## 6 A Probabilistic Model for UT Data Analysis  
Given an ordered set of _ascans_ taken as a UT probe passes over the surface of a test subject. Let dataset \(\mathcal{X}\) be the similarly ordered peak amplitude response time of flight of each _ascan_ in the set. Let the ordering provide a one-to-one mapping between each \(x\in\mathcal{X}\) and a unique time and position of data acquisition. Define \(\mathcal{X}\) as the union of nominal and anomalous data subsets \(\mathcal{X}^{n}\) and \(\mathcal{X}^{a}\) so that \(\mathcal{X}\) is partitioned:  
\[\mathcal{X}=\mathcal{X}^{n}\cup\mathcal{X}^{a} \tag{2}\]  
Then according to our understanding of data generating factors \(x\) is distributed as joint probability  
\[x\sim P(x^{n},x^{a}) \tag{3}\]  
Figure 2: The plot in the first row gives the peak amplitude response time of flight for a series of _ascans_ taken by a rotating head inside a PT. The data shown in the first row was acquired over two rotations of the UT probe and the x-axis order the \(t_{pq}\) by time of acquisition. Detail of the two irregular excursions in the curve is given in the second row. The histogram gives the log mean squared error of reconstruction of a VAE trained on the set of _ascans_ from which the data in the plots were taken. The red and cyan lines on the vertical plot are at the mean log MSE and \(2.75\times 10^{-6}\) less than the mean, respectively. Using these two log MSE values as thresholds to partition the data, the red and cyan traces are fit to all data less than the similarly color-coded threshold value using a Savitzky–Golay Filter (SGF). We observe that the traces are quite similar across the first peak, but do not track the nominal trend well. On the second peak, the fits diverge so that the cyan trace appears to represent the nominal trend. However, without ground truth, there is no clear method to select the partition leading to the best line of best fit and no clear path to fault diagnosis. where \(x^{n}\in\mathcal{X}^{n}\) and \(x^{a}\in\mathcal{X}^{a}\). We define dataset \(\mathcal{X}^{u}\) with elements \(x^{u}\in\mathcal{X}^{u}\) to allow reference to unobserved nominal data. The \(x^{u}\) are the nominal signals that would have been observed were it not for the occurrence of an anomalous data generating factor. In terms of causality we might conceive of \(\mathcal{X}^{u}\) as the observations that would have occurred had we intervened to constrain the effect of anomalous data-generating factors. We desire estimates for \(x^{u}\in\mathcal{X}^{u}\). An analyst estimates the unobserved nominal data by first classifying observed data as either nominal or anomalous and then estimating unobserved nominal data on the basis of observed local nominal signal. This estimate relies on an inference process that makes strong use of the observed element wise continuity in \(\mathcal{X}^{n}\). We model this estimation process as sampling the unobserved nominal with a conditional dependence on the observed nominal:  
\[x^{u}\sim P(x^{u}|\mathcal{X}^{n}) \tag{4}\]  
In the unsupervised learning setting we are given no knowledge of membership in \(\mathcal{X}^{n}\), \(\mathcal{X}^{a}\), or \(\mathcal{X}^{u}\). The best we can do directly from the dataset is sample \(x^{u}\sim P(x|\mathcal{X})\). We believe that with some engineering we can do better than this, and make the reasonable assumption that unobserved nominal data are distributed as the observed nominal so that  
\[x^{u}\sim P(\mathcal{X}^{n}) \tag{5}\]  
Substituting \(x^{n}\) for \(x^{u}\) in Expression (4) gives  
\[x^{u}\sim P(x^{n}|\mathcal{X}^{n}) \tag{6}\]  
which, given the assumptions, allows the estimation of \(\mathcal{X}^{u}\) by way of some unsupervised generative model trained to sample from \(P(x^{n}|\mathcal{X}^{n})\) by training on partitioned dataset \(\mathcal{X}\). Because PT are well maintained, and data are acquired with very strict quality controls, we expect that cardinality \(|\mathcal{X}^{n}|\gg|\mathcal{X}^{a}|\).