Spectral Ranking of Causal Influence in Complex Systems.

Errol Zalmijn,Tom Heskes,Tom Claassen

doi:10.3390/e23030369

Abstract

Similar to natural complex systems, such as the Earth’s climate or a living cell, semiconductor lithography systems are characterized by nonlinear dynamics across more than a dozen orders of magnitude in space and time. Thousands of sensors measure relevant process variables at appropriate sampling rates, to provide time series as primary sources for system diagnostics. However, high-dimensionality, non-linearity and non-stationarity of the data are major challenges to efficiently, yet accurately, diagnose rare or new system issues by merely using model-based approaches. To reliably narrow down the causal search space, we validate a ranking algorithm that applies transfer entropy for bivariate interaction analysis of a system’s multivariate time series to obtain a weighted directed graph, and graph eigenvector centrality to identify the system’s most important sources of original information or causal influence. The results suggest that this approach robustly identifies the true drivers or causes of a complex system’s deviant behavior, even when its reconstructed information transfer network includes redundant edges.

Highlights

Semiconductor lithography systems are extremely complicated electromechanical systems, capable of sub-nanometer positioning and sub-milliKelvin temperature control, while generating extreme ultraviolet light from laser-pulsed tin plasma
To fully understand a complex system’s dynamical behavior, it is essential to identify its main sources of causal influence affecting downstream elements throughout the system
We empirically show that spectral centrality analysis of its causal network as approximated by standard transfer entropy allows one to accurately and consistently identify the most important node(s) of original information representing the most probable cause(s) or driver(s) of disturbance in the system

Summary

Introduction

Semiconductor lithography systems are extremely complicated electromechanical systems, capable of sub-nanometer positioning and sub-milliKelvin temperature control, while generating extreme ultraviolet light from laser-pulsed tin plasma. Multivariate approaches use conditional transfer entropy [4] to separate true cause–effect relations from mere correlations, i.e., direct from so-called transitive indirect or semi-metric, and redundant relations, by iteratively conditioning out (subsets of) all other time series. Such an information decomposition is infeasible in (real time) diagnosis or prognosis of high-dimensional technological complex systems, due to its exponential scaling of computational costs with time-series dimension. We use a ranking algorithm that relies on standard transfer entropy in exhaustive, but computationally feasible bivariate interaction analysis of a complex system’s multivariate time series, resulting in an information transfer network that is likely to contain redundant edges. We assess the ranking algorithm in diagnosing a real-world industrial system issue, using a higher-dimensional time series

Transfer Entropy

Eigenvector Centrality

Validation

Coupled Lorenz Systems

Technological Complex Systems

Conclusions