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1 \documentclass[a4paper]{article} 2 3 \usepackage[T1]{fontenc} 4 \usepackage[utf8]{inputenc} 5 6 \usepackage{mathptmx} 7 8 \usepackage[a4paper, total={6in, 8in}]{geometry} 9 \usepackage{subcaption} 10 \usepackage[shortlabels]{enumitem} 11 \usepackage{amsmath,amssymb} 12 \usepackage{amsthm} 13 \usepackage{bbm} 14 \usepackage{graphicx} 15 \usepackage{float} 16 \usepackage[colorlinks=true,naturalnames=true,plainpages=false,pdfpagelabels=true]{hyperref} 17 \usepackage[parfill]{parskip} 18 \usepackage[backend=biber, sorting=none]{biblatex} 19 \addbibresource{uni.bib} 20 \pagestyle{myheadings} 21 \markright{Popovic, Vogel\hfill Unbiased Fitting \hfill} 22 23 \title{Theoretical Physics Lab-Course 2021S\\ University of Vienna \vspace{1.25cm}\\ Unbiased Fitting} 24 \author{Milutin Popovic \\ Tim Vogel \vspace{1cm}\\ Supervisor: Peter Stoffer} 25 \date{April 18, 2021} 26 27 \begin{document} 28 29 \maketitle 30 \begin{abstract} 31 When provided with data that doesn't only come with statistical 32 uncertainties but also with systematic uncertainties, the least square 33 method will not be sufficient enough and will provide false results. 34 This report will emphasize on the D'Agostini bias, which explains the 35 corrrelation between these uncertainties and give an example of data 36 provided by particle physics measurements where the bias is present. In 37 this regard the report will also explain how to avoid the D'Agostini 38 bias with the $t_0$-method. 39 \end{abstract} 40 41 \thispagestyle{empty} 42 \tableofcontents 43 44 \newpage 45 \section{Introduction and Motivation} 46 In physics, we often come across the situation, where measured data needs to be 47 approximated by a theoretical model, which requires a set amount of parameters. 48 To determine these parameters a so called "data fit" is required, which can be 49 done via different techniques. One of the most widely used fit-techniques is 50 "Least-squares fitting", but as soon as we deal with correlated data points, 51 which means, each data point is not a completely independent measurement, a 52 wrong application of the Least-squares fit, can lead to a bias, which, in 53 return, will affect the accuracy of the fit in a negative way. This so called 54 D'Agostini bias, albeit a very situational phenomenon, has to be considered 55 when dealing with correlated data from one or even more experiments. It can be 56 avoided by implementing a iterative fit method and we will consider this in an 57 example from particle physics. The pion Vector Form Factor is a perfect example 58 for this bias, as the experimental (even after many exact experiments) still 59 doesn't fit the predictions of the theoretical model perfectly. In this report 60 we will fit experimental data to the theoretical model, by implementing the so 61 called "$t_0-model$" into the fit. 62 \section{Physical background and Findings} %wird aus mehreren Teilen bestehen, nur als Platzhalter% 63 \subsection{The Vector Form Factor of Pions} 64 65 In particle physics, one of the best to study reactions of elementary 66 particles, is the collision between an electron $(e^-)$ and it's anti-particle, 67 the positron ($e^+)$. When these two particle collide, they annihilate each 68 other and produce new types of particles. In these experiments very precise 69 measurements can be taken and as such, be a very valuable base of empirical 70 data of the Standard model of