# Publications by Type: Journal Article

2015
Mukhin, Dmitry, Dmitri Kondrashov, Evgeny Loskutov, Andrey Gavrilov, Alexander Feigin, and Michael Ghil. 2015. “Predicting critical transitions in ENSO models. Part II: Spatially dependent models.” Journal of Climate 28 (5): 1962–1976. Abstract
The present paper is the second part of a two-part study on empirical modeling and prediction of climate variability. This paper deals with spatially distributed data, as opposed to the univariate data of Part I. The choice of a basis for effective data compression becomes of the essence. In many applications, it is the set of spatial empirical orthogonal functions that provides the uncorrelated time series of principal components (PCs) used in the learning set. In this paper, the basis of the learning set is obtained instead by applying multichannel singular-spectrum analysis to climatic time series and using the leading spatiotemporal PCs to construct a reduced stochastic model. The effectiveness of this approach is illustrated by predicting the behavior of the Jin–Neelin–Ghil (JNG) hybrid seasonally forced coupled ocean–atmosphere model of El Niño–Southern Oscillation. The JNG model produces spatially distributed and weakly nonstationary time series to which the model reduction and prediction methodology is applied. Critical transitions in the hybrid periodically forced coupled model are successfully predicted on time scales that are substantially longer than the duration of the learning sample.
Vannitsem, Stéphane, Jonathan Demaeyer, Lesley De Cruz, and Michael Ghil. 2015. “Low-frequency variability and heat transport in a low-order nonlinear coupled ocean–atmosphere model.” Physica D: Nonlinear Phenomena 309. Elsevier: 71–85. Abstract
We formulate and study a low-order nonlinear coupled ocean–atmosphere model with an emphasis on the impact of radiative and heat fluxes and of the frictional coupling between the two components. This model version extends a previous 24-variable version by adding a dynamical equation for the passive advection of temperature in the ocean, together with an energy balance model. The bifurcation analysis and the numerical integration of the model reveal the presence of low-frequency variability (LFV) concentrated on and near a long-periodic, attracting orbit. This orbit combines atmospheric and oceanic modes, and it arises for large values of the meridional gradient of radiative input and of frictional coupling. Chaotic behavior develops around this orbit as it loses its stability; this behavior is still dominated by the LFV on decadal and multi-decadal time scales that is typical of oceanic processes. Atmospheric diagnostics also reveals the presence of predominant low- and high-pressure zones, as well as of a subtropical jet; these features recall realistic climatological properties of the oceanic atmosphere. Finally, a predictability analysis is performed. Once the decadal-scale periodic orbits develop, the coupled system’s short-term instabilities–as measured by its Lyapunov exponents–are drastically reduced, indicating the ocean’s stabilizing role on the atmospheric dynamics. On decadal time scales, the recurrence of the solution in a certain region of the invariant subspace associated with slow modes displays some extended predictability, as reflected by the oscillatory behavior of the error for the atmospheric variables at long lead times.
Ghil, Michael, Mickaël D. Chekroun, and Gabor Stepan. 2015. “A collection on Climate Dynamics: Multiple Scales and Memory Effects'.” Proceedings of the Royal Society A 471 (20150097). Royal Society London. Publisher's Version
Kondrashov, Dmitri, Mickaël D. Chekroun, and Michael Ghil. 2015. “Data-driven non-Markovian closure models.” Physica D: Nonlinear Phenomena 297. Elsevier: 33–55. Abstract

