mhjensen
342 lines
18 KiB
HTML
342 lines
18 KiB
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'sections': [('Plans for week 38', 2, None, '___sec0'),
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('Singular Value decomposition', 2, None, '___sec13'),
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('Performance as function of the regularization parameter',
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('Basics', 2, None, '___sec23'),
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('The logistic function', 2, None, '___sec28'),
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('Two parameters', 2, None, '___sec30'),
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('Maximum likelihood', 2, None, '___sec31'),
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('The cost function rewritten', 2, None, '___sec32'),
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('Minimizing the cross entropy', 2, None, '___sec33'),
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('A more compact expression', 2, None, '___sec34'),
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('Extending to more predictors', 2, None, '___sec35'),
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('Including more classes', 2, None, '___sec36'),
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('More classes', 2, None, '___sec37'),
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('Wisconsin Cancer Data', 2, None, '___sec38'),
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('Using the correlation matrix', 2, None, '___sec39'),
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<!-- navigation toc: --> <li><a href="._week38-bs009.html#___sec8" style="font-size: 80%;">Another Example from Scikit-Learn's Repository</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs010.html#___sec9" style="font-size: 80%;">Cross-validation with Ridge</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs011.html#___sec10" style="font-size: 80%;">The Ising model</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs012.html#___sec11" style="font-size: 80%;">Reformulating the problem to suit regression</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs013.html#___sec12" style="font-size: 80%;">Linear regression</a></li>
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<!-- navigation toc: --> <li><a href="#___sec13" style="font-size: 80%;">Singular Value decomposition</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs020.html#___sec19" style="font-size: 80%;">Friday September 18: Intro to Logistic Regression</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs021.html#___sec20" style="font-size: 80%;">Logistic Regression</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs022.html#___sec21" style="font-size: 80%;">Classification problems</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs024.html#___sec23" style="font-size: 80%;">Basics</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs025.html#___sec24" style="font-size: 80%;">Linear classifier</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs026.html#___sec25" style="font-size: 80%;">Some selected properties</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs027.html#___sec26" style="font-size: 80%;">Simple example</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs028.html#___sec27" style="font-size: 80%;">Plotting the mean value for each group</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs029.html#___sec28" style="font-size: 80%;">The logistic function</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs030.html#___sec29" style="font-size: 80%;">Examples of likelihood functions used in logistic regression and nueral networks</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs031.html#___sec30" style="font-size: 80%;">Two parameters</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs034.html#___sec33" style="font-size: 80%;">Minimizing the cross entropy</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs035.html#___sec34" style="font-size: 80%;">A more compact expression</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs036.html#___sec35" style="font-size: 80%;">Extending to more predictors</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs037.html#___sec36" style="font-size: 80%;">Including more classes</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs038.html#___sec37" style="font-size: 80%;">More classes</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs039.html#___sec38" style="font-size: 80%;">Wisconsin Cancer Data</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs040.html#___sec39" style="font-size: 80%;">Using the correlation matrix</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs041.html#___sec40" style="font-size: 80%;">Discussing the correlation data</a></li>
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<!-- navigation toc: --> <li><a href="._week38-bs042.html#___sec41" style="font-size: 80%;">Other measures in classification studies: Cancer Data again</a></li>
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<a name="part0014"></a>
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<!-- !split -->
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<h2 id="___sec13" class="anchor">Singular Value decomposition </h2>
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<p>
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Doing the inversion directly turns out to be a bad idea since the matrix
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\( \boldsymbol{X}^T\boldsymbol{X} \) is singular. An alternative approach is to use the <b>singular
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value decomposition</b>. Using the definition of the Moore-Penrose
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pseudoinverse we can write the equation for \( \boldsymbol{\beta} \) as
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$$
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\boldsymbol{\beta} = \boldsymbol{X}^{+}\boldsymbol{y},
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$$
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<p>
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where the pseudoinverse of \( \boldsymbol{X} \) is given by
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$$
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\boldsymbol{X}^{+} = \frac{\boldsymbol{X}^T}{\boldsymbol{X}^T\boldsymbol{X}}.
