mhjensen
253 lines
12 KiB
HTML
253 lines
12 KiB
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{'highest level': 2,
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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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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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('Basic math of the SVD', 2, None, '___sec6'),
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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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('Introducing the Covariance and Correlation functions',
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('Rewriting the Covariance and/or Correlation Matrix',
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<a class="navbar-brand" href="week37-bs.html">Week 37: Ridge and Lasso Regression</a>
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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="._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-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="#___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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</li>
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</ul>
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<a name="part0021"></a>
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<!-- !split -->
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<h2 id="___sec20" class="anchor">Covariance Matrix Examples </h2>
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<p>
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The Numpy function <b>np.cov</b> calculates the covariance elements using
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the factor \( 1/(n-1) \) instead of \( 1/n \) since it assumes we do not have
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the exact mean values. The following simple function uses the
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<b>np.vstack</b> function which takes each vector of dimension \( 1\times n \)
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and produces a \( 2\times n \) matrix \( \boldsymbol{W} \)
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$$
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\boldsymbol{W} = \begin{bmatrix} x_0 & y_0 \\
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x_1 & y_1 \\
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x_2 & y_2\\
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\dots & \dots \\
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x_{n-2} & y_{n-2}\\
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x_{n-1} & y_{n-1} &
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\end{bmatrix},
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$$
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<p>
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which in turn is converted into into the \( 2\times 2 \) covariance matrix
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\( \boldsymbol{C} \) via the Numpy function <b>np.cov()</b>. We note that we can also calculate
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the mean value of each set of samples \( \boldsymbol{x} \) etc using the Numpy
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function <b>np.mean(x)</b>. We can also extract the eigenvalues of the
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covariance matrix through the <b>np.linalg.eig()</b> function.
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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"># Importing various packages</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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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>random<span style="color: #666666">.</span>normal(size<span style="color: #666666">=</span>n)
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<span style="color: #008000; font-weight: bold">print</span>(np<span style="color: #666666">.</span>mean(x))
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y <span style="color: #666666">=</span> <span style="color: #666666">4+3*</span>x<span style="color: #666666">+</span>np<span style="color: #666666">.</span>random<span style="color: #666666">.</span>normal(size<span style="color: #666666">=</span>n)
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<span style="color: #008000; font-weight: bold">print</span>(np<span style="color: #666666">.</span>mean(y))
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W <span style="color: #666666">=</span> np<span style="color: #666666">.</span>vstack((x, y))
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C <span style="color: #666666">=</span> np<span style="color: #666666">.</span>cov(W)
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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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<p>
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