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<!-- ------------------- main content ---------------------- -->
<center><h1 style="text-align: center;">Data Analysis and Machine Learning: Preprocessing and Dimensionality Reduction</h1></center> <!-- document title -->
<p>
<!-- author(s): Morten Hjorth-Jensen -->
<center>
<b>Morten Hjorth-Jensen</b> [1, 2]
</center>
<p>&nbsp;<br>
<!-- institution(s) -->
<center>[1] <b>Department of Physics, University of Oslo</b></center>
<center>[2] <b>Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University</b></center>
<br>
<p>&nbsp;<br>
<center><h4>Oct 19, 2019</h4></center> <!-- date -->
<br>
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<center style="font-size:80%">
<!-- copyright --> &copy; 1999-2019, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license
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</section>
<section>
<h2 id="___sec0">Reducing the number of degrees of freedom, overarching view </h2>
<div class="alert alert-block alert-block alert-text-normal">
<b></b>
<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>
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.
</div>
</section>
<section>
<h2 id="___sec1">Preprocessing our data </h2>
<div class="alert alert-block alert-block alert-text-normal">
<b></b>
<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>
<b>Scikit-Learn</b> has several functions which allow us to rescale the
data, normally resulting in much better results in terms of various
accuracy scores. The <b>StandardScaler</b> function in <b>Scikit-Learn</b>
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 <b>Scikit-Learn</b> is the <b>MinMaxScaler</b> which
ensures that all features are exactly between \( 0 \) and \( 1 \). The
</section>
<section>
<h2 id="___sec2">More preprocessing </h2>
<p>
The <b>Normalizer</b> 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&#8217;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>
The <b>RobustScaler</b> 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.
</div>
</section>
<section>
<h2 id="___sec3">Simple preprocessing examples, Franke function and regression </h2>
<p>
<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
<div class="highlight" style="background: #eeeedd"><pre style="font-size: 80%; line-height: 125%"><span></span><span style="color: #228B22"># Common imports</span>
<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">os</span>
<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">numpy</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">np</span>
<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">pandas</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">pd</span>
<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">matplotlib.pyplot</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">plt</span>
<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">sklearn.linear_model</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">skl</span>
<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">sklearn.metrics</span> <span style="color: #8B008B; font-weight: bold">import</span> mean_squared_error
<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">sklearn.model_selection</span> <span style="color: #8B008B; font-weight: bold">import</span> train_test_split
<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">sklearn.preprocessing</span> <span style="color: #8B008B; font-weight: bold">import</span> MinMaxScaler, StandardScaler, Normalizer
<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">sklearn.svm</span> <span style="color: #8B008B; font-weight: bold">import</span> SVR
<span style="color: #228B22"># Where to save the figures and data files</span>
PROJECT_ROOT_DIR = <span style="color: #CD5555">&quot;Results&quot;</span>
FIGURE_ID = <span style="color: #CD5555">&quot;Results/FigureFiles&quot;</span>
DATA_ID = <span style="color: #CD5555">&quot;DataFiles/&quot;</span>
<span style="color: #8B008B; font-weight: bold">if</span> <span style="color: #8B008B">not</span> os.path.exists(PROJECT_ROOT_DIR):
os.mkdir(PROJECT_ROOT_DIR)
