mhjensen
267 lines
14 KiB
HTML
267 lines
14 KiB
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'sections': [('Plans for week 37', 2, None, '___sec0'),
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('Thursday September 10', 2, None, '___sec1'),
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('A Bayesian approach to develop intuition about skrinkage '
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'methods',
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2,
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None,
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('The singular value decomposition', 2, None, '___sec3'),
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('Linear Regression Problems', 2, None, '___sec4'),
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('Fixing the singularity', 2, None, '___sec5'),
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('The SVD, a Fantastic Algorithm', 2, None, '___sec7'),
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('Economy-size SVD', 2, None, '___sec8'),
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('Codes for the SVD', 2, None, '___sec9'),
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('Mathematical Properties', 2, None, '___sec10'),
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('Friday September 12', 2, None, '___sec11'),
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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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('More interpretations', 2, None, '___sec15'),
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('A better understanding of regularization', 2, None, '___sec16'),
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('Decomposing the OLS and Ridge expressions',
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2,
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None,
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'___sec17'),
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('Introducing the Covariance and Correlation functions',
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2,
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('Correlation Function and Design/Feature Matrix',
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('Correlation Matrix with Pandas and the Franke function',
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2,
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None,
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'___sec23'),
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('Rewriting the Covariance and/or Correlation Matrix',
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2,
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None,
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'___sec24'),
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('Linking with SVD', 2, None, '___sec25')]}
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<a class="navbar-brand" href="week37-bs.html">Week 37: Ridge and Lasso Regression</a>
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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="#___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-bs012.html#___sec11" style="font-size: 80%;">Friday September 12</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-bs014.html#___sec13" style="font-size: 80%;">More on Ridge 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-bs016.html#___sec15" style="font-size: 80%;">More interpretations</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-bs022.html#___sec21" style="font-size: 80%;">Correlation Matrix</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="._week37-bs024.html#___sec23" style="font-size: 80%;">Correlation Matrix with Pandas and the Franke function</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs025.html#___sec24" style="font-size: 80%;">Rewriting the Covariance and/or Correlation Matrix</a></li>
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<!-- navigation toc: --> <li><a href="._week37-bs026.html#___sec25" style="font-size: 80%;">Linking with SVD</a></li>
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<a name="part0010"></a>
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<!-- !split -->
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<h2 id="___sec9" class="anchor">Codes for the SVD </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: #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: #408080; font-style: italic"># SVD inversion</span>
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<span style="color: #008000; font-weight: bold">def</span> <span style="color: #0000FF">SVDinv</span>(A):
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<span style="color: #BA2121; font-style: italic">''' Takes as input a numpy matrix A and returns inv(A) based on singular value decomposition (SVD).</span>
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<span style="color: #BA2121; font-style: italic"> SVD is numerically more stable than the inversion algorithms provided by</span>
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<span style="color: #BA2121; font-style: italic"> numpy and scipy.linalg at the cost of being slower.</span>
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<span style="color: #BA2121; font-style: italic"> '''</span>
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U, s, VT <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linalg<span style="color: #666666">.</span>svd(A)
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<span style="color: #408080; font-style: italic"># print('test U')</span>
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<span style="color: #408080; font-style: italic"># print( (np.transpose(U) @ U - U @np.transpose(U)))</span>
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<span style="color: #408080; font-style: italic"># print('test VT')</span>
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<span style="color: #408080; font-style: italic"># print( (np.transpose(VT) @ VT - VT @np.transpose(VT)))</span>
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<span style="color: #008000; font-weight: bold">print</span>(U)
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<span style="color: #008000; font-weight: bold">print</span>(s)
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<span style="color: #008000; font-weight: bold">print</span>(VT)
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D <span style="color: #666666">=</span> np<span style="color: #666666">.</span>zeros((<span style="color: #008000">len</span>(U),<span style="color: #008000">len</span>(VT)))
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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">0</span>,<span style="color: #008000">len</span>(VT)):
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D[i,i]<span style="color: #666666">=</span>s[i]
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UT <span style="color: #666666">=</span> np<span style="color: #666666">.</span>transpose(U); V <span style="color: #666666">=</span> np<span style="color: #666666">.</span>transpose(VT); invD <span style="color: #666666">=</span> np<span style="color: #666666">.</span>linalg<span style="color: #666666">.</span>inv(D)
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<span style="color: #008000; font-weight: bold">return</span> np<span style="color: #666666">.</span>matmul(V,np<span style="color: #666666">.</span>matmul(invD,UT))
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X <span style="color: #666666">=</span> np<span style="color: #666666">.</span>array([ [<span style="color: #666666">1.0</span>, <span style="color: #666666">-1.0</span>, <span style="color: #666666">2.0</span>], [<span style="color: #666666">1.0</span>, <span style="color: #666666">0.0</span>, <span style="color: #666666">1.0</span>], [<span style="color: #666666">1.0</span>, <span style="color: #666666">2.0</span>, <span style="color: #666666">-1.0</span>], [<span style="color: #666666">1.0</span>, <span style="color: #666666">1.0</span>, <span style="color: #666666">0.0</span>] ])
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<span style="color: #008000; font-weight: bold">print</span>(X)
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A <span style="color: #666666">=</span> np<span style="color: #666666">.</span>transpose(X) @ X
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<span style="color: #008000; font-weight: bold">print</span>(A)
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<span style="color: #408080; font-style: italic"># Brute force inversion of super-collinear matrix</span>
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<span style="color: #408080; font-style: italic">#B = np.linalg.inv(A)</span>
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<span style="color: #408080; font-style: italic">#print(B)</span>
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C <span style="color: #666666">=</span> SVDinv(A)
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<span style="color: #008000; font-weight: bold">print</span>(C)
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</pre></div>
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<p>
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The matrix \( \boldsymbol{X} \) has columns that are linearly dependent. The first
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column is the row-wise sum of the other two columns. The rank of a
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matrix (the column rank) is the dimension of space spanned by the
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column vectors. The rank of the matrix is the number of linearly
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independent columns, in this case just \( 2 \). We see this from the
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singular values when running the above code. Running the standard
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inversion algorithm for matrix inversion with \( \boldsymbol{X}^T\boldsymbol{X} \) results
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in the program terminating due to a singular matrix.
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