Federated Granger Causality Learning For Interdependent Clients With State Space Representation

Advanced sensors and IoT devices have improved the monitoring and control of complex industrial enterprises. They have also created an interdependent fabric of geographically distributed process operations (clients) across these enterprises. Granger causality is an effective approach to detect and quantify interdependencies by examining how the state of one client affects the states of others over time. Understanding these interdependencies helps capture how localized events, such as faults and disruptions, can propagate throughout the system, potentially leading to widespread operational impacts. However, the large volume and complexity of industrial data present significant challenges in effectively modeling these interdependencies. This paper develops a federated approach to learning Granger causality. We utilize a linear state space system framework that leverages low-dimensional state estimates to analyze interdependencies. This helps address bandwidth limitations and the computational burden commonly associated with centralized data processing. We propose augmenting the client models with the Granger causality information learned by the server through a Machine Learning (ML) function. We examine the co-dependence between the augmented client and server models and reformulate the framework as a standalone ML algorithm providing conditions for its sublinear and linear convergence rates. We also study the convergence of the framework to a centralized oracle model. Moreover, we include a differential privacy analysis to ensure data security while preserving causal insights. Using synthetic data, we conduct comprehensive experiments to demonstrate the robustness of our approach to perturbations in causality, the scalability to the size of communication, number of clients, and the dimensions of raw data. We also evaluate the performance on two real-world industrial control system datasets by reporting the volume of data saved by decentralization. * Equal contribution Published as a conference paper at ICLR 2025 modeling and causal analysis to better understand their cause-and-effect relationships. Granger causality Granger ( 1969 ) is an effective approach to detect and quantify interdependencies by examining how the state of one client affects the states of others over time. This approach captures how localized events, such as faults or disruptions, can propagate throughout the system, potentially leading to widespread operational impacts. The decentralized nature of data, coupled with its large volume and high dimensionality, presents significant challenges in establishing causality through centralized data analysis. Aggregating data from multiple sources in a central server can become inefficient and impractical as the scale and complexity of the data increase. However, in many applications, it is possible to represent highdimensional data using low-dimensional state. In the context of causality analysis, low-dimensional state enable the identification of critical interdependencies without aggregating raw data. In this work, we use linear time-invariant (LTI) state space representation for individual models of a multi-client system. Clients operate independently using only their client-specific information. The measurements (i.e., raw data) at each client are assumed to be high-dimensional. Clients cannot share their measurements, but can only share their low-dimensional state with a central server. Our goal is to develop a federated learning framework that allows a decentralized system of clients to collaboratively learn the off-diagonal blocks of the system's state matrix that represent the crossclient Granger causality-by sharing only their state with a central server. To achieve this, we propose augmenting client models with the off-diagonal information of state matrix through a Machine Learning (ML) based function. To the best of our knowledge, this is the first study on federated granger causality learning. Please refer to Appendix A.1 for preliminaries on state space modeling, Kalman filter, and Granger causality, along their brief mathematical representations. Research Objective: Our objective is to develop a federated learning framework in which the augmented state gradually converges to the centralized state, thus achieving parity between a local and a centralized (oracle) model. Through this process, the decentralized system learns the offdiagonal blocks of the system's state matrix, which capture client interactions by sharing only their states with a central server rather than large volumes of high-dimensional measurements. Main Contributions: Our key technical contributions can be delineated as follows: A discussion on the possible choices of f c , f a , f M L along with the rationale behind our models, is provided in the Appendix A.2. A simplified pictorial description of the aforementioned problem setting is shown in figure 1 . A pseudocode for our proposed framework is given in Appendix