# PV Fault Detection Using Positive Unlabeled Learning
## 4 Discussion  
In this section, we discuss the results provided in Section 3 in some depth. As each graph in Figure 2 provides results for a different fault type and model, we break these results down separately and scrutinize each individually. All Faults vs. No Faults  
In the top graph in Figure 2a, we see that all algorithms including the MLR, MLRf, and \"tiny\" supervised learning algorithm behaved well and provided similar results, even with only 2% of total fault data labeled. As illustrated in Table 2, 344 datapoints out of 21,485 total datapoints were labeled at this lowest \(c\) value. It should be noted that treating unlabeled samples as negatives, as illustrated by the naive PU algorithm, is ineffective unless nearly all faulty points (17,188 total) are labeled. In the bottom graph, it is clear that both the MLR and MLRf algorithms slightly and consistently outperform the oracle and the \"tiny\" supervised learning algorithms when at least 10% of the positive samples are labeled. We believe this may be due in part to the non-linear classification capabilities of both algorithms. This is discussed further in the next bullet point. Shaded vs. All Others  
Figure 2b demonstrates a situation where the nonlinear nature of the MLRf algorithm provides a clear advantage. The poor results of the supervised linear oracle model (with an F-Score of approximately 0.64) indicate that the faulty and non-faulty data for this problem are non-separable in the given feature space. The oracle model is underfitting the data. The improvement gained by the MLRf algorithm with its much higher feature dimensionality confirms this. If the decision boundary is substantially nonlinear, this could explain the noticeable F-score improvement of the non-linear MLRf classifier. A future test should be performed against a nonlinear oracle model for confirmation. The more surprising result in this graph is the improvement made possible by the simpler MLR algorithm. The MLR algorithm has one additional variable over the oracle (the \(b\) variable described in Section 2.2), and we surmise that this slight nonlinearity may be contributing to its success. It is remarkable that these high scores are possible even when only 4% of the faulty data is labeled, or 172 datapoints. Soiled vs. All Others  
The soilage detection models in Figure 2c act similarly to those in Figure 2a in that the MLR, MLRf, and the \"tiny\" supervised algorithm are similar to that of the Oracle except at low values of \(c\). Unlike the other graphs in Figure 2, MLRf performance increases noticeably above the Oracle only when the label frequency \(c\) is around 70%--much higher than in other graphs. However, the actual difference is slight and may simply indicate an upward trend like that of the MLR algorithm. Due to the small number (five) of runs that we were able to do for this algorithm on this problem at that \(c\) value, the jump at \(c=0.7\) may be a random outlier. Additional simulations would need to be performed to test this hypothesis. between algorithms at all. Despite the obvious separability of this problem, neither the naive PU algorithm nor the unsupervised k-means algorithm were useful. * Short Circuit vs. All Others  
Figure 2e illustrates our most interesting and enigmatic result. The oracle algorithm and related \"tiny\" supervised algorithm are unable to classify short circuit faults in the given feature space. The MLR and MLRf algorithms, while performing poorly with an F-score between 0.4 and 0.6, are nevertheless substantially more effective. Unlike the Shaded problem shown in Figure 2b, higher feature dimensionality alone is not sufficient to explain this discrepancy as the lower dimensional MLR algorithm performed better than the higher dimensional MLR algorithm. One thought is that the MLRf algorithm with \(p=3\) and \(k=3\) expanded features may be overfitting the problem while the slight increase in dimensionality provided by the MLR algorithm may be preferable, though this conclusion is not particularly satisfying.