<span style="color: #8B008B; font-weight: bold">if</span> <span style="color: #8B008B">not</span> os.path.exists(FIGURE_ID):
os.makedirs(FIGURE_ID)
<span style="color: #8B008B; font-weight: bold">if</span> <span style="color: #8B008B">not</span> os.path.exists(DATA_ID):
os.makedirs(DATA_ID)
<span style="color: #8B008B; font-weight: bold">def</span> <span style="color: #008b45">image_path</span>(fig_id):
<span style="color: #8B008B; font-weight: bold">return</span> os.path.join(FIGURE_ID, fig_id)
<span style="color: #8B008B; font-weight: bold">def</span> <span style="color: #008b45">data_path</span>(dat_id):
<span style="color: #8B008B; font-weight: bold">return</span> os.path.join(DATA_ID, dat_id)
<span style="color: #8B008B; font-weight: bold">def</span> <span style="color: #008b45">save_fig</span>(fig_id):
plt.savefig(image_path(fig_id) + <span style="color: #CD5555">&quot;.png&quot;</span>, format=<span style="color: #CD5555">&#39;png&#39;</span>)
<span style="color: #8B008B; font-weight: bold">def</span> <span style="color: #008b45">FrankeFunction</span>(x,y):
term1 = <span style="color: #B452CD">0.75</span>*np.exp(-(<span style="color: #B452CD">0.25</span>*(<span style="color: #B452CD">9</span>*x-<span style="color: #B452CD">2</span>)**<span style="color: #B452CD">2</span>) - <span style="color: #B452CD">0.25</span>*((<span style="color: #B452CD">9</span>*y-<span style="color: #B452CD">2</span>)**<span style="color: #B452CD">2</span>))
term2 = <span style="color: #B452CD">0.75</span>*np.exp(-((<span style="color: #B452CD">9</span>*x+<span style="color: #B452CD">1</span>)**<span style="color: #B452CD">2</span>)/<span style="color: #B452CD">49.0</span> - <span style="color: #B452CD">0.1</span>*(<span style="color: #B452CD">9</span>*y+<span style="color: #B452CD">1</span>))
term3 = <span style="color: #B452CD">0.5</span>*np.exp(-(<span style="color: #B452CD">9</span>*x-<span style="color: #B452CD">7</span>)**<span style="color: #B452CD">2</span>/<span style="color: #B452CD">4.0</span> - <span style="color: #B452CD">0.25</span>*((<span style="color: #B452CD">9</span>*y-<span style="color: #B452CD">3</span>)**<span style="color: #B452CD">2</span>))
term4 = -<span style="color: #B452CD">0.2</span>*np.exp(-(<span style="color: #B452CD">9</span>*x-<span style="color: #B452CD">4</span>)**<span style="color: #B452CD">2</span> - (<span style="color: #B452CD">9</span>*y-<span style="color: #B452CD">7</span>)**<span style="color: #B452CD">2</span>)
<span style="color: #8B008B; font-weight: bold">return</span> term1 + term2 + term3 + term4
<span style="color: #8B008B; font-weight: bold">def</span> <span style="color: #008b45">create_X</span>(x, y, n ):
<span style="color: #8B008B; font-weight: bold">if</span> <span style="color: #658b00">len</span>(x.shape) &gt; <span style="color: #B452CD">1</span>:
x = np.ravel(x)
y = np.ravel(y)
N = <span style="color: #658b00">len</span>(x)
l = <span style="color: #658b00">int</span>((n+<span style="color: #B452CD">1</span>)*(n+<span style="color: #B452CD">2</span>)/<span style="color: #B452CD">2</span>) <span style="color: #228B22"># Number of elements in beta</span>
X = np.ones((N,l))
<span style="color: #8B008B; font-weight: bold">for</span> i <span style="color: #8B008B">in</span> <span style="color: #658b00">range</span>(<span style="color: #B452CD">1</span>,n+<span style="color: #B452CD">1</span>):
q = <span style="color: #658b00">int</span>((i)*(i+<span style="color: #B452CD">1</span>)/<span style="color: #B452CD">2</span>)
<span style="color: #8B008B; font-weight: bold">for</span> k <span style="color: #8B008B">in</span> <span style="color: #658b00">range</span>(i+<span style="color: #B452CD">1</span>):
X[:,q+k] = (x**(i-k))*(y**k)
<span style="color: #8B008B; font-weight: bold">return</span> X
<span style="color: #228B22"># Making meshgrid of datapoints and compute Franke&#39;s function</span>
n = <span style="color: #B452CD">5</span>
N = <span style="color: #B452CD">1000</span>
x = np.sort(np.random.uniform(<span style="color: #B452CD">0</span>, <span style="color: #B452CD">1</span>, N))