physics. A central point of study, of these 71 electron-positron-collisions has been the anomalous magnetic moment g-2 of the 72 muon. The anomaly of this number comes from the fact, that the measured data 73 differs to the theoretical model by quite a large margin. As such it could be 74 the source of exciting discoveries. The theoretical value of the g-2 momentum 75 relies on data from the aforementioned collisions, which is used to reconstruct 76 the so called hadronic vacuum polarization. The hadronic vacuum polarization 77 itself comes from the hadronic final states. About 70\% of the contribution to 78 the g-2 momentum comes from the annihilation of an electron and a positron into 79 two pions. The probability of this happening is dependent on the energy of the 80 two particles. The strong interaction between these two pions is given by the 81 so called pion vector form factor (VFF, $F_\pi^V$). To obtain the pion VFF, we 82 start with a classical damped driven oscillator, which can be described as 83 84 \begin{align} 85 \frac{d^2x}{dt^2}+\gamma\frac{dx}{dt}+\omega_0^2x=A\cos{(\omega t)} 86 \end{align} 87 88 that has a solution $x(t)=K\cos{(\omega t-\Phi)}$ The coefficient K describes the 89 amplitude, which is a function of the natural frequence $\omega_0$ of the 90 oscillator, the damping coefficient $\gamma$, the driving frequency $\omega$ 91 and the amplitude A: 92 93 \begin{align} 94 K^2=\frac{A^2}{(\omega^2-\omega_0^2)^2+\gamma^2\omega^2} 95 \end{align} 96 97 If now $\omega_0^2>\frac{\gamma^2}{2}$, the peak of the ampöitudes appears at the 98 resonance frequency \newline 99 $\omega=\omega_R:=\sqrt{\omega_0^2-\frac{\gamma^2}{2}}$. This phenomenon can be 100 transferred to the pion VFF, as a very similar effect happens there. An 101 unstable particle, the $\rho$ meson with a mass of $M_\rho=0.77GeV$ and a decay 102 width $\Gamma_\rho=0.15GeV$, acts as a resonance. Now the parameters need to be 103 transcribed to the relativistic particle-physics context. The driving frequency 104 is replaced by the invariant squared energy s, the resonance mass takes the 105 place of the natural frequency and the decay width acts as the damping 106 coefficient. As such, we obtain the Breit-Wigner form of the VFF: 107 \begin{align} 108 |F_\pi^V(s)|^2_{BW,\rho}=\frac{M_\rho^4}{(M_\rho^2-s)^2+\Gamma_\rho^2M_\rho^2} 109 \end{align} 110 This can also be written in complex form, as: 111 \begin{align} 112 F_\pi^V(s)_{BW,\rho}=\frac{M_\rho^2}{M_\rho^2-s-iM_\rho\Gamma_\rho} 113 \end{align} 114 Although representing the VFF quite well, it is still very simplistic and has 115 to be modified. By implementing another resonance contribution, the Vector Form 116 Factor can be brought into the form: 117 \begin{align} 118 F_\pi^V(s)_{BW,\rho+\omega}=\frac{M_\rho^2}{M_\rho^2-s-iM_\rho\Gamma_\rho(s)}\times(1+\epsilon_\omega\frac{s}{M_\omega^2-s-iM_\omega\Gamma_\omega})\times(1+as+bs^2+cs^3) 119 \end{align} 120 or the modulus given as: 121 \begin{align} 122 |F_\pi^V(s)|^2=\frac{M_\rho^4}{(M_\rho^2-s)^2+M_\rho^2\Gamma_\rho(s)^2}\times(1+\epsilon_\omega\frac{2s(M_\omega^2-s)}{(M_\rho^2-s)^2+M_\rho^2\Gamma_\rho(s)^2}\times(1+as+bs^2+cs^3) 123 \end{align} 124 \subsection{The D'Agostini bias} 125 126 The D'Agostini bias was first introduced by Giuilo D'Agostini in 1994. It 127 describes a problem with data-fits, when considering data with overall 128 systematic errors, that share a uncertainty on the normalization factor. In 129 such a situation, if the