mhjensen
284 lines
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HTML
284 lines
16 KiB
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<a class="navbar-brand" href="DimRed-bs.html">Data Analysis and Machine Learning: Preprocessing and Dimensionality Reduction</a>
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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>
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<!-- navigation toc: --> <li><a href="._DimRed-bs002.html#___sec1" style="font-size: 80%;">Preprocessing our data</a></li>
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<!-- navigation toc: --> <li><a href="._DimRed-bs003.html#___sec2" style="font-size: 80%;">More preprocessing</a></li>
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<!-- navigation toc: --> <li><a href="._DimRed-bs004.html#___sec3" style="font-size: 80%;">Simple preprocessing examples, Franke function and regression</a></li>
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<!-- 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>
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<!-- navigation toc: --> <li><a href="._DimRed-bs006.html#___sec5" style="font-size: 80%;">More on Cancer Data, now with Logistic Regression</a></li>
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<!-- navigation toc: --> <li><a href="._DimRed-bs007.html#___sec6" style="font-size: 80%;">Why should we think of reducing the dimensionality</a></li>
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<!-- navigation toc: --> <li><a href="._DimRed-bs008.html#___sec7" style="font-size: 80%;">Basic ideas of the Principal Component Analysis (PCA)</a></li>
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<!-- navigation toc: --> <li><a href="._DimRed-bs009.html#___sec8" style="font-size: 80%;">Introducing the Covariance and Correlation functions</a></li>
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<!-- navigation toc: --> <li><a href="._DimRed-bs010.html#___sec9" style="font-size: 80%;">Correlation Function and Design/Feature Matrix</a></li>
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<!-- navigation toc: --> <li><a href="._DimRed-bs011.html#___sec10" style="font-size: 80%;">Covariance Matrix Examples</a></li>
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<!-- navigation toc: --> <li><a href="._DimRed-bs012.html#___sec11" style="font-size: 80%;">Correlation Matrix</a></li>
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<!-- navigation toc: --> <li><a href="._DimRed-bs013.html#___sec12" style="font-size: 80%;">Correlation Matrix with Pandas</a></li>
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<!-- navigation toc: --> <li><a href="._DimRed-bs014.html#___sec13" style="font-size: 80%;">Correlation Matrix with Pandas and the Franke function</a></li>
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<!-- navigation toc: --> <li><a href="._DimRed-bs015.html#___sec14" style="font-size: 80%;">Classical PCA Theorem</a></li>
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<!-- navigation toc: --> <li><a href="._DimRed-bs016.html#___sec15" style="font-size: 80%;">Prof of the PCA Theorem</a></li>
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<!-- navigation toc: --> <li><a href="._DimRed-bs017.html#___sec16" style="font-size: 80%;">Getting started with PCA</a></li>
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<!-- navigation toc: --> <li><a href="._DimRed-bs018.html#___sec17" style="font-size: 80%;">Principal Component Analysis</a></li>
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<!-- navigation toc: --> <li><a href="._DimRed-bs019.html#___sec18" style="font-size: 80%;">PCA and scikit-learn</a></li>
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<!-- navigation toc: --> <li><a href="._DimRed-bs020.html#___sec19" style="font-size: 80%;">More on the PCA</a></li>
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<!-- navigation toc: --> <li><a href="._DimRed-bs021.html#___sec20" style="font-size: 80%;">Incremental PCA</a></li>
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<!-- navigation toc: --> <li><a href="._DimRed-bs022.html#___sec21" style="font-size: 80%;">Randomized PCA</a></li>
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<!-- navigation toc: --> <li><a href="._DimRed-bs023.html#___sec22" style="font-size: 80%;">Kernel PCA</a></li>
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<!-- navigation toc: --> <li><a href="._DimRed-bs024.html#___sec23" style="font-size: 80%;">LLE</a></li>
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<!-- navigation toc: --> <li><a href="#___sec24" style="font-size: 80%;">Other techniques</a></li>
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<a name="part0025"></a>
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<!-- !split -->
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<h2 id="___sec24" class="anchor">Other techniques </h2>
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<p>
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There are many other dimensionality reduction techniques, several of which are available in Scikit-Learn.
