Background Analysis

Background Analysis needs in general to be performed for each new file in order to be able to do a burst search and compute background-corrected burst statistics. The tools available on this page provide a number of options to accomplish this task.

Note that it is also possible to manually enter background rate values for all photon streams in the Burst Search page or in the Settings >> Background Rates tab.

Background Analysis involves several steps, some of which are optional (indicated in brackets in the list below), depending on the chosen approach. This tools offers two approaches to estimate background rates:

• one based on the photon interval histograms (PIHs) computation and and their subsequent tail fit by exponential functions,

• the other based on computing only photon interval arrays and performing a maximum likelihood estimation (MLE) of the exponential parameters characterizing these arrays.

The first approach has the advantage of providing visual feedback and to be adjustable by using more complex fit models, but requires a few extra computation steps which can be time consuming for large files. The second approach is faster, but uses assumptions that may not be always verified. The results of both approaches are available when choosing the PIH fitting approach, which allows comparing them.

Both approaches involve the following steps (optional ones are indicated in brackets):

1. Photon Interval Arrays [and Histograms] Calculation

2. [Photon Interval Histogram Fit]

3. Validation of Results

4. [Definition as Uncorrelated Background Rates]

All steps can be automatically performed successively using the Background Rate Analysis command of the Analysis menu (shortcut: Ctrl-B) or by pressing the Background Analysis (Background) button on the top left of the page.

1. Photon Interval Arrays and Histograms Calculation

1a. MLE Approach

To use the MLE approach, press the Bkgd R MLE button (Green 1a label in the Figure above).

This computes the photon interval arrays of the following photon streams:  FDD, FDA, FAD, FAA, FD, FA and All Photons.

In this notation, the subscript indicates the excitation period and the superscript, the detection channel. The "All Photons" stream comprises all D and A photons, irrespective of when they were emitted (in particular, there could be more of them than in the sum of FD and  FA).

From these photon interval arrays, the average count rates (Average Rates (Hz)) as well as the maximum likelihood estimate (MLE) of the background rates (MLE Bkgd R) are computed. The resulting values are reported in the arrays identified in the green box on the top right of the Figure above.

The calculation of the Background Rates MLE is controlled by the Min Time Lag parameter shown in the green box at the top left of the Figure above.

This minimum time lag parameter allows rejecting very short time lags which originate from photons emitted during single-molecule bursts and should not be taken into account for the calculation of the background rate. Since this value depends on the typical count rate in each stream, it is in general preferable to choose the "automatic" option (checkbox to the right), which computes this value on a stream-by-stream basis, the minimum time time lag being set to k/R, where R is the photon stream's average count rate and k is a scaling factor defined in the ALiX Settings>>Background Analysis page.

The MLE estimate of the background rate b is simply given by:

where the 

's are the time lags larger than the min time lag  defined by the user or automatically.

The average rates are computed using all intervals computed for each stream while the MLE background rate estimates are computed using all time lags larger than the Min Time Lag parameter. In particular, if the "Automatic" option discussed above to define the min time lag has been chosen, it will be used for the MLE calculation.

Another option can be selected in the Background Rates tab of the Settings window. This option, called FRETBursts, for the software implementing it in Python (see Ref. 9 in the Bibliography), first computes a rate according to the formula shown above, setting 

 to the value of MLE Min Time Lag defined in the Settings window. In a second step, this rate b is used to compute a second min time lag as  = 1.7/b. Finally, the MLE estimate of the background rate is calculated according to the formula using the newly defined 

. 

Once done with the background rates MLE calculation, skip to Section 3 below for further instructions.

1b. PIH Approach

To compute the photon interval histograms (PIHs), press the PIH button (number 1b in green in the Figure above).

For further analysis (namely fitting), histograms with equidistant bins are required, which corresponds to the unchecked "Logarithmic Binning" option shown in the figure above. The other histogramming parameters are:

• Min Time Lag (or "Automatic" option)

• Max Time Lag

• Time Bin (Linear)

• Bin Location

The min time lag parameter has already been discussed in section 1a.

The max time lag parameter (Max Time Lag) simply limits the number of bins needed for the histogram. The default value should work in most cases.

The value of the time bin parameter (Time Bin (Linear)) depends on the data file and the purpose of the analysis. In particular, very small time lag bins do not make sense for background rate analysis, as they reduce the bin content of large time lag bins, which are those used for the background rate fit. They will also make the laser alternation visible if chosen smaller than the alternation period use for us ALEX experiments. In those cases, the recommended value is twice the alternation period.

