A Recipe for the Estimation of Information Flow in a Dynamical System

Information-theoretic quantities, such as entropy and mutual information (MI), can be used to quantify the amount of information needed to describe a dataset or the information shared between two datasets. In the case of a dynamical system, the behavior of the relevant variables can be tightly coupled, such that information about one variable at a given instance in time may provide information about other variables at later instances in time. This is often viewed as a flow of information, and tracking such a flow can reveal relationships among the system variables. Since the MI is a symmetric quantity; an asymmetric quantity, called Transfer Entropy (TE), has been proposed to estimate the directionality of the coupling. However, accurate estimation of entropy-based measures is notoriously difficult. Every method has its own free tuning parameter(s) and there is no consensus on an optimal way of estimating the TE from a dataset. We propose a new methodology to estimate TE and apply a set of methods together as an accuracy cross-check to provide a reliable mathematical tool for any given data set. We demonstrate both the variability in TE estimation across techniques as well as the benefits of the proposed methodology to reliably estimate the directionality of coupling among variables

A Recipe for the Estimation of Information Flow in a Dynamical System

Information-theoretic quantities, such as entropy and mutual information (MI), can be used to quantify the amount of information needed to describe a dataset or the information shared between two datasets. In the case of a dynamical system, the behavior of the relevant variables can be tightly coupled, such that information about one variable at a given instance in time may provide information about other variables at later instances in time. This is often viewed as a flow of information, and tracking such a flow can reveal relationships among the system variables. Since the MI is a symmetric quantity; an asymmetric quantity, called Transfer Entropy (TE), has been proposed to estimate the directionality of the coupling. However, accurate estimation of entropy-based measures is notoriously difficult. Every method has its own free tuning parameter(s) and there is no consensus on an optimal way of estimating the TE from a dataset. We propose a new methodology to estimate TE and apply a set of methods together as an accuracy cross-check to provide a reliable mathematical tool for any given data set. We demonstrate both the variability in TE estimation across techniques as well as the benefits of the proposed methodology to reliably estimate the directionality of coupling among variables

Editorial Comment on the Special Issue of "Information in Dynamical Systems and Complex Systems"

This special issue collects contributions from the participants of the "Information in Dynamical Systems and Complex Systems" workshop, which cover a wide range of important problems and new approaches that lie in the intersection of information theory and dynamical systems. The contributions include theoretical characterization and understanding of the different types of information flow and causality in general stochastic processes, inference and identification of coupling structure and parameters of system dynamics, rigorous coarse-grain modeling of network dynamical systems, and exact statistical testing of fundamental information-theoretic quantities such as the mutual information. The collective efforts reported herein reflect a modern perspective of the intimate connection between dynamical systems and information flow, leading to the promise of better understanding and modeling of natural complex systems and better/optimal design of engineering systems

Reducing "Structure From Motion": a General Framework for Dynamic Vision - Part 2: Experimental Evaluation

A number of methods have been proposed in the literature for estimating scene-structure and ego-motion from a sequence of images using dynamical models. Although all methods may be derived from a "natural" dynamical model within a unified framework, from an engineering perspective there are a number of trade-offs that lead to different strategies depending upon the specific applications and the goals one is targeting. Which one is the winning strategy? In this paper we analyze the properties of the dynamical models that originate from each strategy under a variety of experimental conditions. For each model we assess the accuracy of the estimates, their robustness to measurement noise, sensitivity to initial conditions and visual angle, effects of the bas-relief ambiguity and occlusions, dependence upon the number of image measurements and their sampling rate

Reducing “Structure from Motion”: a general framework for dynamic vision. 2. Implementation and experimental assessment

For pt.1 see ibid., p.933-42 (1998). A number of methods have been proposed in the literature for estimating scene-structure and ego-motion from a sequence of images using dynamical models. Despite the fact that all methods may be derived from a “natural” dynamical model within a unified framework, from an engineering perspective there are a number of trade-offs that lead to different strategies depending upon the applications and the goals one is targeting. We want to characterize and compare the properties of each model such that the engineer may choose the one best suited to the specific application. We analyze the properties of filters derived from each dynamical model under a variety of experimental conditions, assess the accuracy of the estimates, their robustness to measurement noise, sensitivity to initial conditions and visual angle, effects of the bas-relief ambiguity and occlusions, dependence upon the number of image measurements and their sampling rate

