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Federated Granger Causality Learning for Interdependent Clients with State Space Representation

Ayush Mohanty, Nazal Mohamed, Paritosh Ramanan, Nagi Gebraeel

TL;DR

This work addresses learning Granger causality in distributed industrial systems without centralizing high-dimensional measurements. It introduces a federated framework built on a linear state-space representation, where cross-client causality is encoded in off-diagonal state blocks and clients progressively augment their local models with an ML-driven causality term. The authors establish convergence to a centralized oracle, provide differential privacy guarantees for both client-to-server and server-to-client communications, and validate the approach with synthetic data and real ICS datasets, demonstrating data-volume savings and robustness. The framework enables scalable, privacy-preserving inference of interdependencies among geographically distributed process operations, with theoretical guarantees and practical relevance for industrial monitoring and control.

Abstract

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 one client's state affects others over time. Understanding these interdependencies captures how localized events, such as faults and disruptions, can propagate throughout the system, possibly causing widespread operational impacts. However, the large volume and complexity of industrial data pose challenges in 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 addresses 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.

Federated Granger Causality Learning for Interdependent Clients with State Space Representation

TL;DR

This work addresses learning Granger causality in distributed industrial systems without centralizing high-dimensional measurements. It introduces a federated framework built on a linear state-space representation, where cross-client causality is encoded in off-diagonal state blocks and clients progressively augment their local models with an ML-driven causality term. The authors establish convergence to a centralized oracle, provide differential privacy guarantees for both client-to-server and server-to-client communications, and validate the approach with synthetic data and real ICS datasets, demonstrating data-volume savings and robustness. The framework enables scalable, privacy-preserving inference of interdependencies among geographically distributed process operations, with theoretical guarantees and practical relevance for industrial monitoring and control.

Abstract

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 one client's state affects others over time. Understanding these interdependencies captures how localized events, such as faults and disruptions, can propagate throughout the system, possibly causing widespread operational impacts. However, the large volume and complexity of industrial data pose challenges in 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 addresses 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.
Paper Structure (39 sections, 15 theorems, 79 equations, 3 figures, 7 tables, 2 algorithms)

This paper contains 39 sections, 15 theorems, 79 equations, 3 figures, 7 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Problem Setting
  4. Federated Granger Causality Framework
  5. Augmented Client Model
  6. Server Model
  7. Understanding Decentralization Through a Centralized Lens
  8. Asymptotic Convergence to the Centralized Oracle
  9. Privacy Analysis
  10. Experiments: Synthetic Dataset
  11. Experiments: Real World Datasets
  12. Conclusion and Limitations
  13. Appendix
  14. Preliminaries
  15. Defining Functions $f_c(.), f_a(.), f_{ML}(.)$
  16. ...and 24 more sections

Key Result

Theorem 5.1

At the $(k+1)^{th}$ iteration, the augmented client model's parameter i.e., $\theta_{m}^{k+1}$ depends on the $k^{th}$ iter. of the server model's parameter i.e., $\hat{A}_{mn}^{k}, \space n \neq m$, and vice versa.

Figures (3)

  • Figure 1: Federated cross-client Granger causality learning framework
  • Figure 2: Loss functions at (a) client 1, (b) client 2, and (c) server during the first mean-shift. (e) $l_2$ norm diff. between states of centralized oracle, server, client, augmented client models of client 2 (d) and evolution of Frobenius norm difference between estimation and ground-truth value of $A_{21}$
  • Figure 3: (a) Server loss and (b) Client 4's loss for HAI dataset, and (c) Sever loss and (d) Client 4's loss for SWaT dataset at a randomly chosen time

Theorems & Definitions (29)

  • Claim 4.3
  • Claim 4.5
  • Theorem 5.1: Co-dependence
  • Corollary 5.2
  • Proposition 5.3: Optimal model parameters
  • Theorem 5.4: Unified framework
  • Lemma 5.5: Convergence of framework
  • Theorem 5.6: Sub linear conv.
  • Theorem 5.7: Linear conv.
  • Theorem 6.1
  • ...and 19 more