From 5cd9a2f27968fcc495ec0e094ae99219034acf49 Mon Sep 17 00:00:00 2001 From: Morten Hjorth-Jensen Date: Tue, 17 Oct 2017 23:35:46 +0200 Subject: [PATCH] Updated regression analysis, sad to come --- .../Regression/html/._Regression-bs000.html | 23 +- .../Regression/html/._Regression-bs001.html | 21 +- .../Regression/html/._Regression-bs002.html | 21 +- .../Regression/html/._Regression-bs003.html | 209 +++++++++++++++++ .../Regression/html/._Regression-bs004.html | 201 ++++++++++++++++ .../Regression/html/._Regression-bs005.html | 196 ++++++++++++++++ .../Regression/html/._Regression-bs006.html | 198 ++++++++++++++++ .../Regression/html/._Regression-bs007.html | 202 +++++++++++++++++ .../Regression/html/._Regression-bs008.html | 170 ++++++++++++++ doc/pub/Regression/html/Regression-bs.html | 23 +- .../Regression/html/Regression-reveal.html | 214 +++++++++++++----- .../Regression/html/Regression-solarized.html | 200 +++++++++++----- doc/pub/Regression/html/Regression.html | 200 +++++++++++----- .../ipynb/ipynb-Regression-src.tar.gz | Bin 212 -> 215 bytes .../pdf/Regression-beamer-handouts2x3.pdf | Bin 154215 -> 162715 bytes doc/pub/Regression/pdf/Regression-beamer.pdf | Bin 154007 -> 159689 bytes doc/pub/Regression/pdf/Regression-minted.pdf | Bin 200842 -> 207106 bytes doc/src/Regression/Regression.do.txt | 73 +++++- 18 files changed, 1774 insertions(+), 177 deletions(-) create mode 100644 doc/pub/Regression/html/._Regression-bs003.html create mode 100644 doc/pub/Regression/html/._Regression-bs004.html create mode 100644 doc/pub/Regression/html/._Regression-bs005.html create mode 100644 doc/pub/Regression/html/._Regression-bs006.html create mode 100644 doc/pub/Regression/html/._Regression-bs007.html create mode 100644 doc/pub/Regression/html/._Regression-bs008.html diff --git a/doc/pub/Regression/html/._Regression-bs000.html b/doc/pub/Regression/html/._Regression-bs000.html index 476aaadbb..8fe63757f 100644 --- a/doc/pub/Regression/html/._Regression-bs000.html +++ b/doc/pub/Regression/html/._Regression-bs000.html @@ -50,7 +50,16 @@ Automatically generated HTML file from DocOnce source 2, None, '___sec3'), - ('Optimizing our parameters', 2, None, '___sec4')]} + ('Optimizing our parameters', 2, None, '___sec4'), + ('Interpretations and optimizing our parameters', + 2, + None, + '___sec5'), + ('Interpretations and optimizing our parameters', + 2, + None, + '___sec6'), + ('The singular value decompostion', 2, None, '___sec7')]} end of tocinfo --> @@ -65,8 +74,8 @@ MathJax.Hub.Config({ } }); - @@ -93,6 +102,9 @@ MathJax.Hub.Config({
  • Rewriting the fitting procedure as a linear algebra problem
  • Generalizing the fitting procedure as a linear algebra problem
  • Optimizing our parameters
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  • Interpretations and optimizing our parameters
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  • Interpretations and optimizing our parameters
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  • The singular value decompostion
  • @@ -127,7 +139,7 @@ MathJax.Hub.Config({
    [2] Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University

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    Oct 14, 2017

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    Oct 17, 2017


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  • Rewriting the fitting procedure as a linear algebra problem
  • Generalizing the fitting procedure as a linear algebra problem
  • Optimizing our parameters
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  • Interpretations and optimizing our parameters
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  • Interpretations and optimizing our parameters
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  • The singular value decompostion
  • @@ -142,6 +154,9 @@ A regression model aims at finding a likelihood function \( p(y\vert \hat{x}) \)
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  • Rewriting the fitting procedure as a linear algebra problem
  • Generalizing the fitting procedure as a linear algebra problem
  • Optimizing our parameters
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  • Interpretations and optimizing our parameters
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  • Interpretations and optimizing our parameters
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  • The singular value decompostion
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    + +

     

     

     

    + + + + +

    Rewriting the fitting procedure as a linear algebra problem

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    +For every set of values \( y_i,x_i \) we have thus the corresponding set of equations +$$ +\begin{align*} +y_0&=\beta_0+\beta_1x_0^1+\beta_2x_0^2+\dots+\beta_{n-1}x_0^{n-1}+\epsilon_0\\ +y_1&=\beta_0+\beta_1x_1^1+\beta_2x_1^2+\dots+\beta_{n-1}x_1^{n-1}+\epsilon_1\\ +y_2&=\beta_0+\beta_1x_2^1+\beta_2x_2^2+\dots+\beta_{n-1}x_2^{n-1}+\epsilon_2\\ +\dots & \dots \\ +y_{n-1}&=\beta_0+\beta_1x_{n-1}^1+\beta_2x_{n-1}^2+\dots+\beta_1x_{n-1}^{n-1}+\epsilon_{n-1}.\\ +\end{align*} +$$ + +Defining the vectors +$$ +\hat{y} = [y_0,y_1, y_2,\dots, y_{n-1}]^T, +$$ + +$$ +\hat{\beta} = [\beta_0,\beta_1, \beta_2,\dots, \beta_{n-1}]^T, +$$ + +$$ +\hat{\epsilon} = [\epsilon_0,\epsilon_1, \epsilon_2,\dots, \epsilon_{n-1}]^T, +$$ + +and the matrix +$$ +\hat{X}= +\begin{bmatrix} +1& x_{0}^1 &x_{0}^2& \dots & \dots &x_{0}^{n-1}\\ +1& x_{1}^1 &x_{1}^2& \dots & \dots &x_{1}^{n-1}\\ +1& x_{2}^1 &x_{2}^2& \dots & \dots &x_{2}^{n-1}\\ +\dots& \dots &\dots& \dots & \dots &\dots\\ +1& x_{n-1}^1 &x_{n-1}^2& \dots & \dots &x_{n-1}^{n-1}\\ +\end{bmatrix} +$$ + +we can rewrite our equations as +$$ +\hat{y} = \hat{X}\hat{\beta}+\hat{\epsilon}. +$$ +

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    + + + + + + diff --git a/doc/pub/Regression/html/._Regression-bs004.html b/doc/pub/Regression/html/._Regression-bs004.html new file mode 100644 index 000000000..4404cc416 --- /dev/null +++ b/doc/pub/Regression/html/._Regression-bs004.html @@ -0,0 +1,201 @@ + + + + + + + +Data Analysis and Machine Learning: Linear Regression and more Advanced Regression Analysis + + + + + + + + + + + + + + + + + + + + + + + + + + +
    + +

     

     

     

