Abnormality detection is essential to the performance of safety-critical and latency-constrained systems. However, as systems are becoming increasingly complicated with a large quantity of heterogeneous data, conventional statistical change point detection methods are becoming less effective and efficient. Although Deep Learning (DL) and Deep Neural Networks (DNNs) are increasingly employed to handle heterogeneous data, they still lack theoretic assurable performance and explainability. This paper integrates zero-bias DNN and Quickest Event Detection algorithms to provide a holistic framework for quick and reliable detection of both abnormalities and time-dependent abnormal events in the Internet of Things (IoT). We first use the zero-bias dense layer to increase the explainability of DNN. We provide a solution to convert zero-bias DNN classifiers into performance assured binary abnormality detectors. Using the converted abnormality detector, we then present a sequential quickest detection scheme that provides the theoretically assured lowest abnormal event detection delay under false alarm constraints. Finally, we demonstrate the effectiveness of the framework using both massive signal records from real-world aviation communication systems and simulated data. Code and data of our work is available at url{https://github.com/pcwhy/AbnormalityDetectionInZbDNN}
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