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The frequency-domain technique, commonly used in analyzing and designing feedback control systems, is effective for linear, time-invariant systems. However, it falls short when dealing with nonlinear, time-varying, and multiple-input multiple-output systems. The time-domain or state-space approach addresses these limitations by utilizing state variables to construct simultaneous, first-order differential equations, known as state equations, for an nth-order system.
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Spectral State Compression of Markov Processes.

Anru Zhang1, Mengdi Wang2

  • 1Department of Statistics, University of Wisconsin-Madison, Madison, WI 53706.

IEEE Transactions on Information Theory
|March 22, 2021
PubMed
Summary
This summary is machine-generated.

We developed spectral methods to statistically compress Markov chains from data, enabling efficient modeling of complex systems like taxi trip patterns by identifying key states and reducing model complexity.

Keywords:
Computational complexitymaximum likelihood estimationminimax techniquessignal denoisingtensor SVD

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Area of Science:

  • Statistical modeling
  • Control theory
  • Machine learning

Background:

  • Model reduction is crucial for analyzing complex state-transition systems.
  • State aggregation is a key concept from control theory for simplifying models.
  • Markov chains are widely used to model systems with discrete states and transitions.

Purpose of the Study:

  • To develop statistical methods for state compression of discrete-state Markov chains using empirical data.
  • To leverage spectral decomposition for analyzing Markov process properties like rank, features, aggregability, and lumpability.
  • To recover latent structures, including state aggregation and lumpable partitions, from observed trajectories.

Main Methods:

  • Spectral decomposition of Markov processes.
  • Estimation of low-rank transition matrices.
  • Identification of Markov features and their spanning subspace.
  • Development of algorithms for state aggregation and lumpable partition recovery.

Main Results:

  • Statistical upper bounds on estimation errors for spectral methods.
  • Nearly matching minimax lower bounds, establishing theoretical performance limits.
  • Successful application of methods to synthetic data and real-world New York City taxi trip data.

Conclusions:

  • Spectral methods provide a robust framework for statistically compressing Markov chains.
  • The developed techniques effectively identify underlying structures and reduce model complexity.
  • This work offers significant advancements in the statistical analysis and practical application of Markov models.