deletions | additions       diff --git a/Background estimation.tex b/Background estimation.tex  index 51094d0..0c70d62 100644  --- a/Background estimation.tex  +++ b/Background estimation.tex 
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\subsection{Background estimation}  \label{sec:bg_calc}  The first step of smFRET analysis involves estimating the background level. In fact, even when no molecule is crossing the excitation volume, there are "counts" originating from detectors dark counting rates (DCR), samples scattering and auto-fluorescence. Figure~\ref{fig:bgdist} shows the typical distribution of timestamps delays (waiting times between two subsequent timestamps) As noted  in a smFRET measurement. The "tail" of the distribution (a line in semi-log scale) corresponds to exponentially-distributed delays, indicating that a Poisson process is generating those counts. At short timescales, the distribution departs from the exponential due to the bursts of photons from diffusing single-molecules (the signal).  To estimate section~\ref{sec:bg_intro}, estimating  the background rate, (i.e. rate involves estimating  the exponential time constant) we need to select rate of  a minimum threshold above which the distribution can be considered exponential. Poisson process.  In practice we need to estimate the background for all the different photon streams. Furthermore, since the background rate can typically change during the measurement on time scales of tens of seconds or minutes, we want to estimate it periodically.   In FRETBursts we call \textit{background period} the time duration for each background estimation. For example, to To  compute the background every 30s, using a threshold of 2 ms for all the photon streams we execute: \begin{verbatim}  d.calc_bg(bg.exp_fit, time_s=30, tail_min_us=2000) 
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(MLE) of the delays distribution. More fitting functions are available in   \verb|bg| namespace (see the   \href{http://fretbursts.readthedocs.org/en/latest/background.html}  {\texttt{background} module}). The second argument, \verb|time_s|, is the background   period \textit{background period} (section~\ref{sec:bg_intro})  and the third argument is the threshold above which the distribution issupposed to be  exponential. It is possible to use a different threshold for each photon stream passing a tuple as \verb|tail_min_us| (instead   of a scalar). For an ALEX measurement the tuple needs to have 5 values   corresponding to thresholds for the 5 photon streams. The list of photon   streams for a \verb|Data| object can be found in the \verb|ph_streams|   attribute (in the present example \verb|d.ph_streams|). Finally, is possible to use a heuristic estimation of the threshold using   \verb|tail_min_us='auto'|. For more details refer to the   \href{http://fretbursts.readthedocs.org/en/latest/data_class.html#fretbursts.burstlib.Data.calc_bg}{\texttt{calc\_bg} documentation}.  \subsubsection{Choice of the \subsubsection{Optimal  threshold} \begin{itemize}  \item heuristic Kolgomorov-Smirnov  \item brute force Cramer von Mise  \end{itemize}