\relax \@writefile{toc}{\contentsline {section}{\numberline {1}Introduction}{1}} \@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces {\relax \fontsize {10}{12}\selectfont \abovedisplayskip 10\p@ plus2\p@ minus5\p@ \abovedisplayshortskip \z@ plus3\p@ \belowdisplayshortskip 6\p@ plus3\p@ minus3\p@ \def \leftmargin \leftmargini \parsep 4.5\p@ plus2\p@ minus\p@ \topsep 9\p@ plus3\p@ minus5\p@ \itemsep 4.5\p@ plus2\p@ minus\p@ {\leftmargin \leftmargini \topsep 6\p@ plus2\p@ minus2\p@ \parsep 3\p@ plus2\p@ minus\p@ \itemsep \parsep }\belowdisplayskip \abovedisplayskip Typical data values for the least-squares problem: measurements $y$ with error bars $\sigma $ are carried out at known values of a control variable $x$. The curve has been fitted to the data.} \label {fig:basicData}}}{1}} \newlabel{fig:basicData}{{1}{1}} \@writefile{toc}{\contentsline {section}{\numberline {2}The method of least squares}{1}} \newlabel{sec:ls}{{2}{1}} \newlabel{eq:straightline}{{1}{2}} \@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces {\relax \fontsize {10}{12}\selectfont \abovedisplayskip 10\p@ plus2\p@ minus5\p@ \abovedisplayshortskip \z@ plus3\p@ \belowdisplayshortskip 6\p@ plus3\p@ minus3\p@ \def \leftmargin \leftmargini \parsep 4.5\p@ plus2\p@ minus\p@ \topsep 9\p@ plus3\p@ minus5\p@ \itemsep 4.5\p@ plus2\p@ minus\p@ {\leftmargin \leftmargini \topsep 6\p@ plus2\p@ minus2\p@ \parsep 3\p@ plus2\p@ minus\p@ \itemsep \parsep }\belowdisplayskip \abovedisplayskip The line corresponding to an arbitrarily chosen point in the parameter space $(\theta _0, \theta _1)$ together with the data points. The thick vertical bars show the residuals $y_i - f(x_i; {\unhbox \voidb@x \hbox {\relax \mathversion {bold}$\bf \theta $}})$.} \label {fig:arbLine}}}{2}} \newlabel{fig:arbLine}{{2}{2}} \citation{bib:Cowan98} \newlabel{eq:normresid}{{2}{3}} \newlabel{eq:chi2basic}{{3}{3}} \@writefile{lof}{\contentsline {figure}{\numberline {3}{\ignorespaces {\relax \fontsize {10}{12}\selectfont \abovedisplayskip 10\p@ plus2\p@ minus5\p@ \abovedisplayshortskip \z@ plus3\p@ \belowdisplayshortskip 6\p@ plus3\p@ minus3\p@ \def \leftmargin \leftmargini \parsep 4.5\p@ plus2\p@ minus\p@ \topsep 9\p@ plus3\p@ minus5\p@ \itemsep 4.5\p@ plus2\p@ minus\p@ {\leftmargin \leftmargini \topsep 6\p@ plus2\p@ minus2\p@ \parsep 3\p@ plus2\p@ minus\p@ \itemsep \parsep }\belowdisplayskip \abovedisplayskip Least-squares fit of a straight line (see text).} \label {fig:lineFit}}}{3}} \newlabel{fig:lineFit}{{3}{3}} \@writefile{toc}{\contentsline {section}{\numberline {3}Finding the minimum of $\chi ^2({\unhbox \voidb@x \hbox {\relax \mathversion {bold}$\bf \theta $}})$}{3}} \newlabel{sec:minim}{{3}{3}} \@writefile{toc}{\contentsline {subsection}{\numberline {3.1}Finding LS estimators in closed