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$$
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<p>
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Using singular value decomposition we can decompose the matrix \( \boldsymbol{X} = \boldsymbol{U}\boldsymbol{\Sigma} \boldsymbol{V}^T \),
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where \( \boldsymbol{U} \) and \( \boldsymbol{V} \) are orthogonal(unitary) matrices and \( \boldsymbol{\Sigma} \) contains the singular values (more details below).
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where \( X^{+} = V\Sigma^{+} U^T \). This reduces the equation for
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\( \omega \) to
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$$
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\begin{align}
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\boldsymbol{\beta} = \boldsymbol{V}\boldsymbol{\Sigma}^{+} \boldsymbol{U}^T \boldsymbol{y}.
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\tag{6}
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\end{align}
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$$
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<p>
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Note that solving this equation by actually doing the pseudoinverse
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(which is what we will do) is not a good idea as this operation scales
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as \( \mathcal{O}(n^3) \), where \( n \) is the number of elements in a
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general matrix. Instead, doing \( QR \)-factorization and solving the
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linear system as an equation would reduce this down to
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\( \mathcal{O}(n^2) \) 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">def</span> <span style="color: #0000FF">ols_svd</span>(x: np<span style="color: #666666">.</span>ndarray, y: np<span style="color: #666666">.</span>ndarray) <span style="color: #666666">-></span> np<span style="color: #666666">.</span>ndarray:
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u, s, v <span style="color: #666666">=</span> scl<span style="color: #666666">.</span>svd(x)
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<span style="color: #008000; font-weight: bold">return</span> v<span style="color: #666666">.</span>T <span style="color: #666666">@</span> scl<span style="color: #666666">.</span>pinv(scl<span style="color: #666666">.</span>diagsvd(s, u<span style="color: #666666">.</span>shape[<span style="color: #666666">0</span>], v<span style="color: #666666">.</span>shape[<span style="color: #666666">0</span>])) <span style="color: #666666">@</span> u<span style="color: #666666">.</span>T <span style="color: #666666">@</span> y
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</pre></div>
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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>beta <span style="color: #666666">=</span> ols_svd(X_train_own,y_train)
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</pre></div>
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<p>
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When extracting the \( J \)-matrix we need to make sure that we remove the intercept, as is done here
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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>J <span style="color: #666666">=</span> beta[<span style="color: #666666">1</span>:]<span style="color: #666666">.</span>reshape(L, L)
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</pre></div>
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<p>
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A way of looking at the coefficients in \( J \) is to plot the matrices as images.
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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>fig <span style="color: #666666">=</span> plt<span style="color: #666666">.</span>figure(figsize<span style="color: #666666">=</span>(<span style="color: #666666">20</span>, <span style="color: #666666">14</span>))
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im <span style="color: #666666">=</span> plt<span style="color: #666666">.</span>imshow(J, <span style="color: #666666">**</span>cmap_args)
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plt<span style="color: #666666">.</span>title(<span style="color: #BA2121">"OLS"</span>, fontsize<span style="color: #666666">=18</span>)
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plt<span style="color: #666666">.</span>xticks(fontsize<span style="color: #666666">=18</span>)
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plt<span style="color: #666666">.</span>yticks(fontsize<span style="color: #666666">=18</span>)
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cb <span style="color: #666666">=</span> fig<span style="color: #666666">.</span>colorbar(im)
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cb<span style="color: #666666">.</span>ax<span style="color: #666666">.</span>set_yticklabels(cb<span style="color: #666666">.</span>ax<span style="color: #666666">.</span>get_yticklabels(), fontsize<span style="color: #666666">=18</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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It is interesting to note that OLS
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considers both \( J_{j, j + 1} = -0.5 \) and \( J_{j, j - 1} = -0.5 \) as
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valid matrix elements for \( J \).
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In our discussion below on hyperparameters and Ridge and Lasso regression we will see that
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this problem can be removed, partly and only with Lasso regression.
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<p>
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In this case our matrix inversion was actually possible. The obvious question now is what is the mathematics behind the SVD?
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<p>
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<p>
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</center>
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</body>
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</html>
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