An Interventional Perspective on Identifiability in Gaussian LTI Systems with Independent Component Analysis

TL;DR

The paper studies identifiability in Gaussian LTI systems from an interventional viewpoint, showing that diverse control signals across multiple environments suffice to identify the Markov parameter matrix up to permutations and diagonal scaling, and hence recover the transfer function. It proposes a learning approach based on maximizing a multi-environment log-likelihood and proves a connection to a continuous-parameter causal de Finetti theorem for Gaussian trajectories. Empirical validation on a real DC motor and synthetic Gaussian systems demonstrates that increasing environmental diversity improves identifiability and the quality of learned dynamics. By situating system identification within an active, causal framework, the work provides practical guidance for designing interventions and links to broader causal inference theory.

Abstract

We investigate the relationship between system identification and intervention design in dynamical systems. While previous research demonstrated how identifiable representation learning methods, such as Independent Component Analysis (ICA), can reveal cause-effect relationships, it relied on a passive perspective without considering how to collect data. Our work shows that in Gaussian Linear Time-Invariant (LTI) systems, the system parameters can be identified by introducing diverse intervention signals in a multi-environment setting. By harnessing appropriate diversity assumptions motivated by the ICA literature, our findings connect experiment design and representational identifiability in dynamical systems. We corroborate our findings on synthetic and (simulated) physical data. Additionally, we show that Hidden Markov Models, in general, and (Gaussian) LTI systems, in particular, fulfil a generalization of the Causal de Finetti theorem with continuous parameters.

Figures (4)

• Figure 1: Left: methods categorized based on active data collection (interventions) and identifiability. Right: components of the training pipeline for each method on the left. General methods use pre-collected data to learn a representation (black components only); Reinforcement Learning (RL) additionally leverages interventions via agency (i.e., interactions with the world; black+red); Independent Component Analysis (ICA) uses pre-collected data with underlying assumptions to achieve identifiability (blue+black); whereas our method uses assumptions about the system to design interventions, i.e., actively collecting data to achieve identifiability (red+blue+black+green)
• Figure 2: An example for a single environment (i.e., a single set of parameters $\theta, \psi$) and two trajectories (denoted via superscripts). $\theta$ determines the state transition probabilities $p\left(_{t+1}\xspace|_{t}\xspace\right)$, whereas $\psi$ the conditional probabilities for the observations $p\left(_{t}\xspace|_{t}\xspace\right)$ (note that $\psi$ is also the same for both trajectories)
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Theorems & Definitions (34)

• definition 1: Discrete System
• definition 2: Transfer function
• definition 3: Markov parameter matrix
• definition 4: Environment variability matrix
• theorem 3
• proof : Sketch
• definition 5: Hankel matrix
• corollary 1
• proof
• corollary 2