21,259 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
Understanding Complex Systems: From Networks to Optimal Higher-Order Models
To better understand the structure and function of complex systems,
researchers often represent direct interactions between components in complex
systems with networks, assuming that indirect influence between distant
components can be modelled by paths. Such network models assume that actual
paths are memoryless. That is, the way a path continues as it passes through a
node does not depend on where it came from. Recent studies of data on actual
paths in complex systems question this assumption and instead indicate that
memory in paths does have considerable impact on central methods in network
science. A growing research community working with so-called higher-order
network models addresses this issue, seeking to take advantage of information
that conventional network representations disregard. Here we summarise the
progress in this area and outline remaining challenges calling for more
research.Comment: 8 pages, 4 figure
Overcoming non-Markovian dephasing in single photon sources through post-selection
We study the effects of realistic dephasing environments on a pair of
solid-state single-photon sources in the context of the Hong-Ou-Mandel dip. By
means of solutions for the Markovian or exact non-Markovian dephasing dynamics
of the sources, we show that the resulting loss of visibility depends crucially
on the timing of photon detection events. Our results demonstrate that the
effective visibility can be improved via temporal post-selection, and also that
time-resolved interference can be a useful probe of the interaction between the
emitter and its host environment.Comment: 5 pages, 2 figures, published version, title changed, references
update
Markovian Testing Equivalence and Exponentially Timed Internal Actions
In the theory of testing for Markovian processes developed so far,
exponentially timed internal actions are not admitted within processes. When
present, these actions cannot be abstracted away, because their execution takes
a nonzero amount of time and hence can be observed. On the other hand, they
must be carefully taken into account, in order not to equate processes that are
distinguishable from a timing viewpoint. In this paper, we recast the
definition of Markovian testing equivalence in the framework of a Markovian
process calculus including exponentially timed internal actions. Then, we show
that the resulting behavioral equivalence is a congruence, has a sound and
complete axiomatization, has a modal logic characterization, and can be decided
in polynomial time
Optimal arbitrage under model uncertainty
In an equity market model with "Knightian" uncertainty regarding the relative
risk and covariance structure of its assets, we characterize in several ways
the highest return relative to the market that can be achieved using
nonanticipative investment rules over a given time horizon, and under any
admissible configuration of model parameters that might materialize. One
characterization is in terms of the smallest positive supersolution to a fully
nonlinear parabolic partial differential equation of the
Hamilton--Jacobi--Bellman type. Under appropriate conditions, this smallest
supersolution is the value function of an associated stochastic control
problem, namely, the maximal probability with which an auxiliary
multidimensional diffusion process, controlled in a manner which affects both
its drift and covariance structures, stays in the interior of the positive
orthant through the end of the time-horizon. This value function is also
characterized in terms of a stochastic game, and can be used to generate an
investment rule that realizes such best possible outperformance of the market.Comment: Published in at http://dx.doi.org/10.1214/10-AAP755 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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