    + + + + +

    Generalizing the fitting procedure as a linear algebra problem

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    +We are obviously not limited to the above polynomial. We could replace the various powers of \( x \) with elements of Fourier series, that is, instead of \( x_i^j \) we could have \( \cos{(j x_i)} \) or \( \sin{(j x_i)} \), or time series or other orthogonal functions. +For every set of values \( y_i,x_i \) we can then generalize the equations to +$$ +\begin{align*} +y_0&=\beta_0x_{00}+\beta_1x_{01}+\beta_2x_{02}+\dots+\beta_{n-1}x_{0n-1}+\epsilon_0\\ +y_1&=\beta_0x_{10}+\beta_1x_{11}+\beta_2x_{12}+\dots+\beta_{n-1}x_{1n-1}+\epsilon_1\\ +y_2&=\beta_0x_{20}+\beta_1x_{21}+\beta_2x_{22}+\dots+\beta_{n-1}x_{2n-1}+\epsilon_2\\ +\dots & \dots \\ +y_{i}&=\beta_0x_{i0}+\beta_1x_{i1}+\beta_2x_{i2}+\dots+\beta_{n-1}x_{in-1}+\epsilon_i\\ +\dots & \dots \\ +y_{n-1}&=\beta_0x_{n-1,0}+\beta_1x_{n-1,2}+\beta_2x_{n-1,2}+\dots+\beta_1x_{n-1}^{n-1,n-1}+\epsilon_{n-1}.\\ +\end{align*} +$$ + +We redefine in turn the matrix \( \hat{X} \) as +$$ +\hat{X}= +\begin{bmatrix} +x_{00}& x_{01} &x_{02}& \dots & \dots &x_{0,n-1}\\ +x_{10}& x_{11} &x_{12}& \dots & \dots &x_{1,n-1}\\ +x_{20}& x_{21}^1 &x_{22}2& \dots & \dots &x_{2,n-1}\\ +\dots& \dots &\dots& \dots & \dots &\dots\\ +x_{n-1,00}& x_{n-1,1} &x_{n-1,2}& \dots & \dots &x_{n-1,n-1}\\ +\end{bmatrix} +$$ + +and without loss of generality we rewrite again our equations as +$$ +\hat{y} = \hat{X}\hat{\beta}+\hat{\epsilon}. +$$ + +The left-hand side of this equation forms know. Our error vector \( \hat{\epsilon} \) and the parameter vector \( \hat{\beta} \) are our unknow quantities. How can we obtain the optimal set of \( \beta_i \) values? +

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    + + + + + + diff --git a/doc/pub/Regression/html/._Regression-bs005.html b/doc/pub/Regression/html/._Regression-bs005.html new file mode 100644 index 000000000..a611e211b --- /dev/null +++ b/doc/pub/Regression/html/._Regression-bs005.html @@ -0,0 +1,196 @@ + + + + + + + +Data Analysis and Machine Learning: Linear Regression and more Advanced Regression Analysis + + + + + + + + + + + + + + + + + + + + + + + + + + +
    + +

     

     

     

    + + + + +

    Optimizing our parameters

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    +We have defined the matrix \( \hat{X} \) +$$ +\begin{align*} +y_0&=\beta_0x_{00}+\beta_1x_{01}+\beta_2x_{02}+\dots+\beta_{n-1}x_{0n-1}+\epsilon_0\\ +y_1&=\beta_0x_{10}+\beta_1x_{11}+\beta_2x_{12}+\dots+\beta_{n-1}x_{1n-1}+\epsilon_1\\ +y_2&=\beta_0x_{20}+\beta_1x_{21}+\beta_2x_{22}+\dots+\beta_{n-1}x_{2n-1}+\epsilon_1\\ +\dots & \dots \\ +y_{i}&=\beta_0x_{i0}+\beta_1x_{i1}+\beta_2x_{i2}+\dots+\beta_{n-1}x_{in-1}+\epsilon_1\\ +\dots & \dots \\ +y_{n-1}&=\beta_0x_{n-1,0}+\beta_1x_{n-1,2}+\beta_2x_{n-1,2}+\dots+\beta_1x_{n-1,n-1}+\epsilon_{n-1}.\\ +\end{align*} +$$ + +We well use this matrix to define the approximation \( \hat{\tilde{y}} \) via the unknown quantity \( \hat{\beta} \) as +$$ +\hat{\tilde{y}}= \hat{X}\hat{\beta}, +$$ + +and in order to find the optimal parameters \( \beta_i \) instead of solving the above linear algebra problem, we define a function which gives a measure of the spread between the values \( y_i \) (which represent hopefully the exact values) and the parametrized values \( \tilde{y}_i \), namely +$$ +Q(\hat{\beta})=\sum_{i=0}^{n-1}\left(y_i-\tilde{y}_i\right)^2=\left(\hat{y}-\hat{\tilde{y}}\right)^T\left(\hat{y}-\hat{\tilde{y}}\right), +$$ + +or using the matrix \( \hat{X} \) as +$$ +Q(\hat{\beta})=\left(\hat{y}-\hat{X}\hat{\beta}\right)^T\left(\hat{y}-\hat{X}\hat{\beta}\right). +$$ +

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    + +

     

     

     

    + + + + +

    Interpretations and optimizing our parameters

    +
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    +The function +$$ +Q(\hat{\beta})=\left(\hat{y}-\hat{X}\hat{\beta}\right)^T\left(\hat{y}-\hat{X}\hat{\beta}\right), +$$ + +can be linked to the variance of the quantity \( y_i \) if we interpret the latter as the mean value of for example a numerical experiment. When linking below with the maximum likelihood approach below, we will indeed interpret \( y_i \) as a mean value +$$ +y_{i}=\langle y_i \rangle = \beta_0x_{i,0}+\beta_1x_{i,1}+\beta_2x_{i,2}+\dots+\beta_{n-1}x_{i,n-1}+\epsilon_i, +$$ + +where \( \langle y_i \rangle \) is the mean value. Keep in mind also that till now we have treated \( y_i \) as the exact value. 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. + +

    +In order to find the parameters \( \beta_i \) we will then minimize the spread of \( Q(\hat{\beta}) \) by requiring +$$ +\frac{\partial Q(\hat{\beta})}{\partial \beta_j} = \frac{\partial }{\partial \beta_j}\left[ \sum_{i=0}^{n-1}\left(y_i-\beta_0x_{i,0}+\beta_1x_{i,1}+\beta_2x_{i,2}+\dots+\beta_{n-1}x_{i,n-1}\right)^2\right]=0, +$$ + +which results in +$$ +\frac{\partial Q(\hat{\beta})}{\partial \beta_j} = -2\left[ \sum_{i=0}^{n-1}x_{ij}\left(y_i-\beta_0x_{i,0}+\beta_1x_{i,1}+\beta_2x_{i,2}+\dots+\beta_{n-1}x_{i,n-1}\right)\right]=0, +$$ + +or in a matrix-vector form as +$$ +\frac{\partial Q(\hat{\beta})}{\partial \hat{\beta}} = 0 = \hat{X}^T\left( \hat{y}-\hat{X}\hat{\beta}\right). +$$ + +

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    + +

     

     

     

    + + + + +

    Interpretations and optimizing our parameters

    +
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    +We can rewrite +$$ +\frac{\partial Q(\hat{\beta})}{\partial \hat{\beta}} = 0 = \hat{X}^T\left( \hat{y}-\hat{X}\hat{\beta}\right), +$$ + +as +$$ +\hat{X}^T\hat{y} = \hat{X}^T\hat{X}\hat{\beta}, +$$ + +and if the matrix \( \hat{X}^T\hat{X} \) is invertible we have the solution +$$ +\hat{\beta} =\left(\hat{X}^T\hat{X}\right)^{-1}\hat{X}^T\hat{y}. +$$ + +The residuals \( \hat{\epsilon} \) are in turn given by +$$ +\hat{\epsilon} = \hat{y}-\hat{\tilde{y}} = \hat{y}-\hat{X}\hat{\beta}, +$$ + +and with +$$ +\hat{X}^T\left( \hat{y}-\hat{X}\hat{\beta}\right)= 0, +$$ + +we have +$$ +\hat{X}^T\hat{\epsilon}=\hat{X}^T\left( \hat{y}-\hat{X}\hat{\beta}\right)= 0, +$$ + +meaning that the solution for \( \hat{\beta} \) is the one which minimizes the residuals. Later we will link this with the maximum likelihood approach. + +

