deletions | additions       diff --git a/background-estimation.tex b/background-estimation.tex  index 01d83aa..d4fd787a 100644  --- a/background-estimation.tex  +++ b/background-estimation.tex 
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\label{sec:bg_calc}  The first step of smFRET analysis involves estimating background rates.  For example, to compute the background every 30 s, using a minimal timestamp inter-photon  delay threshold of 2 ms 2~ms  for all the photon streams, execute: photon, we use:  \begin{lstlisting}  d.calc_bg(bg.exp_fit, time_s=30, tail_min_us=2000) 
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The first argument (\verb|bg.exp_fit|) is the underlying function used to fit the  background in each period and for each photon stream (see section~\ref{sec:bg_intro}).  The function  \verb|bg.exp_fit| estimates the background using a Maximum Likelihood Estimation maximum likelihood estimation  (MLE) of the delays distribution. Additional fitting functions are available in  \verb|bg| namespace (see (i.e.  the \verb|background| modulw,  \href{http://fretbursts.readthedocs.org/en/latest/background.html} {\texttt{background} module}). {link}).  The second argument, \verb|time_s|, is the \textit{background period} (section~\ref{sec:bg_intro}) and the third argument third, \verb|tail_min_us|,  is the timestamp inter-photon  delay threshold above which the distribution is assumed exponential. It is possible to use different thresholds for each photon stream stream,  passing a\href{https://docs.python.org/2/tutorial/datastructures.html#tuples-and-sequences}{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 (i.e. a comma-separated  list of photon  streams for values, \href{https://docs.python.org/3.5/tutorial/datastructures.html#tuples-and-sequences}{link}) instead of  a \verb|Data| object can be found in the \verb|ph_streams|  attribute (in the present example \verb|d.ph_streams|). scalar.  Finally, it 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}. \verb|calc_bg| documentation  (\href{http://fretbursts.readthedocs.org/en/latest/data\_class.html#fretbursts.burstlib.Data.calc_bg}{link}).  After the background has been estimated, it is useful to compare the histogram of   photon waiting times inter-photon delays  with an the fitted exponential distribution as shown in figure~\ref{fig:bg_dist_all} (see section~\ref{sec:bg_intro}). This plot is performed with:  \begin{lstlisting} 
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and these impurities tend to bleach on timescales of minutes resulting in  background variations such as the one shown in figure~\ref{fig:bg_timetrace}a.  Another source of background variation, visible in figure~\ref{fig:bg_timetrace}b, is   the evaporation in case the sample was is  placed on the coverglass without an enclosure (gasket). \paragraph{Python details} For an ALEX measurement, the tuple passed to  \verb|tail_min_us| to define the thresholds, is required to have have   5 values corresponding the 5 photon streams.   The ordering of the photon streams can be obtained from  the \verb|Data.ph_streams| attributes (i.e. \verb|d.ph_streams| in our example).  The estimated background rates are stored in the \verb|Data| attributes  \verb|bg_dd|, \verb|bg_ad| and \verb|bg_aa|, corresponding to the photon  streams \verb|Ph_sel(Dex='Dem')|, \verb|Ph_sel(Dex='Aem')| and \verb|Ph_sel(Aex='Aem')|  respectively. These attributes are lists of arrays (one array per excitation spot).  The arrays contain the estimated background rates in the different background periods.  \subsubsection{Error metric and optimal threshold}