Index: /trunk/MagicSoft/TDAS-Extractor/Algorithms.tex =================================================================== --- /trunk/MagicSoft/TDAS-Extractor/Algorithms.tex (revision 6377) +++ /trunk/MagicSoft/TDAS-Extractor/Algorithms.tex (revision 6378) @@ -599,4 +599,5 @@ \begin{equation} \tau=\frac{(e\tau)_{i_0^*}}{e_{i_0^*}} +\label{eq:offsettau} \end{equation} @@ -883,3 +884,4 @@ %%% TeX-master: "MAGIC_signal_reco.te" %%% TeX-master: "MAGIC_signal_reco.te" +%%% TeX-master: "MAGIC_signal_reco" %%% End: Index: /trunk/MagicSoft/TDAS-Extractor/Pedestal.tex =================================================================== --- /trunk/MagicSoft/TDAS-Extractor/Pedestal.tex (revision 6377) +++ /trunk/MagicSoft/TDAS-Extractor/Pedestal.tex (revision 6378) @@ -9,56 +9,70 @@ \par -By definition, the noise autocorrelation matrix $B$ and thus the ``pedestal RMS'' +By definition, the $\boldsymbol{B}$ and thus the ``pedestal RMS'' is independent from the signal extractor. -\subsection{Bias and Error} - -Consider a large number of signals (FADC spectra), all with the same -integrated charge $ST$ (true signal). By applying a signal extractor -we obtain a distribution of extracted signals $SE$ (for fixed $ST$ and +\subsection{Bias and Mean-squared Error} + +Consider a large number of same signals $S$. By applying a signal extractor +we obtain a distribution of estimated signals $\widehat{S}$ (for fixed $S$ and fixed background fluctuations $BG$). The distribution of the quantity \begin{equation} -X = SE-ST -\end{equation} - -has the mean $B$ and the RMS $R$ defined by: +X = \widehat{S}-S +\end{equation} + +has the mean $B$ and the Variance $MSE$ defined as: \begin{eqnarray} - B &=& \\ - R &=& \sqrt{<(X-B)^2>} + B \ \ \ \ = \ \ \ \ \ \ \ \ \ \ \ &=& \ \ <\widehat{S}> \ -\ S\\ + R \ \ \ \ = \ <(X-B)^2> &=& \ Var[\widehat{S}]\\ + MSE \ = \ \ \ \ \ \ \ \ \ &=& \ Var[\widehat{S}] +\ B^2 \end{eqnarray} -The parameter $B$ can be called the {\textit{\bf bias}} of the pedestal extractor and $R$ -the RMS of the distribution of $X$ which -depend generally on the size of $ST$ and the size of the background fluctuations $BG$. - -\par - -For the normal image cleaning, knowledge of $B$ is sufficient and the -error $R$ should be known in order to calculate a correct background probability. -\par -Also for the model analysis, $B$ and $R$ are needed if one wants to keep small -signals. +The parameter $B$ is also called the {\textit{\bf BIAS}} of the estimator and $MSE$ +the {\textit{\bf MEAN-SQUARED ERROR}} which combines the variance of $\widehat{S}$ and +the bias. Both depend generally on the size of $S$ and the background fluctuations $BG$, +thus: $B = B(S,BG)$ and $MSE = MSE(S,BG)$. + +\par +Usually, one measures easily the parameter $R$, but needs the $MSE$ for statistical analysis (e.g. +in the image cleaning). +However, only in case of a vanishing bias $B$, the two numbers are equal. Otherwise, +the bias $B$ has to be known beforehand. Note that every sliding window extractor has a +bias, especially at low or vanishing signals $S$. \subsection{Pedestal Fluctuations as Contribution to the Signal Fluctuations} -In case of the calibration with the F-Factor methoid, -the basic relation is: - -\begin{equation} -\frac{(\Delta ST)^2}{^2} = \frac{1}{} * F^2 -\end{equation} - -Here $\Delta ST$ is the fluctuation of the true signal $ST$ due to the -fluctuation of the number of photo-electrons. $ST$ is obtained from the -measured fluctuations of $SE$ ($RMS_{SE}$) subtracting those contributions to the -fluctuations of the -extracted signal which are due to the fluctuation of the pedestal\footnote{% +A photo-multiplier signal yields, to a very good approximation, the +following relation: + +\begin{equation} +\frac{Var[Q]}{^2} = \frac{1}{} * F^2 +\end{equation} + +Here, $Q$ is the signal fluctuation due to the number of signal photo-electrons +(equiv. to the signal $S$), and $Var[Q]$ the fluctuations of the true signal $Q$ +due to the Poisson fluctuations of the number of photo-electrons. Because of: + +\begin{eqnarray} +\widehat{Q} &=& Q + X \\ +Var(\widehat{Q}) &=& Var(Q) + Var(X) \\ +Var(Q) &=& Var(\widehat{Q}) - Var(X) +\end{eqnarray} + +$Var[Q]$ can be obtained from: + +\begin{eqnarray} +Var(Q) &\approx& Var(\widehat{Q}) - Var(\widehat{Q}=0) +\label{eq:rmssubtraction} +\end{eqnarray} + +In the last line of eq.