mhjensen
270 lines
16 KiB
HTML
270 lines
16 KiB
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('Ridge and LASSO Regression', 2, None, '___sec12'),
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('More on Ridge Regression', 2, None, '___sec13'),
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('Interpreting the Ridge results', 2, None, '___sec14'),
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<!-- navigation toc: --> <li><a href="._week37-bs001.html#___sec0" style="font-size: 80%;">Plans for week 37</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs002.html#___sec1" style="font-size: 80%;">Thursday September 10</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs003.html#___sec2" style="font-size: 80%;">A Bayesian approach to develop intuition about skrinkage methods</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs004.html#___sec3" style="font-size: 80%;">The singular value decomposition</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs005.html#___sec4" style="font-size: 80%;">Linear Regression Problems</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs006.html#___sec5" style="font-size: 80%;">Fixing the singularity</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs007.html#___sec6" style="font-size: 80%;">Basic math of the SVD</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs008.html#___sec7" style="font-size: 80%;">The SVD, a Fantastic Algorithm</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs009.html#___sec8" style="font-size: 80%;">Economy-size SVD</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs010.html#___sec9" style="font-size: 80%;">Codes for the SVD</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs011.html#___sec10" style="font-size: 80%;">Mathematical Properties</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs013.html#___sec12" style="font-size: 80%;">Ridge and LASSO Regression</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs015.html#___sec14" style="font-size: 80%;">Interpreting the Ridge results</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs017.html#___sec16" style="font-size: 80%;">A better understanding of regularization</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs018.html#___sec17" style="font-size: 80%;">Decomposing the OLS and Ridge expressions</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs019.html#___sec18" style="font-size: 80%;">Introducing the Covariance and Correlation functions</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs020.html#___sec19" style="font-size: 80%;">Correlation Function and Design/Feature Matrix</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs021.html#___sec20" style="font-size: 80%;">Covariance Matrix Examples</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs023.html#___sec22" style="font-size: 80%;">Correlation Matrix with Pandas</a></li>
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<!-- navigation toc: --> <li><a href="#___sec23" style="font-size: 80%;">Correlation Matrix with Pandas and the Franke function</a></li>
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<h2 id="___sec23" class="anchor">Correlation Matrix with Pandas and the Franke function </h2>
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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: #408080; font-style: italic"># Common imports</span>
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<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">def</span> <span style="color: #0000FF">FrankeFunction</span>(x,y):
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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>))
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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>))
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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>))
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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>)
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<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
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<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">create_X</span>(x, y, n ):
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<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">></span> <span style="color: #666666">1</span>:
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x <span style="color: #666666">=</span> np<span style="color: #666666">.</span>ravel(x)
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y <span style="color: #666666">=</span> np<span style="color: #666666">.</span>ravel(y)
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N <span style="color: #666666">=</span> <span style="color: #008000">len</span>(x)
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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>
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X <span style="color: #666666">=</span> np<span style="color: #666666">.</span>ones((N,l))
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<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>):
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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>)
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<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>):
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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)
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<span style="color: #008000; font-weight: bold">return</span> X
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<span style="color: #408080; font-style: italic"># Making meshgrid of datapoints and compute Franke's function</span>
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n <span style="color: #666666">=</span> <span style="color: #666666">4</span>
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N <span style="color: #666666">=</span> <span style="color: #666666">100</span>
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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))
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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))
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z <span style="color: #666666">=</span> FrankeFunction(x, y)
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X <span style="color: #666666">=</span> create_X(x, y, n<span style="color: #666666">=</span>n)
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Xpd <span style="color: #666666">=</span> pd<span style="color: #666666">.</span>DataFrame(X)
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<span style="color: #408080; font-style: italic"># subtract the mean values and set up the covariance matrix</span>
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Xpd <span style="color: #666666">=</span> Xpd <span style="color: #666666">-</span> Xpd<span style="color: #666666">.</span>mean()
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covariance_matrix <span style="color: #666666">=</span> Xpd<span style="color: #666666">.</span>cov()
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<span style="color: #008000; font-weight: bold">print</span>(covariance_matrix)
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</pre></div>
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<p>
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We note here that the covariance is zero for the first rows and
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columns since all matrix elements in the design matrix were set to one
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(we are fitting the function in terms of a polynomial of degree \( n \)).
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<p>
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This means that the variance for these elements will be zero and will
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cause problems when we set up the correlation matrix. We can simply
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drop these elements and construct a correlation
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matrix without these elements.
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<p>
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