mhjensen
291 lines
14 KiB
HTML
291 lines
14 KiB
HTML
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'sections': [('Plans for week 35, August 24-28', 2, None, '___sec0'),
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('Thursday August 27', 2, None, '___sec1'),
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('Why Linear Regression (aka Ordinary Least Squares and family)',
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('Regression analysis, overarching aims', 2, None, '___sec3'),
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('Rewriting the fitting procedure as a linear algebra problem',
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('Generalizing the fitting procedure as a linear algebra problem',
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('Generalizing the fitting procedure as a linear algebra problem',
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('Optimizing our parameters', 2, None, '___sec11'),
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('Our model for the nuclear binding energies',
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('Optimizing our parameters, more details', 2, None, '___sec13'),
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('Interpretations and optimizing our parameters',
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('Some useful matrix and vector expressions',
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'___sec16'),
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('Interpretations and optimizing our parameters',
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2,
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'___sec17'),
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('Own code for Ordinary Least Squares', 2, None, '___sec18'),
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('Adding error analysis and training set up',
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('The $\\chi^2$ function', 2, None, '___sec20'),
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('The $\\chi^2$ function', 2, None, '___sec21'),
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('The $\\chi^2$ function', 2, None, '___sec22'),
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('The $\\chi^2$ function', 2, None, '___sec23'),
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('The $\\chi^2$ function', 2, None, '___sec24'),
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'___sec26'),
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('The code', 2, None, '___sec27'),
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('Splitting our Data in Training and Test data',
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('The Boston housing data example', 2, None, '___sec29'),
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('Reducing the number of degrees of freedom, overarching view',
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('Preprocessing our data', 2, None, '___sec32'),
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('Simple preprocessing examples, Franke function and regression',
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<a class="navbar-brand" href="week35-bs.html">Week 35: Linear Regression and Review of Statistical Analysis and Probability Theory</a>
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</div>
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<ul class="nav navbar-nav navbar-right">
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<a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
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<ul class="dropdown-menu">
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<!-- navigation toc: --> <li><a href="._week35-bs001.html#___sec0" style="font-size: 80%;">Plans for week 35, August 24-28</a></li>
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|
<!-- navigation toc: --> <li><a href="._week35-bs002.html#___sec1" style="font-size: 80%;">Thursday August 27</a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs003.html#___sec2" style="font-size: 80%;">Why Linear Regression (aka Ordinary Least Squares and family)</a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs004.html#___sec3" style="font-size: 80%;">Regression analysis, overarching aims</a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs005.html#___sec4" style="font-size: 80%;">Regression analysis, overarching aims II</a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs006.html#___sec5" style="font-size: 80%;">Examples</a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs007.html#___sec6" style="font-size: 80%;">General linear models</a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs008.html#___sec7" style="font-size: 80%;">Rewriting the fitting procedure as a linear algebra problem</a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs009.html#___sec8" style="font-size: 80%;">Rewriting the fitting procedure as a linear algebra problem, more details</a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs010.html#___sec9" style="font-size: 80%;">Generalizing the fitting procedure as a linear algebra problem</a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs011.html#___sec10" style="font-size: 80%;">Generalizing the fitting procedure as a linear algebra problem</a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs012.html#___sec11" style="font-size: 80%;">Optimizing our parameters</a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs013.html#___sec12" style="font-size: 80%;">Our model for the nuclear binding energies</a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs014.html#___sec13" style="font-size: 80%;">Optimizing our parameters, more details</a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs015.html#___sec14" style="font-size: 80%;">Interpretations and optimizing our parameters</a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs016.html#___sec15" style="font-size: 80%;">Interpretations and optimizing our parameters</a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs017.html#___sec16" style="font-size: 80%;">Some useful matrix and vector expressions</a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs018.html#___sec17" style="font-size: 80%;">Interpretations and optimizing our parameters</a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs019.html#___sec18" style="font-size: 80%;">Own code for Ordinary Least Squares</a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs020.html#___sec19" style="font-size: 80%;">Adding error analysis and training set up</a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs021.html#___sec20" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs022.html#___sec21" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs023.html#___sec22" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs024.html#___sec23" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs025.html#___sec24" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs026.html#___sec25" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs027.html#___sec26" style="font-size: 80%;">Fitting an Equation of State for Dense Nuclear Matter</a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs028.html#___sec27" style="font-size: 80%;">The code</a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs029.html#___sec28" style="font-size: 80%;">Splitting our Data in Training and Test data</a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs030.html#___sec29" style="font-size: 80%;">The Boston housing data example</a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs031.html#___sec30" style="font-size: 80%;">Housing data, the code</a></li>
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<!-- navigation toc: --> <li><a href="#___sec31" style="font-size: 80%;">Reducing the number of degrees of freedom, overarching view</a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs033.html#___sec32" style="font-size: 80%;">Preprocessing our data</a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs034.html#___sec33" style="font-size: 80%;">More preprocessing</a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs035.html#___sec34" style="font-size: 80%;">Simple preprocessing examples, Franke function and regression</a></li>
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<!-- navigation toc: --> <li><a href="._week35-bs036.html#___sec35" style="font-size: 80%;">Friday August 28</a></li>
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</ul>
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</li>
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<p> </p><p> </p><p> </p> <!-- add vertical space -->
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<a name="part0032"></a>
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<!-- !split -->
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<h2 id="___sec31" class="anchor">Reducing the number of degrees of freedom, overarching view </h2>
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<div class="panel panel-default">
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<div class="panel-body">
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<p> <!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
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<p>
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Many Machine Learning problems involve thousands or even millions of
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features for each training instance. Not only does this make training
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extremely slow, it can also make it much harder to find a good
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solution, as we will see. This problem is often referred to as the
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curse of dimensionality. Fortunately, in real-world problems, it is
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often possible to reduce the number of features considerably, turning
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an intractable problem into a tractable one.
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<p>
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Later we will discuss some of the most popular dimensionality reduction
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techniques: the principal component analysis (PCA), Kernel PCA, and
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Locally Linear Embedding (LLE).
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<p>
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Principal component analysis and its various variants deal with the
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problem of fitting a low-dimensional <a href="https://en.wikipedia.org/wiki/Affine_space" target="_self">affine
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subspace</a> to a set of of
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data points in a high-dimensional space. With its family of methods it
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is one of the most used tools in data modeling, compression and
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visualization.
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<p>
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</div>
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</div>
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<p>
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<p>
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<li><a href="._week35-bs031.html">«</a></li>
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<li><a href="._week35-bs026.html">27</a></li>
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<li><a href="._week35-bs027.html">28</a></li>
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