updating regression analysis slides by adding info on chi2
This commit is contained in:
Morten Hjorth-Jensen
@@ -72,7 +72,13 @@ Automatically generated HTML file from DocOnce source
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2,
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None,
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'___sec10'),
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('The singular value decompostion', 2, None, '___sec11')]}
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('The $\\chi^2$ function', 2, None, '___sec11'),
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('The $\\chi^2$ function', 2, None, '___sec12'),
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('The $\\chi^2$ function', 2, None, '___sec13'),
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('The $\\chi^2$ function', 2, None, '___sec14'),
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('The $\\chi^2$ function', 2, None, '___sec15'),
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('The $\\chi^2$ function', 2, None, '___sec16'),
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('The singular value decompostion', 2, None, '___sec17')]}
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end of tocinfo -->
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<body>
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@@ -87,8 +93,8 @@ MathJax.Hub.Config({
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}
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});
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</script>
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<script type="text/javascript" async
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src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS-MML_HTMLorMML">
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<script type="text/javascript"
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src="http://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML">
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</script>
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@@ -121,7 +127,13 @@ MathJax.Hub.Config({
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<!-- navigation toc: --> <li><a href="._Regression-bs009.html#___sec8" style="font-size: 80%;">Interpretations and optimizing our parameters</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs010.html#___sec9" style="font-size: 80%;">Interpretations and optimizing our parameters</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs011.html#___sec10" style="font-size: 80%;">Interpretations and optimizing our parameters</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs012.html#___sec11" style="font-size: 80%;">The singular value decompostion</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs012.html#___sec11" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs013.html#___sec12" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs014.html#___sec13" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs015.html#___sec14" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs016.html#___sec15" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs017.html#___sec16" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs018.html#___sec17" style="font-size: 80%;">The singular value decompostion</a></li>
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</ul>
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</li>
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@@ -156,7 +168,7 @@ MathJax.Hub.Config({
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<center>[2] <b>Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University</b></center>
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<br>
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<p>
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<center><h4>Oct 18, 2017</h4></center> <!-- date -->
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<center><h4>Oct 24, 2017</h4></center> <!-- date -->
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<br>
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<p>
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@@ -180,7 +192,7 @@ MathJax.Hub.Config({
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<li><a href="._Regression-bs008.html">9</a></li>
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<li><a href="._Regression-bs009.html">10</a></li>
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<li><a href="">...</a></li>
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<li><a href="._Regression-bs012.html">13</a></li>
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<li><a href="._Regression-bs018.html">19</a></li>
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<li><a href="._Regression-bs001.html">»</a></li>
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</ul>
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<!-- ------------------- end of main content --------------- -->
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@@ -72,7 +72,13 @@ Automatically generated HTML file from DocOnce source
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2,
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None,
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'___sec10'),
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('The singular value decompostion', 2, None, '___sec11')]}
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('The $\\chi^2$ function', 2, None, '___sec11'),
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('The $\\chi^2$ function', 2, None, '___sec12'),
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('The $\\chi^2$ function', 2, None, '___sec13'),
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('The $\\chi^2$ function', 2, None, '___sec14'),
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('The $\\chi^2$ function', 2, None, '___sec15'),
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('The $\\chi^2$ function', 2, None, '___sec16'),
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('The singular value decompostion', 2, None, '___sec17')]}
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end of tocinfo -->
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<body>
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@@ -87,8 +93,8 @@ MathJax.Hub.Config({
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}
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});
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</script>
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<script type="text/javascript" async
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src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS-MML_HTMLorMML">
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<script type="text/javascript"
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src="http://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML">
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</script>
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@@ -121,7 +127,13 @@ MathJax.Hub.Config({
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<!-- navigation toc: --> <li><a href="._Regression-bs009.html#___sec8" style="font-size: 80%;">Interpretations and optimizing our parameters</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs010.html#___sec9" style="font-size: 80%;">Interpretations and optimizing our parameters</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs011.html#___sec10" style="font-size: 80%;">Interpretations and optimizing our parameters</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs012.html#___sec11" style="font-size: 80%;">The singular value decompostion</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs012.html#___sec11" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs013.html#___sec12" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs014.html#___sec13" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs015.html#___sec14" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs016.html#___sec15" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs017.html#___sec16" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs018.html#___sec17" style="font-size: 80%;">The singular value decompostion</a></li>
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</ul>
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</li>
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@@ -177,7 +189,7 @@ A regression model aims at finding a likelihood function \( p(y\vert \hat{x}) \)
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<li><a href="._Regression-bs009.html">10</a></li>
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<li><a href="._Regression-bs010.html">11</a></li>
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<li><a href="">...</a></li>
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<li><a href="._Regression-bs012.html">13</a></li>
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<li><a href="._Regression-bs018.html">19</a></li>
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<li><a href="._Regression-bs002.html">»</a></li>
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</ul>
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<!-- ------------------- end of main content --------------- -->
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@@ -72,7 +72,13 @@ Automatically generated HTML file from DocOnce source
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2,
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None,
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'___sec10'),
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('The singular value decompostion', 2, None, '___sec11')]}
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('The $\\chi^2$ function', 2, None, '___sec11'),
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('The $\\chi^2$ function', 2, None, '___sec12'),
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('The $\\chi^2$ function', 2, None, '___sec13'),
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('The $\\chi^2$ function', 2, None, '___sec14'),
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('The $\\chi^2$ function', 2, None, '___sec15'),
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('The $\\chi^2$ function', 2, None, '___sec16'),
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('The singular value decompostion', 2, None, '___sec17')]}
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end of tocinfo -->
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<body>
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@@ -87,8 +93,8 @@ MathJax.Hub.Config({
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}
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});
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</script>
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<script type="text/javascript" async
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src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS-MML_HTMLorMML">
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<script type="text/javascript"
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src="http://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML">
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</script>
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@@ -121,7 +127,13 @@ MathJax.Hub.Config({
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<!-- navigation toc: --> <li><a href="._Regression-bs009.html#___sec8" style="font-size: 80%;">Interpretations and optimizing our parameters</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Regression-bs010.html#___sec9" style="font-size: 80%;">Interpretations and optimizing our parameters</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Regression-bs011.html#___sec10" style="font-size: 80%;">Interpretations and optimizing our parameters</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Regression-bs012.html#___sec11" style="font-size: 80%;">The singular value decompostion</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs012.html#___sec11" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs013.html#___sec12" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs014.html#___sec13" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs015.html#___sec14" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs016.html#___sec15" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs017.html#___sec16" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs018.html#___sec17" style="font-size: 80%;">The singular value decompostion</a></li>
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</ul>
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</li>
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@@ -174,7 +186,7 @@ where \( \epsilon_i \) is the error in our approximation.
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<li><a href="._Regression-bs010.html">11</a></li>
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<li><a href="._Regression-bs011.html">12</a></li>
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<li><a href="">...</a></li>
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<li><a href="._Regression-bs012.html">13</a></li>
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<li><a href="._Regression-bs018.html">19</a></li>
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<li><a href="._Regression-bs003.html">»</a></li>
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</ul>
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<!-- ------------------- end of main content --------------- -->
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@@ -72,7 +72,13 @@ Automatically generated HTML file from DocOnce source
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2,
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None,
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'___sec10'),
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('The singular value decompostion', 2, None, '___sec11')]}
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('The $\\chi^2$ function', 2, None, '___sec11'),
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('The $\\chi^2$ function', 2, None, '___sec12'),
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('The $\\chi^2$ function', 2, None, '___sec13'),
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('The $\\chi^2$ function', 2, None, '___sec14'),
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('The $\\chi^2$ function', 2, None, '___sec15'),
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('The $\\chi^2$ function', 2, None, '___sec16'),
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('The singular value decompostion', 2, None, '___sec17')]}
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end of tocinfo -->
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||||
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<body>
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||||
@@ -87,8 +93,8 @@ MathJax.Hub.Config({
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||||
}
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||||
});
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</script>
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<script type="text/javascript" async
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src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS-MML_HTMLorMML">
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||||
<script type="text/javascript"
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src="http://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML">
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</script>
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@@ -121,7 +127,13 @@ MathJax.Hub.Config({
