## 3 Materials and Methods
### Application and Implementation Details
#### 3.7.1 APS Failure at Scania Trucks Dataset  
The APS Failure at Scania Trucks dataset [61] is an imbalanced dataset meant for failure prediction in the Air Pressure System (APS) of Scania Trucks. APS is an essential part of the vehicle pneumatic system in which pressurized, compressed air is used for power distribution [62]. The APS control unit intelligently manages pressurized air, engaging and disengaging the compressor to regulate energy during functions such as braking and gear changes. APS is particularly useful in compressed air brake systems found in large commercial and passenger vehicles such as trucks, busses, trailers, and railroad trains. As a result, FD systems involving APS are safety-related, owing to the catastrophic consequences of road accidents resulting from brake system failures. Notably, safety-related FD systems exhibiting high rates of false negatives are counterproductive, resulting in faults not only going undetected but incorrectly declared normal. The APS Failure at Scania Trucks dataset [61] has a misclassification cost metric whose objective seeks to minimize the number of false negatives. In particular, false positives (Type I errors) associated with unnecessary checks performed by a mechanic at the workshop bear a penalty cost of 10 each while on the other hand, false negatives (Type IIerrors), associated with missing a faulty truck and eventually leading to a breakdown, bear a penalty cost of 500 each. The dataset consists of sensor data collected from APS equipment in heavy Scania trucks mapping out their everyday usage. The result is a labeled dataset in which the positive class represents failures from a specific component in the APS [61]. However, the dataset is extensively imbalanced, with the negative class vastly outnumbering the positive class (see Figure 4b). In addition, the dataset features missing values for some attributes due to sensor failures, making it unsuitable for most of the Multivariate Statistical Process Control (MSPC) methods [63, 64, 65]. Therefore, we implement a neural network-based FD system, which uses the available mapping between process variables and faults to identify the system faults [66, 67, 68, 69]. The neural network can model complex nonlinear dynamic processes through a learning algorithm that extracts features from historical training data and subsequently apply pattern classification to detect the faults. This feature makes the neural networks robust to datasets with missing and incorrect data. In particular, deep learning technologies can harness the vast amounts of industrial process data made available through 14.0 technologies such as the Industrial Internet of Things (IIoT) and IBD, effectively extracting features required to create robust FD systems [70]. For experiments on the APS Failure at Scania Trucks dataset [61], we utilize a deep feed forward neural network. The architecture is made up of three fully-connected layers (492, 328, and 82 output features), each layer followed by ReLU nonlinearity [71], Batch Normalization layer [72] and a Dropout layer [73]. The loss layer for the neural network is the DLWP loss or one of the other SOTA methods depending on the evaluation being carried out. We train for 200 epochs using the Adam optimizer [74] and a base learning rate of 0.1, which is through a learning rate scheduler adaptively changed to 0.01 at epoch 150 and 0.001 at epoch 180 during training. As pointed out in [75], tuning the Adam optimizer \(\epsilon\) hyperparameter can improve performance. Upon further tuning, we ultimately settle on Adam with \(\epsilon\) values of \(10^{-5}\) for the APS Failure at Scania Trucks dataset. We use a larger batch size of 256 for all experiments as we are dealing with imbalanced datasets, and this increases the chances of having samples from the minority classes included in each batch during training.