Prediction regions based on dissimilarity functions.

Area of Science:

• Dynamical Systems and Control Theory
• Data-Driven Modeling
• Optimization Methods

Background:

• Accurate prediction regions are crucial for understanding and controlling dynamical systems.
• Existing methods often require detailed system models or are computationally intensive.
• A data-based approach offers flexibility and reduces reliance on prior system knowledge.

Purpose of the Study:

• To develop a novel, entirely data-based methodology for generating prediction regions of dynamical system outputs.
• To provide methods for optimally estimating the two necessary hyperparameters.
• To ensure the generated prediction regions are computationally tractable and useful for analysis.

Main Methods:

• Utilizing stored past outputs of the system for a data-driven approach.
• Minimizing prediction region size while satisfying empirical probability constraints using hyperparameter optimization.
• Employing convex optimization for checking point membership within prediction regions.
• Developing approximation methods for ellipsoidal prediction regions.

Main Results:

• A new methodology for constructing prediction regions for dynamical systems has been successfully developed.
• Optimal estimation methods for the two key hyperparameters were derived and presented.
• The prediction regions are convex, and membership checking is efficient via convex optimization.
• Approximation methods for ellipsoidal regions were provided for explicit descriptions.

Conclusions:

• The proposed data-based methodology effectively generates prediction regions for dynamical systems.
• The method's reliance on only two hyperparameters simplifies its application.
• Convexity and efficient checking mechanisms make the prediction regions practical for real-world use.
• Numerical examples demonstrate the methodology's effectiveness, particularly for non-linear uncertain systems.