# An Interpretable Method for Anomaly Detection in Multivariate Time Series Predictions
## 1 Introduction  
With the widespread application of artificial intelligence, anomaly detection based on deep learning has become an important means of detecting network attacks in industrial control systems (ICSs). However, deep learning models are typically composed of multiple layers, thousands of neurons, and a large number of parameters, and their complexity makes it difficult for people to intuitively understand how these parameters affect the decision results of the model, leading to the emergence of a \"black-box\" problem. This issue makes the decision-making process of the model opaque, which in turn affects the credibility and usability of the model [1]. The necessity of introducing interpretability in deep learning has been extensively and deeply discussed in both academia and industry. On the one hand, increasing the interpretability of a model usually accompanies more model complexity, which may not only lead to an increase in computing resources [2] and a decrease in inference speed but also inevitably introduce a certain degree of subjectivity, and may even sacrifice the ultimate performance of the model. These factors must be carefully considered when using interpretable models. On the other hand, the current situation urgently requires the emergence of interpretable methods. With the development of deep learning technology, models are becoming increasingly \"black\", and the requirements for the credibility, security, and other aspects of deep learning are also increasing. The enhancement of interpretable performance increases trust in these applications. Most of the existing ICS anomaly detection research focuses on evaluating detection accuracy [3] but rarely explains why data is marked as abnormal and which physical components have been attacked. This 'black-box' decision-making process makes it difficult to gain the trust of operators. In terms of network security, exceptional interpretability helps users to quickly identify attack points or targets and adopt attacking strategies to prevent attacks in a timely manner, ultimately achieving the goal of reducing attack losses [4]. Most of the anomaly interpretable methods based on deep learning are closely related to the original model. These methods perturb input data and use techniques such as reverse gradient and graph neural networks to interpret anomalies based on changes in the internal parameters of the model. This process can help developers to gain a deeper understanding of the decision-making mechanism of the model, effectively identify and fix potential problems, and improve the accuracy and effectiveness of the detection system [5]. This type of explanation is more suitable for developers to use when debugging anomaly detection models. However, these methods may lead to the exposure of the internal structure of the model, thereby increasing the risk of the model being attacked. Some anomaly interpretation methods are independent of the original model and can achieve interpretation without relying on the structure of the original model, which can help operation and maintenance personnel to understand the decision results and attribution interpretation of anomaly detection and accurately locate potential attacks. This type of interpretability does not make the model more reliable or perform better, but it is an important component in building highly reliable systems [6]. Although these methods are flexible, they are designed for supervised tasks [7], while unsupervised methods are commonly used in anomaly detection tasks in ICSs. Interpretable methods based on gradient optimization are effective means of explaining decision attribution problems in deep learning models. Their theoretical basis mainly comes from the gradient analysis theory in differential calculus, aiming to transform the explanation of anomalous features into solving an optimization problem in a normal \"reference\" state. Based on this idea, we propose an interpretable method for anomaly detection models based on gradient optimization.