FUNCTIONAL RELATION FIELD: A MODEL-AGNOSTIC FRAMEWORK FOR MULTIVARIATE TIME SERIES FORE-CASTING

Abstract

In multivariate time series forecasting, the most popular strategy for modeling the relationship between multiple time series is the construction of graph, where each time series is represented as a node and related nodes are connected by edges, i.e. spatial-temporal graph neural networks. The graph structure is either given apriori or learned based the similarity between nodes. However, the relationship between multiple time series is typically complicated, for instance, the sum of outflows from upstream nodes may be equal to the inflows of downstream nodes. Such relations widely exist in many real-world multivariate time series forecasting scenarios, yet are far from well studied. In these cases, graph might only be a crude description on the dependency between nodes. To this end, we explore a new framework to model the inter-node relationship in a more precise way based our proposed inductive bias for graphs, Functional Relation Field, where a group of functions parameterized by neural networks are learned to characterize the dependency between multiple time series. These learned functions are versatile: they can then be used to discover the underlying graph structure by identifying the most relevant neighbors of the target node; and on the other hand, the learned functions will form a "field" where the nodes in the backbone prediction networks are enforced to satisfy the constraints defined by these functions. The experiment is conducted on one toy dataset to show our approach can well recover the true constraint relationship between nodes. And two real-world MiniApp calling traffic and road networks datasets are also considered with various different backbone networks. Results show that the prediction error can be reduced remarkably with the aid of the proposed functional relation field framework.

1. INTRODUCTION

Multivariate time series forecasting has surged recently due to its strong expressiveness of the spatio-temporal dependence among the data and its enormous popularity in vast application areas, such as the prediction of urban traffic, computer network flow, cloud micro-services calling flow, and rigid body motion, to name a few (Li et al., 2018; Yu et al., 2018; Bai et al., 2020; Yan et al., 2018; Liu et al., 2020) . The most popular and straightforward strategy for modeling the relationship between multiple time series is the introduction of graph, where each time series is represented as a node and related nodes are connected by edges. This particular inductive bias for multivariate time series prediction results in the so called spatial-temporal graph neural networks (Yu et al., 2018) . The graph structure is either given apriori (e.g. in traffic flow prediction, each road as a node has connected roads forming the graph.) or learned based the similarity between nodes (Yu et al., 2019; Bai et al., 2020; Shang et al., 2021) . However, in practice, the relationship



Figure 1: Comparison between traditional graphbased modeling and our approach. (a) Graph structure learning with time-series similarity; (b) Functional relation field, modeling inter-node functional relationship in a linear form, where D is the target node and {A, B, C} represent dependent nodes.