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<!-- navigation toc: --> <li><a href="._Regression-bs009.html#___sec8" style="font-size: 80%;">Interpretations and optimizing our parameters</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Regression-bs010.html#___sec9" style="font-size: 80%;">Interpretations and optimizing our parameters</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Regression-bs011.html#___sec10" style="font-size: 80%;">Interpretations and optimizing our parameters</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Regression-bs012.html#___sec11" style="font-size: 80%;">The singular value decompostion</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Regression-bs012.html#___sec11" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs013.html#___sec12" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs014.html#___sec13" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs015.html#___sec14" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs016.html#___sec15" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs017.html#___sec16" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs018.html#___sec17" style="font-size: 80%;">The singular value decompostion</a></li>
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</ul>
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</li>
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@@ -173,6 +185,8 @@ $$
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<li><a href="._Regression-bs010.html">11</a></li>
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<li><a href="._Regression-bs011.html">12</a></li>
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<li><a href="._Regression-bs012.html">13</a></li>
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<li><a href="">...</a></li>
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<li><a href="._Regression-bs018.html">19</a></li>
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<li><a href="._Regression-bs004.html">»</a></li>
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</ul>
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<!-- ------------------- end of main content --------------- -->
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@@ -72,7 +72,13 @@ Automatically generated HTML file from DocOnce source
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2,
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None,
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'___sec10'),
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('The singular value decompostion', 2, None, '___sec11')]}
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('The $\\chi^2$ function', 2, None, '___sec11'),
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('The $\\chi^2$ function', 2, None, '___sec12'),
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('The $\\chi^2$ function', 2, None, '___sec13'),
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('The $\\chi^2$ function', 2, None, '___sec14'),
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('The $\\chi^2$ function', 2, None, '___sec15'),
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('The $\\chi^2$ function', 2, None, '___sec16'),
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('The singular value decompostion', 2, None, '___sec17')]}
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end of tocinfo -->
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||||
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<body>
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||||
@@ -87,8 +93,8 @@ MathJax.Hub.Config({
|
||||
}
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||||
});
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</script>
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<script type="text/javascript" async
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||||
src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS-MML_HTMLorMML">
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||||
<script type="text/javascript"
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||||
src="http://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML">
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</script>
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@@ -121,7 +127,13 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs009.html#___sec8" style="font-size: 80%;">Interpretations and optimizing our parameters</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs010.html#___sec9" style="font-size: 80%;">Interpretations and optimizing our parameters</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs011.html#___sec10" style="font-size: 80%;">Interpretations and optimizing our parameters</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs012.html#___sec11" style="font-size: 80%;">The singular value decompostion</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs012.html#___sec11" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs013.html#___sec12" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs014.html#___sec13" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Regression-bs015.html#___sec14" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
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||||
<!-- navigation toc: --> <li><a href="._Regression-bs016.html#___sec15" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs017.html#___sec16" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs018.html#___sec17" style="font-size: 80%;">The singular value decompostion</a></li>
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||||
</ul>
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</li>
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@@ -192,6 +204,9 @@ $$
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<li><a href="._Regression-bs010.html">11</a></li>
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||||
<li><a href="._Regression-bs011.html">12</a></li>
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||||
<li><a href="._Regression-bs012.html">13</a></li>
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||||
<li><a href="._Regression-bs013.html">14</a></li>
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<li><a href="">...</a></li>
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<li><a href="._Regression-bs018.html">19</a></li>
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<li><a href="._Regression-bs005.html">»</a></li>
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||||
</ul>
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||||
<!-- ------------------- end of main content --------------- -->
|
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@@ -72,7 +72,13 @@ Automatically generated HTML file from DocOnce source
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2,
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None,
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'___sec10'),
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('The singular value decompostion', 2, None, '___sec11')]}
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('The $\\chi^2$ function', 2, None, '___sec11'),
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('The $\\chi^2$ function', 2, None, '___sec12'),
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('The $\\chi^2$ function', 2, None, '___sec13'),
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('The $\\chi^2$ function', 2, None, '___sec14'),
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('The $\\chi^2$ function', 2, None, '___sec15'),
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('The $\\chi^2$ function', 2, None, '___sec16'),
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('The singular value decompostion', 2, None, '___sec17')]}
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||||
end of tocinfo -->
|
||||
|
||||
<body>
|
||||
@@ -87,8 +93,8 @@ MathJax.Hub.Config({
|
||||
}
|
||||
});
|
||||
</script>
|
||||
<script type="text/javascript" async
|
||||
src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS-MML_HTMLorMML">
|
||||
<script type="text/javascript"
|
||||
src="http://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML">
|
||||
</script>
|
||||
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@@ -121,7 +127,13 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs009.html#___sec8" style="font-size: 80%;">Interpretations and optimizing our parameters</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs010.html#___sec9" style="font-size: 80%;">Interpretations and optimizing our parameters</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs011.html#___sec10" style="font-size: 80%;">Interpretations and optimizing our parameters</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs012.html#___sec11" style="font-size: 80%;">The singular value decompostion</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs012.html#___sec11" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs013.html#___sec12" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs014.html#___sec13" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs015.html#___sec14" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs016.html#___sec15" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs017.html#___sec16" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs018.html#___sec17" style="font-size: 80%;">The singular value decompostion</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -176,6 +188,10 @@ $$
|
||||
<li><a href="._Regression-bs010.html">11</a></li>
|
||||
<li><a href="._Regression-bs011.html">12</a></li>
|
||||
<li><a href="._Regression-bs012.html">13</a></li>
|
||||
<li><a href="._Regression-bs013.html">14</a></li>
|
||||
<li><a href="._Regression-bs014.html">15</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._Regression-bs018.html">19</a></li>
|
||||
<li><a href="._Regression-bs006.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -72,7 +72,13 @@ Automatically generated HTML file from DocOnce source
|
||||
2,
|
||||
None,
|
||||
'___sec10'),
|
||||
('The singular value decompostion', 2, None, '___sec11')]}
|
||||
('The $\\chi^2$ function', 2, None, '___sec11'),
|
||||
('The $\\chi^2$ function', 2, None, '___sec12'),
|
||||
('The $\\chi^2$ function', 2, None, '___sec13'),
|
||||
('The $\\chi^2$ function', 2, None, '___sec14'),
|
||||
('The $\\chi^2$ function', 2, None, '___sec15'),
|
||||
('The $\\chi^2$ function', 2, None, '___sec16'),
|
||||
('The singular value decompostion', 2, None, '___sec17')]}
|
||||
end of tocinfo -->
|
||||
|
||||
<body>
|
||||
@@ -87,8 +93,8 @@ MathJax.Hub.Config({
|
||||
}
|
||||
});
|
||||
</script>
|
||||
<script type="text/javascript" async
|
||||
src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS-MML_HTMLorMML">
|
||||
<script type="text/javascript"
|
||||
src="http://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML">
|
||||
</script>
|
||||
|
||||
|
||||
@@ -121,7 +127,13 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs009.html#___sec8" style="font-size: 80%;">Interpretations and optimizing our parameters</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs010.html#___sec9" style="font-size: 80%;">Interpretations and optimizing our parameters</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs011.html#___sec10" style="font-size: 80%;">Interpretations and optimizing our parameters</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs012.html#___sec11" style="font-size: 80%;">The singular value decompostion</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs012.html#___sec11" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs013.html#___sec12" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs014.html#___sec13" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs015.html#___sec14" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs016.html#___sec15" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs017.html#___sec16" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs018.html#___sec17" style="font-size: 80%;">The singular value decompostion</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -181,6 +193,11 @@ The left-hand side of this equation forms know. Our error vector \( \hat{\epsilo
|
||||
<li><a href="._Regression-bs010.html">11</a></li>
|
||||
<li><a href="._Regression-bs011.html">12</a></li>
|
||||
<li><a href="._Regression-bs012.html">13</a></li>
|
||||
<li><a href="._Regression-bs013.html">14</a></li>
|
||||
<li><a href="._Regression-bs014.html">15</a></li>
|
||||
<li><a href="._Regression-bs015.html">16</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._Regression-bs018.html">19</a></li>
|
||||
<li><a href="._Regression-bs007.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -72,7 +72,13 @@ Automatically generated HTML file from DocOnce source
|
||||
2,
|
||||
None,
|
||||
'___sec10'),
|
||||
('The singular value decompostion', 2, None, '___sec11')]}
|
||||
('The $\\chi^2$ function', 2, None, '___sec11'),
|
||||
('The $\\chi^2$ function', 2, None, '___sec12'),
|
||||
('The $\\chi^2$ function', 2, None, '___sec13'),
|
||||
('The $\\chi^2$ function', 2, None, '___sec14'),
|
||||
('The $\\chi^2$ function', 2, None, '___sec15'),
|
||||
('The $\\chi^2$ function', 2, None, '___sec16'),
|
||||
('The singular value decompostion', 2, None, '___sec17')]}
|
||||
end of tocinfo -->
|
||||
|
||||
<body>
|
||||
@@ -87,8 +93,8 @@ MathJax.Hub.Config({
|
||||
}
|
||||
});
|
||||
</script>
|
||||
<script type="text/javascript" async
|
||||
src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS-MML_HTMLorMML">
|
||||
<script type="text/javascript"
|
||||
src="http://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML">
|
||||
</script>
|
||||
|
||||
|
||||
@@ -121,7 +127,13 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs009.html#___sec8" style="font-size: 80%;">Interpretations and optimizing our parameters</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs010.html#___sec9" style="font-size: 80%;">Interpretations and optimizing our parameters</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs011.html#___sec10" style="font-size: 80%;">Interpretations and optimizing our parameters</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs012.html#___sec11" style="font-size: 80%;">The singular value decompostion</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs012.html#___sec11" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs013.html#___sec12" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs014.html#___sec13" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs015.html#___sec14" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs016.html#___sec15" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs017.html#___sec16" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs018.html#___sec17" style="font-size: 80%;">The singular value decompostion</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -175,6 +187,12 @@ $$
|
||||
<li><a href="._Regression-bs010.html">11</a></li>
|
||||
<li><a href="._Regression-bs011.html">12</a></li>
|
||||
<li><a href="._Regression-bs012.html">13</a></li>
|
||||
<li><a href="._Regression-bs013.html">14</a></li>
|
||||
<li><a href="._Regression-bs014.html">15</a></li>
|
||||
<li><a href="._Regression-bs015.html">16</a></li>
|
||||
<li><a href="._Regression-bs016.html">17</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._Regression-bs018.html">19</a></li>
|
||||
<li><a href="._Regression-bs008.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -72,7 +72,13 @@ Automatically generated HTML file from DocOnce source
|
||||
2,
|
||||
None,
|
||||
'___sec10'),
|
||||
('The singular value decompostion', 2, None, '___sec11')]}
|
||||
('The $\\chi^2$ function', 2, None, '___sec11'),
|
||||
('The $\\chi^2$ function', 2, None, '___sec12'),
|
||||
('The $\\chi^2$ function', 2, None, '___sec13'),
|
||||
('The $\\chi^2$ function', 2, None, '___sec14'),
