Path Signatures and Graph Neural Networks for Slow Earthquake Analysis: Better Together?

TL;DR

This work tackles the analysis of slow earthquake signals from irregular GPS time series by introducing PS-GCNN, a pipeline that fuses path signatures with graph convolutional networks to capture temporal geometry and spatial interactions among sensors. Path signatures provide a theoretically grounded feature representation for irregular paths, while GCNNs model neighborhood interactions on proximity graphs, enabling node-level classification and regression. The authors demonstrate the approach on simulated stochastic reaction-diffusion systems and real GeoNET GPS data, showing improved performance over a summary-statistics baseline and identifying an optimal radius for graph construction. The method holds promise for robust SSE analysis, sensor placement, and broader sensor-network tasks involving irregular time series on graphs, with potential extensions to self-supervised learning.

Abstract

The path signature, having enjoyed recent success in the machine learning community, is a theoretically-driven method for engineering features from irregular paths. On the other hand, graph neural networks (GNN), neural architectures for processing data on graphs, excel on tasks with irregular domains, such as sensor networks. In this paper, we introduce a novel approach, Path Signature Graph Convolutional Neural Networks (PS-GCNN), integrating path signatures into graph convolutional neural networks (GCNN), and leveraging the strengths of both path signatures, for feature extraction, and GCNNs, for handling spatial interactions. We apply our method to analyze slow earthquake sequences, also called slow slip events (SSE), utilizing data from GPS timeseries, with a case study on a GPS sensor network on the east coast of New Zealand's north island. We also establish benchmarks for our method on simulated stochastic differential equations, which model similar reaction-diffusion phenomenon. Our methodology shows promise for future advancement in earthquake prediction and sensor network analysis.

Figures (6)

• Figure 1: Our PS-GCNN pipeline consists of two modules: a Path Signatures module (left) and a Graph Convolutional Neural Network (GCNN) module (right). In our experiments with real data, GPS signals measuring the movement of the earth's crust are collected at stations (GISB and MAHI pictured), interpolated as $3$-dimensional paths, and transformed into node features. For a given radius $\rho\geqslant0$, the feature matrix $\mathbf{X}$ is fed into a GCNN which is customized to perform either a node classification task or a node regression task.
• Figure 2: Lead-lag behavior aparent in the east displacement signal for nearby stations GISB and CNST.
• Figure 3: Graph convolutional neural network (GCNN) node classifier with three convolutional layers. Outputs are binary labels which convey membership in a hypothesis class (yellow or black).
• Figure 4: Tuning the radius hyperparameter $\rho$.
• Figure 5: Ground truth labels for our semi-supervised node classification task. Of the $80$ stations considered, $64$ are labeled; nonlinearity is detected in $11$ of them, while the remaining $53$ stations had no nonlinearity detected. The proximity graph pictured has a threshold radius $\rho = 40$ km.

Theorems & Definitions (1)

• Definition 2.1: Path Signature