This paper has two interrelated foci: (i) obtaining stable and efficient data-driven closure models by using a multivariate time series of partial observations from a large-dimensional system; and (ii) comparing these closure models with the optimal closures predicted by the Mori–Zwanzig (MZ) formalism of statistical physics. Multilayer stochastic models (MSMs) are introduced as both a generalization and a time-continuous limit of existing multilevel, regression-based approaches to closure in a data-driven setting; these approaches include empirical model reduction (EMR), as well as more recent multi-layer modeling. It is shown that the multilayer structure of MSMs can provide a natural Markov approximation to the generalized Langevin equation (GLE) of the MZ formalism. A simple correlation-based stopping criterion for an EMR–MSM model is derived to assess how well it approximates the GLE solution. Sufficient conditions are derived on the structure of the nonlinear cross-interactions between the constitutive layers of a given MSM to guarantee the existence of a global random attractor. This existence ensures that no blow-up can occur for a broad class of MSM applications, a class that includes non-polynomial predictors and nonlinearities that do not necessarily preserve quadratic energy invariants. The EMR–MSM methodology is first applied to a conceptual, nonlinear, stochastic climate model of coupled slow and fast variables, in which only slow variables are observed. It is shown that the resulting closure model with energy-conserving nonlinearities efficiently captures the main statistical features of the slow variables, even when there is no formal scale separation and the fast variables are quite energetic. Second, an MSM is shown to successfully reproduce the statistics of a partially observed, generalized Lotka–Volterra model of population dynamics in its chaotic regime. The challenges here include the rarity of strange attractors in the model’s parameter space and the existence of multiple attractor basins with fractal boundaries. The positivity constraint on the solutions’ components replaces here the quadratic-energy–preserving constraint of fluid-flow problems and it successfully prevents blow-up.

Chekroun, Mickaël D., and Honghu Liu. 2015. “Finite-horizon parameterizing manifolds, and applications to suboptimal control of nonlinear parabolic PDEs.” Acta Applicandae Mathematicae 135 (1): 81–144.
Groth, Andreas, and Michael Ghil. 2015. “Monte Carlo Singular Spectrum Analysis (SSA) revisited: Detecting oscillator clusters in multivariate datasets.” Journal of Climate 28 (19): 7873–7893. Abstract

Singular spectrum analysis (SSA) along with its multivariate extension (M-SSA) provides an efficient way to identify weak oscillatory behavior in high-dimensional data. To prevent the misinterpretation of stochastic fluctuations in short time series as oscillations, Monte Carlo (MC)–type hypothesis tests provide objective criteria for the statistical significance of the oscillatory behavior. Procrustes target rotation is introduced here as a key method for refining previously available MC tests. The proposed modification helps reduce the risk of type-I errors, and it is shown to improve the test’s discriminating power. The reliability of the proposed methodology is examined in an idealized setting for a cluster of harmonic oscillators immersed in red noise. Furthermore, the common method of data compression into a few leading principal components, prior to M-SSA, is reexamined, and its possibly negative effects are discussed. Finally, the generalized Procrustes test is applied to the analysis of interannual variability in the North Atlantic’s sea surface temperature and sea level pressure fields. The results of this analysis provide further evidence for shared mechanisms of variability between the Gulf Stream and the North Atlantic Oscillation in the interannual frequency band.

Groth, Andreas, Michael Ghil, Stéphane Hallegatte, and Patrice Dumas. 2015. “The Role of Oscillatory Modes in U.S. Business Cycles.” OECD Journal: Journal of Business Cycle Measurement and Analysis, no. 2015/1: 63–81. Abstract

We apply multivariate singular spectrum analysis to the study of U.S. business cycle dynamics. This method provides a robust way to identify and reconstruct oscillations, whether intermittent or modulated. We show such oscillations to be associated with comovements across the entire economy. The problem of spurious cycles generated by the use of detrending filters is addressed and we present a Monte Carlo test to extract significant oscillations. The behavior of the U.S. economy is shown to change significantly from one phase of the business cycle to another: the recession phase is dominated by a five-year mode, while the expansion phase exhibits more complex dynamics, with higher-frequency modes coming into play. We show that the variations so identified cannot be generated by random shocks alone, as assumed in ‘real’ business-cycle models, and that endogenous, deterministically generated variability has to be involved.

