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Direct System Identification of Dynamical Networks with Partial Measurements: a Maximum Likelihood Approach

João Victor Galvão da Mata, Anders Hansson, Martin S. Andersen

TL;DR

This work tackles the problem of identifying dynamic networks with latent variables by formulating a direct maximum-likelihood approach that can handle missing data. By exploiting known interconnections and applying a sequence of linear transformations, the authors convert an inherently singular pdf into a nonsingular one, enabling ML estimation for networked ARMAX models. The key contribution is a practical transformation framework that yields a reduced, parameter-invariant, nonsingular problem (via quantities like $\Phi$ and $\Gamma$) suitable for direct ML optimization. Numerical experiments on a 3-system network demonstrate that, with appropriate initialization or global optimization, the direct method can accurately recover both observed and latent states and outperform indirect transfer-function methods in terms of estimation robustness and state recovery.

Abstract

This paper introduces a novel direct approach to system identification of dynamic networks with missing data based on maximum likelihood estimation. Dynamic networks generally present a singular probability density function, which poses a challenge in the estimation of their parameters. By leveraging knowledge about the network's interconnections, we show that it is possible to transform the problem into a more tractable form by applying linear transformations. This results in a nonsingular probability density function, enabling the application of maximum likelihood estimation techniques. Our preliminary numerical results suggest that when combined with global optimization algorithms or a suitable initialization strategy, we are able to obtain a good estimate of the dynamics of the internal systems.

Direct System Identification of Dynamical Networks with Partial Measurements: a Maximum Likelihood Approach

TL;DR

This work tackles the problem of identifying dynamic networks with latent variables by formulating a direct maximum-likelihood approach that can handle missing data. By exploiting known interconnections and applying a sequence of linear transformations, the authors convert an inherently singular pdf into a nonsingular one, enabling ML estimation for networked ARMAX models. The key contribution is a practical transformation framework that yields a reduced, parameter-invariant, nonsingular problem (via quantities like and ) suitable for direct ML optimization. Numerical experiments on a 3-system network demonstrate that, with appropriate initialization or global optimization, the direct method can accurately recover both observed and latent states and outperform indirect transfer-function methods in terms of estimation robustness and state recovery.

Abstract

This paper introduces a novel direct approach to system identification of dynamic networks with missing data based on maximum likelihood estimation. Dynamic networks generally present a singular probability density function, which poses a challenge in the estimation of their parameters. By leveraging knowledge about the network's interconnections, we show that it is possible to transform the problem into a more tractable form by applying linear transformations. This results in a nonsingular probability density function, enabling the application of maximum likelihood estimation techniques. Our preliminary numerical results suggest that when combined with global optimization algorithms or a suitable initialization strategy, we are able to obtain a good estimate of the dynamics of the internal systems.
Paper Structure (9 sections, 40 equations, 5 figures, 1 table)

This paper contains 9 sections, 40 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Nonsingular Gaussian Distribution and Maximum Likelihood Estimation
  3. Singular Gaussian Distribution
  4. Dynamic Networks
  5. Numerical Experiment
  6. Direct Approach
  7. Indirect Approach
  8. Results
  9. Conclusion

Figures (5)

  • Figure 1: Block diagram for a dynamic network.
  • Figure 2: Estimated and true observable variables.
  • Figure 3: Comparison between the true missing variables and the estimate obtained using the proposed direct approach.
  • Figure 4: Comparison of indirect and direct approach: magnitude and phase Bode plots for each internal system.
  • Figure 5: Magnitude and phase Bode plots for each internal system, considering the 50 best solutions, when observing both $u^3$ and $u_1$ using the proposed direct approach.