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    + + + + + + diff --git a/doc/pub/Regression/html/._Regression-bs008.html b/doc/pub/Regression/html/._Regression-bs008.html new file mode 100644 index 000000000..aeda2f187 --- /dev/null +++ b/doc/pub/Regression/html/._Regression-bs008.html @@ -0,0 +1,170 @@ + + + + + + + +Data Analysis and Machine Learning: Linear Regression and more Advanced Regression Analysis + + + + + + + + + + + + + + + + + + + + + + + + + + +
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    + + + + +

    The singular value decompostion

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    +Here we derive the equations for the SVD. +

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  • Rewriting the fitting procedure as a linear algebra problem
  • Generalizing the fitting procedure as a linear algebra problem
  • Optimizing our parameters
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  • Interpretations and optimizing our parameters
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  • Interpretations and optimizing our parameters
  • +
  • The singular value decompostion
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    [2] Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University

     
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    @@ -211,19 +210,13 @@ where \( \epsilon_i \) is the error in our approximation. For every set of values \( y_i,x_i \) we have thus the corresponding set of equations

     
    $$ -\begin{align} -y_0&=\beta_0+\beta_1x_0^1+\beta_2x_0^2+\dots+\beta_{n-1}x_0^{n-1}+\epsilon_0 -\tag{1}\\ -y_1&=\beta_0+\beta_1x_1^1+\beta_2x_1^2+\dots+\beta_{n-1}x_1^{n-1}+\epsilon_1 -\tag{2}\\ -y_2&=\beta_0+\beta_1x_2^1+\beta_2x_2^2+\dots+\beta_{n-1}x_2^{n-1}+\epsilon_2 -\tag{3}\\ -\dots & \dots -\tag{4}\\ -y_{n-1}&=\beta_0+\beta_1x_{n-1}^1+\beta_2x_{n-1}^2+\dots+\beta_1x_{n-1}^{n-1}+\epsilon_{n-1}. -\tag{5}\\ -\tag{6} -\end{align} +\begin{align*} +y_0&=\beta_0+\beta_1x_0^1+\beta_2x_0^2+\dots+\beta_{n-1}x_0^{n-1}+\epsilon_0\\ +y_1&=\beta_0+\beta_1x_1^1+\beta_2x_1^2+\dots+\beta_{n-1}x_1^{n-1}+\epsilon_1\\ +y_2&=\beta_0+\beta_1x_2^1+\beta_2x_2^2+\dots+\beta_{n-1}x_2^{n-1}+\epsilon_2\\ +\dots & \dots \\ +y_{n-1}&=\beta_0+\beta_1x_{n-1}^1+\beta_2x_{n-1}^2+\dots+\beta_1x_{n-1}^{n-1}+\epsilon_{n-1}.\\ +\end{align*} $$

     
    @@ -279,23 +272,15 @@ We are obviously not limited to the above polynomial. We could replace the vario For every set of values \( y_i,x_i \) we can then generalize the equations to

     
    $$ -\begin{align} -y_0&=\beta_0x_{00}+\beta_1x_{01}+\beta_2x_{02}+\dots+\beta_{n-1}x_{0n-1}+\epsilon_0 -\tag{7}\\ -y_1&=\beta_0x_{10}+\beta_1x_{11}+\beta_2x_{12}+\dots+\beta_{n-1}x_{1n-1}+\epsilon_1 -\tag{8}\\ -y_2&=\beta_0x_{20}+\beta_1x_{21}+\beta_2x_{22}+\dots+\beta_{n-1}x_{2n-1}+\epsilon_1 -\tag{9}\\ -\dots & \dots -\tag{10}\\ -y_{i}&=\beta_0x_{i0}+\beta_1x_{i1}+\beta_2x_{i2}+\dots+\beta_{n-1}x_{in-1}+\epsilon_1 -\tag{11}\\ -\dots & \dots -\tag{12}\\ -y_{n-1}&=\beta_0x_{n-1,0}+\beta_1x_{n-1,2}+\beta_2x_{n-1,2}+\dots+\beta_1x_{n-1}^{n-1,n-1}+\epsilon_{n-1}. -\tag{13}\\ -\tag{14} -\end{align} +\begin{align*} +y_0&=\beta_0x_{00}+\beta_1x_{01}+\beta_2x_{02}+\dots+\beta_{n-1}x_{0n-1}+\epsilon_0\\ +y_1&=\beta_0x_{10}+\beta_1x_{11}+\beta_2x_{12}+\dots+\beta_{n-1}x_{1n-1}+\epsilon_1\\ +y_2&=\beta_0x_{20}+\beta_1x_{21}+\beta_2x_{22}+\dots+\beta_{n-1}x_{2n-1}+\epsilon_2\\ +\dots & \dots \\ +y_{i}&=\beta_0x_{i0}+\beta_1x_{i1}+\beta_2x_{i2}+\dots+\beta_{n-1}x_{in-1}+\epsilon_i\\ +\dots & \dots \\ +y_{n-1}&=\beta_0x_{n-1,0}+\beta_1x_{n-1,2}+\beta_2x_{n-1,2}+\dots+\beta_1x_{n-1}^{n-1,n-1}+\epsilon_{n-1}.\\ +\end{align*} $$

     
    @@ -329,27 +314,154 @@ The left-hand side of this equation forms know. Our error vector \( \hat{\epsilo

    Optimizing our parameters

    +

    +We have defined the matrix \( \hat{X} \)

     
    $$ -\begin{align} -y_0&=\beta_0x_{00}+\beta_1x_{01}+\beta_2x_{02}+\dots+\beta_{n-1}x_{0n-1}+\epsilon_0 -\tag{15}\\ -y_1&=\beta_0x_{10}+\beta_1x_{11}+\beta_2x_{12}+\dots+\beta_{n-1}x_{1n-1}+\epsilon_1 -\tag{16}\\ -y_2&=\beta_0x_{20}+\beta_1x_{21}+\beta_2x_{22}+\dots+\beta_{n-1}x_{2n-1}+\epsilon_1 -\tag{17}\\ -\dots & \dots -\tag{18}\\ -y_{i}&=\beta_0x_{i0}+\beta_1x_{i1}+\beta_2x_{i2}+\dots+\beta_{n-1}x_{in-1}+\epsilon_1 -\tag{19}\\ -\dots & \dots -\tag{20}\\ -y_{n-1}&=\beta_0x_{n-1,0}+\beta_1x_{n-1,2}+\beta_2x_{n-1,2}+\dots+\beta_1x_{n-1}^{n-1,n-1}+\epsilon_{n-1}. -\tag{21}\\ -\tag{22} -\end{align} +\begin{align*} +y_0&=\beta_0x_{00}+\beta_1x_{01}+\beta_2x_{02}+\dots+\beta_{n-1}x_{0n-1}+\epsilon_0\\ +y_1&=\beta_0x_{10}+\beta_1x_{11}+\beta_2x_{12}+\dots+\beta_{n-1}x_{1n-1}+\epsilon_1\\ +y_2&=\beta_0x_{20}+\beta_1x_{21}+\beta_2x_{22}+\dots+\beta_{n-1}x_{2n-1}+\epsilon_1\\ +\dots & \dots \\ +y_{i}&=\beta_0x_{i0}+\beta_1x_{i1}+\beta_2x_{i2}+\dots+\beta_{n-1}x_{in-1}+\epsilon_1\\ +\dots & \dots \\ +y_{n-1}&=\beta_0x_{n-1,0}+\beta_1x_{n-1,2}+\beta_2x_{n-1,2}+\dots+\beta_1x_{n-1,n-1}+\epsilon_{n-1}.\\ +\end{align*} $$

