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Ambiguity rate of hidden Markov processes
Alexandra M Jurgens1, James P Crutchfield1
1Complexity Sciences Center, Physics Department University of California at Davis Davis, California 95616, USA.
Physical Review. E
|January 15, 2022
Summary
The ε-machine, an optimal predictive model, requires infinite features. Mapping it to a stochastic dynamical system and introducing the ambiguity rate allows calculation of statistical complexity for complex processes.
Area of Science:
- Complex Systems
- Information Theory
- Dynamical Systems
Background:
- The ε-machine is the optimal predictive and minimal model for stochastic processes.
- Optimal prediction often necessitates probabilistic models, including ε-machines, with infinite features.
- Constructive work with infinite feature sets requires mapping ε-machines to place-dependent iterated function systems (IFS).
Purpose of the Study:
- To develop a method for constructively working with infinite feature sets in ε-machines.
- To introduce the ambiguity rate and its relationship with Shannon entropy rate.
- To enable calculation of statistical complexity dimension for complex processes.
Main Methods:
- Mapping the ε-machine to a place-dependent iterated function system (IFS).
- Introducing the ambiguity rate.
- Relating ambiguity rate and Shannon entropy rate to the growth of predictive features.
- Calculating the ambiguity rate as a correction to the Lyapunov dimension of an IFS.
Main Results:
- A method to constructively handle infinite feature sets in ε-machines was developed.
- The ambiguity rate was introduced and shown to determine the growth rate of predictive features.
- The ambiguity rate was identified as the missing correction to the Lyapunov dimension of an IFS.
- A method to calculate the statistical complexity dimension for complex processes was demonstrated.
Conclusions:
- The developed framework provides a computable method for analyzing complex stochastic processes.
- The ambiguity rate is a key metric for understanding the complexity and predictive power of stochastic systems.
- This work offers a new way to calculate the statistical complexity dimension, relevant for various complex systems.
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