y = np.sort(np.random.uniform(<span style="color: #B452CD">0</span>, <span style="color: #B452CD">1</span>, N))
z = FrankeFunction(x, y)
X = create_X(x, y, n=n)
<span style="color: #228B22"># split in training and test data</span>
X_train, X_test, y_train, y_test = train_test_split(X,z,test_size=<span style="color: #B452CD">0.2</span>)
svm = SVR(gamma=<span style="color: #CD5555">&#39;auto&#39;</span>,C=<span style="color: #B452CD">10.0</span>)
svm.fit(X_train, y_train)
<span style="color: #228B22"># The mean squared error and R2 score</span>
<span style="color: #8B008B; font-weight: bold">print</span>(<span style="color: #CD5555">&quot;MSE before scaling: {:.2f}&quot;</span>.format(mean_squared_error(svm.predict(X_test), y_test)))
<span style="color: #8B008B; font-weight: bold">print</span>(<span style="color: #CD5555">&quot;R2 score before scaling {:.2f}&quot;</span>.format(svm.score(X_test,y_test)))
scaler = StandardScaler()
scaler.fit(X_train)
X_train_scaled = scaler.transform(X_train)
X_test_scaled = scaler.transform(X_test)
<span style="color: #8B008B; font-weight: bold">print</span>(<span style="color: #CD5555">&quot;Feature min values before scaling:\n {}&quot;</span>.format(X_train.min(axis=<span style="color: #B452CD">0</span>)))
<span style="color: #8B008B; font-weight: bold">print</span>(<span style="color: #CD5555">&quot;Feature max values before scaling:\n {}&quot;</span>.format(X_train.max(axis=<span style="color: #B452CD">0</span>)))
<span style="color: #8B008B; font-weight: bold">print</span>(<span style="color: #CD5555">&quot;Feature min values after scaling:\n {}&quot;</span>.format(X_train_scaled.min(axis=<span style="color: #B452CD">0</span>)))
<span style="color: #8B008B; font-weight: bold">print</span>(<span style="color: #CD5555">&quot;Feature max values after scaling:\n {}&quot;</span>.format(X_train_scaled.max(axis=<span style="color: #B452CD">0</span>)))
svm = SVR(gamma=<span style="color: #CD5555">&#39;auto&#39;</span>,C=<span style="color: #B452CD">10.0</span>)
svm.fit(X_train_scaled, y_train)
<span style="color: #8B008B; font-weight: bold">print</span>(<span style="color: #CD5555">&quot;MSE after scaling: {:.2f}&quot;</span>.format(mean_squared_error(svm.predict(X_test_scaled), y_test)))
<span style="color: #8B008B; font-weight: bold">print</span>(<span style="color: #CD5555">&quot;R2 score for scaled data: {:.2f}&quot;</span>.format(svm.score(X_test_scaled,y_test)))
</pre></div>
</section>
<section>
<h2 id="___sec4">Simple preprocessing examples, breast cancer data and classification, Support Vector Machines </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>
<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
<div class="highlight" style="background: #eeeedd"><pre style="font-size: 80%; line-height: 125%"><span></span><span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">matplotlib.pyplot</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">plt</span>
<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">numpy</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">np</span>
<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">sklearn.model_selection</span> <span style="color: #8B008B; font-weight: bold">import</span> train_test_split
<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">sklearn.datasets</span> <span style="color: #8B008B; font-weight: bold">import</span> load_breast_cancer
<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">sklearn.svm</span> <span style="color: #8B008B; font-weight: bold">import</span> SVC
cancer = load_breast_cancer()
X_train, X_test, y_train, y_test = train_test_split(cancer.data,cancer.target,random_state=<span style="color: #B452CD">0</span>)
<span style="color: #8B008B; font-weight: bold">print</span>(X_train.shape)
<span style="color: #8B008B; font-weight: bold">print</span>(X_test.shape)
svm = SVC(C=<span style="color: #B452CD">100</span>)
svm.fit(X_train, y_train)
<span style="color: #8B008B; font-weight: bold">print</span>(<span style="color: #CD5555">&quot;Test set accuracy: {:.2f}&quot;</span>.format(svm.score(X_test,y_test)))