     
    + +We well use this matrix to define the approximation \( \hat{\tilde{y}} \) via the unknown quantity \( \hat{\beta} \) as +

     
    +$$ +\hat{\tilde{y}}= \hat{X}\hat{\beta}, +$$ +

     
    + +and in order to find the optimal parameters \( \beta_i \) instead of solving the above linear algebra problem, we define a function which gives a measure of the spread between the values \( y_i \) (which represent hopefully the exact values) and the parametrized values \( \tilde{y}_i \), namely +

     
    +$$ +Q(\hat{\beta})=\sum_{i=0}^{n-1}\left(y_i-\tilde{y}_i\right)^2=\left(\hat{y}-\hat{\tilde{y}}\right)^T\left(\hat{y}-\hat{\tilde{y}}\right), +$$ +

     
    + +or using the matrix \( \hat{X} \) as +

     
    +$$ +Q(\hat{\beta})=\left(\hat{y}-\hat{X}\hat{\beta}\right)^T\left(\hat{y}-\hat{X}\hat{\beta}\right). +$$ +

     
    +

    + + + +
    +

    Interpretations and optimizing our parameters

    +
    + +

    +The function +

     
    +$$ +Q(\hat{\beta})=\left(\hat{y}-\hat{X}\hat{\beta}\right)^T\left(\hat{y}-\hat{X}\hat{\beta}\right), +$$ +

     
    + +can be linked to the variance of the quantity \( y_i \) if we interpret the latter as the mean value of for example a numerical experiment. When linking below with the maximum likelihood approach below, we will indeed interpret \( y_i \) as a mean value +

     
    +$$ +y_{i}=\langle y_i \rangle = \beta_0x_{i,0}+\beta_1x_{i,1}+\beta_2x_{i,2}+\dots+\beta_{n-1}x_{i,n-1}+\epsilon_i, +$$ +

     
    + +where \( \langle y_i \rangle \) is the mean value. Keep in mind also that till now we have treated \( y_i \) as the exact value. 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. + +

    +In order to find the parameters \( \beta_i \) we will then minimize the spread of \( Q(\hat{\beta}) \) by requiring +

     
    +$$ +\frac{\partial Q(\hat{\beta})}{\partial \beta_j} = \frac{\partial }{\partial \beta_j}\left[ \sum_{i=0}^{n-1}\left(y_i-\beta_0x_{i,0}+\beta_1x_{i,1}+\beta_2x_{i,2}+\dots+\beta_{n-1}x_{i,n-1}\right)^2\right]=0, +$$ +

     
    + +which results in +

     
    +$$ +\frac{\partial Q(\hat{\beta})}{\partial \beta_j} = -2\left[ \sum_{i=0}^{n-1}x_{ij}\left(y_i-\beta_0x_{i,0}+\beta_1x_{i,1}+\beta_2x_{i,2}+\dots+\beta_{n-1}x_{i,n-1}\right)\right]=0, +$$ +

     
    + +or in a matrix-vector form as +

     
    +$$ +\frac{\partial Q(\hat{\beta})}{\partial \hat{\beta}} = 0 = \hat{X}^T\left( \hat{y}-\hat{X}\hat{\beta}\right). +$$ +

     
    + + +

    +
    + + +
    +

    Interpretations and optimizing our parameters

    +
    + +

    +We can rewrite +

     
    +$$ +\frac{\partial Q(\hat{\beta})}{\partial \hat{\beta}} = 0 = \hat{X}^T\left( \hat{y}-\hat{X}\hat{\beta}\right), +$$ +

     
    + +as +

     
    +$$ +\hat{X}^T\hat{y} = \hat{X}^T\hat{X}\hat{\beta}, +$$ +

     
    + +and if the matrix \( \hat{X}^T\hat{X} \) is invertible we have the solution +

     
    +$$ +\hat{\beta} =\left(\hat{X}^T\hat{X}\right)^{-1}\hat{X}^T\hat{y}. +$$ +

     
    + +The residuals \( \hat{\epsilon} \) are in turn given by +

     
    +$$ +\hat{\epsilon} = \hat{y}-\hat{\tilde{y}} = \hat{y}-\hat{X}\hat{\beta}, +$$ +

     
    + +and with +

     
    +$$ +\hat{X}^T\left( \hat{y}-\hat{X}\hat{\beta}\right)= 0, +$$ +

     
    + +we have +

     
    +$$ +\hat{X}^T\hat{\epsilon}=\hat{X}^T\left( \hat{y}-\hat{X}\hat{\beta}\right)= 0, +$$ +

     
    + +meaning that the solution for \( \hat{\beta} \) is the one which minimizes the residuals. Later we will link this with the maximum likelihood approach. + + +

    +
    + + +
    +

    The singular value decompostion

    +
    + +

    +Here we derive the equations for the SVD.

    diff --git a/doc/pub/Regression/html/Regression-solarized.html b/doc/pub/Regression/html/Regression-solarized.html index 795182b89..1bf399f0a 100644 --- a/doc/pub/Regression/html/Regression-solarized.html +++ b/doc/pub/Regression/html/Regression-solarized.html @@ -70,7 +70,16 @@ div { text-align: justify; text-justify: inter-word; } 2, None, '___sec3'), - ('Optimizing our parameters', 2, None, '___sec4')]} + ('Optimizing our parameters', 2, None, '___sec4'), + ('Interpretations and optimizing our parameters', + 2, + None, + '___sec5'), + ('Interpretations and optimizing our parameters', + 2, + None, + '___sec6'), + ('The singular value decompostion', 2, None, '___sec7')]} end of tocinfo --> @@ -85,8 +94,8 @@ MathJax.Hub.Config({ } }); - @@ -112,7 +121,7 @@ MathJax.Hub.Config({
    [2] Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University

    -

    Oct 14, 2017

    +

    Oct 17, 2017












    @@ -167,19 +176,13 @@ where \( \epsilon_i \) is the error in our approximation.