On the other hand, if the purpose is to analyze the effect of laser alternation, a bin significantly smaller than the period is of course preferable.

The bin location parameter (Bin Location) is a matter of personal preference. In general, most histograms in ALiX are represented using a left justified bin, meaning that the coordinate of the bin is equal to its leftmost bound. Whatever the choice, the histogram fits take this into account. displa

In addition to linear bin histograms, it is possible to use logarithmic binning (Logarithmic Binning option), which is characterized by the number of bins per decade (Bin/Decade parameter). A particular case of logarithmic representation is that introduced by Sigworth & Sine, where the bin size is logarithmically arranged, but the bin content is not normalized.

2. Photon Interval Histograms Fit

To fit the PIH, press the Fit button (number 2 in red in the Figure above).

A limited set of fit models is offered:

In general, the expected distribution of background photon intervals is exponential (1-Exp), but this is only true of the long range intervals. If the histogram includes short time lags, the distribution might be multi-exponential (2-Exp, 3-Exp) or even a sum of exponential and a power law (1-Exp + Power Law) as discussed by Gopich & Szabo.

In addition to choosing the fit model, two other options are available:

• Fit Method

• Weighted Fit

A description of these options can be found on the Fit Algorithms page of the manual.

Once the fits are performed (and successful), the resulting curves will be displayed on the graph. Fitted parameters will be added to the Notebook and displayed in the parameter array located to the right of the respective plot legend (red box in the Figure above). Since more than one parameter are available, the different parameters can be accessed using the component selector (up and down arrow to the left on the snapshot below) and the amplitude/rate selector (list shown below). When the symbol R is visible, the parameter to the right is the Rate value (in Hz), while the A symbol indicates an Amplitude parameter. For a single component fit (1-Exp), index 1 should be selected (all other components will be equal to zero). For n components (e.g. 2-Exp or 1-Exp + Power Law: n =2, 3-Exp: n = 3), the index can run from 1 to n.

Multiple fit options

In some cases, using a fixed Min Time Lag parameter (or a single model, or any other of the combination of options) for all streams might not yield satisfactory results for all histograms. In this case, it is possible to keep the fitted parameter of selected curves fixed and change the options in order to fit the remaining histograms again with a different set of options.

Fixing the fit parameters of a curve is done by selecting the checkbox to the right of the fitted parameters. Inversely, to allow an histogram to be fitted again, uncheck that checkbox.

To check all checkboxes, press the cross button below the checkboxes. Inversely, to uncheck all checkboxes, press the red square button below the checkboxes. 

3. Validation of Results

Once all background rates have been determined, it is necessary to define which values will be used for burst search. Two options are available: Fitted Rate and MLE Rate, as show in the snapshot below.

If the first option is used (Fitted Rate), pressing on the Accept All Background Rates button (marked 3 in blue on the top Figure) will use the rates displayed in the fit parameters array next to the graph legends. The corresponding values will be copied in the fitted background rates array (Fitted Bkgd R: blue box on the top right of the Figure at the top of the page) and in the Background Rates (Hz) array on the Burst Search page.

If the second option is used (MLE Rate), only the photon interval histograms need to be calculated, as described in Section 1a. The MLE rates calculated in that step and displayed in the MLE Bkgd R (green box in the Figure at the top right of the page) will be used. Note that the background rates MLE are also computed when computing the PIHs, therefore there is no need to do both PIH calculation AND background rate MLE calculation if this option is eventually chosen after a PIH fit.

In both cases, the selected background rates will be stored and displayed in the Background Rates (Hz) array in the ALiX Settings >> Background Rates tab and used for all subsequent operations in ALiX (e.g. burst statistics corrections).

When background rates have been defined, the 4th LED of the Task Completion Indicator at the bottom right of the page lights up.

4. Definition as Uncorrelated Background Rates

If the data set corresponds to a buffer sample, the computed background rates can be saved in memory for future correction of correlation functions (see the corresponding manual page) by right-clicking on either of the rate arrays on the top right of the window and choosing Use as Uncorrelated Bkgd Rate:

This copies the selected array into the Uncorrelated Bkgd R array. Alternatively, background rates can be manually entered in this array.