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

Core Concepts
The author proposes a direct approach based on maximum likelihood estimation to identify dynamic networks with missing data, transforming the problem into a more tractable form by leveraging knowledge about network interconnections.
The paper introduces a novel direct approach to system identification of dynamic networks with missing data using maximum likelihood estimation. It addresses challenges in estimating parameters due to singular probability density functions and offers insights into transforming these networks for better estimations. The study compares the proposed direct approach with an indirect method, highlighting advantages such as improved estimation accuracy and reduced sensitivity to initialization strategies. The content discusses the technical challenges in controlling large-scale complex systems and the limitations of existing methods in scaling with increasing complexity. It emphasizes the need for systematic approaches for designing and analyzing distributed control structures in various applications. The study delves into model-based controller design methodologies, emphasizing the importance of efficient modeling for scalable control methods. Furthermore, it explores different approaches to estimate parameters in interconnected systems, comparing direct and indirect methods. The analysis reveals drawbacks of the indirect approach, such as unstable models and increased variance in estimates. In contrast, the proposed direct method shows promise in obtaining accurate estimates by observing fewer variables while maintaining stability. The paper includes a numerical experiment illustrating properties of the proposed method using a network of three systems. It discusses results from both direct and indirect approaches, showcasing improvements in estimation accuracy when observing additional signals. The study concludes by suggesting future work on applying the approach to larger network architectures and exploring different optimization methods.
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 internal systems. We generated N = 500 measurements where the error (e1, e2, e3) is a realization of a zero-mean Gaussian random variable with covariance 0.1 · I. The true systems used for generating observable and missing states are zero-order hold discretizations of continuous systems. For most estimates using the direct approach, observed states xo were well estimated while missing states xm were poorly estimated. Using gradient descent combined with backtracking line search led to local minima corresponding to stable systems. The indirect approach presented unstable zeros that required adjustments for stability. Fit values for observable and missing variables showed improvements using both direct and indirect approaches. Observing additional signals led to better estimation accuracy while reducing sensitivity to initialization.
"The obtained results suggest that our approach is suitable for estimating parameters of dynamic networks when combined with global optimization or suitable initialization strategy." "Our implementation is based on Python library JAX which uses automatic differentiation to compute partial derivatives." "The proposed direct approach can benefit from additional observed variables while improving estimation accuracy."

Deeper Inquiries

How can global optimization techniques enhance parameter estimation in dynamic networks

Global optimization techniques can significantly enhance parameter estimation in dynamic networks by helping to overcome the issue of local minima. In complex systems like dynamic networks, the cost functions involved in parameter estimation are often non-convex, leading to multiple local optima. Global optimization algorithms explore a broader search space and aim to find the global optimum solution rather than settling for a suboptimal local minimum. By considering a wider range of possibilities, these techniques increase the likelihood of converging towards the best possible estimates for system parameters in dynamic networks.

What are potential drawbacks or limitations associated with relying on an indirect system identification approach

Relying on an indirect system identification approach comes with several potential drawbacks and limitations. One major limitation is related to stability issues that may arise when estimating subsystems using this method. The indirect approach might lead to unstable models even if individual subsystems are stable, which can result in inaccurate or unreliable parameter estimates. Additionally, this method often requires more parameters to describe closed-loop systems compared to direct approaches, increasing model complexity and potentially introducing higher variance into the estimations. Furthermore, not being able to recover certain components like Ci (in ARMAX models) limits the completeness and accuracy of the estimated models obtained through indirect methods.

How might advancements in second-order optimization methods impact system identification processes

Advancements in second-order optimization methods have a significant impact on system identification processes by offering more efficient ways to optimize complex cost functions commonly encountered in parameter estimation tasks. These methods utilize second-order information such as gradients and Hessians to guide optimization algorithms towards optimal solutions faster than traditional first-order methods like gradient descent alone. By incorporating curvature information from second derivatives, algorithms like AdaHessian or AdaSub can navigate complex landscapes more effectively, leading to quicker convergence and improved accuracy in estimating model parameters for dynamic systems.