Kondrashov, Dmitri, and Pavel S. Berloff. 2015. “Stochastic modeling of decadal variability in ocean gyres.” Geophysical Research Letters 42: 1543–1553.
2014
Podladchikova, T. V., Y. Y. Shprits, Dmitri Kondrashov, and A. C. Kellerman. 2014. “Noise statistics identification for Kalman filtering of the electron radiation belt observations I: Model errors.” Journal of Geophysical Research: Space Physics 119 (7): 5700–5724. Publisher's Version
Podladchikova, T. V., Y. Y. Shprits, A. C. Kellerman, and Dmitri Kondrashov. 2014. “Noise statistics identification for Kalman filtering of the electron radiation belt observations: 2. Filtration and smoothing.” Journal of Geophysical Research: Space Physics 119 (7): 5725–5743. Publisher's Version
Kondrashov, Dmitri, R. Denton, Y. Y. Shprits, and H. J. Singer. 2014. “Reconstruction of gaps in the past history of solar wind parameters.” Geophysical Research Letters 41 (8): 2702–2707. Publisher's Version
Kellerman, A. C., Y. Y. Shprits, Dmitri Kondrashov, D. Subbotin, R. A. Makarevich, E. Donovan, and T. Nagai. 2014. “Three-dimensional data assimilation and reanalysis of radiation belt electrons: Observations of a four-zone structure using five spacecraft and the VERB code.” Journal of Geophysical Research: Space Physics 119 (11): 8764–8783. Publisher's Version
Roques, Lionel, Mickaël D. Chekroun, Michel Cristofol, Samuel Soubeyrand, and Michael Ghil. 2014. “Parameter estimation for energy balance models with memory.” Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences 470 (2169). The Royal Society. Publisher's Version Abstract
We study parameter estimation for one-dimensional energy balance models with memory (EBMMs) given localized and noisy temperature measurements. Our results apply to a wide range of nonlinear, parabolic partial differential equations with integral memory terms. First, we show that a space-dependent parameter can be determined uniquely everywhere in the PDE’s domain of definition D, using only temperature information in a small subdomain E⊂D. This result is valid only when the data correspond to exact measurements of the temperature. We propose a method for estimating a model parameter of the EBMM using more realistic, error-contaminated temperature data derived, for example, from ice cores or marine-sediment cores. Our approach is based on a so-called mechanistic-statistical model that combines a deterministic EBMM with a statistical model of the observation process. Estimating a parameter in this setting is especially challenging, because the observation process induces a strong loss of information. Aside from the noise contained in past temperature measurements, an additional error is induced by the age-dating method, whose accuracy tends to decrease with a sample’s remoteness in time. Using a Bayesian approach, we show that obtaining an accurate parameter estimate is still possible in certain cases.
Chekroun, Mickaël D., J. David Neelin, Dmitri Kondrashov, James C. McWilliams, and Michael Ghil. 2014. “Rough parameter dependence in climate models and the role of Ruelle-Pollicott resonances.” Proceedings of the National Academy of Sciences 111 (5): 1684-1690. Abstract

Despite the importance of uncertainties encountered in climate model simulations, the fundamental mechanisms at the origin of sensitive behavior of long-term model statistics remain unclear. Variability of turbulent flows in the atmosphere and oceans exhibits recurrent large-scale patterns. These patterns, while evolving irregularly in time, manifest characteristic frequencies across a large range of time scales, from intraseasonal through interdecadal. Based on modern spectral theory of chaotic and dissipative dynamical systems, the associated low-frequency variability may be formulated in terms of Ruelle-Pollicott (RP) resonances. RP resonances encode information on the nonlinear dynamics of the system, and an approach for estimating them—as filtered through an observable of the system—is proposed. This approach relies on an appropriate Markov representation of the dynamics associated with a given observable. It is shown that, within this representation, the spectral gap—defined as the distance between the subdominant RP resonance and the unit circle—plays a major role in the roughness of parameter dependences. The model statistics are the most sensitive for the smallest spectral gaps; such small gaps turn out to correspond to regimes where the low-frequency variability is more pronounced, whereas autocorrelations decay more slowly. The present approach is applied to analyze the rough parameter dependence encountered in key statistics of an El-Niño–Southern Oscillation model of intermediate complexity. Theoretical arguments, however, strongly suggest that such links between model sensitivity and the decay of correlation properties are not limited to this particular model and could hold much more generally.