error matrix $V$ of the data points is known, one 130 would normally minimize the $\chi^2$, which can be obtained by 131 \begin{align} 132 \chi^2=\vec{\Lambda}^T\cdot V^{-1}\cdot\vec{\Lambda} 133 \end{align} 134 In this formula, $\Lambda$ denotes the vector between the values of the theoretical model and 135 the measured ones. But, after carrying out such a fit, one often obtains 136 results, which contradict expectations. For example, if we got the results 137 $8.0\pm 2\%$ and $8.5\pm 2\%$, from a measurement, which share a $10\%$ 138 normalization error, if we minimized the $\chi^2$ as described-with the matrix 139 $V$ estimated by the data, we would obtain the value $7.87\pm 0.81$. This 140 result should immediately take attention, as the result with the highest 141 probability, lies outside the range of the measured values. This error also 142 occurs in a situation, where data is taken from two or more independently 143 conducted experiments, which are afflicted by an additional systematic 144 normalization error, even though the dimensions of the error are not quite as 145 severe as in the situation described before. 146 147 \subsection{Iterative solution to the D'Agostini bias} 148 The proposed solution of the D'Agostini bias, that avoids problems with 149 150 multiple experiments or quadraticity with the parameters, is as follows. The 151 covariance matrix is constructed, not by the fit result, but by a fixed guessed 152 value $y_0=f(x,\Vec{p_0})$. For one experiment, we then obtain: The covariance 153 matrix calculated by: 154 \begin{align} 155 [Cov_{ij}]_{syst}=\zeta_{ij}^2|F_pi^V(s_i)|^2|F_pi^V(s_j)|^2 156 \end{align} 157 can be wrtitten as: 158 \begin{align} 159 [Cov(y_i,y_j)]_{syst}=\begin{pmatrix}{rr} 160 \zeta^2y_0^2 & \zeta^2y_0^2 \\ 161 \zeta^2y_0^2 & \zeta^2y_0^2 162 \end{pmatrix}=\begin{pmatrix}{rr} 163 \zeta^2f(x_1,\Vec{p_0})^2 & \zeta^2f(x_1,\Vec{p_0})f(x_2,\Vec{p_0}) \\ 164 \zeta^2f(x_1,\Vec{p_0})f(x_2,\Vec{p_0}) & \zeta^2f(x_2,\Vec{p_0})^2 165 \end{pmatrix} 166 \end{align} 167 168 or for two independent experiments: 169 170 \begin{align} 171 [Cov(y_i,y_j)]_{syst}=\begin{pmatrix}{rr} 172 \zeta_1^2y_0^2 & 0 \\ 173 0 & \zeta_2^2y_0^2 174 \end{pmatrix}=\begin{pmatrix}{rr} 175 \zeta_1^2f(x_1,\Vec{p_0})^2 & 0 \\ 176 0 & \zeta_2^2f(x_2,\Vec{p_0})^2 177 \end{pmatrix} 178 \end{align} 179 180 This conserves the quadraticity of the error functions parameters, that appear 181 in the linear model. After solving this model, one obtains new estimates for 182 the parameters $\Vec{p}$. With these solutions a new systematic covariance can 183 be constructed and with each iteration, the new values are used for the next 184 construction. This iterative solutions is called the "$t_0-method$". 185 \subsection{Code Structure} 186 Here the logical structure of the code is shown. 187 \begin{itemize}[noitemsep] 188 \item Construct statistical covariance matrix 189 \item Construct Jacobi matrix of model function in terms of the parameters 190 \item Guess initial parameters $\vec{p_0}$ 191 \item Iterate 192 \begin{itemize}[noitemsep] 193 \item[-] Construct System covariance matrix with $\vec{p_{i}}$ 194 \item[-] Fill Jacobi matrix with $\vec{p_{i}}$ which is the Design Matrix 195 \item[-] Calculate step $\delta \vec{p_{i}}$ 196 \item[-] update initial parameters 197 \begin{align*} 198 \vec{p}_{i+1} = \vec{p_{i}} + \alpha \cdot \delta \vec{p_{i}} \;\;\;\;\; \alpha = 0.1 199 \end{align*} 200 \end{itemize} 201 \item Calculate errors 202 \item Calculate $\chi^2_{min}$ 203 \end{itemize} 204 205 The guess used for all fits was determined by standard least-square fit 206 provided by \texttt{scipy}. 