|
||||
('The $\\chi^2$ function', 2, None, '___sec15'),
|
||||
('The $\\chi^2$ function', 2, None, '___sec16'),
|
||||
('The singular value decompostion', 2, None, '___sec17')]}
|
||||
end of tocinfo -->
|
||||
|
||||
<body>
|
||||
@@ -87,8 +93,8 @@ MathJax.Hub.Config({
|
||||
}
|
||||
});
|
||||
</script>
|
||||
<script type="text/javascript" async
|
||||
src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS-MML_HTMLorMML">
|
||||
<script type="text/javascript"
|
||||
src="http://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML">
|
||||
</script>
|
||||
|
||||
|
||||
@@ -121,7 +127,13 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs009.html#___sec8" style="font-size: 80%;">Interpretations and optimizing our parameters</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs010.html#___sec9" style="font-size: 80%;">Interpretations and optimizing our parameters</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs011.html#___sec10" style="font-size: 80%;">Interpretations and optimizing our parameters</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs012.html#___sec11" style="font-size: 80%;">The singular value decompostion</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs012.html#___sec11" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs013.html#___sec12" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs014.html#___sec13" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs015.html#___sec14" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs016.html#___sec15" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs017.html#___sec16" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs018.html#___sec17" style="font-size: 80%;">The singular value decompostion</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -177,6 +189,13 @@ $$
|
||||
<li><a href="._Regression-bs010.html">11</a></li>
|
||||
<li><a href="._Regression-bs011.html">12</a></li>
|
||||
<li><a href="._Regression-bs012.html">13</a></li>
|
||||
<li><a href="._Regression-bs013.html">14</a></li>
|
||||
<li><a href="._Regression-bs014.html">15</a></li>
|
||||
<li><a href="._Regression-bs015.html">16</a></li>
|
||||
<li><a href="._Regression-bs016.html">17</a></li>
|
||||
<li><a href="._Regression-bs017.html">18</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._Regression-bs018.html">19</a></li>
|
||||
<li><a href="._Regression-bs009.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -72,7 +72,13 @@ Automatically generated HTML file from DocOnce source
|
||||
2,
|
||||
None,
|
||||
'___sec10'),
|
||||
('The singular value decompostion', 2, None, '___sec11')]}
|
||||
('The $\\chi^2$ function', 2, None, '___sec11'),
|
||||
('The $\\chi^2$ function', 2, None, '___sec12'),
|
||||
('The $\\chi^2$ function', 2, None, '___sec13'),
|
||||
('The $\\chi^2$ function', 2, None, '___sec14'),
|
||||
('The $\\chi^2$ function', 2, None, '___sec15'),
|
||||
('The $\\chi^2$ function', 2, None, '___sec16'),
|
||||
('The singular value decompostion', 2, None, '___sec17')]}
|
||||
end of tocinfo -->
|
||||
|
||||
<body>
|
||||
@@ -87,8 +93,8 @@ MathJax.Hub.Config({
|
||||
}
|
||||
});
|
||||
</script>
|
||||
<script type="text/javascript" async
|
||||
src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS-MML_HTMLorMML">
|
||||
<script type="text/javascript"
|
||||
src="http://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML">
|
||||
</script>
|
||||
|
||||
|
||||
@@ -121,7 +127,13 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="#___sec8" style="font-size: 80%;">Interpretations and optimizing our parameters</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs010.html#___sec9" style="font-size: 80%;">Interpretations and optimizing our parameters</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs011.html#___sec10" style="font-size: 80%;">Interpretations and optimizing our parameters</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs012.html#___sec11" style="font-size: 80%;">The singular value decompostion</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs012.html#___sec11" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs013.html#___sec12" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs014.html#___sec13" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs015.html#___sec14" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs016.html#___sec15" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs017.html#___sec16" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs018.html#___sec17" style="font-size: 80%;">The singular value decompostion</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -156,12 +168,12 @@ where \( \langle y_i \rangle \) is the mean value. Keep in mind also that till n
|
||||
<p>
|
||||
In order to find the parameters \( \beta_i \) we will then minimize the spread of \( Q(\hat{\beta}) \) by requiring
|
||||
$$
|
||||
\frac{\partial Q(\hat{\beta})}{\partial \beta_j} = \frac{\partial }{\partial \beta_j}\left[ \sum_{i=0}^{n-1}\left(y_i-\beta_0x_{i,0}+\beta_1x_{i,1}+\beta_2x_{i,2}+\dots+\beta_{n-1}x_{i,n-1}\right)^2\right]=0,
|
||||
\frac{\partial Q(\hat{\beta})}{\partial \beta_j} = \frac{\partial }{\partial \beta_j}\left[ \sum_{i=0}^{n-1}\left(y_i-\beta_0x_{i,0}-\beta_1x_{i,1}-\beta_2x_{i,2}-\dots-\beta_{n-1}x_{i,n-1}\right)^2\right]=0,
|
||||
$$
|
||||
|
||||
which results in
|
||||
$$
|
||||
\frac{\partial Q(\hat{\beta})}{\partial \beta_j} = -2\left[ \sum_{i=0}^{n-1}x_{ij}\left(y_i-\beta_0x_{i,0}+\beta_1x_{i,1}+\beta_2x_{i,2}+\dots+\beta_{n-1}x_{i,n-1}\right)\right]=0,
|
||||
\frac{\partial Q(\hat{\beta})}{\partial \beta_j} = -2\left[ \sum_{i=0}^{n-1}x_{ij}\left(y_i-\beta_0x_{i,0}-\beta_1x_{i,1}-\beta_2x_{i,2}-\dots-\beta_{n-1}x_{i,n-1}\right)\right]=0,
|
||||
$$
|
||||
|
||||
or in a matrix-vector form as
|
||||
@@ -192,6 +204,12 @@ $$
|
||||
<li><a href="._Regression-bs010.html">11</a></li>
|
||||
<li><a href="._Regression-bs011.html">12</a></li>
|
||||
<li><a href="._Regression-bs012.html">13</a></li>
|
||||
<li><a href="._Regression-bs013.html">14</a></li>
|
||||
<li><a href="._Regression-bs014.html">15</a></li>
|
||||
<li><a href="._Regression-bs015.html">16</a></li>
|
||||
<li><a href="._Regression-bs016.html">17</a></li>
|
||||
<li><a href="._Regression-bs017.html">18</a></li>
|
||||
<li><a href="._Regression-bs018.html">19</a></li>
|
||||
<li><a href="._Regression-bs010.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -72,7 +72,13 @@ Automatically generated HTML file from DocOnce source
|
||||
2,
|
||||
None,
|
||||
'___sec10'),
|
||||
('The singular value decompostion', 2, None, '___sec11')]}
|
||||
('The $\\chi^2$ function', 2, None, '___sec11'),
|
||||
('The $\\chi^2$ function', 2, None, '___sec12'),
|
||||
('The $\\chi^2$ function', 2, None, '___sec13'),
|
||||
('The $\\chi^2$ function', 2, None, '___sec14'),
|
||||
('The $\\chi^2$ function', 2, None, '___sec15'),
|
||||
('The $\\chi^2$ function', 2, None, '___sec16'),
|
||||
('The singular value decompostion', 2, None, '___sec17')]}
|
||||
end of tocinfo -->
|
||||
|
||||
<body>
|
||||
@@ -87,8 +93,8 @@ MathJax.Hub.Config({
|
||||
}
|
||||
});
|
||||
</script>
|
||||
<script type="text/javascript" async
|
||||
src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS-MML_HTMLorMML">
|
||||
<script type="text/javascript"
|
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src="http://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML">
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<!-- navigation toc: --> <li><a href="._Regression-bs009.html#___sec8" style="font-size: 80%;">Interpretations and optimizing our parameters</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs012.html#___sec11" style="font-size: 80%;">The singular value decompostion</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs018.html#___sec17" style="font-size: 80%;">The singular value decompostion</a></li>
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@@ -180,6 +192,12 @@ meaning that the solution for \( \hat{\beta} \) is the one which minimizes the r
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<li><a href="._Regression-bs018.html">19</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs009.html#___sec8" style="font-size: 80%;">Interpretations and optimizing our parameters</a></li>
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|
||||
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|
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|
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<!-- navigation toc: --> <li><a href="._Regression-bs014.html#___sec13" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
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|
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<!-- navigation toc: --> <li><a href="._Regression-bs017.html#___sec16" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs018.html#___sec17" style="font-size: 80%;">The singular value decompostion</a></li>
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<a name="part0012"></a>
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<h2 id="___sec11" class="anchor">The singular value decompostion </h2>
|
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<h2 id="___sec11" class="anchor">The \( \chi^2 \) function </h2>
|
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|
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|
||||
How can we use the singular value decomposition to find the parameters \( \beta_j \)? More details will come. We first note that a general \( m\times n \) matrix \( \hat{A} \) can be written in terms of a diagonal matrix \( \hat{\Sigma} \) of dimensionality \( n\times n \) and two orthognal matrices \( \hat{U} \) and \( \hat{V} \), where the first has dimensionality \( m \times n \) and the last dimensionality \( n\times n \). We have then
|
||||
|
||||
<p>
|
||||
Normally, the response (dependent or outcome) variable \( y_i \) the outcome of a numerical experiment or another type of experiment and is thus only an approximation to the true value. It is then always accompanied by an error estimate, often limited to a statistical error estimate given by the standard deviation discussed earlier. In the discussion here we will treat \( y_i \) as our exact value for the response variable.
|
||||
|
||||
<p>
|
||||
Introducing the standard deviation \( \sigma_i \) for each measurement \( y_i \), we define now the \( \chi^2 \) function as
|
||||
$$
|
||||
\hat{A} = \hat{U}\hat{\Sigma}\hat{V}
|
||||
\chi^2(\hat{\beta})=\sum_{i=0}^{n-1}\frac{\left(y_i-\tilde{y}_i\right)^2}{\sigma_i^2}=\left(\hat{y}-\hat{\tilde{y}}\right)^T\frac{1}{\hat{\Sigma^2}}\left(\hat{y}-\hat{\tilde{y}}\right),
|
||||
$$
|
||||
|
||||
where the matrix \( \hat{\Sigma} \) is a diagonal matrix with \( \sigma_i \) as matrix elements.
|
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<p>
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@@ -166,6 +186,13 @@ $$
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<li><a href="._Regression-bs011.html">12</a></li>
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<li class="active"><a href="._Regression-bs012.html">13</a></li>
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<li><a href="._Regression-bs013.html">14</a></li>
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<li><a href="._Regression-bs015.html">16</a></li>
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<li><a href="._Regression-bs017.html">18</a></li>
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<li><a href="._Regression-bs018.html">19</a></li>
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|
||||
|
||||
<p> </p><p> </p><p> </p> <!-- add vertical space -->
|
||||
|
||||
<a name="part0013"></a>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="___sec12" class="anchor">The \( \chi^2 \) function </h2>
|
||||
<div class="panel panel-default">
|
||||
<div class="panel-body">
|
||||
<p> <!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
|
||||
|
||||
<p>
|
||||
In order to find the parameters \( \beta_i \) we will then minimize the spread of \( \chi^2(\hat{\beta}) \) by requiring
|
||||
$$
|
||||
\frac{\partial \chi^2(\hat{\beta})}{\partial \beta_j} = \frac{\partial }{\partial \beta_j}\left[ \sum_{i=0}^{n-1}\left(\frac{y_i-\beta_0x_{i,0}-\beta_1x_{i,1}-\beta_2x_{i,2}-\dots-\beta_{n-1}x_{i,n-1}}{\sigma_i}\right)^2\right]=0,
|
||||
$$
|
||||
|
||||
which results in
|
||||
$$
|
||||
\frac{\partial \chi^2(\hat{\beta})}{\partial \beta_j} = -2\left[ \sum_{i=0}^{n-1}\frac{x_{ij}}{\sigma_i}\left(\frac{y_i-\beta_0x_{i,0}-\beta_1x_{i,1}-\beta_2x_{i,2}-\dots-\beta_{n-1}x_{i,n-1}}{\sigma_i}\right)\right]=0,
|
||||
$$
|
||||
|
||||
or in a matrix-vector form as
|
||||
$$
|
||||
\frac{\partial \chi^2(\hat{\beta})}{\partial \hat{\beta}} = 0 = \hat{A}^T\left( \hat{b}-\hat{A}\hat{\beta}\right).
|
||||
$$
|
||||
|
||||
where we have defined the matrix \( \hat{A} =\hat{X}/\hat{\Sigma} \) with matrix elements \( a_{ij} = x_{ij}/\sigma_i \) and the vector \( \hat{b} \) with elements \( b_i = y_i/\sigma_i \).
|
||||
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<!-- navigation toc: --> <li><a href="._Regression-bs012.html#___sec11" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs018.html#___sec17" style="font-size: 80%;">The singular value decompostion</a></li>
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<a name="part0014"></a>
|
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<!-- !split -->
|
||||
|
||||
<h2 id="___sec13" class="anchor">The \( \chi^2 \) function </h2>
|
||||
<div class="panel panel-default">
|
||||
<div class="panel-body">
|
||||
<p> <!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
|
||||
|
||||
<p>
|
||||
We can rewrite
|
||||
$$
|
||||
\frac{\partial \chi^2(\hat{\beta})}{\partial \hat{\beta}} = 0 = \hat{A}^T\left( \hat{b}-\hat{A}\hat{\beta}\right),
|
||||
$$
|
||||
|
||||
as
|
||||
$$
|
||||
\hat{A}^T\hat{b} = \hat{A}^T\hat{A}\hat{\beta},
|
||||
$$
|
||||
|
||||
and if the matrix \( \hat{A}^T\hat{A} \) is invertible we have the solution
|
||||
$$
|
||||
\hat{\beta} =\left(\hat{A}^T\hat{A}\right)^{-1}\hat{A}^T\hat{b}.
|
||||
$$
|
||||
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|
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<!-- navigation toc: --> <li><a href="._Regression-bs012.html#___sec11" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs013.html#___sec12" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
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|
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||||
<!-- navigation toc: --> <li><a href="._Regression-bs017.html#___sec16" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs018.html#___sec17" style="font-size: 80%;">The singular value decompostion</a></li>
|
||||
|
||||
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<a name="part0015"></a>
|
||||
<!-- !split -->
|
||||
|
||||
<h2 id="___sec14" class="anchor">The \( \chi^2 \) function </h2>
|
||||
<div class="panel panel-default">
|
||||
<div class="panel-body">
|
||||
<p> <!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
|
||||
|
||||
<p>
|
||||
If we then introduce the matrix
|
||||
$$
|
||||
\hat{H} = \hat{A}^T\hat{A},
|
||||
$$
|
||||
|
||||
we have then the following expression for the parameters \( \beta_j \) (the matrix elements of \( \hat{H} \) are \( h_{ij} \))
|
||||
$$
|
||||
\beta_j = \sum_{k=0}^{p-1}h_{jk}\sum_{i=0}^{n-1}\frac{y_i}{\sigma_i}\frac{x_{ik}}{\sigma_i} = \sum_{k=0}^{p-1}h_{jk}\sum_{i=0}^{n-1}b_ia_{ik}
|
||||
$$
|
||||
|
||||
We state without proof the expression for the uncertainty in the parameters \( \beta_j \) as
|
||||
$$
|
||||
\sigma^2(\beta_j) = \sum_{i=0}^{n-1}\sigma_i^2\left( \frac{\partial \beta_j}{\partial y_i}\right)^2,
|
||||
$$
|
||||
|
||||
resulting in
|
||||
$$
|
||||
\sigma^2(\beta_j) = \left(\sum_{k=0}^{p-1}h_{jk}\sum_{i=0}^{n-1}a_{ik}\right)\left(\sum_{l=0}^{p-1}h_{jl}\sum_{m=0}^{n-1}a_{ml}\right) = h_{jj}!