Mass estimation in the outer regions of galaxy clusters

We present a technique for estimating the mass in the outskirts of galaxy clusters where the usual assumption of dynamical equilibrium is not valid. The method assumes that clusters form through hierarchical clustering and requires only galaxy redshifts and positions on the sky. We apply the method to dissipationless cosmological N-body simulations where galaxies form and evolve according to semi-analytic modelling. The method recovers the actual cluster mass profile within a factor of two to several megaparsecs from the cluster centre. This error originates from projection effects, sparse sampling, and contamination by foreground and background galaxies. In the absence of velocity biases, this method can provide an estimate of the mass-to-light ratio on scales ~1-10 Mpc/h where this quantity is still poorly known.Comment: 14 pages, 7 figures, MN LaTeX style, MNRAS, in pres

Measuring the escape velocity and mass profiles of galaxy clusters beyond their virial radius

The caustic technique uses galaxy redshifts alone to measure the escape velocity and mass profiles of galaxy clusters to clustrocentric distances well beyond the virial radius, where dynamical equilibrium does not necessarily hold. We provide a detailed description of this technique and analyse its possible systematic errors. We apply the caustic technique to clusters with mass M_200>=10^{14}h^{-1} M_sun extracted from a cosmological hydrodynamic simulation of a LambdaCDM universe. With a few tens of redshifts per squared comoving megaparsec within the cluster, the caustic technique, on average, recovers the profile of the escape velocity from the cluster with better than 10 percent accuracy up to r~4 r_200. The caustic technique also recovers the mass profile with better than 10 percent accuracy in the range (0.6-4) r_200, but it overestimates the mass up to 70 percent at smaller radii. This overestimate is a consequence of neglecting the radial dependence of the filling function F_beta(r). The 1-sigma uncertainty on individual escape velocity profiles increases from ~20 to ~50 percent when the radius increases from r~0.1 r_200 to ~4 r_200. Individual mass profiles have 1-sigma uncertainty between 40 and 80 percent within the radial range (0.6-4) r_200. We show that the amplitude of these uncertainties is completely due to the assumption of spherical symmetry, which is difficult to drop. Alternatively, we can apply the technique to synthetic clusters obtained by stacking individual clusters: in this case, the 1-sigma uncertainty on the escape velocity profile is smaller than 20 percent out to 4 r_200. The caustic technique thus provides reliable average profiles which extend to regions difficult or impossible to probe with other techniques.Comment: MNRAS accepted, 20 page

Estimating long term behavior of flows without trajectory integration: the infinitesimal generator approach

The long-term distributions of trajectories of a flow are described by invariant densities, i.e. fixed points of an associated transfer operator. In addition, global slowly mixing structures, such as almost-invariant sets, which partition phase space into regions that are almost dynamically disconnected, can also be identified by certain eigenfunctions of this operator. Indeed, these structures are often hard to obtain by brute-force trajectory-based analyses. In a wide variety of applications, transfer operators have proven to be very efficient tools for an analysis of the global behavior of a dynamical system. The computationally most expensive step in the construction of an approximate transfer operator is the numerical integration of many short term trajectories. In this paper, we propose to directly work with the infinitesimal generator instead of the operator, completely avoiding trajectory integration. We propose two different discretization schemes; a cell based discretization and a spectral collocation approach. Convergence can be shown in certain circumstances. We demonstrate numerically that our approach is much more efficient than the operator approach, sometimes by several orders of magnitude

Markov chain sampling of the O(n)O(n) loop models on the infinite plane

It was recently proposed in https://journals.aps.org/pre/abstract/10.1103/PhysRevE.94.043322 [Herdeiro & Doyon Phys.,Rev.,E (2016)] a numerical method showing a precise sampling of the infinite plane 2d critical Ising model for finite lattice subsections. The present note extends the method to a larger class of models, namely the $O(n)$ loop gas models for $n \in (1,2]$. We argue that even though the Gibbs measure is non local, it is factorizable on finite subsections when sufficient information on the loops touching the boundaries is stored. Our results attempt to show that provided an efficient Markov chain mixing algorithm and an improved discrete lattice dilation procedure the planar limit of the $O(n)$ models can be numerically studied with efficiency similar to the Ising case. This confirms that scale invariance is the only requirement for the present numerical method to work.Comment: v2: added conclusion section, changes in introduction and appendice