    For every set of values \( y_i,x_i \) we have thus the corresponding set of equations $$ -\begin{align} -y_0&=\beta_0+\beta_1x_0^1+\beta_2x_0^2+\dots+\beta_{n-1}x_0^{n-1}+\epsilon_0 -\label{_auto1}\\ -y_1&=\beta_0+\beta_1x_1^1+\beta_2x_1^2+\dots+\beta_{n-1}x_1^{n-1}+\epsilon_1 -\label{_auto2}\\ -y_2&=\beta_0+\beta_1x_2^1+\beta_2x_2^2+\dots+\beta_{n-1}x_2^{n-1}+\epsilon_2 -\label{_auto3}\\ -\dots & \dots -\label{_auto4}\\ -y_{n-1}&=\beta_0+\beta_1x_{n-1}^1+\beta_2x_{n-1}^2+\dots+\beta_1x_{n-1}^{n-1}+\epsilon_{n-1}. -\label{_auto5}\\ -\label{_auto6} -\end{align} +\begin{align*} +y_0&=\beta_0+\beta_1x_0^1+\beta_2x_0^2+\dots+\beta_{n-1}x_0^{n-1}+\epsilon_0\\ +y_1&=\beta_0+\beta_1x_1^1+\beta_2x_1^2+\dots+\beta_{n-1}x_1^{n-1}+\epsilon_1\\ +y_2&=\beta_0+\beta_1x_2^1+\beta_2x_2^2+\dots+\beta_{n-1}x_2^{n-1}+\epsilon_2\\ +\dots & \dots \\ +y_{n-1}&=\beta_0+\beta_1x_{n-1}^1+\beta_2x_{n-1}^2+\dots+\beta_1x_{n-1}^{n-1}+\epsilon_{n-1}.\\ +\end{align*} $$ Defining the vectors @@ -224,23 +227,15 @@ $$ We are obviously not limited to the above polynomial. We could replace the various powers of \( x \) with elements of Fourier series, that is, instead of \( x_i^j \) we could have \( \cos{(j x_i)} \) or \( \sin{(j x_i)} \), or time series or other orthogonal functions. For every set of values \( y_i,x_i \) we can then generalize the equations to $$ -\begin{align} -y_0&=\beta_0x_{00}+\beta_1x_{01}+\beta_2x_{02}+\dots+\beta_{n-1}x_{0n-1}+\epsilon_0 -\label{_auto7}\\ -y_1&=\beta_0x_{10}+\beta_1x_{11}+\beta_2x_{12}+\dots+\beta_{n-1}x_{1n-1}+\epsilon_1 -\label{_auto8}\\ -y_2&=\beta_0x_{20}+\beta_1x_{21}+\beta_2x_{22}+\dots+\beta_{n-1}x_{2n-1}+\epsilon_1 -\label{_auto9}\\ -\dots & \dots -\label{_auto10}\\ -y_{i}&=\beta_0x_{i0}+\beta_1x_{i1}+\beta_2x_{i2}+\dots+\beta_{n-1}x_{in-1}+\epsilon_1 -\label{_auto11}\\ -\dots & \dots -\label{_auto12}\\ -y_{n-1}&=\beta_0x_{n-1,0}+\beta_1x_{n-1,2}+\beta_2x_{n-1,2}+\dots+\beta_1x_{n-1}^{n-1,n-1}+\epsilon_{n-1}. -\label{_auto13}\\ -\label{_auto14} -\end{align} +\begin{align*} +y_0&=\beta_0x_{00}+\beta_1x_{01}+\beta_2x_{02}+\dots+\beta_{n-1}x_{0n-1}+\epsilon_0\\ +y_1&=\beta_0x_{10}+\beta_1x_{11}+\beta_2x_{12}+\dots+\beta_{n-1}x_{1n-1}+\epsilon_1\\ +y_2&=\beta_0x_{20}+\beta_1x_{21}+\beta_2x_{22}+\dots+\beta_{n-1}x_{2n-1}+\epsilon_2\\ +\dots & \dots \\ +y_{i}&=\beta_0x_{i0}+\beta_1x_{i1}+\beta_2x_{i2}+\dots+\beta_{n-1}x_{in-1}+\epsilon_i\\ +\dots & \dots \\ +y_{n-1}&=\beta_0x_{n-1,0}+\beta_1x_{n-1,2}+\beta_2x_{n-1,2}+\dots+\beta_1x_{n-1}^{n-1,n-1}+\epsilon_{n-1}.\\ +\end{align*} $$ We redefine in turn the matrix \( \hat{X} \) as @@ -271,28 +266,131 @@ The left-hand side of this equation forms know. Our error vector \( \hat{\epsilo

    +We have defined the matrix \( \hat{X} \) $$ -\begin{align} -y_0&=\beta_0x_{00}+\beta_1x_{01}+\beta_2x_{02}+\dots+\beta_{n-1}x_{0n-1}+\epsilon_0 -\label{_auto15}\\ -y_1&=\beta_0x_{10}+\beta_1x_{11}+\beta_2x_{12}+\dots+\beta_{n-1}x_{1n-1}+\epsilon_1 -\label{_auto16}\\ -y_2&=\beta_0x_{20}+\beta_1x_{21}+\beta_2x_{22}+\dots+\beta_{n-1}x_{2n-1}+\epsilon_1 -\label{_auto17}\\ -\dots & \dots -\label{_auto18}\\ -y_{i}&=\beta_0x_{i0}+\beta_1x_{i1}+\beta_2x_{i2}+\dots+\beta_{n-1}x_{in-1}+\epsilon_1 -\label{_auto19}\\ -\dots & \dots -\label{_auto20}\\ -y_{n-1}&=\beta_0x_{n-1,0}+\beta_1x_{n-1,2}+\beta_2x_{n-1,2}+\dots+\beta_1x_{n-1}^{n-1,n-1}+\epsilon_{n-1}. -\label{_auto21}\\ -\label{_auto22} -\end{align} +\begin{align*} +y_0&=\beta_0x_{00}+\beta_1x_{01}+\beta_2x_{02}+\dots+\beta_{n-1}x_{0n-1}+\epsilon_0\\ +y_1&=\beta_0x_{10}+\beta_1x_{11}+\beta_2x_{12}+\dots+\beta_{n-1}x_{1n-1}+\epsilon_1\\ +y_2&=\beta_0x_{20}+\beta_1x_{21}+\beta_2x_{22}+\dots+\beta_{n-1}x_{2n-1}+\epsilon_1\\ +\dots & \dots \\ +y_{i}&=\beta_0x_{i0}+\beta_1x_{i1}+\beta_2x_{i2}+\dots+\beta_{n-1}x_{in-1}+\epsilon_1\\ +\dots & \dots \\ +y_{n-1}&=\beta_0x_{n-1,0}+\beta_1x_{n-1,2}+\beta_2x_{n-1,2}+\dots+\beta_1x_{n-1,n-1}+\epsilon_{n-1}.\\ +\end{align*} +$$ + +We well use this matrix to define the approximation \( \hat{\tilde{y}} \) via the unknown quantity \( \hat{\beta} \) as +$$ +\hat{\tilde{y}}= \hat{X}\hat{\beta}, +$$ + +and in order to find the optimal parameters \( \beta_i \) instead of solving the above linear algebra problem, we define a function which gives a measure of the spread between the values \( y_i \) (which represent hopefully the exact values) and the parametrized values \( \tilde{y}_i \), namely +$$ +Q(\hat{\beta})=\sum_{i=0}^{n-1}\left(y_i-\tilde{y}_i\right)^2=\left(\hat{y}-\hat{\tilde{y}}\right)^T\left(\hat{y}-\hat{\tilde{y}}\right), +$$ + +or using the matrix \( \hat{X} \) as +$$ +Q(\hat{\beta})=\left(\hat{y}-\hat{X}\hat{\beta}\right)^T\left(\hat{y}-\hat{X}\hat{\beta}\right). $$

    +

    +









    + +

    Interpretations and optimizing our parameters

    +
    + +

    +The function +$$ +Q(\hat{\beta})=\left(\hat{y}-\hat{X}\hat{\beta}\right)^T\left(\hat{y}-\hat{X}\hat{\beta}\right), +$$ + +can be linked to the variance of the quantity \( y_i \) if we interpret the latter as the mean value of for example a numerical experiment. When linking below with the maximum likelihood approach below, we will indeed interpret \( y_i \) as a mean value +$$ +y_{i}=\langle y_i \rangle = \beta_0x_{i,0}+\beta_1x_{i,1}+\beta_2x_{i,2}+\dots+\beta_{n-1}x_{i,n-1}+\epsilon_i, +$$ + +where \( \langle y_i \rangle \) is the mean value. Keep in mind also that till now we have treated \( y_i \) as the exact value. 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. + +

    +In order to find the parameters \( \beta_i \) we will then minimize the spread of \( Q(\hat{\beta}) \) by requiring +$$ +\frac{\partial Q(\hat{\beta})}{\partial \beta_j} = \frac{\partial }{\partial \beta_j}\left[ \sum_{i=0}^{n-1}\left(y_i-\beta_0x_{i,0}+\beta_1x_{i,1}+\beta_2x_{i,2}+\dots+\beta_{n-1}x_{i,n-1}\right)^2\right]=0, +$$ + +which results in +$$ +\frac{\partial Q(\hat{\beta})}{\partial \beta_j} = -2\left[ \sum_{i=0}^{n-1}x_{ij}\left(y_i-\beta_0x_{i,0}+\beta_1x_{i,1}+\beta_2x_{i,2}+\dots+\beta_{n-1}x_{i,n-1}\right)\right]=0, +$$ + +or in a matrix-vector form as +$$ +\frac{\partial Q(\hat{\beta})}{\partial \hat{\beta}} = 0 = \hat{X}^T\left( \hat{y}-\hat{X}\hat{\beta}\right). +$$ + + +

