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs006.html#___sec5" style="font-size: 80%;">More on Cancer Data, now with Logistic Regression</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs007.html#___sec6" style="font-size: 80%;">Why should we think of reducing the dimensionality</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs008.html#___sec7" style="font-size: 80%;">Basic ideas of the Principal Component Analysis (PCA)</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs009.html#___sec8" style="font-size: 80%;">Introducing the Covariance and Correlation functions</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs010.html#___sec9" style="font-size: 80%;">Correlation Function and Design/Feature Matrix</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs011.html#___sec10" style="font-size: 80%;">Covariance Matrix Examples</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs012.html#___sec11" style="font-size: 80%;">Correlation Matrix</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs013.html#___sec12" style="font-size: 80%;">Correlation Matrix with Pandas</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs014.html#___sec13" style="font-size: 80%;">Correlation Matrix with Pandas and the Franke function</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs015.html#___sec14" style="font-size: 80%;">Rewriting the Covariance and/or Correlation Matrix</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs016.html#___sec15" style="font-size: 80%;">Towards the PCA theorem</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs017.html#___sec16" style="font-size: 80%;">The Algorithm before the Theorem</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs018.html#___sec17" style="font-size: 80%;">Classical PCA Theorem</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">Proof of the PCA Theorem</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">PCA Proof continued</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">The final step</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">Principal Component Analysis</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">PCA and scikit-learn</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">Back to the Cancer Data</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs025.html#___sec24" style="font-size: 80%;">More on the PCA</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs026.html#___sec25" style="font-size: 80%;">Incremental PCA</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs027.html#___sec26" style="font-size: 80%;">Randomized PCA</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs028.html#___sec27" style="font-size: 80%;">Kernel PCA</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs029.html#___sec28" style="font-size: 80%;">LLE</a></li>
<!-- navigation toc: --> <li><a href="._DimRed-bs030.html#___sec29" style="font-size: 80%;">Other techniques</a></li>
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<h2 id="___sec3" class="anchor">Simple preprocessing examples, Franke function and regression </h2>
<p>
<!-- code=python (!bc pycod) typeset with pygments style "default" -->
<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span><span style="color: #408080; font-style: italic"># Common imports</span>
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">os</span>
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">numpy</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">np</span>
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">pandas</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">pd</span>
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">matplotlib.pyplot</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">plt</span>
<span style="color: #008000; font-weight: bold">import</span> <span style="color: #0000FF; font-weight: bold">sklearn.linear_model</span> <span style="color: #008000; font-weight: bold">as</span> <span style="color: #0000FF; font-weight: bold">skl</span>
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.metrics</span> <span style="color: #008000; font-weight: bold">import</span> mean_squared_error
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.model_selection</span> <span style="color: #008000; font-weight: bold">import</span> train_test_split
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.preprocessing</span> <span style="color: #008000; font-weight: bold">import</span> MinMaxScaler, StandardScaler, Normalizer
<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">sklearn.svm</span> <span style="color: #008000; font-weight: bold">import</span> SVR
<span style="color: #408080; font-style: italic"># Where to save the figures and data files</span>
PROJECT_ROOT_DIR <span style="color: #666666">=</span> <span style="color: #BA2121">&quot;Results&quot;</span>
FIGURE_ID <span style="color: #666666">=</span> <span style="color: #BA2121">&quot;Results/FigureFiles&quot;</span>
DATA_ID <span style="color: #666666">=</span> <span style="color: #BA2121">&quot;DataFiles/&quot;</span>
<span style="color: #008000; font-weight: bold">if</span> <span style="color: #AA22FF; font-weight: bold">not</span> os<span style="color: #666666">.</span>path<span style="color: #666666">.</span>exists(PROJECT_ROOT_DIR):
