Symbolic transfer entropy rate is equal to transfer entropy rate for bivariate finite-alphabet stationary ergodic Markov processes

Transfer entropy is a measure of the magnitude and the direction of information flow between jointly distributed stochastic processes. In recent years, its permutation analogues are considered in the literature to estimate the transfer entropy by counting the number of occurrences of orderings of values, not the values themselves. It has been suggested that the method of permutation is easy to implement, computationally low cost and robust to noise when applying to real world time series data. In this paper, we initiate a theoretical treatment of the corresponding rates. In particular, we consider the transfer entropy rate and its permutation analogue, the symbolic transfer entropy rate, and show that they are equal for any bivariate finite-alphabet stationary ergodic Markov process. This result is an illustration of the duality method introduced in [T. Haruna and K. Nakajima, Physica D 240, 1370 (2011)]. We also discuss the relationship among the transfer entropy rate, the time-delayed mutual information rate and their permutation analogues.Comment: 18 page

Symbolic local information transfer

Recently, the permutation-information theoretic approach has been used in a broad range of research fields. In particular, in the study of highdimensional dynamical systems, it has been shown that this approach can be effective in characterizing global properties, including the complexity of their spatiotemporal dynamics. Here, we show that this approach can also be applied to reveal local spatiotemporal profiles of distributed computations existing at each spatiotemporal point in the system. J. T. Lizier et al. have recently introduced the concept of local information dynamics, which consists of information storage, transfer, and modification. This concept has been intensively studied with regard to cellular automata, and has provided quantitative evidence of several characteristic behaviors observed in the system. In this paper, by focusing on the local information transfer, we demonstrate that the application of the permutation-information theoretic approach, which introduces natural symbolization methods, makes the concept easily extendible to systems that have continuous states. We propose measures called symbolic local transfer entropies, and apply these measures to two test models, the coupled map lattice (CML) system and the Bak-Sneppen model (BS-model), to show their relevance to spatiotemporal systems that have continuous states.Comment: 20 pages, 7 figure

Exploiting short-term memory in soft body dynamics as a computational resource

Soft materials are not only highly deformable but they also possess rich and diverse body dynamics. Soft body dynamics exhibit a variety of properties, including nonlinearity, elasticity, and potentially infinitely many degrees of freedom. Here we demonstrate that such soft body dynamics can be employed to conduct certain types of computation. Using body dynamics generated from a soft silicone arm, we show that they can be exploited to emulate functions that require memory and to embed robust closed-loop control into the arm. Our results suggest that soft body dynamics have a short-term memory and can serve as a computational resource. This finding paves the way toward exploiting passive body dynamics for control of a large class of underactuated systems.Comment: 22 pages, 11 figures; email address correcte

Boosting computational power through spatial multiplexing in quantum reservoir computing

Quantum reservoir computing provides a framework for exploiting the natural dynamics of quantum systems as a computational resource. It can implement real-time signal processing and solve temporal machine learning problems in general, which requires memory and nonlinear mapping of the recent input stream using the quantum dynamics in computational supremacy region, where the classical simulation of the system is intractable. A nuclear magnetic resonance spin-ensemble system is one of the realistic candidates for such physical implementations, which is currently available in laboratories. In this paper, considering these realistic experimental constraints for implementing the framework, we introduce a scheme, which we call a spatial multiplexing technique, to effectively boost the computational power of the platform. This technique exploits disjoint dynamics, which originate from multiple different quantum systems driven by common input streams in parallel. Accordingly, unlike designing a single large quantum system to increase the number of qubits for computational nodes, it is possible to prepare a huge number of qubits from multiple but small quantum systems, which are operationally easy to handle in laboratory experiments. We numerically demonstrate the effectiveness of the technique using several benchmark tasks and quantitatively investigate its specifications, range of validity, and limitations in detail.Comment: 15 page