    + + +

    +









    + +

    Interpretations and optimizing our parameters

    +
    + +

    +We can rewrite +$$ +\frac{\partial Q(\hat{\beta})}{\partial \hat{\beta}} = 0 = \hat{X}^T\left( \hat{y}-\hat{X}\hat{\beta}\right), +$$ + +as +$$ +\hat{X}^T\hat{y} = \hat{X}^T\hat{X}\hat{\beta}, +$$ + +and if the matrix \( \hat{X}^T\hat{X} \) is invertible we have the solution +$$ +\hat{\beta} =\left(\hat{X}^T\hat{X}\right)^{-1}\hat{X}^T\hat{y}. +$$ + +The residuals \( \hat{\epsilon} \) are in turn given by +$$ +\hat{\epsilon} = \hat{y}-\hat{\tilde{y}} = \hat{y}-\hat{X}\hat{\beta}, +$$ + +and with +$$ +\hat{X}^T\left( \hat{y}-\hat{X}\hat{\beta}\right)= 0, +$$ + +we have +$$ +\hat{X}^T\hat{\epsilon}=\hat{X}^T\left( \hat{y}-\hat{X}\hat{\beta}\right)= 0, +$$ + +meaning that the solution for \( \hat{\beta} \) is the one which minimizes the residuals. Later we will link this with the maximum likelihood approach. + + +

    + + +

    +









    + +

    The singular value decompostion

    +
    + +

    +Here we derive the equations for the SVD. +

    + + +

    + diff --git a/doc/pub/Regression/html/Regression.html b/doc/pub/Regression/html/Regression.html index ab2d9e268..8c775ba7f 100644 --- a/doc/pub/Regression/html/Regression.html +++ b/doc/pub/Regression/html/Regression.html @@ -75,7 +75,16 @@ div { text-align: justify; text-justify: inter-word; } 2, None, '___sec3'), - ('Optimizing our parameters', 2, None, '___sec4')]} + ('Optimizing our parameters', 2, None, '___sec4'), + ('Interpretations and optimizing our parameters', + 2, + None, + '___sec5'), + ('Interpretations and optimizing our parameters', + 2, + None, + '___sec6'), + ('The singular value decompostion', 2, None, '___sec7')]} end of tocinfo --> @@ -90,8 +99,8 @@ MathJax.Hub.Config({ } }); - @@ -117,7 +126,7 @@ MathJax.Hub.Config({

    [2] Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University

    -

    Oct 14, 2017

    +

    Oct 17, 2017












    @@ -172,19 +181,13 @@ where \( \epsilon_i \) is the error in our approximation.

    For every set of values \( y_i,x_i \) we have thus the corresponding set of equations $$ -\begin{align} -y_0&=\beta_0+\beta_1x_0^1+\beta_2x_0^2+\dots+\beta_{n-1}x_0^{n-1}+\epsilon_0 -\label{_auto1}\\ -y_1&=\beta_0+\beta_1x_1^1+\beta_2x_1^2+\dots+\beta_{n-1}x_1^{n-1}+\epsilon_1 -\label{_auto2}\\ -y_2&=\beta_0+\beta_1x_2^1+\beta_2x_2^2+\dots+\beta_{n-1}x_2^{n-1}+\epsilon_2 -\label{_auto3}\\ -\dots & \dots -\label{_auto4}\\ -y_{n-1}&=\beta_0+\beta_1x_{n-1}^1+\beta_2x_{n-1}^2+\dots+\beta_1x_{n-1}^{n-1}+\epsilon_{n-1}. -\label{_auto5}\\ -\label{_auto6} -\end{align} +\begin{align*} +y_0&=\beta_0+\beta_1x_0^1+\beta_2x_0^2+\dots+\beta_{n-1}x_0^{n-1}+\epsilon_0\\ +y_1&=\beta_0+\beta_1x_1^1+\beta_2x_1^2+\dots+\beta_{n-1}x_1^{n-1}+\epsilon_1\\ +y_2&=\beta_0+\beta_1x_2^1+\beta_2x_2^2+\dots+\beta_{n-1}x_2^{n-1}+\epsilon_2\\ +\dots & \dots \\ +y_{n-1}&=\beta_0+\beta_1x_{n-1}^1+\beta_2x_{n-1}^2+\dots+\beta_1x_{n-1}^{n-1}+\epsilon_{n-1}.\\ +\end{align*} $$ Defining the vectors @@ -229,23 +232,15 @@ $$ We are obviously not limited to the above polynomial. We could replace the various powers of \( x \) with elements of Fourier series, that is, instead of \( x_i^j \) we could have \( \cos{(j x_i)} \) or \( \sin{(j x_i)} \), or time series or other orthogonal functions. For every set of values \( y_i,x_i \) we can then generalize the equations to $$ -\begin{align} -y_0&=\beta_0x_{00}+\beta_1x_{01}+\beta_2x_{02}+\dots+\beta_{n-1}x_{0n-1}+\epsilon_0 -\label{_auto7}\\ -y_1&=\beta_0x_{10}+\beta_1x_{11}+\beta_2x_{12}+\dots+\beta_{n-1}x_{1n-1}+\epsilon_1 -\label{_auto8}\\ -y_2&=\beta_0x_{20}+\beta_1x_{21}+\beta_2x_{22}+\dots+\beta_{n-1}x_{2n-1}+\epsilon_1 -\label{_auto9}\\ -\dots & \dots -\label{_auto10}\\ -y_{i}&=\beta_0x_{i0}+\beta_1x_{i1}+\beta_2x_{i2}+\dots+\beta_{n-1}x_{in-1}+\epsilon_1 -\label{_auto11}\\ -\dots & \dots -\label{_auto12}\\ -y_{n-1}&=\beta_0x_{n-1,0}+\beta_1x_{n-1,2}+\beta_2x_{n-1,2}+\dots+\beta_1x_{n-1}^{n-1,n-1}+\epsilon_{n-1}. -\label{_auto13}\\ -\label{_auto14} -\end{align} +\begin{align*} +y_0&=\beta_0x_{00}+\beta_1x_{01}+\beta_2x_{02}+\dots+\beta_{n-1}x_{0n-1}+\epsilon_0\\ +y_1&=\beta_0x_{10}+\beta_1x_{11}+\beta_2x_{12}+\dots+\beta_{n-1}x_{1n-1}+\epsilon_1\\ +y_2&=\beta_0x_{20}+\beta_1x_{21}+\beta_2x_{22}+\dots+\beta_{n-1}x_{2n-1}+\epsilon_2\\ +\dots & \dots \\ +y_{i}&=\beta_0x_{i0}+\beta_1x_{i1}+\beta_2x_{i2}+\dots+\beta_{n-1}x_{in-1}+\epsilon_i\\ +\dots & \dots \\ +y_{n-1}&=\beta_0x_{n-1,0}+\beta_1x_{n-1,2}+\beta_2x_{n-1,2}+\dots+\beta_1x_{n-1}^{n-1,n-1}+\epsilon_{n-1}.\\ +\end{align*} $$ We redefine in turn the matrix \( \hat{X} \) as @@ -276,28 +271,131 @@ The left-hand side of this equation forms know. Our error vector \( \hat{\epsilo

