The complex analysis of the data measured with the TAS can be reduced to the solution of a linear problem:
d = Ri
given that it relates the measured data (d) with the feedings (i) from which the beta intensity distribution I can be obtained.
R is the response matrix of the detector (meaning the probability that a decay that feeds a certain level gives a count in certain bin of the spectrum). The function R depends on the detector but also of the particular level scheme that is being measured. To be able to extract the value of i from the data d the equation has to be inverted (this equation is also called the "inverse problem").
Unfortunately this can not be done easily because there is similar response to the feeding of adjacent levels when they are at high excitation energies where the level density is high. In other words, this is one of the so-called "ill-posed" problems, for which several sets of parameters can reproduce closely the same data set. Then, to find i, the response has to be obtained for which the branching ratios and a precise simulation of the geometry of the detector are needed. The higher the efficiency of the TAS used, the lower the dependence of the response on the branching ratios will be. Then it is possible to introduce the unknown branching ratios by hand from a plausible guess. A good guess can be calculated by means of the Statistical Model.
Then the procedure to find the feedings is iterative: using the expectation-maximization algorithm to solve the inverse problem, Then the procedure to find the feedings is iterative: using the expectation-maximization algorithm to solve the inverse problem, the feedings are extracted; if they don't reproduce the experimental data, it means that the initial guess of the branching ratios is wrong and has to be changed (of course, it is possible to play with other parameters of the analysis). Repeating this procedure iteratively in a reduced number of steps, the data is finally reproduced.