<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">sklearn.preprocessing</span> <span style="color: #8B008B; font-weight: bold">import</span> MinMaxScaler, StandardScaler
scaler = MinMaxScaler()
scaler.fit(X_train)
X_train_scaled = scaler.transform(X_train)
X_test_scaled = scaler.transform(X_test)
<span style="color: #8B008B; font-weight: bold">print</span>(<span style="color: #CD5555">&quot;Feature min values before scaling:\n {}&quot;</span>.format(X_train.min(axis=<span style="color: #B452CD">0</span>)))
<span style="color: #8B008B; font-weight: bold">print</span>(<span style="color: #CD5555">&quot;Feature max values before scaling:\n {}&quot;</span>.format(X_train.max(axis=<span style="color: #B452CD">0</span>)))
<span style="color: #8B008B; font-weight: bold">print</span>(<span style="color: #CD5555">&quot;Feature min values before scaling:\n {}&quot;</span>.format(X_train_scaled.min(axis=<span style="color: #B452CD">0</span>)))
<span style="color: #8B008B; font-weight: bold">print</span>(<span style="color: #CD5555">&quot;Feature max values before scaling:\n {}&quot;</span>.format(X_train_scaled.max(axis=<span style="color: #B452CD">0</span>)))
svm.fit(X_train_scaled, y_train)
<span style="color: #8B008B; font-weight: bold">print</span>(<span style="color: #CD5555">&quot;Test set accuracy scaled data with Min-Max scaling: {:.2f}&quot;</span>.format(svm.score(X_test_scaled,y_test)))
scaler = StandardScaler()
scaler.fit(X_train)
X_train_scaled = scaler.transform(X_train)
X_test_scaled = scaler.transform(X_test)
svm.fit(X_train_scaled, y_train)
<span style="color: #8B008B; font-weight: bold">print</span>(<span style="color: #CD5555">&quot;Test set accuracy scaled data with Standar Scaler: {:.2f}&quot;</span>.format(svm.score(X_test_scaled,y_test)))
</pre></div>
</section>
<section>
<h2 id="___sec5">More on Cancer Data, now with Logistic Regression </h2>
<p>
<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
<div class="highlight" style="background: #eeeedd"><pre style="font-size: 80%; line-height: 125%"><span></span><span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">matplotlib.pyplot</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">plt</span>
<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">numpy</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">np</span>
<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">sklearn.model_selection</span> <span style="color: #8B008B; font-weight: bold">import</span> train_test_split
<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">sklearn.datasets</span> <span style="color: #8B008B; font-weight: bold">import</span> load_breast_cancer
<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">sklearn.linear_model</span> <span style="color: #8B008B; font-weight: bold">import</span> LogisticRegression
cancer = load_breast_cancer()
<span style="color: #228B22"># Set up training data</span>
X_train, X_test, y_train, y_test = train_test_split(cancer.data,cancer.target,random_state=<span style="color: #B452CD">0</span>)
logreg = LogisticRegression()
logreg.fit(X_train, y_train)
<span style="color: #8B008B; font-weight: bold">print</span>(<span style="color: #CD5555">&quot;Test set accuracy: {:.2f}&quot;</span>.format(logreg.score(X_test,y_test)))
<span style="color: #228B22"># Scale data</span>
<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">sklearn.preprocessing</span> <span style="color: #8B008B; font-weight: bold">import</span> StandardScaler
scaler = StandardScaler()
scaler.fit(X_train)
X_train_scaled = scaler.transform(X_train)
X_test_scaled = scaler.transform(X_test)
logreg.fit(X_train_scaled, y_train)
<span style="color: #8B008B; font-weight: bold">print</span>(<span style="color: #CD5555">&quot;Test set accuracy scaled data: {:.2f}&quot;</span>.format(logreg.score(X_test_scaled,y_test)))
</pre></div>
</section>
<section>
<h2 id="___sec6">Why should we think of reducing the dimensionality </h2>
<p>
In addition to the plot of the features, we study now also the covariance (or rather the correlation matrix).