    +We have defined the matrix \( \hat{X} \) $$ -\begin{align} -y_0&=\beta_0x_{00}+\beta_1x_{01}+\beta_2x_{02}+\dots+\beta_{n-1}x_{0n-1}+\epsilon_0 -\label{_auto15}\\ -y_1&=\beta_0x_{10}+\beta_1x_{11}+\beta_2x_{12}+\dots+\beta_{n-1}x_{1n-1}+\epsilon_1 -\label{_auto16}\\ -y_2&=\beta_0x_{20}+\beta_1x_{21}+\beta_2x_{22}+\dots+\beta_{n-1}x_{2n-1}+\epsilon_1 -\label{_auto17}\\ -\dots & \dots -\label{_auto18}\\ -y_{i}&=\beta_0x_{i0}+\beta_1x_{i1}+\beta_2x_{i2}+\dots+\beta_{n-1}x_{in-1}+\epsilon_1 -\label{_auto19}\\ -\dots & \dots -\label{_auto20}\\ -y_{n-1}&=\beta_0x_{n-1,0}+\beta_1x_{n-1,2}+\beta_2x_{n-1,2}+\dots+\beta_1x_{n-1}^{n-1,n-1}+\epsilon_{n-1}. -\label{_auto21}\\ -\label{_auto22} -\end{align} +\begin{align*} +y_0&=\beta_0x_{00}+\beta_1x_{01}+\beta_2x_{02}+\dots+\beta_{n-1}x_{0n-1}+\epsilon_0\\ +y_1&=\beta_0x_{10}+\beta_1x_{11}+\beta_2x_{12}+\dots+\beta_{n-1}x_{1n-1}+\epsilon_1\\ +y_2&=\beta_0x_{20}+\beta_1x_{21}+\beta_2x_{22}+\dots+\beta_{n-1}x_{2n-1}+\epsilon_1\\ +\dots & \dots \\ +y_{i}&=\beta_0x_{i0}+\beta_1x_{i1}+\beta_2x_{i2}+\dots+\beta_{n-1}x_{in-1}+\epsilon_1\\ +\dots & \dots \\ +y_{n-1}&=\beta_0x_{n-1,0}+\beta_1x_{n-1,2}+\beta_2x_{n-1,2}+\dots+\beta_1x_{n-1,n-1}+\epsilon_{n-1}.\\ +\end{align*} +$$ + +We well use this matrix to define the approximation \( \hat{\tilde{y}} \) via the unknown quantity \( \hat{\beta} \) as +$$ +\hat{\tilde{y}}= \hat{X}\hat{\beta}, +$$ + +and in order to find the optimal parameters \( \beta_i \) instead of solving the above linear algebra problem, we define a function which gives a measure of the spread between the values \( y_i \) (which represent hopefully the exact values) and the parametrized values \( \tilde{y}_i \), namely +$$ +Q(\hat{\beta})=\sum_{i=0}^{n-1}\left(y_i-\tilde{y}_i\right)^2=\left(\hat{y}-\hat{\tilde{y}}\right)^T\left(\hat{y}-\hat{\tilde{y}}\right), +$$ + +or using the matrix \( \hat{X} \) as +$$ +Q(\hat{\beta})=\left(\hat{y}-\hat{X}\hat{\beta}\right)^T\left(\hat{y}-\hat{X}\hat{\beta}\right). $$

    +

    +









    + +

    Interpretations and optimizing our parameters

    +
    + +

    +The function +$$ +Q(\hat{\beta})=\left(\hat{y}-\hat{X}\hat{\beta}\right)^T\left(\hat{y}-\hat{X}\hat{\beta}\right), +$$ + +can be linked to the variance of the quantity \( y_i \) if we interpret the latter as the mean value of for example a numerical experiment. When linking below with the maximum likelihood approach below, we will indeed interpret \( y_i \) as a mean value +$$ +y_{i}=\langle y_i \rangle = \beta_0x_{i,0}+\beta_1x_{i,1}+\beta_2x_{i,2}+\dots+\beta_{n-1}x_{i,n-1}+\epsilon_i, +$$ + +where \( \langle y_i \rangle \) is the mean value. Keep in mind also that till now we have treated \( y_i \) as the exact value. 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. + +

    +In order to find the parameters \( \beta_i \) we will then minimize the spread of \( Q(\hat{\beta}) \) by requiring +$$ +\frac{\partial Q(\hat{\beta})}{\partial \beta_j} = \frac{\partial }{\partial \beta_j}\left[ \sum_{i=0}^{n-1}\left(y_i-\beta_0x_{i,0}+\beta_1x_{i,1}+\beta_2x_{i,2}+\dots+\beta_{n-1}x_{i,n-1}\right)^2\right]=0, +$$ + +which results in +$$ +\frac{\partial Q(\hat{\beta})}{\partial \beta_j} = -2\left[ \sum_{i=0}^{n-1}x_{ij}\left(y_i-\beta_0x_{i,0}+\beta_1x_{i,1}+\beta_2x_{i,2}+\dots+\beta_{n-1}x_{i,n-1}\right)\right]=0, +$$ + +or in a matrix-vector form as +$$ +\frac{\partial Q(\hat{\beta})}{\partial \hat{\beta}} = 0 = \hat{X}^T\left( \hat{y}-\hat{X}\hat{\beta}\right). +$$ + + +

    + + +

    +









    + +

    Interpretations and optimizing our parameters

    +
    + +

    +We can rewrite +$$ +\frac{\partial Q(\hat{\beta})}{\partial \hat{\beta}} = 0 = \hat{X}^T\left( \hat{y}-\hat{X}\hat{\beta}\right), +$$ + +as +$$ +\hat{X}^T\hat{y} = \hat{X}^T\hat{X}\hat{\beta}, +$$ + +and if the matrix \( \hat{X}^T\hat{X} \) is invertible we have the solution +$$ +\hat{\beta} =\left(\hat{X}^T\hat{X}\right)^{-1}\hat{X}^T\hat{y}. +$$ + +The residuals \( \hat{\epsilon} \) are in turn given by +$$ +\hat{\epsilon} = \hat{y}-\hat{\tilde{y}} = \hat{y}-\hat{X}\hat{\beta}, +$$ + +and with +$$ +\hat{X}^T\left( \hat{y}-\hat{X}\hat{\beta}\right)= 0, +$$ + +we have +$$ +\hat{X}^T\hat{\epsilon}=\hat{X}^T\left( \hat{y}-\hat{X}\hat{\beta}\right)= 0, +$$ + +meaning that the solution for \( \hat{\beta} \) is the one which minimizes the residuals. Later we will link this with the maximum likelihood approach. + + +