os<span style="color: #666666">.</span>mkdir(PROJECT_ROOT_DIR)
<span style="color: #008000; font-weight: bold">if</span> <span style="color: #AA22FF; font-weight: bold">not</span> os<span style="color: #666666">.</span>path<span style="color: #666666">.</span>exists(FIGURE_ID):
os<span style="color: #666666">.</span>makedirs(FIGURE_ID)
<span style="color: #008000; font-weight: bold">if</span> <span style="color: #AA22FF; font-weight: bold">not</span> os<span style="color: #666666">.</span>path<span style="color: #666666">.</span>exists(DATA_ID):
os<span style="color: #666666">.</span>makedirs(DATA_ID)
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">image_path</span>(fig_id):
<span style="color: #008000; font-weight: bold">return</span> os<span style="color: #666666">.</span>path<span style="color: #666666">.</span>join(FIGURE_ID, fig_id)
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">data_path</span>(dat_id):
<span style="color: #008000; font-weight: bold">return</span> os<span style="color: #666666">.</span>path<span style="color: #666666">.</span>join(DATA_ID, dat_id)
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">save_fig</span>(fig_id):
plt<span style="color: #666666">.</span>savefig(image_path(fig_id) <span style="color: #666666">+</span> <span style="color: #BA2121">&quot;.png&quot;</span>, format<span style="color: #666666">=</span><span style="color: #BA2121">&#39;png&#39;</span>)
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">FrankeFunction</span>(x,y):
term1 <span style="color: #666666">=</span> <span style="color: #666666">0.75*</span>np<span style="color: #666666">.</span>exp(<span style="color: #666666">-</span>(<span style="color: #666666">0.25*</span>(<span style="color: #666666">9*</span>x<span style="color: #666666">-2</span>)<span style="color: #666666">**2</span>) <span style="color: #666666">-</span> <span style="color: #666666">0.25*</span>((<span style="color: #666666">9*</span>y<span style="color: #666666">-2</span>)<span style="color: #666666">**2</span>))
term2 <span style="color: #666666">=</span> <span style="color: #666666">0.75*</span>np<span style="color: #666666">.</span>exp(<span style="color: #666666">-</span>((<span style="color: #666666">9*</span>x<span style="color: #666666">+1</span>)<span style="color: #666666">**2</span>)<span style="color: #666666">/49.0</span> <span style="color: #666666">-</span> <span style="color: #666666">0.1*</span>(<span style="color: #666666">9*</span>y<span style="color: #666666">+1</span>))
term3 <span style="color: #666666">=</span> <span style="color: #666666">0.5*</span>np<span style="color: #666666">.</span>exp(<span style="color: #666666">-</span>(<span style="color: #666666">9*</span>x<span style="color: #666666">-7</span>)<span style="color: #666666">**2/4.0</span> <span style="color: #666666">-</span> <span style="color: #666666">0.25*</span>((<span style="color: #666666">9*</span>y<span style="color: #666666">-3</span>)<span style="color: #666666">**2</span>))
term4 <span style="color: #666666">=</span> <span style="color: #666666">-0.2*</span>np<span style="color: #666666">.</span>exp(<span style="color: #666666">-</span>(<span style="color: #666666">9*</span>x<span style="color: #666666">-4</span>)<span style="color: #666666">**2</span> <span style="color: #666666">-</span> (<span style="color: #666666">9*</span>y<span style="color: #666666">-7</span>)<span style="color: #666666">**2</span>)
<span style="color: #008000; font-weight: bold">return</span> term1 <span style="color: #666666">+</span> term2 <span style="color: #666666">+</span> term3 <span style="color: #666666">+</span> term4
<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">create_X</span>(x, y, n ):
<span style="color: #008000; font-weight: bold">if</span> <span style="color: #008000">len</span>(x<span style="color: #666666">.</span>shape) <span style="color: #666666">&gt;</span> <span style="color: #666666">1</span>:
x <span style="color: #666666">=</span> np<span style="color: #666666">.</span>ravel(x)
y <span style="color: #666666">=</span> np<span style="color: #666666">.</span>ravel(y)
N <span style="color: #666666">=</span> <span style="color: #008000">len</span>(x)
l <span style="color: #666666">=</span> <span style="color: #008000">int</span>((n<span style="color: #666666">+1</span>)<span style="color: #666666">*</span>(n<span style="color: #666666">+2</span>)<span style="color: #666666">/2</span>) <span style="color: #408080; font-style: italic"># Number of elements in beta</span>
X <span style="color: #666666">=</span> np<span style="color: #666666">.</span>ones((N,l))
<span style="color: #008000; font-weight: bold">for</span> i <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(<span style="color: #666666">1</span>,n<span style="color: #666666">+1</span>):
q <span style="color: #666666">=</span> <span style="color: #008000">int</span>((i)<span style="color: #666666">*</span>(i<span style="color: #666666">+1</span>)<span style="color: #666666">/2</span>)
<span style="color: #008000; font-weight: bold">for</span> k <span style="color: #AA22FF; font-weight: bold">in</span> <span style="color: #008000">range</span>(i<span style="color: #666666">+1</span>):