|
||||
$$
|
||||
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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<!-- navigation toc: --> <li><a href="._Regression-bs012.html#___sec11" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
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<!-- navigation toc: --> <li><a href="._Regression-bs018.html#___sec17" style="font-size: 80%;">The singular value decompostion</a></li>
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<a name="part0016"></a>
|
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<!-- !split -->
|
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|
||||
<h2 id="___sec15" class="anchor">The \( \chi^2 \) function </h2>
|
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<div class="panel panel-default">
|
||||
<div class="panel-body">
|
||||
<p> <!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
|
||||
The first step here is to approximate the function \( y \) with a first-order polynomial, that is we write
|
||||
$$
|
||||
y=y(x) \rightarrow y(x_i) \approx \beta_0+\beta_1 x_i.
|
||||
$$
|
||||
|
||||
By computing the derivatives of \( \chi^2 \) with respect to \( \beta_0 \) and \( \beta_1 \) show that these are given by
|
||||
$$
|
||||
\frac{\partial \chi^2(\hat{\beta})}{\partial \beta_0} = -2\left[ \sum_{i=0}^{1}\left(\frac{y_i-\beta_0-\beta_1x_{i}}{\sigma_i^2}\right)\right]=0,
|
||||
$$
|
||||
|
||||
and
|
||||
$$
|
||||
\frac{\partial \chi^2(\hat{\beta})}{\partial \beta_0} = -2\left[ \sum_{i=0}^{1}x_i\left(\frac{y_i-\beta_0-\beta_1x_{i}}{\sigma_i^2}\right)\right]=0.
|
||||
$$
|
||||
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|
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<h2 id="___sec16" class="anchor">The \( \chi^2 \) function </h2>
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|
||||
<p> <!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
|
||||
|
||||
<p>
|
||||
We define then
|
||||
$$
|
||||
\gamma = \sum_{i=0}^{1}\frac{1}{\sigma_i^2},
|
||||
$$
|
||||
|
||||
|
||||
$$
|
||||
\gamma_x = \sum_{i=0}^{1}\frac{x_{i}}{\sigma_i^2},
|
||||
$$
|
||||
|
||||
$$
|
||||
\gamma_y = \sum_{i=0}^{1}\left(\frac{y_i}{\sigma_i^2}\right),
|
||||
$$
|
||||
|
||||
$$
|
||||
\gamma_{xx} = \sum_{i=0}^{1}\frac{x_ix_{i}}{\sigma_i^2},
|
||||
$$
|
||||
|
||||
$$
|
||||
\gamma_{xy} = \sum_{i=0}^{1}\frac{y_ix_{i}}{\sigma_i^2},
|
||||
$$
|
||||
|
||||
and show that
|
||||
$$
|
||||
\beta_0 = \frac{\gamma_{xx}\gamma_y-\gamma_x\gamma_y}{\gamma\gamma_{xx}-\gamma_x^2},
|
||||
$$
|
||||
|
||||
$$
|
||||
\beta_1 = \frac{\gamma_{xy}\gamma-\gamma_x\gamma_y}{\gamma\gamma_{xx}-\gamma_x^2}.
|
||||
$$
|
||||
|
||||
<p>
|
||||
The LSM suffers often from both being underdetermined and overdetermined in the unknown coefficients \( \beta_i \). A better approach is to use the Singular Value Decomposition (SVD) method discussed below.
|
||||
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<h2 id="___sec17" class="anchor">The singular value decompostion </h2>
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<p> <!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
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How can we use the singular value decomposition to find the parameters \( \beta_j \)? More details will come. We first note that a general \( m\times n \) matrix \( \hat{A} \) can be written in terms of a diagonal matrix \( \hat{\Sigma} \) of dimensionality \( n\times n \) and two orthognal matrices \( \hat{U} \) and \( \hat{V} \), where the first has dimensionality \( m \times n \) and the last dimensionality \( n\times n \). We have then
|
||||
$$
|
||||
\hat{A} = \hat{U}\hat{\Sigma}\hat{V}
|
||||
$$
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-->
|
||||
|
||||
|
||||
<center style="font-size:80%">
|
||||
<!-- copyright only on the titlepage -->
|
||||
</center>
|
||||
|
||||
|
||||
</body>
|
||||
</html>
|
||||
|
||||
|
||||
@@ -72,7 +72,13 @@ Automatically generated HTML file from DocOnce source
|
||||
2,
|
||||
None,
|
||||
'___sec10'),
|
||||
('The singular value decompostion', 2, None, '___sec11')]}
|
||||
('The $\\chi^2$ function', 2, None, '___sec11'),
|
||||
('The $\\chi^2$ function', 2, None, '___sec12'),
|
||||
('The $\\chi^2$ function', 2, None, '___sec13'),
|
||||
('The $\\chi^2$ function', 2, None, '___sec14'),
|
||||
('The $\\chi^2$ function', 2, None, '___sec15'),
|
||||
('The $\\chi^2$ function', 2, None, '___sec16'),
|
||||
('The singular value decompostion', 2, None, '___sec17')]}
|
||||
end of tocinfo -->
|
||||
|
||||
<body>
|
||||
@@ -87,8 +93,8 @@ MathJax.Hub.Config({
|
||||
}
|
||||
});
|
||||
</script>
|
||||
<script type="text/javascript" async
|
||||
src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS-MML_HTMLorMML">
|
||||
<script type="text/javascript"
|
||||
src="http://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML">
|
||||
</script>
|
||||
|
||||
|
||||
@@ -121,7 +127,13 @@ MathJax.Hub.Config({
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs009.html#___sec8" style="font-size: 80%;">Interpretations and optimizing our parameters</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs010.html#___sec9" style="font-size: 80%;">Interpretations and optimizing our parameters</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs011.html#___sec10" style="font-size: 80%;">Interpretations and optimizing our parameters</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs012.html#___sec11" style="font-size: 80%;">The singular value decompostion</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs012.html#___sec11" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs013.html#___sec12" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs014.html#___sec13" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs015.html#___sec14" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs016.html#___sec15" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs017.html#___sec16" style="font-size: 80%;">The \( \chi^2 \) function</a></li>
|
||||
<!-- navigation toc: --> <li><a href="._Regression-bs018.html#___sec17" style="font-size: 80%;">The singular value decompostion</a></li>
|
||||
|
||||
</ul>
|
||||
</li>
|
||||
@@ -156,7 +168,7 @@ MathJax.Hub.Config({
|
||||
<center>[2] <b>Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University</b></center>
|
||||
<br>
|
||||
<p>
|
||||
<center><h4>Oct 18, 2017</h4></center> <!-- date -->
|
||||
<center><h4>Oct 24, 2017</h4></center> <!-- date -->
|
||||
<br>
|
||||
<p>
|
||||
|
||||
@@ -180,7 +192,7 @@ MathJax.Hub.Config({
|
||||
<li><a href="._Regression-bs008.html">9</a></li>
|
||||
<li><a href="._Regression-bs009.html">10</a></li>
|
||||
<li><a href="">...</a></li>
|
||||
<li><a href="._Regression-bs012.html">13</a></li>
|
||||
<li><a href="._Regression-bs018.html">19</a></li>
|
||||
<li><a href="._Regression-bs001.html">»</a></li>
|
||||
</ul>
|
||||
<!-- ------------------- end of main content --------------- -->
|
||||
|
||||
@@ -1,4 +1,3 @@
|
||||
\
|
||||
<!DOCTYPE html>
|
||||
|
||||
<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
|
||||
@@ -117,8 +116,8 @@ MathJax.Hub.Config({
|
||||
}
|
||||
});
|
||||
</script>
|
||||
<script type="text/javascript" async
|
||||
src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS-MML_HTMLorMML">
|
||||
<script type="text/javascript"
|
||||
src="http://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML">
|
||||
</script>
|
||||
|
||||
|
||||
@@ -148,7 +147,7 @@ MathJax.Hub.Config({
|
||||
<center>[2] <b>Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University</b></center>
|
||||
<br>
|
||||
<p> <br>
|
||||
<center><h4>Oct 18, 2017</h4></center> <!-- date -->
|
||||
<center><h4>Oct 24, 2017</h4></center> <!-- date -->
|
||||
<br>
|
||||
<p>
|
||||
|
||||
@@ -404,14 +403,14 @@ where \( \langle y_i \rangle \) is the mean value. Keep in mind also that till n
|
||||
In order to find the parameters \( \beta_i \) we will then minimize the spread of \( Q(\hat{\beta}) \) by requiring
|
||||
<p> <br>
|
||||
$$
|
||||
\frac{\partial Q(\hat{\beta})}{\partial \beta_j} = \frac{\partial }{\partial \beta_j}\left[ \sum_{i=0}^{n-1}\left(y_i-\beta_0x_{i,0}+\beta_1x_{i,1}+\beta_2x_{i,2}+\dots+\beta_{n-1}x_{i,n-1}\right)^2\right]=0,
|
||||
\frac{\partial Q(\hat{\beta})}{\partial \beta_j} = \frac{\partial }{\partial \beta_j}\left[ \sum_{i=0}^{n-1}\left(y_i-\beta_0x_{i,0}-\beta_1x_{i,1}-\beta_2x_{i,2}-\dots-\beta_{n-1}x_{i,n-1}\right)^2\right]=0,
|
||||
$$
|
||||
<p> <br>
|
||||
|
||||
which results in
|
||||
<p> <br>
|
||||
$$
|
||||
\frac{\partial Q(\hat{\beta})}{\partial \beta_j} = -2\left[ \sum_{i=0}^{n-1}x_{ij}\left(y_i-\beta_0x_{i,0}+\beta_1x_{i,1}+\beta_2x_{i,2}+\dots+\beta_{n-1}x_{i,n-1}\right)\right]=0,
|
||||
\frac{\partial Q(\hat{\beta})}{\partial \beta_j} = -2\left[ \sum_{i=0}^{n-1}x_{ij}\left(y_i-\beta_0x_{i,0}-\beta_1x_{i,1}-\beta_2x_{i,2}-\dots-\beta_{n-1}x_{i,n-1}\right)\right]=0,
|
||||
$$
|
||||
<p> <br>
|
||||
|
||||
@@ -492,7 +491,210 @@ meaning that the solution for \( \hat{\beta} \) is the one which minimizes the r
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec11">The singular value decompostion </h2>
|
||||
<h2 id="___sec11">The \( \chi^2 \) function </h2>
|
||||
<div class="alert alert-block alert-block alert-text-normal">
|
||||
<b></b>
|
||||
<p>
|
||||
Normally, the response (dependent or outcome) variable \( y_i \) the outcome of a numerical experiment or another type of experiment and is thus only an approximation to the true value. It is then always accompanied by an error estimate, often limited to a statistical error estimate given by the standard deviation discussed earlier. In the discussion here we will treat \( y_i \) as our exact value for the response variable.
|
||||
|
||||
<p>
|
||||
Introducing the standard deviation \( \sigma_i \) for each measurement \( y_i \), we define now the \( \chi^2 \) function as
|
||||
|
||||
<p> <br>
|
||||
$$
|
||||
\chi^2(\hat{\beta})=\sum_{i=0}^{n-1}\frac{\left(y_i-\tilde{y}_i\right)^2}{\sigma_i^2}=\left(\hat{y}-\hat{\tilde{y}}\right)^T\frac{1}{\hat{\Sigma^2}}\left(\hat{y}-\hat{\tilde{y}}\right),
|
||||
$$
|
||||
<p> <br>
|
||||
|
||||
where the matrix \( \hat{\Sigma} \) is a diagonal matrix with \( \sigma_i \) as matrix elements.