~\ref{eq:rmssubtraction}, it is assumed that $R$ does not dependent +on the signal height\footnote{% A way to check whether the right RMS has been subtracted is to make the ``Razmick''-plot \begin{equation} - \frac{(\Delta ST)^2}{^2} \quad \textit{vs.} \quad \frac{1}{} + \frac{Var[\widehat{Q}]}{<\widehat{Q}>^2} \quad \textit{vs.} \quad \frac{1}{<\widehat{Q}>} \end{equation} @@ -70,40 +84,88 @@ \end{equation} -where $c$ is the photon/ADC conversion factor $/$.}. - -\begin{equation} - (\Delta ST)^2 = RMS_{SE}^2 - R^2 -\label{eq:rmssubtraction} -\end{equation} - -If $R$ does not dependent on the signal height, (as it is the case -for the digital filter, eq.~\ref{eq:of_noise}), then one can retrieve $R$ by -applying the signal extractor on a {\textit{\bf fixed window}} of pedestal events. - -\subsection{Methods to Retrieve Bias $B$ and Errors $R$} - -$R$ is in general different from the pedestal RMS. It cannot be -obtained by applying the signal extractor to pedestal events, especially -for large signals (e.g. calibration signals). -\par -In the case of the digital filter, $R$ is in theory independent from the -signal amplitude $ST$ and depends only on the background $BG$ (eq.~\ref{eq:of_noise}). -It can be obtained from the -fitted error of the extracted signal ($\Delta(SE)_{fitted}$), -which one can calculate for every event or by applying the extractor to a fixed window -of pure background events (``pedestal events''). - -\par - -In order to get the missing information, we did the following investigations: +where $c$ is the photon/ADC conversion factor $/$.} +(as is the case +for the digital filter, eq.~\ref{eq:of_noise}). One can then retrieve $R$ +by applying the signal extractor to a {\textit{\bf fixed window}} of pedestal events, where the +bias vanishes and measure $Var[\widehat{Q}=0]$. + +\subsection{Methods to Retrieve Bias and Mean-Squared Error} + +In general, the extracted signal variance $R$ is different from the pedestal RMS. +It cannot be obtained by applying the signal extractor to pedestal events, because of the +(unknown) bias. +\par +In the case of the digital filter, $R$ is expected to be independent from the +signal amplitude $S$ and depends only on the background $BG$ (eq.~\ref{eq:of_noise}). +It can then be obtained from the calculation of the variance $Var[\widehat{Q}]$ +by applying the extractor to a fixed window of pure background events (``pedestal events'') +and get rid of the bias in that way. Figures~\ref{fig:amp:relmean} through~\ref{fig:df:relmean} +show that the bias vanishes indeed for the used extractors in this TDAS. + +\begin{figure}[htp] +\centering +\includegraphics[height=0.3\textheight]{MExtractTimeAndChargeSpline_Amplitude_Amplitude_Range_01_09_01_10_Run_38993_RelMean.eps} +\vspace{\floatsep} +\includegraphics[height=0.3\textheight]{MExtractTimeAndChargeSpline_Amplitude_Amplitude_Range_01_09_01_10_Run_38995_RelMean.eps} +\vspace{\floatsep} +\includegraphics[height=0.3\textheight]{MExtractTimeAndChargeSpline_Amplitude_Amplitude_Range_01_09_01_10_Run_38996_RelMean.eps} +\caption{MExtractTimeAndChargeSpline with amplitude extraction: +Difference in mean pedestal (per FADC slice) between extraction algorithm +applied on a fixed window of 1 FADC slice (``extractor random'') and a simple addition of +2 FADC slices (``fundamental''). On the top, a run with closed camera has been taken, in the center + an opened camera observing an extra-galactic star field and on the bottom, an open camera being +illuminated by the continuous light of the calibration (level: 100). Every entry corresponds to one +pixel.} +\label{fig:amp:relmean} +\end{figure} + + +\begin{figure}[htp] +\centering +\includegraphics[height=0.3\textheight]{MExtractTimeAndChargeSpline_Rise-and-Fall-Time_0.5_1.5_Range_01_10_02_12_Run_38993_RelMean.eps} +\vspace{\floatsep} +\includegraphics[height=0.3\textheight]{MExtractTimeAndChargeSpline_Rise-and-Fall-Time_0.5_1.5_Range_01_10_02_12_Run_38995_RelMean.eps} +\vspace{\floatsep} +\includegraphics[height=0.3\textheight]{MExtractTimeAndChargeSpline_Rise-and-Fall-Time_0.5_1.5_Range_01_10_02_12_Run_38996_RelMean.eps} +\caption{MExtractTimeAndChargeSpline with integral over 2 slices: +Difference in mean pedestal (per FADC slice) between extraction algorithm +applied on a fixed window of 2 FADC slices (``extractor random'') and a simple addition of +2 FADC slices (``fundamental''). On the top, a run with closed camera has been taken, in the center + an opened camera observing an extra-galactic star field and on the bottom, an open camera being +illuminated by the continuous light of the calibration (level: 100). Every entry corresponds to one +pixel.