Dimitrios Giannakis

Nonlinear Laplacian spectral analysis for time series with intermittency and low-frequency variability

Many processes in science and engineering develop multiscale temporal and spatial patterns, with complex underlying dynamics and time-dependent external forcings. Because of the importance in understanding and predicting these phenomena, extracting the salient modes of variability empirically from incomplete observations is a problem of wide contemporary interest. Here, we present a technique for analyzing high-dimensional, complex time series that exploits the geometrical relationships between the observed data points to recover features characteristic of strongly nonlinear dynamics (such as intermittency and rare events), which are not accessible to classical singular spectrum analysis. The method employs Laplacian eigenmaps, evaluated after suitable time-lagged embedding, to produce a reduced representation of the observed samples, where standard tools of matrix algebra can be used to perform truncated singular-value decomposition despite the nonlinear geometrical structure of the dataset. We illustrate the utility of the technique in capturing intermittent modes associated with the Kuroshio current in the North Pacific sector of a general circulation model and dimensional reduction of a low-order atmospheric model featuring chaotic intermittent regime transitions, where classical singular spectrum analysis is already known to fail dramatically.

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 present an approach to image formation based on the symmetry properties of operations in three-dimensional space. Augmented with graph-theoretic means, this approach can recover the three-dimensional structure of objects from random snapshots of unknown orientation at four orders of magnitude higher complexity than previously demonstrated. This is critical for the burgeoning field of structure recovery by X-ray Free Electron Lasers, as well as the more established electron microscopic techniques, including cryo-electron microscopy of biological systems. In a subsequent paper, we demonstrate the recovery of structure and dynamics from experimental, ultralow-signal random sightings of systems with X-rays, electrons, and photons, with no orientational or timing information.

The symmetries of image formation by scattering. II. Applications

We show that the symmetries of image formation by scattering enable graph-theoretic manifold-embedding techniques to extract structural and timing information from simulated and experimental snapshots at extremely low signal. The approach constitutes a physically-based, computationally efficient, and noise-robust route to analyzing the large and varied datasets generated by existing and emerging methods for studying structure and dynamics by scattering. We demonstrate three-dimensional structure recovery from X-ray diffraction and cryo-electron microscope image snapshots of unknown orientation, the latter at 12 times lower dose than currently in use. We also show that ultra-low-signal, random sightings of dynamically evolving systems can be sequenced into high quality movies to reveal their evolution. Our approach offers a route to recovering timing information in time-resolved experiments, and extracting 3D movies from two-dimensional random sightings of dynamic systems.

Predicting the cloud patterns of the Madden‐Julian Oscillation through a low‐order nonlinear stochastic model

We assess the limits of predictability of the large-scale cloud patterns in the boreal winter Madden-Julian Oscillation (MJO) as measured through outgoing longwave radiation (OLR) alone, a proxy for convective activity. A recent advanced nonlinear time series technique, nonlinear Laplacian spectral analysis, is applied to the OLR data to define two spatial modes with high intermittency associated with the boreal winter MJO. A recent data-driven physics-constrained low-order stochastic modeling procedure is applied to these time series. The result is a four-dimensional nonlinear stochastic model for the two observed OLR variables and two hidden variables involving correlated multiplicative noise defined through energy-conserving nonlinear interaction. Systematic calibration and prediction experiments show the skillful prediction by these models for 40, 25, and 18 days in strong, moderate, and weak MJO winters, respectively. Furthermore, the ensemble spread is an accurate indicator of forecast uncertainty at long lead times.

Reemergence Mechanisms for North Pacific Sea Ice Revealed through Nonlinear Laplacian Spectral Analysis

AbstractThis paper studies spatiotemporal modes of variability of sea ice concentration and sea surface temperature (SST) in the North Pacific sector in a comprehensive climate model and observations. These modes are obtained via nonlinear Laplacian spectral analysis (NLSA), a recently developed data analysis technique for high-dimensional nonlinear datasets. The existing NLSA algorithm is modified to allow for a scale-invariant coupled analysis of multiple variables in different physical units. The coupled NLSA modes are utilized to investigate North Pacific sea ice reemergence: a process in which sea ice anomalies originating in the melt season (spring) are positively correlated with anomalies in the growth season (fall) despite a loss of correlation in the intervening summer months. It is found that a low-dimensional family of NLSA modes is able to reproduce the lagged correlations observed in sea ice data from the North Pacific Ocean. This mode family exists in both model output and observations and i...

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 datasets. 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 Vl 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 Vl required to capture these features while avoiding overfitting is estimated here using spectral entropy criteria.

Nonparametric forecasting of low-dimensional dynamical systems

This paper presents a nonparametric 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 nontrivial 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.

Arctic Sea Ice Reemergence: The Role of Large-Scale Oceanic and Atmospheric Variability*

AbstractArctic sea ice reemergence is a phenomenon in which spring sea ice anomalies are positively correlated with fall anomalies, despite a loss of correlation over the intervening summer months. This work employs a novel data analysis algorithm for high-dimensional multivariate datasets, coupled nonlinear Laplacian spectral analysis (NLSA), to investigate the regional and temporal aspects of this reemergence phenomenon. Coupled NLSA modes of variability of sea ice concentration (SIC), sea surface temperature (SST), and sea level pressure (SLP) are studied in the Arctic sector of a comprehensive climate model and in observations. It is found that low-dimensional families of NLSA modes are able to efficiently reproduce the prominent lagged correlation features of the raw sea ice data. In both the model and observations, these families provide an SST–sea ice reemergence mechanism, in which melt season (spring) sea ice anomalies are imprinted as SST anomalies and stored over the summer months, allowing for...

Quantifying the Predictive Skill in Long-Range Forecasting. Part II: Model Error in Coarse-Grained Markov Models with Application to Ocean-Circulation Regimes

AbstractAn information-theoretic framework is developed to assess the predictive skill and model error in imperfect climate models for long-range forecasting. Here, of key importance is a climate equilibrium consistency test for detecting false predictive skill, as well as an analogous criterion describing model error during relaxation to equilibrium. Climate equilibrium consistency enforces the requirement that long-range forecasting models should reproduce the climatology of prediction observables with high fidelity. If a model meets both climate consistency and the analogous criterion describing model error during relaxation to equilibrium, then relative entropy can be used as an unbiased superensemble measure of the model’s skill in long-range coarse-grained forecasts. As an application, the authors investigate the error in modeling regime transitions in a 1.5-layer ocean model as a Markov process and identify models that are strongly persistent but their predictive skill is false. The general techniq...

Quantifying the Predictive Skill in Long-Range Forecasting. Part I: Coarse-Grained Predictions in a Simple Ocean Model

AbstractAn information-theoretic framework is developed to assess the long-range coarse-grained predictive skill in a perfect-model environment. Central to the scheme is the notion that long-range forecasting involves regimes; specifically, that the appropriate initial data for ensemble prediction is the affiliation of the system to a coarse-grained partition of phase space representing regimes. The corresponding ensemble prediction probabilities, which are computable using ergodic signals from the model, are then used to quantify through relative entropy the information beyond climatology in the partition. As an application, the authors study the predictability of circulation regimes in an equivalent barotropic double-gyre ocean model using a partition algorithm based on K-means clustering and running-average coarse graining. Besides the established rolled up and extensional phases of the eastward jet, optimal partitions for triennial-scale forecasts feature a jet configuration dominated by the second em...