|
||||
|
||||
|
||||
</div>
|
||||
</section>
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec12">The \( \chi^2 \) function </h2>
|
||||
<div class="alert alert-block alert-block alert-text-normal">
|
||||
<b></b>
|
||||
<p>
|
||||
In order to find the parameters \( \beta_i \) we will then minimize the spread of \( \chi^2(\hat{\beta}) \) by requiring
|
||||
<p> <br>
|
||||
$$
|
||||
\frac{\partial \chi^2(\hat{\beta})}{\partial \beta_j} = \frac{\partial }{\partial \beta_j}\left[ \sum_{i=0}^{n-1}\left(\frac{y_i-\beta_0x_{i,0}-\beta_1x_{i,1}-\beta_2x_{i,2}-\dots-\beta_{n-1}x_{i,n-1}}{\sigma_i}\right)^2\right]=0,
|
||||
$$
|
||||
<p> <br>
|
||||
|
||||
which results in
|
||||
<p> <br>
|
||||
$$
|
||||
\frac{\partial \chi^2(\hat{\beta})}{\partial \beta_j} = -2\left[ \sum_{i=0}^{n-1}\frac{x_{ij}}{\sigma_i}\left(\frac{y_i-\beta_0x_{i,0}-\beta_1x_{i,1}-\beta_2x_{i,2}-\dots-\beta_{n-1}x_{i,n-1}}{\sigma_i}\right)\right]=0,
|
||||
$$
|
||||
<p> <br>
|
||||
|
||||
or in a matrix-vector form as
|
||||
<p> <br>
|
||||
$$
|
||||
\frac{\partial \chi^2(\hat{\beta})}{\partial \hat{\beta}} = 0 = \hat{A}^T\left( \hat{b}-\hat{A}\hat{\beta}\right).
|
||||
$$
|
||||
<p> <br>
|
||||
|
||||
where we have defined the matrix \( \hat{A} =\hat{X}/\hat{\Sigma} \) with matrix elements \( a_{ij} = x_{ij}/\sigma_i \) and the vector \( \hat{b} \) with elements \( b_i = y_i/\sigma_i \).
|
||||
</div>
|
||||
</section>
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec13">The \( \chi^2 \) function </h2>
|
||||
<div class="alert alert-block alert-block alert-text-normal">
|
||||
<b></b>
|
||||
<p>
|
||||
We can rewrite
|
||||
<p> <br>
|
||||
$$
|
||||
\frac{\partial \chi^2(\hat{\beta})}{\partial \hat{\beta}} = 0 = \hat{A}^T\left( \hat{b}-\hat{A}\hat{\beta}\right),
|
||||
$$
|
||||
<p> <br>
|
||||
|
||||
as
|
||||
<p> <br>
|
||||
$$
|
||||
\hat{A}^T\hat{b} = \hat{A}^T\hat{A}\hat{\beta},
|
||||
$$
|
||||
<p> <br>
|
||||
|
||||
and if the matrix \( \hat{A}^T\hat{A} \) is invertible we have the solution
|
||||
<p> <br>
|
||||
$$
|
||||
\hat{\beta} =\left(\hat{A}^T\hat{A}\right)^{-1}\hat{A}^T\hat{b}.
|
||||
$$
|
||||
<p> <br>
|
||||
</div>
|
||||
</section>
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec14">The \( \chi^2 \) function </h2>
|
||||
<div class="alert alert-block alert-block alert-text-normal">
|
||||
<b></b>
|
||||
<p>
|
||||
If we then introduce the matrix
|
||||
<p> <br>
|
||||
$$
|
||||
\hat{H} = \hat{A}^T\hat{A},
|
||||
$$
|
||||
<p> <br>
|
||||
|
||||
we have then the following expression for the parameters \( \beta_j \) (the matrix elements of \( \hat{H} \) are \( h_{ij} \))
|
||||
<p> <br>
|
||||
$$
|
||||
\beta_j = \sum_{k=0}^{p-1}h_{jk}\sum_{i=0}^{n-1}\frac{y_i}{\sigma_i}\frac{x_{ik}}{\sigma_i} = \sum_{k=0}^{p-1}h_{jk}\sum_{i=0}^{n-1}b_ia_{ik}
|
||||
$$
|
||||
<p> <br>
|
||||
|
||||
We state without proof the expression for the uncertainty in the parameters \( \beta_j \) as
|
||||
<p> <br>
|
||||
$$
|
||||
\sigma^2(\beta_j) = \sum_{i=0}^{n-1}\sigma_i^2\left( \frac{\partial \beta_j}{\partial y_i}\right)^2,
|
||||
$$
|
||||
<p> <br>
|
||||
|
||||
resulting in
|
||||
<p> <br>
|
||||
$$
|
||||
\sigma^2(\beta_j) = \left(\sum_{k=0}^{p-1}h_{jk}\sum_{i=0}^{n-1}a_{ik}\right)\left(\sum_{l=0}^{p-1}h_{jl}\sum_{m=0}^{n-1}a_{ml}\right) = h_{jj}!
|
||||
$$
|
||||
<p> <br>
|
||||
</div>
|
||||
</section>
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec15">The \( \chi^2 \) function </h2>
|
||||
<div class="alert alert-block alert-block alert-text-normal">
|
||||
<b></b>
|
||||
<p>
|
||||
The first step here is to approximate the function \( y \) with a first-order polynomial, that is we write
|
||||
<p> <br>
|
||||
$$
|
||||
y=y(x) \rightarrow y(x_i) \approx \beta_0+\beta_1 x_i.
|
||||
$$
|
||||
<p> <br>
|
||||
|
||||
By computing the derivatives of \( \chi^2 \) with respect to \( \beta_0 \) and \( \beta_1 \) show that these are given by
|
||||
<p> <br>
|
||||
$$
|
||||
\frac{\partial \chi^2(\hat{\beta})}{\partial \beta_0} = -2\left[ \sum_{i=0}^{1}\left(\frac{y_i-\beta_0-\beta_1x_{i}}{\sigma_i^2}\right)\right]=0,
|
||||
$$
|
||||
<p> <br>
|
||||
|
||||
and
|
||||
<p> <br>
|
||||
$$
|
||||
\frac{\partial \chi^2(\hat{\beta})}{\partial \beta_0} = -2\left[ \sum_{i=0}^{1}x_i\left(\frac{y_i-\beta_0-\beta_1x_{i}}{\sigma_i^2}\right)\right]=0.
|
||||
$$
|
||||
<p> <br>
|
||||
</div>
|
||||
</section>
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec16">The \( \chi^2 \) function </h2>
|
||||
<div class="alert alert-block alert-block alert-text-normal">
|
||||
<b></b>
|
||||
<p>
|
||||
We define then
|
||||
<p> <br>
|
||||
$$
|
||||
\gamma = \sum_{i=0}^{1}\frac{1}{\sigma_i^2},
|
||||
$$
|
||||
<p> <br>
|
||||
|
||||
<p> <br>
|
||||
$$
|
||||
\gamma_x = \sum_{i=0}^{1}\frac{x_{i}}{\sigma_i^2},
|
||||
$$
|
||||
<p> <br>
|
||||
|
||||
<p> <br>
|
||||
$$
|
||||
\gamma_y = \sum_{i=0}^{1}\left(\frac{y_i}{\sigma_i^2}\right),
|
||||
$$
|
||||
<p> <br>
|
||||
|
||||
<p> <br>
|
||||
$$
|
||||
\gamma_{xx} = \sum_{i=0}^{1}\frac{x_ix_{i}}{\sigma_i^2},
|
||||
$$
|
||||
<p> <br>
|
||||
|
||||
<p> <br>
|
||||
$$
|
||||
\gamma_{xy} = \sum_{i=0}^{1}\frac{y_ix_{i}}{\sigma_i^2},
|
||||
$$
|
||||
<p> <br>
|
||||
|
||||
and show that
|
||||
<p> <br>
|
||||
$$
|
||||
\beta_0 = \frac{\gamma_{xx}\gamma_y-\gamma_x\gamma_y}{\gamma\gamma_{xx}-\gamma_x^2},
|
||||
$$
|
||||
<p> <br>
|
||||
|
||||
<p> <br>
|
||||
$$
|
||||
\beta_1 = \frac{\gamma_{xy}\gamma-\gamma_x\gamma_y}{\gamma\gamma_{xx}-\gamma_x^2}.
|
||||
$$
|
||||
<p> <br>
|
||||
|
||||
<p>
|
||||
The LSM suffers often from both being underdetermined and overdetermined in the unknown coefficients \( \beta_i \). A better approach is to use the Singular Value Decomposition (SVD) method discussed below.
|
||||
</div>
|
||||
</section>
|
||||
|
||||
|
||||
<section>
|
||||
<h2 id="___sec17">The singular value decompostion </h2>
|
||||
<div class="alert alert-block alert-block alert-text-normal">
|
||||
<b></b>
|
||||
<p>
|
||||
|
||||
@@ -92,7 +92,13 @@ div { text-align: justify; text-justify: inter-word; }
|
||||
2,
|
||||
None,
|
||||
'___sec10'),
|
||||
('The singular value decompostion', 2, None, '___sec11')]}
|
||||
('The $\\chi^2$ function', 2, None, '___sec11'),
|
||||
('The $\\chi^2$ function', 2, None, '___sec12'),
|
||||
('The $\\chi^2$ function', 2, None, '___sec13'),
|
||||
('The $\\chi^2$ function', 2, None, '___sec14'),
|
||||
('The $\\chi^2$ function', 2, None, '___sec15'),
|
||||
('The $\\chi^2$ function', 2, None, '___sec16'),
|
||||
('The singular value decompostion', 2, None, '___sec17')]}
|
||||
end of tocinfo -->
|
||||
|
||||
<body>
|
||||
@@ -107,8 +113,8 @@ MathJax.Hub.Config({
|
||||
}
|
||||
});
|
||||
</script>
|
||||
<script type="text/javascript" async
|
||||
src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS-MML_HTMLorMML">
|
||||
<script type="text/javascript"
|
||||
src="http://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML">
|
||||
</script>
|
||||
|
||||
|
||||
@@ -134,7 +140,7 @@ MathJax.Hub.Config({
|
||||
<center>[2] <b>Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University</b></center>
|
||||
<br>
|
||||
<p>
|
||||
<center><h4>Oct 18, 2017</h4></center> <!-- date -->
|
||||
<center><h4>Oct 24, 2017</h4></center> <!-- date -->
|
||||
<br>
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
@@ -360,12 +366,12 @@ where \( \langle y_i \rangle \) is the mean value. Keep in mind also that till n
|
||||
<p>
|
||||
In order to find the parameters \( \beta_i \) we will then minimize the spread of \( Q(\hat{\beta}) \) by requiring
|
||||
$$
|
||||
\frac{\partial Q(\hat{\beta})}{\partial \beta_j} = \frac{\partial }{\partial \beta_j}\left[ \sum_{i=0}^{n-1}\left(y_i-\beta_0x_{i,0}+\beta_1x_{i,1}+\beta_2x_{i,2}+\dots+\beta_{n-1}x_{i,n-1}\right)^2\right]=0,
|
||||
\frac{\partial Q(\hat{\beta})}{\partial \beta_j} = \frac{\partial }{\partial \beta_j}\left[ \sum_{i=0}^{n-1}\left(y_i-\beta_0x_{i,0}-\beta_1x_{i,1}-\beta_2x_{i,2}-\dots-\beta_{n-1}x_{i,n-1}\right)^2\right]=0,
|
||||
$$
|
||||
|
||||
which results in
|
||||
$$
|
||||
\frac{\partial Q(\hat{\beta})}{\partial \beta_j} = -2\left[ \sum_{i=0}^{n-1}x_{ij}\left(y_i-\beta_0x_{i,0}+\beta_1x_{i,1}+\beta_2x_{i,2}+\dots+\beta_{n-1}x_{i,n-1}\right)\right]=0,
|
||||
\frac{\partial Q(\hat{\beta})}{\partial \beta_j} = -2\left[ \sum_{i=0}^{n-1}x_{ij}\left(y_i-\beta_0x_{i,0}-\beta_1x_{i,1}-\beta_2x_{i,2}-\dots-\beta_{n-1}x_{i,n-1}\right)\right]=0,
|
||||
$$
|
||||
|
||||
or in a matrix-vector form as
|
||||
@@ -434,7 +440,184 @@ meaning that the solution for \( \hat{\beta} \) is the one which minimizes the r
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec11">The singular value decompostion </h2>
|
||||
<h2 id="___sec11">The \( \chi^2 \) function </h2>
|
||||
<div class="alert alert-block alert-block alert-text-normal">
|
||||
<b></b>
|
||||
<p>
|
||||
|
||||
<p>
|
||||
Normally, the response (dependent or outcome) variable \( y_i \) the outcome of a numerical experiment or another type of experiment and is thus only an approximation to the true value. It is then always accompanied by an error estimate, often limited to a statistical error estimate given by the standard deviation discussed earlier. In the discussion here we will treat \( y_i \) as our exact value for the response variable.
