100 research outputs found
Data-driven spectral decomposition and forecasting of ergodic dynamical systems
We develop a framework for dimension reduction, mode decomposition, and
nonparametric forecasting of data generated by ergodic dynamical systems. This
framework is based on a representation of the Koopman and Perron-Frobenius
groups of unitary operators in a smooth orthonormal basis of the L2 space of
the dynamical system, acquired from time-ordered data through the diffusion
maps algorithm. Using this representation, we compute Koopman eigenfunctions
through a regularized advection-diffusion operator, and employ these
eigenfunctions in dimension reduction maps with projectible dynamics and high
smoothness for the given observation modality. In systems with pure point
spectra, we construct a decomposition of the generator of the Koopman group
into mutually commuting vector fields that transform naturally under changes of
observation modality, which we reconstruct in data space through a
representation of the pushforward map in the Koopman eigenfunction basis. We
also establish a correspondence between Koopman operators and Laplace-Beltrami
operators constructed from data in Takens delay-coordinate space, and use this
correspondence to provide an interpretation of diffusion-mapped delay
coordinates for this class of systems. Moreover, we take advantage of a special
property of the Koopman eigenfunction basis, namely that the basis elements
evolve as simple harmonic oscillators, to build nonparametric forecast models
for probability densities and observables. In systems with more complex
spectral behavior, including mixing systems, we develop a method inspired from
time change in dynamical systems to transform the generator to a new operator
with potentially improved spectral properties, and use that operator for vector
field decomposition and nonparametric forecasting.Comment: 56 pages, 20 figure
Analog Forecasting with Dynamics-Adapted Kernels
Analog forecasting is a nonparametric technique introduced by Lorenz in 1969
which predicts the evolution of states of a dynamical system (or observables
defined on the states) by following the evolution of the sample in a historical
record of observations which most closely resembles the current initial data.
Here, we introduce a suite of forecasting methods which improve traditional
analog forecasting by combining ideas from kernel methods developed in harmonic
analysis and machine learning and state-space reconstruction for dynamical
systems. A key ingredient of our approach is to replace single-analog
forecasting with weighted ensembles of analogs constructed using local
similarity kernels. The kernels used here employ a number of dynamics-dependent
features designed to improve forecast skill, including Takens' delay-coordinate
maps (to recover information in the initial data lost through partial
observations) and a directional dependence on the dynamical vector field
generating the data. Mathematically, our approach is closely related to kernel
methods for out-of-sample extension of functions, and we discuss alternative
strategies based on the Nystr\"om method and the multiscale Laplacian pyramids
technique. We illustrate these techniques in applications to forecasting in a
low-order deterministic model for atmospheric dynamics with chaotic
metastability, and interannual-scale forecasting in the North Pacific sector of
a comprehensive climate model. We find that forecasts based on kernel-weighted
ensembles have significantly higher skill than the conventional approach
following a single analog.Comment: submitted to Nonlinearit
Delay-coordinate maps and the spectra of Koopman operators
The Koopman operator induced by a dynamical system is inherently linear and
provides an alternate method of studying many properties of the system,
including attractor reconstruction and forecasting. Koopman eigenfunctions
represent the non-mixing component of the dynamics. They factor the dynamics,
which can be chaotic, into quasiperiodic rotations on tori. Here, we describe a
method through which these eigenfunctions can be obtained from a kernel
integral operator, which also annihilates the continuous spectrum. We show that
incorporating a large number of delay coordinates in constructing the kernel of
that operator results, in the limit of infinitely many delays, in the creation
of a map into the discrete spectrum subspace of the Koopman operator. This
enables efficient approximation of Koopman eigenfunctions from high-dimensional
data in systems with pure point or mixed spectra
Extraction and Prediction of Coherent Patterns in Incompressible Flows through Space-Time Koopman Analysis
We develop methods for detecting and predicting the evolution of coherent
spatiotemporal patterns in incompressible time-dependent fluid flows driven by