207 \begin{align} 208 \vec{p_0} = 209 \begin{pmatrix} 210 900,\; 200,\; 810,\; 40,\; 20,\; -1000,\; 840,\; 1550 211 \end{pmatrix} 212 \end{align} 213 214 The code can be viewed and/or downloaded from here \cite{code} 215 (including the calculation of the guess parameters). 216 \newpage 217 \section{Findings} 218 219 In this section the results with consideration of the D'Agostini are shown. 220 Furthermore the findings with fits of two experiments together 221 (6 combinations of two) and also a fit with all experiments 222 are shown. In the end of the section the fitted parameters are compared with 223 the literature values. The plots of the given data and their fits can be found in Section \ref{plots}. 224 225 \subsection{Single Experiment Fits under consideration of the D'Agostini bias} 226 In this section the data is fitted under consideration of the D'Agostini bias of all experiments separately is shown. 227 \begin{table}[H] 228 \caption{Results of all experiment data fitted separately\label{tabsingle}} 229 \centering 230 \begin{tabular}{|c|c|c|c|c|} 231 \hline 232 $\vec{p}$ & SND & CMD2 & KLOE & BABAR \\ \hline 233 $M_{\rho}$ [MeV] & $772.72 \pm 0.59$ & $773.93 \pm 0.67$ &$773.91 \pm 0.25 $&$773.33 \pm 0.43$ \\ 234 $\Gamma_{\rho}$ [MeV] & $149.53 \pm 1.15$ & $147.67 \pm 1.32$ &$149.72 \pm 0.37 $&$149.19 \pm 0.81$\\ 235 $M_{\omega}$ [MeV] & $781.94 \pm 0.09$ & $782.32 \pm 0.07$ &$782.44 \pm 0.11 $&$782.18 \pm 0.07$\\ 236 $\Gamma_{\omega}$ [MeV] & $8.55 \pm 0.33 $ & $8.65 \pm 0.44$ &$9.66 \pm 0.33 $&$8.17 \pm 0.16$\\ 237 $\varepsilon_{\omega}$ [] & $2.02 \pm 0.09 $ & $1.92 \pm 0.12$ &$2.07 \pm 0.05 $&$1.95 \pm 0.03$\\ 238 \hline \hline 239 $\chi^2_{min}/dof$ & $1.001$&$ 1.054$&$ 1.443$&$ 1.031$\\ 240 $p\text{-value}$ & $0.530$&$ 0.395$&$ 0.001$&$ 0.377$\\ 241 \hline 242 \end{tabular} 243 \end{table} 244 245 \subsection{Multi Experiment Fits under consideration of the D'Agostini bias} 246 In this section the data is fitted considering the D'Agostini bias, first the data of 247 two experiments together then the data of all experiments is shown. 248 249 \begin{table}[H] 250 \caption{Results of data fits of experimental data fitted in pairs\label{tabtwo1}} 251 \centering 252 \begin{tabular}{|c|c|c|c|} 253 \hline 254 $\vec{p}$ & SND-CMD2 & SND-KLOE & SND-BABAR \\ \hline 255 $M_{\rho}$[MeV] & $772.72 \pm 0.42$ & $773.92 \pm 0.23$ &$773.17 \pm 0.36 $ \\ 256 $\Gamma_{\rho}$[MeV] & $149.53 \pm 0.81$ & $149.42 \pm 0.35$ &$149.70 \pm 0.64 $\\ 257 $M_{\omega}$ [MeV] & $781.95 \pm 0.07$ & $782.39 \pm 0.07$ &$782.07 \pm 0.06 $\\ 258 $\Gamma_{\omega}$ [MeV] & $8.56 \pm 0.24 $ & $9.42 \pm 0.20$ &$8.27 \pm 0.13 $\\ 259 $\varepsilon_{\omega}$[] & $2.02 \pm 0.07 $ & $2.07 \pm 0.05$ &$1.96 \pm 0.03 $\\ 260 \hline \hline 261 $\chi^2_{min}/dof$ & $0.904$&$ 1.839$&$ 0.945$\\ 262 $p\text{-value}$ & $0.754$&$ 0.001$&$ 0.763$\\ 263 \hline 264 265 \end{tabular} 266 \end{table} 267 268 269 \begin{table}[H] 270 \caption{Results of data fits of experimental data fitted in pairs\label{tabtwo2}} 271 \centering 272 \begin{tabular}{|c|c|c|c|} 273 \hline 274 $\vec{p}$ & CMD2-KLOE & CMD2-BABAR & KLOE-BABAR \\ \hline 275 $M_{\rho}$ [MeV] & $773.92 \pm 0.23$ & $773.17 \pm 0.36$ &$773.66 \pm 0.20 $ \\ 276 $\Gamma_{\rho}$ [MeV] & $149.42 \pm 0.35$ & $149.70 \pm 0.64$ &$149.41 \pm $\\ $M_{\omega}$ [MeV] & $782.39 \pm 0.07$ & $782.07 \pm 0.06$ &$782.49 \pm 0.06 $\\ 277 $\Gamma_{\omega}$ [MeV] & $9.42 \pm 0.02 $ & $8.27 \pm 0.13$ &$8.98 \pm 0.12 $\\ 278 $\varepsilon_{\omega}$ [] & $2.07 \pm 0.05 $ & $1.96 \pm 0.03$ &$1.98 \pm 0.02 $\\ 279 \hline \hline 280 $\chi^2_{min}/dof$ & $1.838$&$ 0.943$&$ 1.470$\\ 281 $p\text{-value}$ & $0.001$&$ 0.772$&$ 0.001$\\ 282 \hline 283 284 \end{tabular} 285 \end{table} 286 287 \begin{table}[H] 288 \caption{Results of data fit of all experimental data fitted together\label{tabmulti}} 289 \centering 290 \begin{tabular}{|c|c|} 291 \hline 292 $\vec{p}$ & Multi-Fit\\ 293 \hline 294 $M_{\rho}$ [MeV] & $773.62 \pm 0.18$ \\ 295 $\Gamma_{\rho}$ [MeV] & $149.42 \pm 0.29$ \\ 296 $M_{\omega}$ [MeV] & $782.36 \pm 0.08$ \\ 297 $\Gamma_{\omega}$ [MeV] & $8.75 \pm 0.08 $ \\ 298 $\varepsilon_{\omega}$[] & $1.96 \pm 0.02 $ \\ 299 \hline \hline 300 $\chi^2_{min}/dof$ & $1.735$\\ 301 $p\text{-value}$ & $0.000$\\ 302 \hline 303 304 \end{tabular} 305 \end{table} 306 307 \subsection{Litrature comparison} 308 In this section the fitted parameters of CMD2-BABAR and the literature values\cite{particleref} are compared. 309 The reason why CMD2-BABAR was chosen, is that it has a value of $\chi^2_{min}$ close to $1$. 310 311 \begin{table}[H] 312 \caption{Result comparison with literature\label{tabref}} 313 \centering 314 \begin{tabular}{|l|c|c|c|} 315 \hline 316 $\vec{p}$ & Literature & CMD2-BABAR & Relative error \\ 317 \hline 318 $M_{\rho}$ [MeV] & $775.26 \pm 0.25$ & $773.17 \pm 0.36$ & $ 0.28 \%$\\ 319 $\Gamma_{\rho}$ [MeV] & $147.80 \pm 0.90$ & $149.70 \pm 0.64$ & $ 1.29 \%$ \\ 320 $M_{\omega}$ [MeV] & $782.65 \pm 0.12$ & $782.07 \pm 0.06$ & $ 0.08\% $ \\ 321 $\Gamma_{\omega}$ [MeV] & $8.49 \pm 0.08 $ & $8.27 \pm 0.13$ & $ 2.60 \%$\\ 322 \hline 323 \end{tabular} 324 \end{table} 325 326 \newpage 327 \subsection{Fitting with D'Agostini bias} 328 Furthermore here we compare the results of the t0-method, with the incorrect method of 329 multiplying the relative systematic uncertainties with the data provided, instead of 330 calculating the systematic uncertainties in regards of the newly calculated parameters in 331 every iteration. For demonstrational purposes only the single fitted experiments are pulled. 332 Again we would like to reference the plots in Section \ref{plots}. In the following table 333 the results of the incorrect method applied to fitting are shown. 334 335 \begin{table}[H] 336 \caption{Results of all experiment data wrongly fitted separately\label{tabwrong}} 337 \centering 338 \begin{tabular}{|c|c|c|c|c|} 339 \hline 340 $\vec{p}$ & SND & CMD2 & KLOE & BABAR \\ \hline 341 $M_{\rho}$ [MeV] & $772.21 \pm 4.97$ & $774.51 \pm 2.59$ &$774.09 \pm 0.25 $&$773.42 \pm 0.54$ \\ 342 $\Gamma_{\rho}$ [MeV] & $151.26 \pm 13.11$ & $145.18 \pm 5.83$ &$149.89 \pm 0.84 $&$149.29 \pm 1.01$\\ 343 $M_{\omega}$ [MeV] & $781.03 \pm 0.47$ & $782.09 \pm 0.32$ &$782.79 \pm 0.18 $&$782.18 \pm 0.07$\\ 344 $\Gamma_{\omega}$ [MeV] & $8.96 \pm 2.25 $ & $8.58 \pm 0.15$ &$11.11 \pm 0.70 $&$8.18 \pm 0.18$\\ 345 $\varepsilon_{\omega}$[] & $2.14 \pm 0.68 $ & $1.85 \pm 0.38$ &$2.15 \pm 0.08 $&$1.94 \pm 0.04$\\ 346 \hline \hline 347 $\chi^2_{min}/dof$ & $0.016$&$ 0.125$&$ 0.903$&$ 1.118$\\ 348 $p\text{-value}$ & $1.000$&$ 1.000$&$ 0.839$&$ 0.099$\\ 349 \hline 350 \end{tabular} 351 \end{table} 352 353 354 \subsection{Plots\label{plots}} 355 \begin{figure}[H] 356 \centering 357 \includegraphics[width=0.8\textwidth]{./plots/SND.png} 