|
||||
|
||||
<p>
|
||||
Introducing the standard deviation \( \sigma_i \) for each measurement \( y_i \), we define now the \( \chi^2 \) function as
|
||||
$$
|
||||
\chi^2(\hat{\beta})=\sum_{i=0}^{n-1}\frac{\left(y_i-\tilde{y}_i\right)^2}{\sigma_i^2}=\left(\hat{y}-\hat{\tilde{y}}\right)^T\frac{1}{\hat{\Sigma^2}}\left(\hat{y}-\hat{\tilde{y}}\right),
|
||||
$$
|
||||
|
||||
where the matrix \( \hat{\Sigma} \) is a diagonal matrix with \( \sigma_i \) as matrix elements.
|
||||
|
||||
|
||||
</div>
|
||||
|
||||
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec12">The \( \chi^2 \) function </h2>
|
||||
<div class="alert alert-block alert-block alert-text-normal">
|
||||
<b></b>
|
||||
<p>
|
||||
|
||||
<p>
|
||||
In order to find the parameters \( \beta_i \) we will then minimize the spread of \( \chi^2(\hat{\beta}) \) by requiring
|
||||
$$
|
||||
\frac{\partial \chi^2(\hat{\beta})}{\partial \beta_j} = \frac{\partial }{\partial \beta_j}\left[ \sum_{i=0}^{n-1}\left(\frac{y_i-\beta_0x_{i,0}-\beta_1x_{i,1}-\beta_2x_{i,2}-\dots-\beta_{n-1}x_{i,n-1}}{\sigma_i}\right)^2\right]=0,
|
||||
$$
|
||||
|
||||
which results in
|
||||
$$
|
||||
\frac{\partial \chi^2(\hat{\beta})}{\partial \beta_j} = -2\left[ \sum_{i=0}^{n-1}\frac{x_{ij}}{\sigma_i}\left(\frac{y_i-\beta_0x_{i,0}-\beta_1x_{i,1}-\beta_2x_{i,2}-\dots-\beta_{n-1}x_{i,n-1}}{\sigma_i}\right)\right]=0,
|
||||
$$
|
||||
|
||||
or in a matrix-vector form as
|
||||
$$
|
||||
\frac{\partial \chi^2(\hat{\beta})}{\partial \hat{\beta}} = 0 = \hat{A}^T\left( \hat{b}-\hat{A}\hat{\beta}\right).
|
||||
$$
|
||||
|
||||
where we have defined the matrix \( \hat{A} =\hat{X}/\hat{\Sigma} \) with matrix elements \( a_{ij} = x_{ij}/\sigma_i \) and the vector \( \hat{b} \) with elements \( b_i = y_i/\sigma_i \).
|
||||
</div>
|
||||
|
||||
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec13">The \( \chi^2 \) function </h2>
|
||||
<div class="alert alert-block alert-block alert-text-normal">
|
||||
<b></b>
|
||||
<p>
|
||||
|
||||
<p>
|
||||
We can rewrite
|
||||
$$
|
||||
\frac{\partial \chi^2(\hat{\beta})}{\partial \hat{\beta}} = 0 = \hat{A}^T\left( \hat{b}-\hat{A}\hat{\beta}\right),
|
||||
$$
|
||||
|
||||
as
|
||||
$$
|
||||
\hat{A}^T\hat{b} = \hat{A}^T\hat{A}\hat{\beta},
|
||||
$$
|
||||
|
||||
and if the matrix \( \hat{A}^T\hat{A} \) is invertible we have the solution
|
||||
$$
|
||||
\hat{\beta} =\left(\hat{A}^T\hat{A}\right)^{-1}\hat{A}^T\hat{b}.
|
||||
$$
|
||||
</div>
|
||||
|
||||
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec14">The \( \chi^2 \) function </h2>
|
||||
<div class="alert alert-block alert-block alert-text-normal">
|
||||
<b></b>
|
||||
<p>
|
||||
|
||||
<p>
|
||||
If we then introduce the matrix
|
||||
$$
|
||||
\hat{H} = \hat{A}^T\hat{A},
|
||||
$$
|
||||
|
||||
we have then the following expression for the parameters \( \beta_j \) (the matrix elements of \( \hat{H} \) are \( h_{ij} \))
|
||||
$$
|
||||
\beta_j = \sum_{k=0}^{p-1}h_{jk}\sum_{i=0}^{n-1}\frac{y_i}{\sigma_i}\frac{x_{ik}}{\sigma_i} = \sum_{k=0}^{p-1}h_{jk}\sum_{i=0}^{n-1}b_ia_{ik}
|
||||
$$
|
||||
|
||||
We state without proof the expression for the uncertainty in the parameters \( \beta_j \) as
|
||||
$$
|
||||
\sigma^2(\beta_j) = \sum_{i=0}^{n-1}\sigma_i^2\left( \frac{\partial \beta_j}{\partial y_i}\right)^2,
|
||||
$$
|
||||
|
||||
resulting in
|
||||
$$
|
||||
\sigma^2(\beta_j) = \left(\sum_{k=0}^{p-1}h_{jk}\sum_{i=0}^{n-1}a_{ik}\right)\left(\sum_{l=0}^{p-1}h_{jl}\sum_{m=0}^{n-1}a_{ml}\right) = h_{jj}!
|
||||
$$
|
||||
</div>
|
||||
|
||||
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec15">The \( \chi^2 \) function </h2>
|
||||
<div class="alert alert-block alert-block alert-text-normal">
|
||||
<b></b>
|
||||
<p>
|
||||
The first step here is to approximate the function \( y \) with a first-order polynomial, that is we write
|
||||
$$
|
||||
y=y(x) \rightarrow y(x_i) \approx \beta_0+\beta_1 x_i.
|
||||
$$
|
||||
|
||||
By computing the derivatives of \( \chi^2 \) with respect to \( \beta_0 \) and \( \beta_1 \) show that these are given by
|
||||
$$
|
||||
\frac{\partial \chi^2(\hat{\beta})}{\partial \beta_0} = -2\left[ \sum_{i=0}^{1}\left(\frac{y_i-\beta_0-\beta_1x_{i}}{\sigma_i^2}\right)\right]=0,
|
||||
$$
|
||||
|
||||
and
|
||||
$$
|
||||
\frac{\partial \chi^2(\hat{\beta})}{\partial \beta_0} = -2\left[ \sum_{i=0}^{1}x_i\left(\frac{y_i-\beta_0-\beta_1x_{i}}{\sigma_i^2}\right)\right]=0.
|
||||
$$
|
||||
</div>
|
||||
|
||||
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec16">The \( \chi^2 \) function </h2>
|
||||
<div class="alert alert-block alert-block alert-text-normal">
|
||||
<b></b>
|
||||
<p>
|
||||
|
||||
<p>
|
||||
We define then
|
||||
$$
|
||||
\gamma = \sum_{i=0}^{1}\frac{1}{\sigma_i^2},
|
||||
$$
|
||||
|
||||
|
||||
$$
|
||||
\gamma_x = \sum_{i=0}^{1}\frac{x_{i}}{\sigma_i^2},
|
||||
$$
|
||||
|
||||
$$
|
||||
\gamma_y = \sum_{i=0}^{1}\left(\frac{y_i}{\sigma_i^2}\right),
|
||||
$$
|
||||
|
||||
$$
|
||||
\gamma_{xx} = \sum_{i=0}^{1}\frac{x_ix_{i}}{\sigma_i^2},
|
||||
$$
|
||||
|
||||
$$
|
||||
\gamma_{xy} = \sum_{i=0}^{1}\frac{y_ix_{i}}{\sigma_i^2},
|
||||
$$
|
||||
|
||||
and show that
|
||||
$$
|
||||
\beta_0 = \frac{\gamma_{xx}\gamma_y-\gamma_x\gamma_y}{\gamma\gamma_{xx}-\gamma_x^2},
|
||||
$$
|
||||
|
||||
$$
|
||||
\beta_1 = \frac{\gamma_{xy}\gamma-\gamma_x\gamma_y}{\gamma\gamma_{xx}-\gamma_x^2}.
|
||||
$$
|
||||
|
||||
<p>
|
||||
The LSM suffers often from both being underdetermined and overdetermined in the unknown coefficients \( \beta_i \). A better approach is to use the Singular Value Decomposition (SVD) method discussed below.
