mhjensen
295 lines
14 KiB
HTML
295 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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None,
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('Regression analysis, overarching aims', 2, None, '___sec3'),
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('Regression analysis, overarching aims II', 2, None, '___sec4'),
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('Examples', 2, None, '___sec5'),
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('General linear models', 2, None, '___sec6'),
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('Rewriting the fitting procedure as a linear algebra problem',
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('Rewriting the fitting procedure as a linear algebra problem, '
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None,
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'___sec8'),
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('Generalizing the fitting procedure as a linear algebra problem',
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2,
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None,
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'___sec9'),
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('Generalizing the fitting procedure as a linear algebra problem',
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2,
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None,
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'___sec10'),
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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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2,
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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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2,
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None,
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'___sec14'),
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('Interpretations and optimizing our parameters',
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2,
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None,
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'___sec15'),
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('Some useful matrix and vector expressions',
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2,
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None,
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'___sec16'),
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('Interpretations and optimizing our parameters',
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2,
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None,
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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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2,
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None,
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'___sec19'),
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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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('The $\\chi^2$ function', 2, None, '___sec25'),
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('Fitting an Equation of State for Dense Nuclear Matter',
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2,
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None,
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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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2,
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None,
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'___sec28'),
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('The Boston housing data example', 2, None, '___sec29'),
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('Housing data, the code', 2, None, '___sec30'),
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('Reducing the number of degrees of freedom, overarching view',
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2,
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None,
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'___sec31'),
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('Preprocessing our data', 2, None, '___sec32'),
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('More preprocessing', 2, None, '___sec33'),
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('Simple preprocessing examples, Franke function and regression',
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2,
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None,
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('Friday August 28', 2, None, '___sec35')]}
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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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<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="#___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="._week35-bs032.html#___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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<!-- !split -->
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<h2 id="___sec23" class="anchor">The \( \chi^2 \) function </h2>
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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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If we then introduce the matrix
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$$
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\boldsymbol{H} = \left(\boldsymbol{A}^T\boldsymbol{A}\right)^{-1},
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$$
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we have then the following expression for the parameters \( \beta_j \) (the matrix elements of \( \boldsymbol{H} \) are \( h_{ij} \))
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$$
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\beta_j = \sum_{k=0}^{p-1}h_{jk}\sum_{i=0}^{n-1}\frac{y_i}{\sigma_i}\frac{x_{ik}}{\sigma_i} = \sum_{k=0}^{p-1}h_{jk}\sum_{i=0}^{n-1}b_ia_{ik}
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$$
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We state without proof the expression for the uncertainty in the parameters \( \beta_j \) as (we leave this as an exercise)
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$$
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\sigma^2(\beta_j) = \sum_{i=0}^{n-1}\sigma_i^2\left( \frac{\partial \beta_j}{\partial y_i}\right)^2,
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$$
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resulting in
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$$
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\sigma^2(\beta_j) = \left(\sum_{k=0}^{p-1}h_{jk}\sum_{i=0}^{n-1}a_{ik}\right)\left(\sum_{l=0}^{p-1}h_{jl}\sum_{m=0}^{n-1}a_{ml}\right) = h_{jj}!
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$$
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