I have enjoyed participating in the problem brought up by Paul Nord of analyzing a time series describing the sum of two exponential decays, and a background signal.
I took some time to examine Paul's second data set. As before I used a model which segregated the background count between two time sequences, and swept the partition over time looking for significant steps.I saw no pronounced discontinuity in the background determination of this second time series from Paul. Setting the published Mean lifetime values 'tau' as constants to a non linear regression model provided the following estimates for the remaining three parameters:
Radioactive isotope 1 initial qty or amplitude1/decay const1 = 1163 SE 9.0, Student's t = 130Radioactive isotope 2 initial qty or amplitude2 / decay const2 = 69614 SE 61.8, t = 1126Background count 14.04 counts/min SE 0.03, t = 489
These were close to John's numbers.In addition I provided a slush bucket measure c to provide an indication of any effects otherwise unaccounted for in the model, which I was gratifiedto see was evaluated 0.56, SE 5.8, t = 0.09 which represents a probability of a chance occurance p = 0.93.
the Durbin-Watson statistic of auto-correlation gave 1.128 (where 2 is no correlation and less than 1 is indicative).
The ANOVAR indicated F = 6.8E7 at 3 DOF.
So much for experimental data. I turned now to the literature of multi-exponentials.
I encountered Lanczos' famous example of multi-exponentials. If twenty four data points are fitted by a double exponential f2(t) = 2.202*exp(-4.45*t) + 0.305*exp(-1.58*t) they can also be fit by a triple exponential f3(t) = 0.0951*(exp(-t) + 0.8607*exp( -3*t) + 1.5576*exp(-5t). When graphed, they showed the difference between f2(t) and f3(t) as less than their respective line widths, and so the lines are indistinguishable.
Even worse, resolving parameters of two exponential functions depends on their lambda a/lamda b ratio and on the signal to noise ratio available.
This is mentioned in this excellent review of all methods and difficulties: https://www.researchgate.net/publication/234843836_Exponential_analysis_in_physical_phenomena
Exponential analysis in physical phenomenaFebruary 1999 The Review of scientific instruments 70(2):1233-1257DOI:10.1063/1.1149581Authors: Andrei A. Istratov, Oleg Vyvenko: St.Petersburg State Univ.
There is a pointer to accessible methods for computing solutions here: https://assets.cambridge.org/97805218/80688/frontmatter/9780521880688_frontmatter.pdf