|
||||
</div>
|
||||
|
||||
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec17">The singular value decompostion </h2>
|
||||
<div class="alert alert-block alert-block alert-text-normal">
|
||||
<b></b>
|
||||
<p>
|
||||
|
||||
@@ -97,7 +97,13 @@ div { text-align: justify; text-justify: inter-word; }
|
||||
2,
|
||||
None,
|
||||
'___sec10'),
|
||||
('The singular value decompostion', 2, None, '___sec11')]}
|
||||
('The $\\chi^2$ function', 2, None, '___sec11'),
|
||||
('The $\\chi^2$ function', 2, None, '___sec12'),
|
||||
('The $\\chi^2$ function', 2, None, '___sec13'),
|
||||
('The $\\chi^2$ function', 2, None, '___sec14'),
|
||||
('The $\\chi^2$ function', 2, None, '___sec15'),
|
||||
('The $\\chi^2$ function', 2, None, '___sec16'),
|
||||
('The singular value decompostion', 2, None, '___sec17')]}
|
||||
end of tocinfo -->
|
||||
|
||||
<body>
|
||||
@@ -112,8 +118,8 @@ MathJax.Hub.Config({
|
||||
}
|
||||
});
|
||||
</script>
|
||||
<script type="text/javascript" async
|
||||
src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS-MML_HTMLorMML">
|
||||
<script type="text/javascript"
|
||||
src="http://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML">
|
||||
</script>
|
||||
|
||||
|
||||
@@ -139,7 +145,7 @@ MathJax.Hub.Config({
|
||||
<center>[2] <b>Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University</b></center>
|
||||
<br>
|
||||
<p>
|
||||
<center><h4>Oct 18, 2017</h4></center> <!-- date -->
|
||||
<center><h4>Oct 24, 2017</h4></center> <!-- date -->
|
||||
<br>
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
@@ -365,12 +371,12 @@ where \( \langle y_i \rangle \) is the mean value. Keep in mind also that till n
|
||||
<p>
|
||||
In order to find the parameters \( \beta_i \) we will then minimize the spread of \( Q(\hat{\beta}) \) by requiring
|
||||
$$
|
||||
\frac{\partial Q(\hat{\beta})}{\partial \beta_j} = \frac{\partial }{\partial \beta_j}\left[ \sum_{i=0}^{n-1}\left(y_i-\beta_0x_{i,0}+\beta_1x_{i,1}+\beta_2x_{i,2}+\dots+\beta_{n-1}x_{i,n-1}\right)^2\right]=0,
|
||||
\frac{\partial Q(\hat{\beta})}{\partial \beta_j} = \frac{\partial }{\partial \beta_j}\left[ \sum_{i=0}^{n-1}\left(y_i-\beta_0x_{i,0}-\beta_1x_{i,1}-\beta_2x_{i,2}-\dots-\beta_{n-1}x_{i,n-1}\right)^2\right]=0,
|
||||
$$
|
||||
|
||||
which results in
|
||||
$$
|
||||
\frac{\partial Q(\hat{\beta})}{\partial \beta_j} = -2\left[ \sum_{i=0}^{n-1}x_{ij}\left(y_i-\beta_0x_{i,0}+\beta_1x_{i,1}+\beta_2x_{i,2}+\dots+\beta_{n-1}x_{i,n-1}\right)\right]=0,
|
||||
\frac{\partial Q(\hat{\beta})}{\partial \beta_j} = -2\left[ \sum_{i=0}^{n-1}x_{ij}\left(y_i-\beta_0x_{i,0}-\beta_1x_{i,1}-\beta_2x_{i,2}-\dots-\beta_{n-1}x_{i,n-1}\right)\right]=0,
|
||||
$$
|
||||
|
||||
or in a matrix-vector form as
|
||||
@@ -439,7 +445,184 @@ meaning that the solution for \( \hat{\beta} \) is the one which minimizes the r
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec11">The singular value decompostion </h2>
|
||||
<h2 id="___sec11">The \( \chi^2 \) function </h2>
|
||||
<div class="alert alert-block alert-block alert-text-normal">
|
||||
<b></b>
|
||||
<p>
|
||||
|
||||
<p>
|
||||
Normally, the response (dependent or outcome) variable \( y_i \) the outcome of a numerical experiment or another type of experiment and is thus only an approximation to the true value. It is then always accompanied by an error estimate, often limited to a statistical error estimate given by the standard deviation discussed earlier. In the discussion here we will treat \( y_i \) as our exact value for the response variable.
|
||||
|
||||
<p>
|
||||
Introducing the standard deviation \( \sigma_i \) for each measurement \( y_i \), we define now the \( \chi^2 \) function as
|
||||
$$
|
||||
\chi^2(\hat{\beta})=\sum_{i=0}^{n-1}\frac{\left(y_i-\tilde{y}_i\right)^2}{\sigma_i^2}=\left(\hat{y}-\hat{\tilde{y}}\right)^T\frac{1}{\hat{\Sigma^2}}\left(\hat{y}-\hat{\tilde{y}}\right),
|
||||
$$
|
||||
|
||||
where the matrix \( \hat{\Sigma} \) is a diagonal matrix with \( \sigma_i \) as matrix elements.
|
||||
|
||||
|
||||
</div>
|
||||
|
||||
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec12">The \( \chi^2 \) function </h2>
|
||||
<div class="alert alert-block alert-block alert-text-normal">
|
||||
<b></b>
|
||||
<p>
|
||||
|
||||
<p>
|
||||
In order to find the parameters \( \beta_i \) we will then minimize the spread of \( \chi^2(\hat{\beta}) \) by requiring
|
||||
$$
|
||||
\frac{\partial \chi^2(\hat{\beta})}{\partial \beta_j} = \frac{\partial }{\partial \beta_j}\left[ \sum_{i=0}^{n-1}\left(\frac{y_i-\beta_0x_{i,0}-\beta_1x_{i,1}-\beta_2x_{i,2}-\dots-\beta_{n-1}x_{i,n-1}}{\sigma_i}\right)^2\right]=0,
|
||||
$$
|
||||
|
||||
which results in
|
||||
$$
|
||||
\frac{\partial \chi^2(\hat{\beta})}{\partial \beta_j} = -2\left[ \sum_{i=0}^{n-1}\frac{x_{ij}}{\sigma_i}\left(\frac{y_i-\beta_0x_{i,0}-\beta_1x_{i,1}-\beta_2x_{i,2}-\dots-\beta_{n-1}x_{i,n-1}}{\sigma_i}\right)\right]=0,
|
||||
$$
|
||||
|
||||
or in a matrix-vector form as
|
||||
$$
|
||||
\frac{\partial \chi^2(\hat{\beta})}{\partial \hat{\beta}} = 0 = \hat{A}^T\left( \hat{b}-\hat{A}\hat{\beta}\right).
|
||||
$$
|
||||
|
||||
where we have defined the matrix \( \hat{A} =\hat{X}/\hat{\Sigma} \) with matrix elements \( a_{ij} = x_{ij}/\sigma_i \) and the vector \( \hat{b} \) with elements \( b_i = y_i/\sigma_i \).
|
||||
</div>
|
||||
|
||||
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec13">The \( \chi^2 \) function </h2>
|
||||
<div class="alert alert-block alert-block alert-text-normal">
|
||||
<b></b>
|
||||
<p>
|
||||
|
||||
<p>
|
||||
We can rewrite
|
||||
$$
|
||||
\frac{\partial \chi^2(\hat{\beta})}{\partial \hat{\beta}} = 0 = \hat{A}^T\left( \hat{b}-\hat{A}\hat{\beta}\right),
|
||||
$$
|
||||
|
||||
as
|
||||
$$
|
||||
\hat{A}^T\hat{b} = \hat{A}^T\hat{A}\hat{\beta},
|
||||
$$
|
||||
|
||||
and if the matrix \( \hat{A}^T\hat{A} \) is invertible we have the solution
|
||||
$$
|
||||
\hat{\beta} =\left(\hat{A}^T\hat{A}\right)^{-1}\hat{A}^T\hat{b}.
|
||||
$$
|
||||
</div>
|
||||
|
||||
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec14">The \( \chi^2 \) function </h2>
|
||||
<div class="alert alert-block alert-block alert-text-normal">
|
||||
<b></b>
|
||||
<p>
|
||||
|
||||
<p>
|
||||
If we then introduce the matrix
|
||||
$$
|
||||
\hat{H} = \hat{A}^T\hat{A},
|
||||
$$
|
||||
|
||||
we have then the following expression for the parameters \( \beta_j \) (the matrix elements of \( \hat{H} \) are \( h_{ij} \))
|
||||
$$
|
||||
\beta_j = \sum_{k=0}^{p-1}h_{jk}\sum_{i=0}^{n-1}\frac{y_i}{\sigma_i}\frac{x_{ik}}{\sigma_i} = \sum_{k=0}^{p-1}h_{jk}\sum_{i=0}^{n-1}b_ia_{ik}
|
||||
$$
|
||||
|
||||
We state without proof the expression for the uncertainty in the parameters \( \beta_j \) as
|
||||
$$
|
||||
\sigma^2(\beta_j) = \sum_{i=0}^{n-1}\sigma_i^2\left( \frac{\partial \beta_j}{\partial y_i}\right)^2,
|
||||
$$
|
||||
|
||||
resulting in
|
||||
$$
|
||||
\sigma^2(\beta_j) = \left(\sum_{k=0}^{p-1}h_{jk}\sum_{i=0}^{n-1}a_{ik}\right)\left(\sum_{l=0}^{p-1}h_{jl}\sum_{m=0}^{n-1}a_{ml}\right) = h_{jj}!
|
||||
$$
|
||||
</div>
|
||||
|
||||
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec15">The \( \chi^2 \) function </h2>
|
||||
<div class="alert alert-block alert-block alert-text-normal">
|
||||
<b></b>
|
||||
<p>
|
||||
The first step here is to approximate the function \( y \) with a first-order polynomial, that is we write
|
||||
$$
|
||||
y=y(x) \rightarrow y(x_i) \approx \beta_0+\beta_1 x_i.
|
||||
$$
|
||||
|
||||
By computing the derivatives of \( \chi^2 \) with respect to \( \beta_0 \) and \( \beta_1 \) show that these are given by
|
||||
$$
|
||||
\frac{\partial \chi^2(\hat{\beta})}{\partial \beta_0} = -2\left[ \sum_{i=0}^{1}\left(\frac{y_i-\beta_0-\beta_1x_{i}}{\sigma_i^2}\right)\right]=0,
|
||||
$$
|
||||
|
||||
and
|
||||
$$
|
||||
\frac{\partial \chi^2(\hat{\beta})}{\partial \beta_0} = -2\left[ \sum_{i=0}^{1}x_i\left(\frac{y_i-\beta_0-\beta_1x_{i}}{\sigma_i^2}\right)\right]=0.
|
||||
$$
|
||||
</div>
|
||||
|
||||
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec16">The \( \chi^2 \) function </h2>
|
||||
<div class="alert alert-block alert-block alert-text-normal">
|
||||
<b></b>
|
||||
<p>
|
||||
|
||||
<p>
|
||||
We define then
|
||||
$$
|
||||
\gamma = \sum_{i=0}^{1}\frac{1}{\sigma_i^2},
|
||||
$$
|
||||
|
||||
|
||||
$$
|
||||
\gamma_x = \sum_{i=0}^{1}\frac{x_{i}}{\sigma_i^2},
|
||||
$$
|
||||
|
||||
$$
|
||||
\gamma_y = \sum_{i=0}^{1}\left(\frac{y_i}{\sigma_i^2}\right),
|
||||
$$
|
||||
|
||||
$$
|
||||
\gamma_{xx} = \sum_{i=0}^{1}\frac{x_ix_{i}}{\sigma_i^2},
|
||||
$$
|
||||
|
||||
$$
|
||||
\gamma_{xy} = \sum_{i=0}^{1}\frac{y_ix_{i}}{\sigma_i^2},
|
||||
$$
|
||||
|
||||
and show that
|
||||
$$
|
||||
\beta_0 = \frac{\gamma_{xx}\gamma_y-\gamma_x\gamma_y}{\gamma\gamma_{xx}-\gamma_x^2},
|
||||
$$
|
||||
|
||||
$$
|
||||
\beta_1 = \frac{\gamma_{xy}\gamma-\gamma_x\gamma_y}{\gamma\gamma_{xx}-\gamma_x^2}.
|
||||
$$
|
||||
|
||||
<p>
|
||||
The LSM suffers often from both being underdetermined and overdetermined in the unknown coefficients \( \beta_i \). A better approach is to use the Singular Value Decomposition (SVD) method discussed below.
|
||||
</div>
|
||||
|
||||
|
||||
<p>
|
||||
<!-- !split --><br><br><br><br><br><br><br><br><br><br>
|
||||
|
||||
<h2 id="___sec17">The singular value decompostion </h2>
|
||||
<div class="alert alert-block alert-block alert-text-normal">
|
||||
<b></b>
|
||||
<p>
|
||||
|
||||
Binary file not shown.
