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\input{../../header}

\renewcommand{\exercisesolutions}{here}  % 0: here, 1: chapter, 2: end

\lstset{inputpath=../code}
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\begin{document} 

\include{regression}

\subsection{Notes}
\begin{itemize}
\item Fig 8.2 right: this should be a chi-squared distribution with one degree of freedom!
\end{itemize}

\begin{figure}[t]
  \includegraphics[width=0.75\textwidth]{error_surface}
  \titlecaption{Error surface.}{The two model parameters $m$ and $b$
    define the base area of the surface plot. For each parameter
    combination of slope and intercept the error is calculated. The
    resulting surface has a minimum which indicates the parameter
    combination that best fits the data.}\label{errorsurfacefig}
\end{figure}


\begin{figure}[t]
  \includegraphics[width=0.75\textwidth]{error_gradient}
  \titlecaption{Gradient of the error surface.}  {Each arrow points
    into the direction of the greatest ascend at different positions
    of the error surface shown in \figref{errorsurfacefig}. The
    contour lines in the background illustrate the error surface. Warm
    colors indicate high errors, colder colors low error values. Each
    contour line connects points of equal
    error.}\label{gradientquiverfig}
\end{figure}

\begin{figure}[t]
  \includegraphics[width=0.45\textwidth]{gradient_descent}
  \titlecaption{Gradient descent.}{The algorithm starts at an
    arbitrary position. At each point the gradient is estimated and
    the position is updated as long as the length of the gradient is
    sufficiently large.The dots show the positions after each
    iteration of the algorithm.} \label{gradientdescentfig}
\end{figure}


\subsection{Linear fits}
\begin{itemize}
\item Polyfit is easy: unique solution! $c x^2$ is also a linear fit.
\item Example for overfitting with polyfit of a high order (=number of data points)
\end{itemize}

\section{Fitting in practice}

Fit with matlab functions lsqcurvefit, polyfit


\subsection{Non-linear fits}
\begin{itemize}
\item Example that illustrates the Nebenminima Problem (with error surface)
\item You need initial values for the parameter!
\item Example that fitting gets harder the more parameter you have.
\item Try to fix as many parameter before doing the fit.
\item How to test the quality of a fit? Residuals. $\chi^2$ test. Run-test.
\end{itemize}


\end{document}
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