X[:,q<span style="color: #666666">+</span>k] <span style="color: #666666">=</span> (x<span style="color: #666666">**</span>(i<span style="color: #666666">-</span>k))<span style="color: #666666">*</span>(y<span style="color: #666666">**</span>k)
<span style="color: #008000; font-weight: bold">return</span> X
<span style="color: #408080; font-style: italic"># Making meshgrid of datapoints and compute Franke&#39;s function</span>
n <span style="color: #666666">=</span> <span style="color: #666666">5</span>
N <span style="color: #666666">=</span> <span style="color: #666666">1000</span>
x <span style="color: #666666">=</span> np<span style="color: #666666">.</span>sort(np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>uniform(<span style="color: #666666">0</span>, <span style="color: #666666">1</span>, N))
y <span style="color: #666666">=</span> np<span style="color: #666666">.</span>sort(np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>uniform(<span style="color: #666666">0</span>, <span style="color: #666666">1</span>, N))
z <span style="color: #666666">=</span> FrankeFunction(x, y)
X <span style="color: #666666">=</span> create_X(x, y, n<span style="color: #666666">=</span>n)
<span style="color: #408080; font-style: italic"># split in training and test data</span>
X_train, X_test, y_train, y_test <span style="color: #666666">=</span> train_test_split(X,z,test_size<span style="color: #666666">=0.2</span>)
svm <span style="color: #666666">=</span> SVR(gamma<span style="color: #666666">=</span><span style="color: #BA2121">&#39;auto&#39;</span>,C<span style="color: #666666">=10.0</span>)
svm<span style="color: #666666">.</span>fit(X_train, y_train)
<span style="color: #408080; font-style: italic"># The mean squared error and R2 score</span>
<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">&quot;MSE before scaling: {:.2f}&quot;</span><span style="color: #666666">.</span>format(mean_squared_error(svm<span style="color: #666666">.</span>predict(X_test), y_test)))
<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">&quot;R2 score before scaling {:.2f}&quot;</span><span style="color: #666666">.</span>format(svm<span style="color: #666666">.</span>score(X_test,y_test)))
scaler <span style="color: #666666">=</span> StandardScaler()
scaler<span style="color: #666666">.</span>fit(X_train)
X_train_scaled <span style="color: #666666">=</span> scaler<span style="color: #666666">.</span>transform(X_train)
X_test_scaled <span style="color: #666666">=</span> scaler<span style="color: #666666">.</span>transform(X_test)
<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">&quot;Feature min values before scaling:</span><span style="color: #BB6622; font-weight: bold">\n</span><span style="color: #BA2121"> {}&quot;</span><span style="color: #666666">.</span>format(X_train<span style="color: #666666">.</span>min(axis<span style="color: #666666">=0</span>)))
<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">&quot;Feature max values before scaling:</span><span style="color: #BB6622; font-weight: bold">\n</span><span style="color: #BA2121"> {}&quot;</span><span style="color: #666666">.</span>format(X_train<span style="color: #666666">.</span>max(axis<span style="color: #666666">=0</span>)))
<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">&quot;Feature min values after scaling:</span><span style="color: #BB6622; font-weight: bold">\n</span><span style="color: #BA2121"> {}&quot;</span><span style="color: #666666">.</span>format(X_train_scaled<span style="color: #666666">.</span>min(axis<span style="color: #666666">=0</span>)))
<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">&quot;Feature max values after scaling:</span><span style="color: #BB6622; font-weight: bold">\n</span><span style="color: #BA2121"> {}&quot;</span><span style="color: #666666">.</span>format(X_train_scaled<span style="color: #666666">.</span>max(axis<span style="color: #666666">=0</span>)))
svm <span style="color: #666666">=</span> SVR(gamma<span style="color: #666666">=</span><span style="color: #BA2121">&#39;auto&#39;</span>,C<span style="color: #666666">=10.0</span>)
svm<span style="color: #666666">.</span>fit(X_train_scaled, y_train)
<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">&quot;MSE after scaling: {:.2f}&quot;</span><span style="color: #666666">.</span>format(mean_squared_error(svm<span style="color: #666666">.</span>predict(X_test_scaled), y_test)))
<span style="color: #008000; font-weight: bold">print</span>(<span style="color: #BA2121">&quot;R2 score for scaled data: {:.2f}&quot;</span><span style="color: #666666">.</span>format(svm<span style="color: #666666">.</span>score(X_test_scaled,y_test)))
</pre></div>
<p>
<p>
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<li><a href="._DimRed-bs009.html">10</a></li>
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<li><a href="._DimRed-bs011.html">12</a></li>
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<li><a href="._DimRed-bs013.html">14</a></li>
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<li><a href="._DimRed-bs030.html">31</a></li>
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