358 \caption{SND data fit\label{fig1}} 359 \end{figure} 360 \begin{figure}[H] 361 \centering 362 \includegraphics[width=0.8\textwidth]{./plots/CMD2.png} 363 \caption{CMD2 data fit\label{fig2}} 364 \end{figure} 365 \begin{figure}[H] 366 \centering 367 \includegraphics[width=0.8\textwidth]{./plots/KLOE.png} 368 \caption{KLOE data fit\label{fig3}} 369 \end{figure} 370 \begin{figure}[H] 371 \centering 372 \includegraphics[width=0.8\textwidth]{./plots/BABAR.png} 373 \caption{BABAR data fit\label{fig4}} 374 \end{figure} 375 376 377 378 \begin{figure}[H] 379 \centering 380 \includegraphics[width=0.8\textwidth]{./plots/SND-CMD2.png} 381 \caption{SND and CMD2 fitted togther\label{fig5}} 382 \end{figure} 383 \begin{figure}[H] 384 \centering 385 \includegraphics[width=0.8\textwidth]{./plots/SND-KLOE.png} 386 \caption{SND and KLOE fitted togther \label{fig6}} 387 \end{figure} 388 \begin{figure}[H] 389 \centering 390 \includegraphics[width=0.8\textwidth]{./plots/SND-BABAR.png} 391 \caption{SND and BABAR fitted togther \label{fig7}} 392 \end{figure} 393 394 395 \begin{figure}[H] 396 \centering 397 \includegraphics[width=0.8\textwidth]{./plots/CMD2-KLOE.png} 398 \caption{CMD2 and KLOE fitted togther \label{fig8}} 399 \end{figure} 400 \begin{figure}[H] 401 \centering 402 \includegraphics[width=0.8\textwidth]{./plots/CMD2-BABAR.png} 403 \caption{CMD2 and BABAR fitted togther \label{fig9}} 404 \end{figure} 405 \begin{figure}[H] 406 \centering 407 \includegraphics[width=0.8\textwidth]{./plots/KLOE-BABAR.png} 408 \caption{KLOE and BABAR fitted togther \label{fig10}} 409 \end{figure} 410 411 \begin{figure}[H] 412 \centering 413 \includegraphics[width=0.8\textwidth]{./plots/multi.png} 414 \caption{SND, CMD, KLOE and BABAR fitted togther \label{fig11}} 415 \end{figure} 416 417 \begin{figure}[H] 418 \centering 419 \includegraphics[width=0.8\textwidth]{./plots/wrong-SND.png} 420 \caption{Wrong method, SND data fit\label{fig12}} 421 \end{figure} 422 \begin{figure}[H] 423 \centering 424 \includegraphics[width=0.8\textwidth]{./plots/wrong-CMD2.png} 425 \caption{Wrong method, CMD2 data fit\label{fig13}} 426 \end{figure} 427 \begin{figure}[H] 428 \centering 429 \includegraphics[width=0.8\textwidth]{./plots/wrong-KLOE.png} 430 \caption{Wrong method, KLOE data fit\label{fig14}} 431 \end{figure} 432 \begin{figure}[H] 433 \centering 434 \includegraphics[width=0.8\textwidth]{./plots/wrong-BABAR.png} 435 \caption{Wrong method, BABAR data fit\label{fig15}} 436 \end{figure} 437 438 \section{Conclusion} 439 Under consideration of the D'Agostini bias in the calculations we arrive very close the the literature values in table \ref{tabref}. 440 Furthermore the parameters in table \ref{tabsingle} all make the $\chi^2_{min}/dof$ value converge to $1$, making the goodnes of the fit very 441 good. When fitting two experiments together the results according to the $\chi^2_{min}/dof$ value are only good for SND-CMD2, SND-BABAR 442 and CMD2-BABAR, table \ref{tabtwo1} and \ref{tabtwo2}. Taking this into account we need to choose the experiments fitted together very 443 carefully to arrive at good results, fitting all experiments together for instance does't provide a good $\chi^2_{min}/dof$ value, table 444 \ref{tabmulti}. Furthermore looking at the parameter fits without consideration of the D'Agostini bias we can conclude that the fits are 445 obviously missing something, table \ref{tabwrong}. 446 447 \printbibliography 448 449 \end{document}