## 8 Experiments and Results  
We seek to interpret results in terms of the probabilistic models developed in Section 6. Where alignment between the models and results gleaned from their implementations will be taken to imply that there exists a principled basis to which we may fix an understanding of performance, and that this may in turn provide the basis for a technical justification of the use of diffusion-based methods in safety-critical contexts. To make relevant observations we begin by constructing a number of datasets to train diffusion models on. Dataset \(\mathcal{X}\) that underlies the NPM and OPM of Expressions (6) and (8), respectively, is defined as a partition \(\mathcal{X}^{n}\), \(\mathcal{X}^{a}\) that classifies the data as nominal or anomalous. To test the validity of the probabilistic models, and their DPM based implementations, we generate a collection of datasets each based on a unique partition of the data. We first use manual classification to construct a ground truth dataset \(\mathcal{X}_{gt}\) with partition formed by \(\mathcal{X}^{n}_{gt}\), and \(\mathcal{X}^{a}_{gt}\). We then use a method similar to that described in [22] to construct a series of datasets with partitions generated using output from a VAE. Partitions are constructed by first training a VAE to encode and decode the _ascans_ from which dataset \(\mathcal{X}\) is derived. Then percentiles of the mean squared error of _ascan_ reconstruction are used as thresholds to partition \(\mathcal{X}\) so that the time of flight associated with _ascans_ with a reconstruction error less than the threshold percentile value are classified as nominal, and otherwise anomalous. Thus, given the value of the 90 th MSE percentile we construct dataset \(\mathcal{X}_{0:90}\) the union of \(\mathcal{X}^{n}_{0:90}\), and \(\mathcal{X}^{a}_{0:90}\). Each of the constructed datasets is then a unique partition over the data. We test DDPM and CSDI diffusion models on each of the datasets and train them for five epochs. Our implementations of DDPM and CSDI are based on the code provided with [5]. The training of CSDI models requires a missing ratio hyperparameter which isequivalently expressed as the percentage rate of observed data held back as simulated ground truth during training. This percentage regulates the degree to which the conditional mechanism of the model is exercised on the observed nominal data. DDPM models do not exercise a conditional mechanism when training. We test CSDI models on the range of missing data rates \([10\%,20\%,...,50\%]\). For each trained model we sample five estimates for each data-point in the dataset. We take the mean and standard deviation of each set of five samples and then fit a curve to the high confidence means with a standard deviation \(<1\), using an SGF. We then take the mean squared, mean absolute, mean absolute percentage, and maximum absolute error of each fitted curve against a SGF fit of \(\mathcal{X}_{gt}^{n}\). Tables A1-A5 in Appendix A tabulate the results of a series the experiments conducted as described above; results are also summarized in the series of plots given in Figure 3. The plots in Figure 3 reveal consistent dynamics related to the training of the models over the datasets. In general, the CSDI models outperform their DDPM counterparts and achieve the best performance in each error category with the exception of maximum absolute error. The hatched constant vertical lines in the plots of Figure 3 show the performance of the models when trained on the ground truth dataset \(\mathcal{X}_{gt}\). Although there are some exceptions, generally, models trained on the ground truth dataset out-perform their counterparts trained on the MSE threshold based datasets; this suggests that models trained on the ground truth datasets are indeed learning \(\sim\)\(P(x_{gt}^{n}|\mathcal{X}_{gt}^{n})\), while those trained on datasets whose partitions mix data from nominal and anomalous data generating factors are learning some other similar but different distributions.