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@@ -198,13 +198,13 @@ where $\langle y_i \rangle$ is the mean value. Keep in mind also that till now
|
||||
In order to find the parameters $\beta_i$ we will then minimize the spread of $Q(\hat{\beta})$ by requiring
|
||||
!bt
|
||||
\[
|
||||
\frac{\partial Q(\hat{\beta})}{\partial \beta_j} = \frac{\partial }{\partial \beta_j}\left[ \sum_{i=0}^{n-1}\left(y_i-\beta_0x_{i,0}+\beta_1x_{i,1}+\beta_2x_{i,2}+\dots+\beta_{n-1}x_{i,n-1}\right)^2\right]=0,
|
||||
\frac{\partial Q(\hat{\beta})}{\partial \beta_j} = \frac{\partial }{\partial \beta_j}\left[ \sum_{i=0}^{n-1}\left(y_i-\beta_0x_{i,0}-\beta_1x_{i,1}-\beta_2x_{i,2}-\dots-\beta_{n-1}x_{i,n-1}\right)^2\right]=0,
|
||||
\]
|
||||
!et
|
||||
which results in
|
||||
!bt
|
||||
\[
|
||||
\frac{\partial Q(\hat{\beta})}{\partial \beta_j} = -2\left[ \sum_{i=0}^{n-1}x_{ij}\left(y_i-\beta_0x_{i,0}+\beta_1x_{i,1}+\beta_2x_{i,2}+\dots+\beta_{n-1}x_{i,n-1}\right)\right]=0,
|
||||
\frac{\partial Q(\hat{\beta})}{\partial \beta_j} = -2\left[ \sum_{i=0}^{n-1}x_{ij}\left(y_i-\beta_0x_{i,0}-\beta_1x_{i,1}-\beta_2x_{i,2}-\dots-\beta_{n-1}x_{i,n-1}\right)\right]=0,
|
||||
\]
|
||||
!et
|
||||
or in a matrix-vector form as
|
||||
@@ -267,8 +267,169 @@ meaning that the solution for $\hat{\beta}$ is the one which minimizes the resid
|
||||
|
||||
!eblock
|
||||
|
||||
!split
|
||||
===== The $\chi^2$ function =====
|
||||
!bblock
|
||||
|
||||
Normally, the response (dependent or outcome) variable $y_i$ the outcome of a numerical experiment or another type of experiment and is thus only an approximation to the true value. It is then always accompanied by an error estimate, often limited to a statistical error estimate given by the standard deviation discussed earlier. In the discussion here we will treat $y_i$ as our exact value for the response variable.
|
||||
|
||||
Introducing the standard deviation $\sigma_i$ for each measurement $y_i$, we define now the $\chi^2$ function as
|
||||
!bt
|
||||
\[
|
||||
\chi^2(\hat{\beta})=\sum_{i=0}^{n-1}\frac{\left(y_i-\tilde{y}_i\right)^2}{\sigma_i^2}=\left(\hat{y}-\hat{\tilde{y}}\right)^T\frac{1}{\hat{\Sigma^2}}\left(\hat{y}-\hat{\tilde{y}}\right),
|
||||
\]
|
||||
!et
|
||||
where the matrix $\hat{\Sigma}$ is a diagonal matrix with $\sigma_i$ as matrix elements.
|
||||
|
||||
!eblock
|
||||
|
||||
!split
|
||||
===== The $\chi^2$ function =====
|
||||
!bblock
|
||||
|
||||
In order to find the parameters $\beta_i$ we will then minimize the spread of $\chi^2(\hat{\beta})$ by requiring
|
||||
!bt
|
||||
\[
|
||||
\frac{\partial \chi^2(\hat{\beta})}{\partial \beta_j} = \frac{\partial }{\partial \beta_j}\left[ \sum_{i=0}^{n-1}\left(\frac{y_i-\beta_0x_{i,0}-\beta_1x_{i,1}-\beta_2x_{i,2}-\dots-\beta_{n-1}x_{i,n-1}}{\sigma_i}\right)^2\right]=0,
|
||||
\]
|
||||
!et
|
||||
which results in
|
||||
!bt
|
||||
\[
|
||||
\frac{\partial \chi^2(\hat{\beta})}{\partial \beta_j} = -2\left[ \sum_{i=0}^{n-1}\frac{x_{ij}}{\sigma_i}\left(\frac{y_i-\beta_0x_{i,0}-\beta_1x_{i,1}-\beta_2x_{i,2}-\dots-\beta_{n-1}x_{i,n-1}}{\sigma_i}\right)\right]=0,
|
||||
\]
|
||||
!et
|
||||
or in a matrix-vector form as
|
||||
!bt
|
||||
\[
|
||||
\frac{\partial \chi^2(\hat{\beta})}{\partial \hat{\beta}} = 0 = \hat{A}^T\left( \hat{b}-\hat{A}\hat{\beta}\right).
|
||||
\]
|
||||
!et
|
||||
where we have defined the matrix $\hat{A} =\hat{X}/\hat{\Sigma}$ with matrix elements $a_{ij} = x_{ij}/\sigma_i$ and the vector $\hat{b}$ with elements $b_i = y_i/\sigma_i$.
|
||||
!eblock
|
||||
|
||||
!split
|
||||
===== The $\chi^2$ function =====
|
||||
!bblock
|
||||
|
||||
We can rewrite
|
||||
!bt
|
||||
\[
|
||||
\frac{\partial \chi^2(\hat{\beta})}{\partial \hat{\beta}} = 0 = \hat{A}^T\left( \hat{b}-\hat{A}\hat{\beta}\right),
|
||||
\]
|
||||
!et
|
||||
as
|
||||
!bt
|
||||
\[
|
||||
\hat{A}^T\hat{b} = \hat{A}^T\hat{A}\hat{\beta},
|
||||
\]
|
||||
!et
|
||||
and if the matrix $\hat{A}^T\hat{A}$ is invertible we have the solution
|
||||
!bt
|
||||
\[
|
||||
\hat{\beta} =\left(\hat{A}^T\hat{A}\right)^{-1}\hat{A}^T\hat{b}.
|
||||
\]
|
||||
!et
|
||||
!eblock
|
||||
|
||||
!split
|
||||
===== The $\chi^2$ function =====
|
||||
!bblock
|
||||
|
||||
If we then introduce the matrix
|
||||
!bt
|
||||
\[
|
||||
\hat{H} = \hat{A}^T\hat{A},
|
||||
\]
|
||||
!et
|
||||
we have then the following expression for the parameters $\beta_j$ (the matrix elements of $\hat{H}$ are $h_{ij}$)
|
||||
!bt
|
||||
\[
|
||||
\beta_j = \sum_{k=0}^{p-1}h_{jk}\sum_{i=0}^{n-1}\frac{y_i}{\sigma_i}\frac{x_{ik}}{\sigma_i} = \sum_{k=0}^{p-1}h_{jk}\sum_{i=0}^{n-1}b_ia_{ik}
|
||||
\]
|
||||
!et
|
||||
We state without proof the expression for the uncertainty in the parameters $\beta_j$ as
|
||||
!bt
|
||||
\[
|
||||
\sigma^2(\beta_j) = \sum_{i=0}^{n-1}\sigma_i^2\left( \frac{\partial \beta_j}{\partial y_i}\right)^2,
|
||||
\]
|
||||
!et
|
||||
resulting in
|
||||
!bt
|
||||
\[
|
||||
\sigma^2(\beta_j) = \left(\sum_{k=0}^{p-1}h_{jk}\sum_{i=0}^{n-1}a_{ik}\right)\left(\sum_{l=0}^{p-1}h_{jl}\sum_{m=0}^{n-1}a_{ml}\right) = h_{jj}!
|
||||
\]
|
||||
!et
|
||||
!eblock
|
||||
|
||||
!split
|
||||
===== The $\chi^2$ function =====
|
||||
!bblock
|
||||
The first step here is to approximate the function $y$ with a first-order polynomial, that is we write
|
||||
!bt
|
||||
\[
|
||||
y=y(x) \rightarrow y(x_i) \approx \beta_0+\beta_1 x_i.
|
||||
\]
|
||||
!et
|
||||
By computing the derivatives of $\chi^2$ with respect to $\beta_0$ and $\beta_1$ show that these are given by
|
||||
!bt
|
||||
\[
|
||||
\frac{\partial \chi^2(\hat{\beta})}{\partial \beta_0} = -2\left[ \sum_{i=0}^{1}\left(\frac{y_i-\beta_0-\beta_1x_{i}}{\sigma_i^2}\right)\right]=0,
|
||||
\]
|
||||
!et
|
||||
and
|
||||
!bt
|
||||
\[
|
||||
\frac{\partial \chi^2(\hat{\beta})}{\partial \beta_0} = -2\left[ \sum_{i=0}^{1}x_i\left(\frac{y_i-\beta_0-\beta_1x_{i}}{\sigma_i^2}\right)\right]=0.
|
||||
\]
|
||||
!et
|
||||
!eblock
|
||||
|
||||
!split
|
||||
===== The $\chi^2$ function =====
|
||||
!bblock
|
||||
|
||||
We define then
|
||||
!bt
|
||||
\[
|
||||
\gamma = \sum_{i=0}^{1}\frac{1}{\sigma_i^2},
|
||||
\]
|
||||
!et
|
||||
|
||||
!bt
|
||||
\[
|
||||
\gamma_x = \sum_{i=0}^{1}\frac{x_{i}}{\sigma_i^2},
|
||||
\]
|
||||
!et
|
||||
!bt
|
||||
\[
|
||||
\gamma_y = \sum_{i=0}^{1}\left(\frac{y_i}{\sigma_i^2}\right),
|
||||
\]
|
||||
!et
|
||||
!bt
|
||||
\[
|
||||
\gamma_{xx} = \sum_{i=0}^{1}\frac{x_ix_{i}}{\sigma_i^2},
|
||||
\]
|
||||
!et
|
||||
!bt
|
||||
\[
|
||||
\gamma_{xy} = \sum_{i=0}^{1}\frac{y_ix_{i}}{\sigma_i^2},
|
||||
\]
|
||||
!et
|
||||
and show that
|
||||
!bt
|
||||
\[
|
||||
\beta_0 = \frac{\gamma_{xx}\gamma_y-\gamma_x\gamma_y}{\gamma\gamma_{xx}-\gamma_x^2},
|
||||
\]
|
||||
!et
|
||||
!bt
|
||||
\[
|
||||
\beta_1 = \frac{\gamma_{xy}\gamma-\gamma_x\gamma_y}{\gamma\gamma_{xx}-\gamma_x^2}.
|
||||
\]
|
||||
!et
|
||||
|
||||
The LSM suffers often from both being underdetermined and overdetermined in the unknown coefficients $\beta_i$. A better approach is to use the Singular Value Decomposition (SVD) method discussed below.
|
||||
!eblock
|
||||
|
||||
|
||||
|
||||
@@ -287,13 +448,7 @@ How can we use the singular value decomposition to find the parameters $\beta_j$
|
||||
|
||||
|
||||
|
||||
Suppose M is a m × n matrix whose entries come from the field K, which is either the field of real numbers or the field of complex numbers. Then there exists a factorization, called a singular value decomposition of M, of the form
|
||||
|
||||
{\displaystyle \mathbf {M} =\mathbf {U} {\boldsymbol {\Sigma }}\mathbf {V} ^{*}} \mathbf {M} =\mathbf {U} {\boldsymbol {\Sigma }}\mathbf {V} ^{*}
|
||||
where
|
||||
|
||||
U is an m × m unitary matrix (if K = {\displaystyle \mathbb {R} } \mathbb {R} , unitary matrices are orthogonal matrices),
|
||||
Σ is a diagonal m × n matrix with non-negative real numbers on the diagonal,
|
||||
V is an n × n unitary matrix over K, and
|
||||
V∗ is the conjugate transpose of V.
|
||||
The diagonal entries σi of Σ are known as the singular values of M. A common convention is to list the singular values in descending order. In this case, the diagonal matrix, Σ, is uniquely determined by M (though not the matrices U and V, see below).
|
||||
|
||||
|
||||
|
||||
Reference in New Issue
Block a user