ergodic dynamical systems. Our approach is based on representations of the
generators of the Koopman and Perron-Frobenius groups of operators governing
the evolution of observables and probability measures on Lagrangian tracers,
respectively, in a smooth orthonormal basis learned from velocity field
snapshots through the diffusion maps algorithm. These operators are defined on
the product space between the state space of the fluid flow and the spatial
domain in which the flow takes place, and as a result their eigenfunctions
correspond to global space-time coherent patterns under a skew-product
dynamical system. Moreover, using this data-driven representation of the
generators in conjunction with Leja interpolation for matrix exponentiation, we
construct model-free prediction schemes for the evolution of observables and
probability densities defined on the tracers. We present applications to
periodic Gaussian vortex flows and aperiodic flows generated by Lorenz 96
systems.Comment: 65 pages, 17 figures, links to accompanying videos provide
Indo-Pacific variability on seasonal to multidecadal timescales. Part I: Intrinsic SST modes in models and observations
The variability of Indo-Pacific SST on seasonal to multidecadal timescales is
investigated using a recently introduced technique called nonlinear Laplacian
spectral analysis (NLSA). Through this technique, drawbacks associated with ad
hoc pre-filtering of the input data are avoided, enabling recovery of
low-frequency and intermittent modes not previously accessible via classical
approaches. Here, a multiscale hierarchy of spatiotemporal modes is identified
for Indo-Pacific SST in millennial control runs of CCSM4 and CM3 and in HadISST
data. On interannual timescales, a mode with spatiotemporal patterns
corresponding to the fundamental component of ENSO emerges, along with
ENSO-modulated annual modes consistent with combination mode theory. The ENSO
combination modes also feature prominent activity in the Indian Ocean,
explaining significant fraction of the SST variance in regions associated with
the Indian Ocean dipole. A pattern resembling the tropospheric biennial
oscillation emerges in addition to ENSO and the associated combination modes.
On multidecadal timescales, the dominant NLSA mode in the model data is
predominantly active in the western tropical Pacific. The interdecadal Pacific
oscillation also emerges as a distinct NLSA mode, though with smaller explained
variance than the western Pacific multidecadal mode. Analogous modes on
interannual and decadal timescales are also identified in HadISST data for the
industrial era, as well as in model data of comparable timespan, though decadal
modes are either absent or of degraded quality in these datasets.Comment: 85 pages, 19 figures; submitted to Journal of Climat
Nonlinear Laplacian spectral analysis: Capturing intermittent and low-frequency spatiotemporal patterns in high-dimensional data
We present a technique for spatiotemporal data analysis called nonlinear
Laplacian spectral analysis (NLSA), which generalizes singular spectrum
analysis (SSA) to take into account the nonlinear manifold structure of complex
data sets. The key principle underlying NLSA is that the functions used to
represent temporal patterns should exhibit a degree of smoothness on the
nonlinear data manifold M; a constraint absent from classical SSA. NLSA
enforces such a notion of smoothness by requiring that temporal patterns belong
in low-dimensional Hilbert spaces V_l spanned by the leading l Laplace-Beltrami
eigenfunctions on M. These eigenfunctions can be evaluated efficiently in high
ambient-space dimensions using sparse graph-theoretic algorithms. Moreover,
they provide orthonormal bases to expand a family of linear maps, whose
singular value decomposition leads to sets of spatiotemporal patterns at
progressively finer resolution on the data manifold. The Riemannian measure of
M and an adaptive graph kernel width enhances the capability of NLSA to detect
important nonlinear processes, including intermittency and rare events. The
minimum dimension of V_l required to capture these features while avoiding
overfitting is estimated here using spectral entropy criteria.Comment: 39 pages, 8 figures, invited paper under review in Statistical
Analysis and Data Minin
Delay-coordinate maps, coherence, and approximate spectra of evolution operators
The problem of data-driven identification of coherent observables of
measure-preserving, ergodic dynamical systems is studied using kernel integral
operator techniques. An approach is proposed whereby complex-valued observables
with approximately cyclical behavior are constructed from a pair eigenfunctions
of integral operators built from delay-coordinate mapped data. It is shown that
these observables are -approximate eigenfunctions of the Koopman
