4,980 research outputs found
An autoregressive (AR) model based stochastic unknown input realization and filtering technique
This paper studies the state estimation problem of linear discrete-time
systems with stochastic unknown inputs. The unknown input is a wide-sense
stationary process while no other prior informaton needs to be known. We
propose an autoregressive (AR) model based unknown input realization technique
which allows us to recover the input statistics from the output data by solving
an appropriate least squares problem, then fit an AR model to the recovered
input statistics and construct an innovations model of the unknown inputs using
the eigensystem realization algorithm (ERA). An augmented state system is
constructed and the standard Kalman filter is applied for state estimation. A
reduced order model (ROM) filter is also introduced to reduce the computational
cost of the Kalman filter. Two numerical examples are given to illustrate the
procedure.Comment: 14 page
A simple method for detecting chaos in nature
Chaos, or exponential sensitivity to small perturbations, appears everywhere
in nature. Moreover, chaos is predicted to play diverse functional roles in
living systems. A method for detecting chaos from empirical measurements should
therefore be a key component of the biologist's toolkit. But, classic
chaos-detection tools are highly sensitive to measurement noise and break down
for common edge cases, making it difficult to detect chaos in domains, like
biology, where measurements are noisy. However, newer tools promise to overcome
these limitations. Here, we combine several such tools into an automated
processing pipeline, and show that our pipeline can detect the presence (or
absence) of chaos in noisy recordings, even for difficult edge cases. As a
first-pass application of our pipeline, we show that heart rate variability is
not chaotic as some have proposed, and instead reflects a stochastic process in
both health and disease. Our tool is easy-to-use and freely available
Bayesian Nonparametric Inference of Switching Linear Dynamical Systems
Many complex dynamical phenomena can be effectively modeled by a system that
switches among a set of conditionally linear dynamical modes. We consider two
such models: the switching linear dynamical system (SLDS) and the switching
vector autoregressive (VAR) process. Our Bayesian nonparametric approach
utilizes a hierarchical Dirichlet process prior to learn an unknown number of
persistent, smooth dynamical modes. We additionally employ automatic relevance
determination to infer a sparse set of dynamic dependencies allowing us to
learn SLDS with varying state dimension or switching VAR processes with varying
autoregressive order. We develop a sampling algorithm that combines a truncated
approximation to the Dirichlet process with efficient joint sampling of the
mode and state sequences. The utility and flexibility of our model are
demonstrated on synthetic data, sequences of dancing honey bees, the IBOVESPA
stock index, and a maneuvering target tracking application.Comment: 50 pages, 7 figure
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
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