} +\label{fig:int:relmean} +\end{figure} + +\begin{figure}[htp] +\centering +\includegraphics[height=0.3\textheight]{MExtractTimeAndChargeDigitalFilter_Weights_cosmics_weights.dat_Range_01_14_02_14_Run_38993_RelMean.eps} +\vspace{\floatsep} +\includegraphics[height=0.3\textheight]{MExtractTimeAndChargeDigitalFilter_Weights_cosmics_weights.dat_Range_01_14_02_14_Run_38995_RelMean.eps} +\vspace{\floatsep} +\includegraphics[height=0.3\textheight]{MExtractTimeAndChargeDigitalFilter_Weights_cosmics_weights.dat_Range_01_14_02_14_Run_38996_RelMean.eps} +\caption{MExtractTimeAndChargeDigitalFilter: +Difference in mean pedestal (per FADC slice) between extraction algorithm +applied on a fixed window of 6 FADC slices and time-randomized weights (``extractor random'') +and a simple addition of +6 FADC slices (``fundamental''). On the top, a run with closed camera has been taken, in the center + an opened camera observing an extra-galactic star field and on the bottom, an open camera being +illuminated by the continuous light of the calibration (level: 100). Every entry corresponds to one +pixel.} +\label{fig:df:relmean} +\end{figure} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +In order to calculate bias and Mean-squared error, we proceeded in the following ways: \begin{enumerate} \item Determine $R$ by applying the signal extractor to a fixed window of pedestal events. The background fluctuations can be simulated with different levels of night sky background and the continuous light source, but no signal size - dependency can be retrieved with the method. -\item Determine bias $B$ and resolution $R$ from MC events with and without added noise. - Assuming that $R$ and $B$ are negligible for the events without noise, one can + dependency can be retrieved with this method. +\item Determine $B$ and $MSE$ from MC events with and without added noise. + Assuming that $MSE$ and $B$ are negligible for the events without noise, one can get a dependency of both values from the size of the signal. -\item Determine $R$ from the fitted error of $SE$, which is possible for the +\item Determine $MSE$ from the fitted error of $\widehat{S}$, which is possible for the fit and the digital filter (eq.~\ref{eq:of_noise}). In prinicple, all dependencies can be retrieved with this method. @@ -114,20 +176,24 @@ By applying the signal extractor to a fixed window of pedestal events, we -determine the parameter $R$ for the case of no signal ($ST = 0$). In the case of -all extractors using a fixed window from the beginning (extractors nr. \#1 to \#22 -in section~\ref{sec:algorithms}), the results are by construction the same as calculating -the pedestal RMS. -\par -In MARS, this possibility is implemented with a function-call to: \\ - -{\textit{\bf MJPedestal::SetExtractionWithExtractorRndm()}}. \\ - -In the case of the {\textit{\bf amplitude extracting spline}} (extractor nr. \#23), we placed the -spline maximum value (which determines the exact extraction window) at a random place -within the digitizing binning resolution of one central FADC slice. -In the case of the {\textit{\bf digital filter}} (extractor nr. \#28), the time shift was -randomized for each event within a fixed global extraction window. - -\par +determine the parameter $R$ for the case of no signal ($Q = 0$). In the case of +extractors using a fixed window (extractors nr. \#1 to \#22 +in section~\ref{sec:algorithms}), the results are the same by construction +as calculating the pedestal RMS. +\par +In MARS, this functionality is implemented with a function-call to: \\ + +{\textit{\bf MJPedestal::SetExtractionWithExtractorRndm()}} and/or \\ +{\textit{\bf MExtractPedestal::SetRandomCalculation()}}\\ + +Besides fixing the global extraction window, additionally the following steps are undertaken +in order to assure that the bias vanishes: + +\begin{description} +\item[\textit{MExtractTimeAndChargeSpline}:\xspace] The spline +maximum position -- which determines the exact extraction window -- is placed arbitrarily +at a random place within the digitizing binning resolution of one central FADC slice. +\item[{\textit{MExtractTimeAndChargeDigitalFilter}:\xspace] The second step timing +offset $\tau$ (eq.