We use also <b>Pandas</b> to compute the correlation matrix.
<p>
<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
<div class="highlight" style="background: #eeeedd"><pre style="font-size: 80%; line-height: 125%"><span></span><span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">matplotlib.pyplot</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">plt</span>
<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">numpy</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">np</span>
<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">sklearn.model_selection</span> <span style="color: #8B008B; font-weight: bold">import</span> train_test_split
<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">sklearn.datasets</span> <span style="color: #8B008B; font-weight: bold">import</span> load_breast_cancer
<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">sklearn.linear_model</span> <span style="color: #8B008B; font-weight: bold">import</span> LogisticRegression
cancer = load_breast_cancer()
<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">pandas</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">pd</span>
<span style="color: #228B22"># Making a data frame</span>
cancerpd = pd.DataFrame(cancer.data, columns=cancer.feature_names)
fig, axes = plt.subplots(<span style="color: #B452CD">15</span>,<span style="color: #B452CD">2</span>,figsize=(<span style="color: #B452CD">10</span>,<span style="color: #B452CD">20</span>))
malignant = cancer.data[cancer.target == <span style="color: #B452CD">0</span>]
benign = cancer.data[cancer.target == <span style="color: #B452CD">1</span>]
ax = axes.ravel()
<span style="color: #8B008B; font-weight: bold">for</span> i <span style="color: #8B008B">in</span> <span style="color: #658b00">range</span>(<span style="color: #B452CD">30</span>):
_, bins = np.histogram(cancer.data[:,i], bins =<span style="color: #B452CD">50</span>)
ax[i].hist(malignant[:,i], bins = bins, alpha = <span style="color: #B452CD">0.5</span>)
ax[i].hist(benign[:,i], bins = bins, alpha = <span style="color: #B452CD">0.5</span>)
ax[i].set_title(cancer.feature_names[i])
ax[i].set_yticks(())
ax[<span style="color: #B452CD">0</span>].set_xlabel(<span style="color: #CD5555">&quot;Feature magnitude&quot;</span>)
ax[<span style="color: #B452CD">0</span>].set_ylabel(<span style="color: #CD5555">&quot;Frequency&quot;</span>)
ax[<span style="color: #B452CD">0</span>].legend([<span style="color: #CD5555">&quot;Malignant&quot;</span>, <span style="color: #CD5555">&quot;Benign&quot;</span>], loc =<span style="color: #CD5555">&quot;best&quot;</span>)
fig.tight_layout()
plt.show()
<span style="color: #8B008B; font-weight: bold">import</span> <span style="color: #008b45; text-decoration: underline">seaborn</span> <span style="color: #8B008B; font-weight: bold">as</span> <span style="color: #008b45; text-decoration: underline">sns</span>
correlation_matrix = cancerpd.corr().round(<span style="color: #B452CD">1</span>)
<span style="color: #228B22"># use the heatmap function from seaborn to plot the correlation matrix</span>
<span style="color: #228B22"># annot = True to print the values inside the square</span>
sns.heatmap(data=correlation_matrix, annot=<span style="color: #658b00">True</span>)
plt.show()
<span style="color: #228B22">#print eigvalues of correlation matrix</span>
EigValues, EigVectors = np.linalg.eig(correlation_matrix)
<span style="color: #8B008B; font-weight: bold">print</span>(EigValues)