form}{4}} \newlabel{sec:line}{{3.1}{4}} \newlabel{eq:polynomial}{{4}{4}} \newlabel{eq:chi2line}{{5}{4}} \newlabel{eq:dchi2dt0}{{6}{4}} \newlabel{eq:dchi2dt1}{{7}{4}} \newlabel{eq:matrixform}{{8}{4}} \newlabel{eq:matrixform2}{{9}{4}} \newlabel{eq:matrixA}{{10}{4}} \citation{bib:Cowan98} \citation{bib:brandt} \newlabel{eq:theta0hat}{{11}{5}} \newlabel{eq:theta1hat}{{12}{5}} \@writefile{toc}{\contentsline {subsection}{\numberline {3.2}Numerical minimization of $\chi ^2({\unhbox \voidb@x \hbox {\relax \mathversion {bold}$\bf \theta $}})$}{5}} \newlabel{sec:numer}{{3.2}{5}} \@writefile{toc}{\contentsline {section}{\numberline {4}Statistical errors of the fitted parameters}{6}} \newlabel{sec:staterr}{{4}{6}} \@writefile{toc}{\contentsline {subsection}{\numberline {4.1}Meaning of statistical errors and estimation by Monte 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minus\p@ \itemsep \parsep }\belowdisplayskip \abovedisplayskip Fit resulting in a poor level of agreement between the hypothesized fit function and the data.} \label {fig:badFit}}}{11}} \newlabel{fig:badFit}{{6}{11}} \newlabel{eq:chisqmin}{{28}{11}} \newlabel{eq:normresid2}{{29}{12}} \newlabel{eq:chisqpdf}{{30}{12}} \@writefile{lof}{\contentsline {figure}{\numberline {7}{\ignorespaces {\relax \fontsize {10}{12}\selectfont \abovedisplayskip 10\p@ plus2\p@ minus5\p@ \abovedisplayshortskip \z@ plus3\p@ \belowdisplayshortskip 6\p@ plus3\p@ minus3\p@ \def \leftmargin \leftmargini \parsep 4.5\p@ plus2\p@ minus\p@ \topsep 9\p@ plus3\p@ minus5\p@ \itemsep 4.5\p@ plus2\p@ minus\p@ {\leftmargin \leftmargini \topsep 6\p@ plus2\p@ minus2\p@ \parsep 3\p@ plus2\p@ minus\p@ \itemsep \parsep }\belowdisplayskip \abovedisplayskip Distribution of the value of $\chi ^2_{\rm min}$ from Monte Carlo simulations of the measurement and fit of a straight line. The solid curve shows the chi-squared pdf for $n=7$ degrees of freedom.} \label {fig:chisqPlot}}}{12}} \newlabel{fig:chisqPlot}{{7}{12}} \citation{bib:Cowan98} \newlabel{eq:chisqmean}{{31}{13}} \newlabel{eq:pval}{{33}{13}} \newlabel{eq:quadraticf}{{34}{14}} \@writefile{lof}{\contentsline {figure}{\numberline {8}{\ignorespaces {\relax \fontsize {10}{12}\selectfont \abovedisplayskip 10\p@ plus2\p@ minus5\p@ \abovedisplayshortskip \z@ plus3\p@ \belowdisplayshortskip 6\p@ plus3\p@ minus3\p@ \def \leftmargin \leftmargini \parsep 4.5\p@ plus2\p@ minus\p@ \topsep 9\p@ plus3\p@ minus5\p@ \itemsep 4.5\p@ plus2\p@ minus\p@ {\leftmargin \leftmargini \topsep 6\p@ plus2\p@ minus2\p@ \parsep 3\p@ plus2\p@ minus\p@ \itemsep \parsep }\belowdisplayskip \abovedisplayskip Fit to the same data values as in Fig.\nobreakspace {}6\hbox {} but here using a fit function with three adjustable parameters.