evolution operator of the system, with a bound controlled by the
length of the delay-embedding window, the evolution time, and appropriate
spectral gap parameters. In particular, can be made arbitrarily
small as the embedding window increases so long as the corresponding
eigenvalues remain sufficiently isolated in the spectrum of the integral
operator. It is also shown that the time-autocorrelation functions of such
observables are -approximate Koopman eigenvalue, exhibiting a
well-defined characteristic oscillatory frequency (estimated using the Koopman
generator) and a slowly-decaying modulating envelope. The results hold for
measure-preserving, ergodic dynamical systems of arbitrary spectral character,
including mixing systems with continuous spectrum and no non-constant Koopman
eigenfunctions in . Numerical examples reveal a coherent observable of the
Lorenz 63 system whose autocorrelation function remains above 0.5 in modulus
over approximately 10 Lyapunov timescales.Comment: 36 pages, 5 figure
The Symmetries of Image Formation by Scattering. I. Theoretical Framework
We perceive the world through images formed by scattering. The ability to
interpret scattering data mathematically has opened to our scrutiny the
constituents of matter, the building blocks of life, and the remotest corners
of the universe. Here, we deduce for the first time the fundamental symmetries
underlying image formation. Intriguingly, these are similar to those of the
anisotropic "Taub universe"' of general relativity, with eigenfunctions closely
related to spinning tops in quantum mechanics. This opens the possibility to
apply the powerful arsenal of tools developed in two major branches of physics
to new problems. We augment these tools with graph-theoretic means to recover
the three-dimensional structure of objects from random snapshots of unknown
orientation at four orders of magnitude higher complexity than previously
demonstrated. Our theoretical framework offers a potential link to recent
observations on face perception in higher primates. In a later paper, we
demonstrate the recovery of structure and dynamics from ultralow-signal random
sightings of systems with no orientational or timing information.Comment: 16 pages, 65 references, 3 figures, 3 tables. Movies available at
http://www.cims.nyu.edu/~dimitri
Nonparametric forecasting of low-dimensional dynamical systems
This letter presents a non-parametric modeling approach for forecasting
stochastic dynamical systems on low-dimensional manifolds. The key idea is to
represent the discrete shift maps on a smooth basis which can be obtained by
the diffusion maps algorithm. In the limit of large data, this approach
converges to a Galerkin projection of the semigroup solution to the underlying
dynamics on a basis adapted to the invariant measure. This approach allows one
to quantify uncertainties (in fact, evolve the probability distribution) for
non-trivial dynamical systems with equation-free modeling. We verify our
approach on various examples, ranging from an inhomogeneous anisotropic
stochastic differential equation on a torus, the chaotic Lorenz
three-dimensional model, and the Ni\~{n}o-3.4 data set which is used as a proxy
of the El-Ni\~{n}o Southern Oscillation.Comment: Supplemental videos available at: http://personal.psu.edu/thb11
Quantum mechanics and data assimilation
A framework for data assimilation combining aspects of operator-theoretic
ergodic theory and quantum mechanics is developed. This framework adapts the
Dirac--von Neumann formalism of quantum dynamics and measurement to perform
sequential data assimilation (filtering) of a partially observed,
measure-preserving dynamical system, using the Koopman operator on the
space associated with the invariant measure as an analog of the Heisenberg
evolution operator in quantum mechanics. In addition, the state of the data
assimilation system is represented by a trace-class operator analogous to the
density operator in quantum mechanics, and the assimilated observables by
self-adjoint multiplication operators. An averaging approach is also
introduced, rendering the spectrum of the assimilated observables discrete, and
thus amenable to numerical approximation. We present a data-driven formulation
of the quantum mechanical data assimilation approach, utilizing kernel methods
from machine learning and delay-coordinate maps of dynamical systems to
represent the evolution and measurement operators via matrices in a data-driven
basis. The data-driven formulation is structurally similar to its
infinite-dimensional counterpart, and shown to converge in a limit of large
data under mild assumptions. Applications to periodic oscillators and the
Lorenz 63 system demonstrate that the framework is able to naturally handle
highly non-Gaussian statistics, complex state space geometries, and chaotic
dynamics.Comment: 20 pages, 6 figure
- …