Uncertainty Propagation in Stochastic Systems via Mixture Models with Error Quantification

المفاهيم الأساسية

Developing a framework for approximating the distribution of non-linear stochastic dynamical systems with formal guarantees of correctness.

الملخص

The article focuses on uncertainty propagation in non-linear stochastic dynamical systems using mixture models. It introduces a novel approach to approximate system distribution over time with tractable approximations and correctness guarantees. The Total Variation (TV) distance is used to quantify the distance between distributions, allowing for efficient computation and optimization of parameters. The effectiveness of the approach is demonstrated on benchmarks from the control community. The content is structured into sections discussing introduction, problem formulation, system description, total variation bounds, convergence analysis, algorithm details, experimental results, and proofs.

Introduction:

• Uncertainty propagation in complex autonomous systems.
• Importance of considering non-linear stochastic dynamics for safety-critical applications.

Problem Formulation:

• Need for efficient frameworks for uncertainty propagation.
• Various methods proposed but lacking error bounds on approximation error.

System Description:

• Discrete-time stochastic process representation.
• Transition kernel definition and probability distribution description.

Total Variation Bounds:

• Definition of Total Variation (TV) distance.
• Use of TV distance to quantify closeness between distributions.

Convergence Analysis:

• Proof showing approximation error can be minimized by increasing mixture size.

Algorithm Details:

• Iterative process based on mixture distribution approximation.
• Refinement procedure to optimize grid partitions for accuracy.

Experimental Results:

• Evaluation on benchmarks including linear systems and Dubins car model.
• Comparison with standard approaches showing improved TV bounds.

Proofs:

• Detailed proofs provided for Theorem 1, Corollary 1, and Theorem 2.

الإحصائيات

None

اقتباسات

"Modern autonomous systems are becoming increasingly complex."
"Various methods have been proposed to propagate uncertainty in non-linear stochastic systems."

الرؤى الأساسية المستخلصة من

Uncertainty Propagation in Stochastic Systems via Mixture Models with Error Quantification

by Eduardo Figu... في arxiv.org 03-26-2024

https://arxiv.org/pdf/2403.15626.pdf

استفسارات أعمق

How can mixture compression be implemented effectively

Mixture compression can be implemented effectively by utilizing techniques such as clustering algorithms to identify redundant components in the mixture. By grouping similar distributions together, we can reduce the number of components while maintaining a close approximation to the original distribution. This process involves evaluating the contribution of each component to the overall mixture and removing or merging those that have minimal impact on the representation of the system's dynamics. Additionally, techniques like pruning based on statistical significance or information criteria can help streamline the mixture model without sacrificing accuracy.

What are the implications of extending formal guarantees to other metrics like Wasserstein distance

Extending formal guarantees to other metrics like Wasserstein distance would have significant implications for enhancing the robustness and applicability of the framework. The Wasserstein distance provides a more nuanced measure of dissimilarity between probability distributions compared to Total Variation (TV) distance, capturing structural differences beyond just probabilities assigned to events. By incorporating Wasserstein distance into formal guarantees, we could offer more comprehensive assessments of approximation quality and convergence rates for non-linear stochastic systems. This extension would enable a deeper understanding of how well mixtures represent complex distributions over time.

How does the framework address challenges posed by noise variance in real-world applications

The framework addresses challenges posed by noise variance in real-world applications through its adaptive grid refinement algorithm and TV bounds computation methodology. When dealing with low noise variances, which lead to closer-to-delta Dirac behavior in one-step transition kernels, obtaining accurate approximations becomes more difficult due to local variations from mean values within Gaussian mixtures. However, by leveraging closed-form expressions for total variation bounds specific to Gaussian processes with additive noise (as demonstrated in Corollary 1), it is possible to quantify correctness even under challenging conditions where noise variance is small. This approach ensures that despite noisy environments or uncertain dynamics, reliable approximations can still be achieved with formal guarantees using mixture models.