arXiv subjects

Dimitrios Giannakis

Publications and source records attributed to Dimitrios Giannakis.

At least 55 records · Page 3Linked to original sources

arXiv2019

Galerkin Approximation of Dynamical Quantities using Trajectory Data

Understanding chemical mechanisms requires estimating dynamical statistics such as expected hitting times, reaction rates, and committors. Here, we present a general framework for calculating these dynamical quantities by approximating boundary value problems using dynamical operators with a Galerkin expansion. A specific choice of basis set in the expansion corresponds to estimation of dynamical quantities using a Markov state model. More generally, the boundary conditions impose restrictions on the choice of basis sets. We demonstrate how an alternative basis can be constructed using ideas from diffusion maps. In our numerical experiments, this basis gives results of comparable or better accuracy to Markov state models. Additionally, we show that delay embedding can reduce the information lost when projecting the system's dynamics for model construction; this improves estimates of dynamical statistics considerably over the standard practice of increasing the lag time.

physics.data-an↗

arXiv2018

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.

math.DS↗

arXiv2018

Spatiotemporal pattern extraction by spectral analysis of vector-valued observables

We present a data-driven framework for extracting complex spatiotemporal patterns generated by ergodic dynamical systems. Our approach, called Vector-valued Spectral Analysis (VSA), is based on an eigendecomposition of a kernel integral operator acting on a Hilbert space of vector-valued observables of the system, taking values in a space of functions (scalar fields) on a spatial domain. This operator is constructed by combining aspects of the theory of operator-valued kernels for machine learning with delay-coordinate maps of dynamical systems. In contrast to conventional eigendecomposition techniques, which decompose the input data into pairs of temporal and spatial modes with a separable, tensor product structure, the patterns recovered by VSA can be manifestly non-separable, requiring only a modest number of modes to represent signals with intermittency in both space and time. Moreover, the kernel construction naturally quotients out dynamical symmetries in the data, and exhibits an asymptotic commutativity property with the Koopman evolution operator of the system, enabling decomposition of multiscale signals into dynamically intrinsic patterns. Application of VSA to the Kuramoto-Sivashinsky model demonstrates significant performance gains in efficient and meaningful decomposition over eigendecomposition techniques utilizing scalar-valued kernels.

math.DS↗

arXiv2018

Predicting regional and pan-Arctic sea ice anomalies with kernel analog forecasting

Predicting Arctic sea ice extent is a notoriously difficult forecasting problem, even for lead times as short as one month. Motivated by Arctic intraannual variability phenomena such as reemergence of sea surface temperature and sea ice anomalies, we use a prediction approach for sea ice anomalies based on analog forecasting. Traditional analog forecasting relies on identifying a single analog in a historical record, usually by minimizing Euclidean distance, and forming a forecast from the analog's historical trajectory. Here an ensemble of analogs are used to make forecasts, where the ensemble weights are determined by a dynamics-adapted similarity kernel, which takes into account the nonlinear geometry on the underlying data manifold. We apply this method for forecasting pan-Arctic and regional sea ice area and volume anomalies from multi-century climate model data, and in many cases find improvement over the benchmark damped persistence forecast. Examples of success include the 3--6 month lead time prediction of pan-Arctic area, the winter sea ice area prediction of some marginal ice zone seas, and the 3--12 month lead time prediction of sea ice volume anomalies in many central Arctic basins. We discuss possible connections between KAF success and sea ice reemergence, and find KAF to be successful in regions and seasons exhibiting high interannual variability.

physics.ao-ph↗

arXiv2018

The Antarctic circumpolar wave and its seasonality: Intrinsic traveling modes and ENSO teleconnections

Interannual variability in the Southern Ocean is investigated via nonlinear Laplacian spectral analysis (NLSA), an objective eigendecomposition technique for nonlinear dynamical systems that can simultaneously recover multiple timescales from data with high skill. Applied to modeled and observed sea surface temperature and sea ice concentration data, NLSA recovers the wavenumber-2 eastward propagating signal corresponding to the Antarctic circumpolar wave (ACW). During certain phases of its lifecycle, the spatial patterns of this mode display a structure that can explain the statistical origin of the Antarctic dipole pattern. Another group of modes have combination frequencies consistent with the modulation of the annual cycle by the ACW. Further examination of these newly identified modes reveals that they can have either eastward or westward propagation, combined with meridional pulsation reminiscent of sea ice reemergence patterns in the Arctic. Moreover, they exhibit smaller-scale spatial structures, and explain more Indian Ocean variance than the primary ACW modes. We attribute these modes to teleconnections between ACW and the tropical Indo-Pacific Ocean; in particular, fundamental ENSO modes and their associated combination modes with the annual cycle recovered by NLSA. Another mode extracted from the Antarctic variables displays an eastward propagating wavenumber-3 structure over the Southern Ocean, but exhibits no strong correlation to interannual Indo-Pacific variability.

