Example 3: Deconvolution of an IRF, Light-Scattering Correction, and Poisson Statistics

These test data were simulated as Poisson deviates corresponding to biexponential kinetics convoluted with a known (i.e., measured) instrument response function (IRF). In addition, scattered light was added with an amplitude equal to one tenth the amplitude of the pre-convoluted biexponential signal. These simulated exponentials are of the same sign and equal amplitude (50 counts each) and have log lifetimes of -1 and 0, respectively. The IRF was simulated as Poisson deviates corresponding to a Gaussian with a full width at half-maximum of 0.1. Note: An IRF can be deconvoluted from other types of data as well, i.e. from kinetics with phases of opposite sign and/or Gaussian noise.

For a Poisson process, no preliminary estimate of uncertainties is needed; i.e., this analysis requires only one maximum entropy calculation (invert mode) with IGNORE set to 2 in the input parameter file, memdec_poisson.def:

memexp-3.0.exe invert memdec_poisson.def dir3/data.003 100.

Both the noisy data and the noisy IRF must be input to MemExp. Before fitting models to the data, MemExp smooths the IRF and plots the result: irf.003.ps.

Again, the calculations are summarized in data.003.out. The distributed fits are plotted in data.003.out.ps (see Recommendation below). The IRF is plotted in blue in the second column of the first row of plots. The light-scattering parameter $\xi$ is also reported in the second column. In the third column, the Poisson deviance Pd is reported in place of $\chi^2$.

The discrete fits are plotted in data.003.exp.ps.

Note that if a value greater than 100, say 150, is input for D0 at the end of the invert command line above, then the integral of the recovered log-lifetime distribution and the exponential amplitudes will be reduced accordingly. Also, note that the extent of scattered light is estimated well; the value of $\xi$ recovered is reasonable, about 10, or one tenth of the amplitude of the simulated biexponential signal.

Recommendation

As always, the PostScript plots should be studied to evaluate the distributed and discrete fits recommended by MemExp. In this case, the recommendation (iteration 238, denoted by a "*" in data.003.out.ps) appears to be a bit conservative. At this point in the calculation, the values of Pd and $\tau_c$ are still decreasing monotonically without additional peaks appearing in the distribution. The automatic recommendations made by MemExp are based on the changes in $\tau_c$ and Pd (or $\chi^2$) from one plot to the next. Of course, these changes depend on the frequency of plotting specified in memdec_poisson.def. Try rerunning this test case with ITMAX0, NPRINT, and NWRITE increased from 10 to 15. You'll see larger differences in the successive plots and enhanced resolution of the two kinetic phases in the recommended distribution without the addition of spurious maxima. (At the same time, $\xi$ gets closer to the correct value of 10.)