<span style="color: #228B22">#split into train and test and then scale thereafter</span>
X_train, X_test, y_train, y_test = train_test_split(cancer.data,cancer.target,random_state=<span style="color: #B452CD">0</span>)
<span style="color: #8B008B; font-weight: bold">print</span>(X_train.shape)
<span style="color: #8B008B; font-weight: bold">print</span>(X_test.shape)
logreg = LogisticRegression()
logreg.fit(X_train, y_train)
<span style="color: #8B008B; font-weight: bold">print</span>(<span style="color: #CD5555">&quot;Test set accuracy from Logistic Regression: {:.2f}&quot;</span>.format(logreg.score(X_test,y_test)))
<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">sklearn.preprocessing</span> <span style="color: #8B008B; font-weight: bold">import</span> MinMaxScaler, StandardScaler
scaler = StandardScaler()
scaler.fit(X_train)
X_train_scaled = scaler.transform(X_train)
X_test_scaled = scaler.transform(X_test)
logreg.fit(X_train_scaled, y_train)
<span style="color: #8B008B; font-weight: bold">print</span>(<span style="color: #CD5555">&quot;Test set accuracy scaled data: {:.2f}&quot;</span>.format(logreg.score(X_test_scaled,y_test)))
</pre></div>
<p>
<!-- todo: add more text in order to explain what is done here, discuss the correlation matrix -->
</section>
<section>
<h2 id="___sec7">Basic ideas of the Principal Component Analysis (PCA) </h2>
</section>
<section>
<h2 id="___sec8">Introducing the Covariance and Correlation functions </h2>
</section>
<section>
<h2 id="___sec9">Classical PCA Theorem </h2>
</section>
<section>
<h2 id="___sec10">Prof of the PCA Theorem </h2>
</section>
<section>
<h2 id="___sec11">Getting started with PCA </h2>
<p>
<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
<div class="highlight" style="background: #eeeedd"><pre style="font-size: 80%; line-height: 125%"><span></span><span style="color: #228B22"># Now add PCA</span>
<span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">sklearn.decomposition</span> <span style="color: #8B008B; font-weight: bold">import</span> PCA
pca = PCA(n_components = <span style="color: #B452CD">2</span>)
pca.fit(X_train_scaled)
X_pca = pca.transform(X_train_scaled)
</pre></div>
</section>
<section>
<h2 id="___sec12">Principal Component Analysis </h2>
<div class="alert alert-block alert-block alert-text-normal">
<b></b>
<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>
The following Python code uses NumPy&#8217;s <b>svd()</b> function to obtain all the principal components of the
training set, then extracts the first two principal components
<p>
<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
<div class="highlight" style="background: #eeeedd"><pre style="font-size: 80%; line-height: 125%"><span></span>X_centered = X - X.mean(axis=<span style="color: #B452CD">0</span>)
U, s, V = np.linalg.svd(X_centered)
c1 = V.T[:, <span style="color: #B452CD">0</span>]
c2 = V.T[:, <span style="color: #B452CD">1</span>]
</pre></div>
<p>
PCA assumes that the dataset is centered around the origin. Scikit-Learn&#8217;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&#8217;t
forget to center the data first.
<p>
Once you have identified all the principal components, you can reduce the dimensionality of the dataset
down to \( d \) dimensions by projecting it onto the hyperplane defined by the first \( d \) principal components.
Selecting this hyperplane ensures that the projection will preserve as much variance as possible.
