# Application of the Approximate Bayesian Computation Algorithm to Gamma-Ray Spectroscopy
## 6 A New ABC-Based RIID  
During ABC-based RIID development, one can easily experiment with different summary statistics. Here, a very simple RIID was developed and applied as follows; future work will include factors such as \(f_{LPI},f_{DPI}f_{PP},f_{AB}\) from [8] described above as candidate summary statistics. In the simple RIID, smoothed data (using IBR) was used to locate peaks. Then, only those candidate peaks that were statistically distinct from the nearby background were used. Each significant peak's area was estimated using the nonparametric peak area approach described in Section 4. The goal was to distinguish between four groups. Group 1 is simulated Lu-177m (\"m\" denotes a meta-stable isotope) spectra. Group 2 is the corresponding simulated Lu-177m spectra with model bias. Group 3 is the simulated weapons grade WgPu spectra; group 4 is the corresponding simulated WgPu spectra with model bias. The simulated Lu-177m and WgPu spectra are plotted in Figure 9.  
Results of this very simple RIID are illustrated in Figures 10 and 11. This is a simple RIID example because the library consists of only four groups and because the shape of the Lu177m spectra is not that difficult to distinguish from that of WgPu (although Lu177m was chosen because its spectrum is has certain similar features to that of WgPu that can cause some algorithms to misidentify it). In this ABC-based RIID example, simulated Lu-177m and simulated WgPu are to be distinguished. The simulations were done in GADRAS 18.8.10 [21], and simulation bias was estimated by comparing real to simulated spectra as in Figure 6. Four classes are in the training data for ABC. Summary statistics from the scaled count data in Figure 9, and from the scaled count data in Figure 9 that is modified by adding an estimated model bias vector to the Lu177m spectra, and a second estimated model bias vector to the WgPu spectra. Figure 9: Simulated (smoothed) Lu177m and WgPu spectra in (**a**) and scaled to range from 0 to 1 in (**b**). Figure 10 is a principle coordinate (PC) plot using the cmdscale function in [29]. The two plotted PCs can be used to closely approximate the distance between each of the 100 spectra (25 sets from each of the four groups) and give strong qualitative evidence that Lu177m spectra are distinct from the WgPu spectra. Figure 11 plots the results of a simple ABC-based RIID that aims to predict which group a test items is in (group 1 is Lu177m; group 2 is WgPu; group 3 is simulated Lu-177m with model bias; group 4 is simulated WgPu with model bias). As in Figure 1 for I-131, model bias is evident when any of the real spectra are compared to the corresponding simulated data. Therefore, model bias  
Figure 11: The ABC-based posterior probabilities for each of the 4 models (1–4) in Figure 10 using an ABC-based radioisotope identification (RIID) that uses nonparametric estimates of each detected peak’s net area as summary statistics. The four classes (real and simulated Lu-177m and WgPu) are easily distinguished in this simple example with only 4 classes. This figure of model probabilities is for only 1 test case for illustration. Figure 10: Principle coordinate plot obtained using multidimensional scaling applied to real and simulated spectra for Lu-177m and WgPu. was simulated as follows. First, model bias was estimated for 10 real (background subtracted) and corresponding GADRAS-simulated isotopes as with the I-131 example in Figure 1. Second, principal components (the prcomp function in [29]) \(P_{i}\) of the collection of bias vectors were calculated and the dominant five components were retained in order to capture most of the variability observed among the 10 estimated bias vectors. Third, to generate other model bias vectors \(B\), small random variations \(\varepsilon\) (zero mean, 0.05 standard deviation random normal variables) were added to the mean coefficient vector \(C=\{C_{1},C_{2},C_{3},C_{4},C_{5}\}\) to generate additional bias vectors \(B_{new}=\sum\limits_{1=1}^{5}C_{i}(1+\epsilon)P_{i}\).