~\ref{eq:offsettau} gets randomized for each event. +\end{description} The following plots~\ref{fig:sw:distped} through~\ref{fig:amp:relrms:run38996} show results @@ -212,58 +278,5 @@ -\begin{figure}[htp] -\centering -\includegraphics[height=0.3\textheight]{MExtractTimeAndChargeSpline_Amplitude_Amplitude_Range_01_09_01_10_Run_38993_RelMean.eps} -\vspace{\floatsep} -\includegraphics[height=0.3\textheight]{MExtractTimeAndChargeSpline_Amplitude_Amplitude_Range_01_09_01_10_Run_38995_RelMean.eps} -\vspace{\floatsep} -\includegraphics[height=0.3\textheight]{MExtractTimeAndChargeSpline_Amplitude_Amplitude_Range_01_09_01_10_Run_38996_RelMean.eps} -\caption{MExtractTimeAndChargeSpline with amplitude extraction: -Difference in mean pedestal (per FADC slice) between extraction algorithm -applied on a fixed window of 1 FADC slice (``extractor random'') and a simple addition of -2 FADC slices (``fundamental''). On the top, a run with closed camera has been taken, in the center - an opened camera observing an extra-galactic star field and on the bottom, an open camera being -illuminated by the continuous light of the calibration (level: 100). Every entry corresponds to one -pixel.} -\label{fig:amp:relmean} -\end{figure} - - -\begin{figure}[htp] -\centering -\includegraphics[height=0.3\textheight]{MExtractTimeAndChargeSpline_Rise-and-Fall-Time_0.5_1.5_Range_01_10_02_12_Run_38993_RelMean.eps} -\vspace{\floatsep} -\includegraphics[height=0.3\textheight]{MExtractTimeAndChargeSpline_Rise-and-Fall-Time_0.5_1.5_Range_01_10_02_12_Run_38995_RelMean.eps} -\vspace{\floatsep} -\includegraphics[height=0.3\textheight]{MExtractTimeAndChargeSpline_Rise-and-Fall-Time_0.5_1.5_Range_01_10_02_12_Run_38996_RelMean.eps} -\caption{MExtractTimeAndChargeSpline with integral over 2 slices: -Difference in mean pedestal (per FADC slice) between extraction algorithm -applied on a fixed window of 2 FADC slices (``extractor random'') and a simple addition of -2 FADC slices (``fundamental''). On the top, a run with closed camera has been taken, in the center - an opened camera observing an extra-galactic star field and on the bottom, an open camera being -illuminated by the continuous light of the calibration (level: 100). Every entry corresponds to one -pixel.} -\label{fig:int:relmean} -\end{figure} - -\begin{figure}[htp] -\centering -\includegraphics[height=0.3\textheight]{MExtractTimeAndChargeDigitalFilter_Weights_cosmics_weights.dat_Range_01_14_02_14_Run_38993_RelMean.eps} -\vspace{\floatsep} -\includegraphics[height=0.3\textheight]{MExtractTimeAndChargeDigitalFilter_Weights_cosmics_weights.dat_Range_01_14_02_14_Run_38995_RelMean.eps} -\vspace{\floatsep} -\includegraphics[height=0.3\textheight]{MExtractTimeAndChargeDigitalFilter_Weights_cosmics_weights.dat_Range_01_14_02_14_Run_38996_RelMean.eps} -\caption{MExtractTimeAndChargeDigitalFilter: -Difference in mean pedestal (per FADC slice) between extraction algorithm -applied on a fixed window of 6 FADC slices and time-randomized weights (``extractor random'') -and a simple addition of -6 FADC slices (``fundamental''). On the top, a run with closed camera has been taken, in the center - an opened camera observing an extra-galactic star field and on the bottom, an open camera being -illuminated by the continuous light of the calibration (level: 100). Every entry corresponds to one -pixel.} -\label{fig:df:relmean} -\end{figure} - -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%1 \begin{figure}[htp] @@ -324,5 +337,5 @@ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -Figures~\ref{fig:df:distped} +Figures~\ref{fig:df:distped},~\ref{fig:amp:distped} and~\ref{fig:amp:distped} show the extracted pedestal distributions for the digital filter with cosmics weights (extractor~\#28) and the