20,120 research outputs found
Variational approach for learning Markov processes from time series data
Inference, prediction and control of complex dynamical systems from time
series is important in many areas, including financial markets, power grid
management, climate and weather modeling, or molecular dynamics. The analysis
of such highly nonlinear dynamical systems is facilitated by the fact that we
can often find a (generally nonlinear) transformation of the system coordinates
to features in which the dynamics can be excellently approximated by a linear
Markovian model. Moreover, the large number of system variables often change
collectively on large time- and length-scales, facilitating a low-dimensional
analysis in feature space. In this paper, we introduce a variational approach
for Markov processes (VAMP) that allows us to find optimal feature mappings and
optimal Markovian models of the dynamics from given time series data. The key
insight is that the best linear model can be obtained from the top singular
components of the Koopman operator. This leads to the definition of a family of
score functions called VAMP-r which can be calculated from data, and can be
employed to optimize a Markovian model. In addition, based on the relationship
between the variational scores and approximation errors of Koopman operators,
we propose a new VAMP-E score, which can be applied to cross-validation for
hyper-parameter optimization and model selection in VAMP. VAMP is valid for
both reversible and nonreversible processes and for stationary and
non-stationary processes or realizations
Noise reduction in coarse bifurcation analysis of stochastic agent-based models: an example of consumer lock-in
We investigate coarse equilibrium states of a fine-scale, stochastic
agent-based model of consumer lock-in in a duopolistic market. In the model,
agents decide on their next purchase based on a combination of their personal
preference and their neighbours' opinions. For agents with independent
identically-distributed parameters and all-to-all coupling, we derive an
analytic approximate coarse evolution-map for the expected average purchase. We
then study the emergence of coarse fronts when spatial segregation is present
in the relative perceived quality of products. We develop a novel Newton-Krylov
method that is able to compute accurately and efficiently coarse fixed points
when the underlying fine-scale dynamics is stochastic. The main novelty of the
algorithm is in the elimination of the noise that is generated when estimating
Jacobian-vector products using time-integration of perturbed initial
conditions. We present numerical results that demonstrate the convergence
properties of the numerical method, and use the method to show that macroscopic
fronts in this model destabilise at a coarse symmetry-breaking bifurcation.Comment: This version of the manuscript was accepted for publication on SIAD
Understanding solar cycle variability
The level of solar magnetic activity, as exemplified by the number of
sunspots and by energetic events in the corona, varies on a wide range of time
scales. Most prominent is the 11-year solar cycle, which is significantly
modulated on longer time scales. Drawing from dynamo theory together with
empirical results of past solar activity and of similar phenomena on solar-like
stars, we show that the variability of the solar cycle can be essentially
understood in terms of a weakly nonlinear limit cycle affected by random noise.
In contrast to ad-hoc `toy models' for the solar cycle, this leads to a generic
normal-form model, whose parameters are all constrained by observations. The
model reproduces the characteristics of the variable solar activity on time
scales between decades and millennia, including the occurrence and statistics
of extended periods of very low activity (grand minima). Comparison with
results obtained with a Babcock-Leighton-type dynamo model confirms the
validity of the normal-mode approach.Comment: ApJ, accepte
Subdiffusion of nonlinear waves in quasiperiodic potentials
We study the spatio-temporal evolution of wave packets in one-dimensional
quasiperiodic lattices which localize linear waves. Nonlinearity (related to
two-body interactions) has destructive effect on localization, as recently
observed for interacting atomic condensates [Phys. Rev. Lett. 106, 230403
(2011)]. We extend the analysis of the characteristics of the subdiffusive
dynamics to large temporal and spatial scales. Our results for the second
moment consistently reveal an asymptotic and
intermediate laws. At variance to purely random systems
[Europhys. Lett. 91, 30001 (2010)] the fractal gap structure of the linear wave
spectrum strongly favors intermediate self-trapping events. Our findings give a
new dimension to the theory of wave packet spreading in localizing
environments
Slow and fast dynamics in coupled systems: A time series analysis view
We study the dynamics of systems with different time scales, when access only
to the slow variables is allowed. We use the concept of Finite Size Lyapunov
Exponent (FSLE) and consider both the case when the equations of motion for the
slow components are known, and the situation when a scalar time series of one
of the slow variables has been measured. A discussion on the effects of
parameterizing the fast dynamics is given. We show that, although the
computation of the largest Lyapunov exponent can be practically infeasible in
complex dynamical systems, the computation of the FSLE allows to extract
information on the characteristic time and on the predictability of the
large-scale, slow-time dynamics even with moderate statistics and unresolved
small scales.Comment: 17 pages RevTeX, 6 PostScript figures, tarred, gzipped, uuencoded.
Submitted to Physica
Backward Linear Control Systems on Time Scales
We show how a linear control systems theory for the backward nabla
differential operator on an arbitrary time scale can be obtained via Caputo's
duality. More precisely, we consider linear control systems with outputs
defined with respect to the backward jump operator. Kalman criteria of
controllability and observability, as well as realizability conditions, are
proved.Comment: Submitted November 11, 2009; Revised March 28, 2010; Accepted April
03, 2010; for publication in the International Journal of Control
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