79 lines
2.6 KiB
TeX
79 lines
2.6 KiB
TeX
\documentclass[12pt]{book}
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\input{../../header}
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\renewcommand{\exercisesolutions}{here} % 0: here, 1: chapter, 2: end
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\lstset{inputpath=../code}
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\graphicspath{{figures/}}
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\typein[\pagenumber]{Number of first page}
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\typein[\chapternumber]{Chapter number}
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\setcounter{page}{\pagenumber}
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\setcounter{chapter}{\chapternumber}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{document}
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\include{regression}
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\subsection{Notes}
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\begin{itemize}
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\item Fig 8.2 right: this should be a chi-squared distribution with one degree of freedom!
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\end{itemize}
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\begin{figure}[t]
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\includegraphics[width=0.75\textwidth]{error_surface}
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\titlecaption{Error surface.}{The two model parameters $m$ and $b$
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define the base area of the surface plot. For each parameter
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combination of slope and intercept the error is calculated. The
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resulting surface has a minimum which indicates the parameter
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combination that best fits the data.}\label{errorsurfacefig}
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\end{figure}
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\begin{figure}[t]
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\includegraphics[width=0.75\textwidth]{error_gradient}
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\titlecaption{Gradient of the error surface.} {Each arrow points
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into the direction of the greatest ascend at different positions
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of the error surface shown in \figref{errorsurfacefig}. The
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contour lines in the background illustrate the error surface. Warm
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colors indicate high errors, colder colors low error values. Each
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contour line connects points of equal
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error.}\label{gradientquiverfig}
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\end{figure}
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\begin{figure}[t]
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\includegraphics[width=0.45\textwidth]{gradient_descent}
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\titlecaption{Gradient descent.}{The algorithm starts at an
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arbitrary position. At each point the gradient is estimated and
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the position is updated as long as the length of the gradient is
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sufficiently large.The dots show the positions after each
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iteration of the algorithm.} \label{gradientdescentfig}
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\end{figure}
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\subsection{Linear fits}
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\begin{itemize}
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\item Polyfit is easy: unique solution! $c x^2$ is also a linear fit.
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\item Example for overfitting with polyfit of a high order (=number of data points)
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\end{itemize}
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\section{Fitting in practice}
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Fit with matlab functions lsqcurvefit, polyfit
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\subsection{Non-linear fits}
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\begin{itemize}
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\item Example that illustrates the Nebenminima Problem (with error surface)
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\item You need initial values for the parameter!
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\item Example that fitting gets harder the more parameter you have.
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\item Try to fix as many parameter before doing the fit.
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\item How to test the quality of a fit? Residuals. $\chi^2$ test. Run-test.
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\end{itemize}
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\end{document}
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