physics.ao-ph↗

arXiv2018

Vector-Valued Spectral Analysis of Space-Time Data

Identifying coherent spatiotemporal patterns generated by complex dynamical systems is a central problem in many science and engineering disciplines. Here, we combine ideas from the theory of operator-valued kernels with delay-embedding techniques in dynamical systems to develop of a method for objective identification of spatiotemporal coherent patterns, without prior knowledge of the state space and/or the dynamical laws of the system generating the data. A key aspect of this method is that it operates on a space of vector-valued observables using a kernel measure of similarity that takes into account both temporal and spatial degrees of freedom (in contrast, classical kernel techniques such as PCA utilize aggregate measures of similarity between 'snapshots'). As a result, spectral decomposition of data via our approach yields a significantly more efficient and physically meaningful representation of complex spatiotemporal signals than conventional methods based on scalar-valued kernels. We demonstrate this behavior with applications to an oscillator model and sea surface temperature data from a climate model.

physics.data-an↗

arXiv2018

Koopman analysis of the long-term evolution in a turbulent convection cell

We analyse the long-time evolution of the three-dimensional flow in a closed cubic turbulent Rayleigh-Bénard convection cell via a Koopman eigenfunction analysis. A data-driven basis derived from diffusion kernels known in machine learning is employed here to represent a regularized generator of the unitary Koopman group in the sense of a Galerkin approximation. The resulting Koopman eigenfunctions can be grouped into subsets in accordance with the discrete symmetries in a cubic box. In particular, a projection of the velocity field onto the first group of eigenfunctions reveals the four stable large-scale circulation (LSC) states in the convection cell. We recapture the preferential circulation rolls in diagonal corners and the short-term switching through roll states parallel to the side faces which have also been seen in other simulations and experiments. The diagonal macroscopic flow states can last as long as a thousand convective free-fall time units. In addition, we find that specific pairs of Koopman eigenfunctions in the secondary subset obey enhanced oscillatory fluctuations for particular stable diagonal states of the LSC. The corresponding velocity field structures, such as corner vortices and swirls in the midplane, are also discussed via spatiotemporal reconstructions.

physics.flu-dyn↗

arXiv2017

Probabilistic prediction of the AL index with the diffusion forecasting model

We propose a nonparametric approach for probabilistic prediction of the AL index trained with AL and solar wind ($v B_z$) data. Our framework relies on the diffusion forecasting technique, which views AL and $ v B_z $ data as observables of an autonomous, ergodic, stochastic dynamical system operating on a manifold. Diffusion forecasting builds a data-driven representation of the Markov semigroup governing the evolution of probability measures of the dynamical system. In particular, the Markov semigroup operator is represented in an orthonormal basis acquired from data using the diffusion maps algorithm and Takens delay embeddings. This representation of the evolution semigroup is used in conjunction with a Bayesian filtering algorithm for forecast initialization to predict the probability that the AL index is less than a user-selected threshold over arbitrary lead times and without requiring exogenous inputs. We find that the model produces skillful forecasts out to at least two-hour leads despite gaps in the training data.

physics.data-an↗

arXiv2017

Data-Driven Koopman Analysis of Tropical Climate Space-Time Variability

We study nonlinear dynamics of the Earth's tropical climate system. For that, we apply a recently developed technique for feature extraction and mode decomposition of spatiotemporal data generated by ergodic dynamical systems. The method relies on constructing low-dimensional representations (temporal patterns) of signals using eigenfunctions of Koopman operators governing the evolution of observables in ergodic dynamical systems. We apply this technique to a variety of tropical climate datasets and extract a multiscale hierarchy of spatiotemporal patterns on diurnal to interannual timescales. In particular, we detect without prefiltering the input data modes operating on intraseasonal and shorter timescales that correspond to propagation of organized convection. We discuss the salient properties of these propagating features and in particular we focus on how the activity of certain types of these traveling patterns is related to lower-frequency dynamics. As an extension of this work, we discuss their potential predictability based on a range of nonparametric techniques and potential advances related to understanding the deterministic and stochastic aspects of the variability of these modes.