<p>
<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
<div class="highlight" style="background: #eeeedd"><pre style="font-size: 80%; line-height: 125%"><span></span>W2 = V.T[:, :<span style="color: #B452CD">2</span>]
X2D = X_centered.dot(W2)
</pre></div>
</section>
<section>
<h2 id="___sec13">PCA and scikit-learn </h2>
<p>
Scikit-Learn&#8217;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>
<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
<div class="highlight" style="background: #eeeedd"><pre style="font-size: 80%; line-height: 125%"><span></span><span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">sklearn.decomposition</span> <span style="color: #8B008B; font-weight: bold">import</span> PCA
pca = PCA(n_components = <span style="color: #B452CD">2</span>)
X2D = pca.fit_transform(X)
</pre></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>
<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
<div class="highlight" style="background: #eeeedd"><pre style="font-size: 80%; line-height: 125%"><span></span>pca.components_.T[:, <span style="color: #B452CD">0</span>]).
</pre></div>
<p>
Another very useful piece of information is the explained variance ratio of each principal component,
available via the \( explained\_variance\_ratio \) variable. It indicates the proportion of the dataset&#8217;s
variance that lies along the axis of each principal component.
More material to come here.
</section>
<section>
<h2 id="___sec14">More on the PCA </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 &#8212; 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&#8217;s variance:
<p>
<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
<div class="highlight" style="background: #eeeedd"><pre style="font-size: 80%; line-height: 125%"><span></span>pca = PCA()
pca.fit(X)
cumsum = np.cumsum(pca.explained_variance_ratio_)
d = np.argmax(cumsum &gt;= <span style="color: #B452CD">0.95</span>) + <span style="color: #B452CD">1</span>
</pre></div>
<p>
You could then set \( n\_components=d \) 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 \( n\_components \) to be
a float between 0.0 and 1.0, indicating the ratio of variance you wish to preserve:
<p>
<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
<div class="highlight" style="background: #eeeedd"><pre style="font-size: 80%; line-height: 125%"><span></span>pca = PCA(n_components=<span style="color: #B452CD">0.95</span>)
X_reduced = pca.fit_transform(X)
</pre></div>
</section>
<section>
<h2 id="___sec15">Incremental PCA </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).
</section>
<section>
<h2 id="___sec16">Randomized PCA </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 \( O(m \times d^2)+O(d^3) \), instead of \( O(m \times n^2) + O(n^3) \), so it is dramatically faster than the
previous algorithms when \( d \) is much smaller than \( n \).
</div>
</section>
<section>
<h2 id="___sec17">Kernel PCA </h2>
<div class="alert alert-block alert-block alert-text-normal">
<b></b>
<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&#8217;s KernelPCA class to perform kPCA with an
<p>
<!-- code=python (!bc pycod) typeset with pygments style "perldoc" -->
<div class="highlight" style="background: #eeeedd"><pre style="font-size: 80%; line-height: 125%"><span></span><span style="color: #8B008B; font-weight: bold">from</span> <span style="color: #008b45; text-decoration: underline">sklearn.decomposition</span> <span style="color: #8B008B; font-weight: bold">import</span> KernelPCA
rbf_pca = KernelPCA(n_components = <span style="color: #B452CD">2</span>, kernel=<span style="color: #CD5555">&quot;rbf&quot;</span>, gamma=<span style="color: #B452CD">0.04</span>)
X_reduced = rbf_pca.fit_transform(X)
</pre></div>
</div>
</section>
<section>
<h2 id="___sec18">LLE </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).
</section>
<section>
<h2 id="___sec19">Other techniques </h2>
<p>
There are many other dimensionality reduction techniques, several of which are available in Scikit-Learn.
<p>
Here are some of the most popular:
<ul>
<p><li> <b>Multidimensional Scaling (MDS)</b> reduces dimensionality while trying to preserve the distances between the instances.</li>
<p><li> <b>Isomap</b> creates a graph by connecting each instance to its nearest neighbors, then reduces dimensionality while trying to preserve the geodesic distances between the instances.</li>
<p><li> <b>t-Distributed Stochastic Neighbor Embedding</b> (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).</li>
<p><li> 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.</li>
</ul>
</section>
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