    + + +

    +









    + +

    The singular value decompostion

    +
    + +

    +Here we derive the equations for the SVD. +

    + + +

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set of equations !bt -\begin{align} +\begin{align*} y_0&=\beta_0+\beta_1x_0^1+\beta_2x_0^2+\dots+\beta_{n-1}x_0^{n-1}+\epsilon_0\\ y_1&=\beta_0+\beta_1x_1^1+\beta_2x_1^2+\dots+\beta_{n-1}x_1^{n-1}+\epsilon_1\\ y_2&=\beta_0+\beta_1x_2^1+\beta_2x_2^2+\dots+\beta_{n-1}x_2^{n-1}+\epsilon_2\\ \dots & \dots \\ y_{n-1}&=\beta_0+\beta_1x_{n-1}^1+\beta_2x_{n-1}^2+\dots+\beta_1x_{n-1}^{n-1}+\epsilon_{n-1}.\\ -\end{align} +\end{align*} !et Defining the vectors !bt @@ -89,7 +89,7 @@ we can rewrite our equations as We are obviously not limited to the above polynomial. We could replace the various powers of $x$ with elements of Fourier series, that is, instead of $x_i^j$ we could have $\cos{(j x_i)}$ or $\sin{(j x_i)}$, or time series or other orthogonal functions. For every set of values $y_i,x_i$ we can then generalize the equations to !bt -\begin{align} +\begin{align*} y_0&=\beta_0x_{00}+\beta_1x_{01}+\beta_2x_{02}+\dots+\beta_{n-1}x_{0n-1}+\epsilon_0\\ y_1&=\beta_0x_{10}+\beta_1x_{11}+\beta_2x_{12}+\dots+\beta_{n-1}x_{1n-1}+\epsilon_1\\ y_2&=\beta_0x_{20}+\beta_1x_{21}+\beta_2x_{22}+\dots+\beta_{n-1}x_{2n-1}+\epsilon_2\\ @@ -97,7 +97,7 @@ y_2&=\beta_0x_{20}+\beta_1x_{21}+\beta_2x_{22}+\dots+\beta_{n-1}x_{2n-1}+\epsilo y_{i}&=\beta_0x_{i0}+\beta_1x_{i1}+\beta_2x_{i2}+\dots+\beta_{n-1}x_{in-1}+\epsilon_i\\ \dots & \dots \\ y_{n-1}&=\beta_0x_{n-1,0}+\beta_1x_{n-1,2}+\beta_2x_{n-1,2}+\dots+\beta_1x_{n-1}^{n-1,n-1}+\epsilon_{n-1}.\\ -\end{align} +\end{align*} !et We redefine in turn the matrix $\hat{X}$ as !bt @@ -127,7 +127,7 @@ The left-hand side of this equation forms know. Our error vector $\hat{\epsilon} !bblock We have defined the matrix $\hat{X}$ !bt -\begin{align} +\begin{align*} y_0&=\beta_0x_{00}+\beta_1x_{01}+\beta_2x_{02}+\dots+\beta_{n-1}x_{0n-1}+\epsilon_0\\ y_1&=\beta_0x_{10}+\beta_1x_{11}+\beta_2x_{12}+\dots+\beta_{n-1}x_{1n-1}+\epsilon_1\\ y_2&=\beta_0x_{20}+\beta_1x_{21}+\beta_2x_{22}+\dots+\beta_{n-1}x_{2n-1}+\epsilon_1\\ @@ -135,7 +135,7 @@ y_2&=\beta_0x_{20}+\beta_1x_{21}+\beta_2x_{22}+\dots+\beta_{n-1}x_{2n-1}+\epsilo y_{i}&=\beta_0x_{i0}+\beta_1x_{i1}+\beta_2x_{i2}+\dots+\beta_{n-1}x_{in-1}+\epsilon_1\\ \dots & \dots \\ y_{n-1}&=\beta_0x_{n-1,0}+\beta_1x_{n-1,2}+\beta_2x_{n-1,2}+\dots+\beta_1x_{n-1,n-1}+\epsilon_{n-1}.\\ -\end{align} +\end{align*} !et We well use this matrix to define the approximation $\hat{\tilde{y}}$ via the unknown quantity $\hat{\beta}$ as !bt @@ -146,13 +146,13 @@ We well use this matrix to define the approximation $\hat{\tilde{y}}$ via the un and in order to find the optimal parameters $\beta_i$ instead of solving the above linear algebra problem, we define a function which gives a measure of the spread between the values $y_i$ (which represent hopefully the exact values) and the parametrized values $\tilde{y}_i$, namely !bt \[ -Q(\hat{\beta})=\sum_{i=0}^{n-1}\left(y_i-\tilde{y}_i\right)^2=\left \hat{y}-\hat{\tilde{y}}\right)^T\left \hat{y}-\hat{\tilde{y}}\right), +Q(\hat{\beta})=\sum_{i=0}^{n-1}\left(y_i-\tilde{y}_i\right)^2=\left(\hat{y}-\hat{\tilde{y}}\right)^T\left(\hat{y}-\hat{\tilde{y}}\right), \] !et or using the matrix $\hat{X}$ as !bt \[ -Q(\hat{\beta})=\left \hat{y}-\hat{X}\hat{\beta}\right)^T\left \hat{y}-\hat{X}\hat{\beta}\right). +Q(\hat{\beta})=\left(\hat{y}-\hat{X}\hat{\beta}\right)^T\left(\hat{y}-\hat{X}\hat{\beta}\right). \] !et !eblock @@ -164,7 +164,7 @@ Q(\hat{\beta})=\left \hat{y}-\hat{X}\hat{\beta}\right)^T\left \hat{y}-\hat{X}\ha The function !bt \[ -Q(\hat{\beta})=\left \hat{y}-\hat{X}\hat{\beta}\right)^T\left \hat{y}-\hat{X}\hat{\beta}\right), +Q(\hat{\beta})=\left(\hat{y}-\hat{X}\hat{\beta}\right)^T\left(\hat{y}-\hat{X}\hat{\beta}\right), \] !et can be linked to the variance of the quantity $y_i$ if we interpret the latter as the mean value of for example a numerical experiment. When linking below with the maximum likelihood approach below, we will indeed interpret $y_i$ as a mean value @@ -178,7 +178,7 @@ 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 \[ -\frac{\partial Q(\hat{\beta})}{\partial \beta_j} = \frac{ }{\partial \beta_j}\left[ \sum_{i=0}^{n-1}\left(y_i-\beta_0x_{i,0}+\beta_1x_{i,1}+\beta_2x_{i,2}+\dots+\beta_{n-1}x_{i,n-1}\right)^2\right]=0, +\frac{\partial Q(\hat{\beta})}{\partial \beta_j} = \frac{\partial }{\partial \beta_j}\left[ \sum_{i=0}^{n-1}\left(y_i-\beta_0x_{i,0}+\beta_1x_{i,1}+\beta_2x_{i,2}+\dots+\beta_{n-1}x_{i,n-1}\right)^2\right]=0, \] !et which results in @@ -196,3 +196,56 @@ or in a matrix-vector form as !eblock + + +!split +===== Interpretations and optimizing our parameters ===== +!bblock +We can rewrite +!bt +\[ +\frac{\partial Q(\hat{\beta})}{\partial \hat{\beta}} = 0 = \hat{X}^T\left( \hat{y}-\hat{X}\hat{\beta}\right), +\] +!et +as +!bt +\[ +\hat{X}^T\hat{y} = \hat{X}^T\hat{X}\hat{\beta}, +\] +!et +and if the matrix $\hat{X}^T\hat{X}$ is invertible we have the solution +!bt +\[ +\hat{\beta} =\left(\hat{X}^T\hat{X}\right)^{-1}\hat{X}^T\hat{y}. +\] +!et +The residuals $\hat{\epsilon}$ are in turn given by +!bt +\[ +\hat{\epsilon} = \hat{y}-\hat{\tilde{y}} = \hat{y}-\hat{X}\hat{\beta}, +\] +!et +and with +!bt +\[ +\hat{X}^T\left( \hat{y}-\hat{X}\hat{\beta}\right)= 0, +\] +!et +we have +!bt +\[ +\hat{X}^T\hat{\epsilon}=\hat{X}^T\left( \hat{y}-\hat{X}\hat{\beta}\right)= 0, +\] +!et +meaning that the solution for $\hat{\beta}$ is the one which minimizes the residuals. Later we will link this with the maximum likelihood approach. + +!eblock + + +!split +===== The singular value decompostion ===== +!bblock +Here we derive the equations for the SVD. +!eblock + +