physics.ao-ph↗

arXiv2017

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.

math.DS↗

arXiv2017

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.

math.DS↗

arXiv2016

Indo-Pacific variability on seasonal to multidecadal timescales. Part II: Multiscale atmosphere-ocean linkages

The coupled atmosphere-ocean variability of the Indo-Pacific on interannual to multidecadal timescales is investigated in a millennial control run of CCSM4 and in observations using a family of modes recovered in Part~I of this work from unprocessed SST data through nonlinear Laplacian spectral analysis (NLSA). It is found that ENSO and combination modes of ENSO with the annual cycle exhibit a seasonally synchronized southward shift of equatorial surface zonal winds and thermocline adjustment consistent with terminating El Nino and La Nina events. The surface wind patterns associated with these modes also generate teleconnections between the Pacific and Indian Oceans, leading to a pattern of SST anomalies characteristic of the Indian Ocean dipole. Fundamental and combination modes representing the tropospheric biennial oscillation (TBO) in CCSM4 are also found to be consistent with mechanisms for seasonally synchronized biennial variability of the Asian-Australian monsoon and Walker circulation. On longer timescales, the leading multidecadal pattern recovered by NLSA from Indo-Pacific SST data in CCSM4, referred to as west Pacific multidecadal mode (WPMM), is found to significantly modulate ENSO and TBO activity with periods of negative SST anomalies in the western tropical Pacific favoring stronger ENSO and TBO variability. Physically, this behavior is attributed to the fact that cold WPMM phases feature anomalous decadal westerlies in the tropical central Pacific and easterlies in the tropical Indian Ocean, as well as an anomalously deep thermocline in the eastern Pacific cold tongue. Moreover, despite the relatively low SST variance explained by this mode, the WPMM is found to correlate significantly with decadal precipitation over Australia in CCSM4.

physics.ao-ph↗

arXiv2016

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.

physics.geo-ph↗

arXiv2016

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 Nyström 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.

physics.data-an↗

arXiv2015

Data-driven prediction strategies for low-frequency patterns of North Pacific climate variability

The North Pacific exhibits patterns of low-frequency variability on the intra-annual to decadal time scales, which manifest themselves in both model data and the observational record, and prediction of such low-frequency modes of variability is of great interest to the community. While parametric models, such as stationary and non-stationary autoregressive models, possibly including external factors, may perform well in a data-fitting setting, they may perform poorly in a prediction setting. Ensemble analog forecasting, which relies on the historical record to provide estimates of the future based on past trajectories of those states similar to the initial state of interest, provides a promising, nonparametric approach to forecasting that makes no assumptions on the underlying dynamics or its statistics. We apply such forecasting to low-frequency modes of variability for the North Pacific sea surface temperature and sea ice concentration fields extracted through Nonlinear Laplacian Spectral Analysis. We find such methods may outperform parametric methods and simple persistence with increased predictive skill.

math.NA↗

arXiv2015

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 Niño-3.4 data set which is used as a proxy of the El-Niño Southern Oscillation.

math.DS↗

arXiv2014

Dynamics-Adapted Cone Kernels

We present a family of kernels for analysis of data generated by dynamical systems. These so-called cone kernels feature an explicit dependence on the dynamical vector field operating in the phase-space manifold, estimated empirically through finite-differences of time-ordered data samples. In particular, cone kernels assign strong affinity to pairs of samples whose relative displacement vector lies within a narrow cone aligned with the dynamical vector field. As a result, in a suitable asymptotic limit, the associated diffusion operator generates diffusions along the dynamical flow, and is invariant under a weakly restrictive class of transformations of the data, which includes conformal transformations. Moreover, the corresponding Dirichlet form is governed by the directional derivative of functions along the dynamical vector field. The latter feature is metric-independent. The diffusion eigenfunctions obtained via cone kernels are therefore adapted to the dynamics in that they vary predominantly in directions transverse to the flow. We demonstrate the utility of cone kernels in nonlinear flows on the 2-torus and North Pacific sea surface temperature data generated by a comprehensive climate model.

math.DS↗

arXiv2012

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.

physics.data-an↗