5,206 research outputs found
Bayesian inference of chaotic dynamics by merging data assimilation, machine learning and expectation-maximization
The reconstruction from observations of high-dimensional chaotic dynamics such as geophysical flows is hampered by (â…°) the partial and noisy observations that can realistically be obtained, (â…±) the need to learn from long time series of data, and (â…˛) the unstable nature of the dynamics. To achieve such inference from the observations over long time series, it has been suggested to combine data assimilation and machine learning in several ways. We show how to unify these approaches from a Bayesian perspective using expectation-maximization and coordinate descents. In doing so, the model, the state trajectory and model error statistics are estimated all together. Implementations and approximations of these methods are discussed. Finally, we numerically and successfully test the approach on two relevant low-order chaotic models with distinct identifiability
State and parameter estimation using Monte Carlo evaluation of path integrals
Transferring information from observations of a dynamical system to estimate
the fixed parameters and unobserved states of a system model can be formulated
as the evaluation of a discrete time path integral in model state space. The
observations serve as a guiding potential working with the dynamical rules of
the model to direct system orbits in state space. The path integral
representation permits direct numerical evaluation of the conditional mean path
through the state space as well as conditional moments about this mean. Using a
Monte Carlo method for selecting paths through state space we show how these
moments can be evaluated and demonstrate in an interesting model system the
explicit influence of the role of transfer of information from the
observations. We address the question of how many observations are required to
estimate the unobserved state variables, and we examine the assumptions of
Gaussianity of the underlying conditional probability.Comment: Submitted to the Quarterly Journal of the Royal Meteorological
Society, 19 pages, 5 figure
Reconstructing dynamical networks via feature ranking
Empirical data on real complex systems are becoming increasingly available.
Parallel to this is the need for new methods of reconstructing (inferring) the
topology of networks from time-resolved observations of their node-dynamics.
The methods based on physical insights often rely on strong assumptions about
the properties and dynamics of the scrutinized network. Here, we use the
insights from machine learning to design a new method of network reconstruction
that essentially makes no such assumptions. Specifically, we interpret the
available trajectories (data) as features, and use two independent feature
ranking approaches -- Random forest and RReliefF -- to rank the importance of
each node for predicting the value of each other node, which yields the
reconstructed adjacency matrix. We show that our method is fairly robust to
coupling strength, system size, trajectory length and noise. We also find that
the reconstruction quality strongly depends on the dynamical regime
Linear State-Space Model with Time-Varying Dynamics
This paper introduces a linear state-space model with time-varying dynamics.
The time dependency is obtained by forming the state dynamics matrix as a
time-varying linear combination of a set of matrices. The time dependency of
the weights in the linear combination is modelled by another linear Gaussian
dynamical model allowing the model to learn how the dynamics of the process
changes. Previous approaches have used switching models which have a small set
of possible state dynamics matrices and the model selects one of those matrices
at each time, thus jumping between them. Our model forms the dynamics as a
linear combination and the changes can be smooth and more continuous. The model
is motivated by physical processes which are described by linear partial
differential equations whose parameters vary in time. An example of such a
process could be a temperature field whose evolution is driven by a varying
wind direction. The posterior inference is performed using variational Bayesian
approximation. The experiments on stochastic advection-diffusion processes and
real-world weather processes show that the model with time-varying dynamics can
outperform previously introduced approaches.Comment: The final publication is available at Springer via
http://dx.doi.org/10.1007/978-3-662-44851-9_2
Nonlinear time-series analysis revisited
In 1980 and 1981, two pioneering papers laid the foundation for what became
known as nonlinear time-series analysis: the analysis of observed
data---typically univariate---via dynamical systems theory. Based on the
concept of state-space reconstruction, this set of methods allows us to compute
characteristic quantities such as Lyapunov exponents and fractal dimensions, to
predict the future course of the time series, and even to reconstruct the
equations of motion in some cases. In practice, however, there are a number of
issues that restrict the power of this approach: whether the signal accurately
and thoroughly samples the dynamics, for instance, and whether it contains
noise. Moreover, the numerical algorithms that we use to instantiate these
ideas are not perfect; they involve approximations, scale parameters, and
finite-precision arithmetic, among other things. Even so, nonlinear time-series
analysis has been used to great advantage on thousands of real and synthetic
data sets from a wide variety of systems ranging from roulette wheels to lasers
to the human heart. Even in cases where the data do not meet the mathematical
or algorithmic requirements to assure full topological conjugacy, the results
of nonlinear time-series analysis can be helpful in understanding,
characterizing, and predicting dynamical systems
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