} \label {fig:badPoly}}}{14}} \newlabel{fig:badPoly}{{8}{14}} \@writefile{toc}{\contentsline {section}{\numberline {6}Least squares with correlated measurements}{14}} \newlabel{sec:corr}{{6}{14}} \newlabel{eq:chi2basic2}{{35}{14}} \newlabel{eq:corrcoeff}{{36}{14}} \citation{bib:Cowan98} \citation{bib:drake} \newlabel{eq:corrls}{{37}{15}} \newlabel{sec:exer}{{6}{15}} \newlabel{sec:poly}{{6}{15}} \newlabel{eq:deltadef}{{38}{15}} \@writefile{lof}{\contentsline {figure}{\numberline {9}{\ignorespaces {\relax \fontsize {10}{12}\selectfont \abovedisplayskip 10\p@ plus2\p@ minus5\p@ \abovedisplayshortskip \z@ plus3\p@ \belowdisplayshortskip 6\p@ plus3\p@ minus3\p@ \def \leftmargin \leftmargini \parsep 4.5\p@ plus2\p@ minus\p@ \topsep 9\p@ plus3\p@ minus5\p@ \itemsep 4.5\p@ plus2\p@ minus\p@ {\leftmargin \leftmargini \topsep 6\p@ plus2\p@ minus2\p@ \parsep 3\p@ plus2\p@ minus\p@ \itemsep \parsep }\belowdisplayskip \abovedisplayskip Result of least-squares fit with the one-sigma error shown as a shaded band.} \label {fig:errorBand}}}{16}} \newlabel{fig:errorBand}{{9}{16}} \newlabel{sec:gal}{{6}{16}} \@writefile{lof}{\contentsline {figure}{\numberline {10}{\ignorespaces {\relax \fontsize {10}{12}\selectfont \abovedisplayskip 10\p@ plus2\p@ minus5\p@ \abovedisplayshortskip \z@ plus3\p@ \belowdisplayshortskip 6\p@ plus3\p@ minus3\p@ \def \leftmargin \leftmargini \parsep 4.5\p@ plus2\p@ minus\p@ \topsep 9\p@ plus3\p@ minus5\p@ \itemsep 4.5\p@ plus2\p@ minus\p@ {\leftmargin \leftmargini \topsep 6\p@ plus2\p@ minus2\p@ \parsep 3\p@ plus2\p@ minus\p@ \itemsep \parsep }\belowdisplayskip \abovedisplayskip The configuration Galileo's ball and ramp experiment.} \label {fig:ramp}}}{16}} \newlabel{fig:ramp}{{10}{16}} \@writefile{lot}{\contentsline {table}{\numberline {1}{\ignorespaces }}{16}} \newlabel{tab:galdat}{{1}{16}} \citation{bib:pedersen} \newlabel{d_alpha_h}{{39}{17}} \newlabel{d_alpha_beta_h}{{40}{17}} \newlabel{ch7:d_vs_h1}{{41}{17}} \newlabel{sec:ptol}{{6}{17}} \@writefile{lof}{\contentsline {figure}{\numberline {11}{\ignorespaces {\relax \fontsize {10}{12}\selectfont \abovedisplayskip 10\p@ plus2\p@ minus5\p@ \abovedisplayshortskip \z@ plus3\p@ \belowdisplayshortskip 6\p@ plus3\p@ minus3\p@ \def \leftmargin \leftmargini \parsep 4.5\p@ plus2\p@ minus\p@ \topsep 9\p@ plus3\p@ minus5\p@ \itemsep 4.5\p@ plus2\p@ minus\p@ {\leftmargin \leftmargini \topsep 6\p@ plus2\p@ minus2\p@ \parsep 3\p@ plus2\p@ minus\p@ \itemsep \parsep }\belowdisplayskip \abovedisplayskip The apparatus used by Ptolemy to investigate the refraction of light.} \label {fig:ptol}}}{18}} \newlabel{fig:ptol}{{11}{18}} \@writefile{lot}{\contentsline {table}{\numberline {2}{\ignorespaces }}{18}} \newlabel{tab:ptol}{{2}{18}} \newlabel{refrac_prop}{{42}{18}} \newlabel{refrac_quad}{{43}{18}} \newlabel{snell}{{44}{18}} \newlabel{sec:algo}{{6}{19}} \newlabel{eq:taylor2}{{45}{19}} \newlabel{eq:dfdx}{{46}{19}} \newlabel{eq:xmin}{{47}{19}} \citation{bib:brandt} \bibcite{bib:brandt}{1} \bibcite{bib:Cowan98}{2} \bibcite{bib:FormulaSheet}{3} \bibcite{bib:drake}{4} \bibcite{bib:pedersen}{5} \newlabel{eq:gradf}{{48}{20}}