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<p>
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Here are some of the most popular:
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<ul>
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<li> <b>Multidimensional Scaling (MDS)</b> reduces dimensionality while trying to preserve the distances between the instances.</li>
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<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>
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<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>
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<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>
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</ul>
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Here are other examples where we use the <b>DataFrame</b> functionality to handle arrays, now with more interesting features for us, namely numbers. We set up a matrix
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of dimensionality \( 10\times 5 \) and compute the mean value and standard deviation of each column. Similarly, we can perform mathematial operations like squaring the matrix elements and many other operations.
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<p>
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<!-- code=python (!bc pycod) typeset with pygments style "default" -->
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<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></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>
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<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>
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<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">IPython.display</span> <span style="color: #008000; font-weight: bold">import</span> display
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np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>seed(<span style="color: #666666">100</span>)
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<span style="color: #408080; font-style: italic"># setting up a 10 x 5 matrix</span>
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rows <span style="color: #666666">=</span> <span style="color: #666666">10</span>
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cols <span style="color: #666666">=</span> <span style="color: #666666">5</span>
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a <span style="color: #666666">=</span> np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>randn(rows,cols)
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df <span style="color: #666666">=</span> pd<span style="color: #666666">.</span>DataFrame(a)
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display(df)
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<span style="color: #008000; font-weight: bold">print</span>(df<span style="color: #666666">.</span>mean())
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<span style="color: #008000; font-weight: bold">print</span>(df<span style="color: #666666">.</span>std())
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display(df<span style="color: #666666">**2</span>)
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</pre></div>
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<p>
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Thereafter we can select specific columns only and plot final results
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<p>
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<!-- code=python (!bc pycod) typeset with pygments style "default" -->
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<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span>df<span style="color: #666666">.</span>columns <span style="color: #666666">=</span> [<span style="color: #BA2121">'First'</span>, <span style="color: #BA2121">'Second'</span>, <span style="color: #BA2121">'Third'</span>, <span style="color: #BA2121">'Fourth'</span>, <span style="color: #BA2121">'Fifth'</span>]
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df<span style="color: #666666">.</span>index <span style="color: #666666">=</span> np<span style="color: #666666">.</span>arange(<span style="color: #666666">10</span>)
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display(df)
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<span style="color: #008000; font-weight: bold">print</span>(df[<span style="color: #BA2121">'Second'</span>]<span style="color: #666666">.</span>mean() )
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<span style="color: #008000; font-weight: bold">print</span>(df<span style="color: #666666">.</span>info())
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<span style="color: #008000; font-weight: bold">print</span>(df<span style="color: #666666">.</span>describe())
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<span style="color: #008000; font-weight: bold">from</span> <span style="color: #0000FF; font-weight: bold">pylab</span> <span style="color: #008000; font-weight: bold">import</span> plt, mpl
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plt<span style="color: #666666">.</span>style<span style="color: #666666">.</span>use(<span style="color: #BA2121">'seaborn'</span>)
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mpl<span style="color: #666666">.</span>rcParams[<span style="color: #BA2121">'font.family'</span>] <span style="color: #666666">=</span> <span style="color: #BA2121">'serif'</span>
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df<span style="color: #666666">.</span>cumsum()<span style="color: #666666">.</span>plot(lw<span style="color: #666666">=2.0</span>, figsize<span style="color: #666666">=</span>(<span style="color: #666666">10</span>,<span style="color: #666666">6</span>))
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plt<span style="color: #666666">.</span>show()
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df<span style="color: #666666">.</span>plot<span style="color: #666666">.</span>bar(figsize<span style="color: #666666">=</span>(<span style="color: #666666">10</span>,<span style="color: #666666">6</span>), rot<span style="color: #666666">=15</span>)
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plt<span style="color: #666666">.</span>show()
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</pre></div>
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<p>
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We can produce a \( 4\times 4 \) matrix
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<p>
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<!-- code=python (!bc pycod) typeset with pygments style "default" -->
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<div class="highlight" style="background: #f8f8f8"><pre style="line-height: 125%"><span></span>b <span style="color: #666666">=</span> np<span style="color: #666666">.</span>arange(<span style="color: #666666">16</span>)<span style="color: #666666">.</span>reshape((<span style="color: #666666">4</span>,<span style="color: #666666">4</span>))
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<span style="color: #008000; font-weight: bold">print</span>(b)
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df1 <span style="color: #666666">=</span> pd<span style="color: #666666">.</span>DataFrame(b)
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<span style="color: #008000; font-weight: bold">print</span>(df1)
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</pre></div>
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<p>
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and many other operations.
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<p>
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