# CDSTAEP: Cross-Domain Spatial-Temporal Association Learning for Abnormal Events Prediction
###### Abstract
The existing methods of graph representation learning usually use Euclidean space for representation learning, limiting the graph to a single space represented by a single constant curvature [9], ignoring the complexity of the graph structure in the application field, resulting in the inability to truly represent the complex network structure during representation learning. At present, these methods usually achieve good experimental results on a set scene or a specific network structure. For example, Riemannian spherical space with curvature greater than 0 is suitable for characterizing periodic graph structures, such as triangles, polygons [10; 11], etc. The Euclidean space with curvature equal to 0 is suitable for characterizing the network structure [12], such as images, because its scale growth shows the characteristics of polynomial expansion. Since the Riemannian hyperbolic space with curvature less than 0, and has the characteristic of exponential expansion speed far greater than the polynomial expansion speed of Euclidean space, it is suitable for characterizing the hierarchical structure with approximate tree shape. In recent years, the learning of graphical representation in Euclidean space has been extended to hyperbolic space mainly through hyperbolic graphical neural network (HGNN) [13; 14; 15]. In reality, complex network data are usually heterogeneous (that is, there are tree structures and mesh structures simultaneously) [12], and their rich relationship features cannot be effectively captured in a single curvature space. The representation of mixed-curvature space is an excellent work recently [9; 16]. However, in the field of abnormal events or behavior analysis, more effective characterization of spatiotemporal multi-dimensional features (relationships between nodes, attributes of nodes, etc.) is of substantial significance for subsequent prediction and analysis. What is more challenging is how to design and select the optimal hybrid curvature space feature learning and feature fusion methods and provide better feature sequence representation for downstream prediction tasks. **Event Spatiotemporal Association Prediction:** By mining the spatiotemporal correlation between events and behaviors, we can predict abnormal events and improve the ability of anomaly detection and defense. In general, there are three mainstream methods in the field of event prediction. The first is the event prediction method based on association rules. In this category, the events that will happen are predicted according to the event correlation pattern in the past. This method can sort critical events in chronological order, but it assumes that events are related to each other in a fixed order [5; 17; 18]. The second is the event prediction method based on the stochastic process. In this category, the occurrence of future events is predicted based on the random process of time interval and invasion scale [19; 20]. In these studies, due to the autocorrelation between events, the time interval between events is described by the point process, and the evolution of the intrusion scale is described by the autoregressive comprehensive moving average. Such methods can predict the scale, probability, and time interval of events, but they do not make full use of the spatial relationship and temporal order relationship between events. In recent years, a third kind of event prediction method has emerged, that is, the prediction method based on multi-task learning [16; 17; 21; 22]. This kind of method uses representation learning in Euclidean space and non-Euclidean space to form the high-dimensional representation of the relationship graph between nodes and uses the neural network with a time memory function to predict events. Although such methods can explain the dependence of event prediction tasks on spatiotemporal characteristics, they do not consider the impact of correlation between events on event prediction.