2013
Shprits, Yuri, Adam Kellerman, Dmitri Kondrashov, and Dmitriy Subbotin. 2013. “Application of a new data operator-splitting data assimilation technique to the 3-D VERB diffusion code and CRRES measurements.” Geophysical Research Letters 40 (19): 4998–5002. Publisher's Version
Sella, Lisa, Gianna Vivaldo, Andreas Groth, and Michael Ghil. 2013. “Economic Cycles and their Synchronization: A spectral survey.” Fondazione Eni Enrico Mattei (FEEM) 105 (105). Fondazione Eni Enrico Mattei (FEEM): 1. Publisher's Version Abstract

The present work applies several advanced spectral methods to the analysis of macroeconomic fluctuations in three countries of the European Union: Italy, The Netherlands, and the United Kingdom. We focus here in particular on singular-spectrum analysis (SSA), which provides valuable spatial and frequency information of multivariate data and that goes far beyond a pure analysis in the time domain. The spectral methods discussed here are well established in the geosciences and life sciences, but not yet widespread in quantitative economics. In particular, they enable one to identify and describe nonlinear trends and dominant cycles –- including seasonal and interannual components –- that characterize the deterministic behavior of each time series. These tools have already proven their robustness in the application on short and noisy data, and we demonstrate their usefulness in the analysis of the macroeconomic indicators of these three countries. We explore several fundamental indicators of the countries' real aggregate economy in a univariate, as well as a multivariate setting. Starting with individual single-channel analysis, we are able to identify similar spectral components among the analyzed indicators. Next, we consider combinations of indicators and countries, in order to take different effects of comovements into account. Since business cycles are cross-national phenomena, which show common characteristics across countries, our aim is to uncover hidden global behavior across the European economies. Results are compared with previous findings on the U.S. indicators \citepGroth.ea.FEEM.2012. Finally, the analysis is extended to include several indicators from the U.S. economy, in order to examine its influence on the European market.

de Viron, O., J. O. Dickey, and Michael Ghil. 2013. “Global modes of climate variability.” Geophysical Research Letters 40 (9): 1832-1837. Abstract

The atmosphere, hydrosphere and cryosphere form a fully coupled climate system. This system exhibits a number of large-scale phenomena, such as the El Nino Southern Oscillation (ENSO), the Asian Monsoon, the North Atlantic Oscillation (NAO), and the Madden-Julian Oscillation (MJO). While these modes of variability are not exactly periodic, they are oscillatory in character, and their state is monitored using so-called climate indices. Each of these scalar indices is a combination of several climate variables. Here, we use a comprehensive set of 25 climate indices for time intervals that range between 1948 and 2011, and estimate an optimal set of lags between these indices to maximize their correlation. We show that most of the index pairs drawn from this set present a significant correlation on interannual time scales. It is also shown that, on average, about two-thirds of the total variability in each index can be described by using only the four leading principal components of the entire set of lagged indices. Our index set's leading orthogonal modes exhibit several interannual frequencies and capture separately variability associated with the North Atlantic and the North Pacific. These modes are associated, in turn, with large-scale variations of sea surface temperatures.

Kondrashov, Dmitri, Mickaël D. Chekroun, Andrew W. Robertson, and Michael Ghil. 2013. “Low-order stochastic model and `past-noise forecasting" of the Madden-Julian oscillation.” Geophysical Research Letters 40: 5305–5310.
Rousseau, D.-D., Michael Ghil, G. Kukla, A. Sima, P. Antoine, M. Fuchs, C. Hatté, F. Lagroix, M. Debret, and O. Moine. 2013. “Major dust events in Europe during marine isotope stage 5 (130–74 ka): a climatic interpretation of the" markers".” Climate of the Past 9 (5). Copernicus GmbH: 2213–2230.
Feliks, Yizhak, Andreas Groth, Andrew W. Robertson, and Michael Ghil. 2013. “Oscillatory Climate Modes in the Indian Monsoon, North Atlantic and Tropical Pacific.” Journal of Climate 26: 9528-–9544. Abstract

This paper explores the three-way interactions between the Indian monsoon, the North Atlantic and the Tropical Pacific. Four climate records were analyzed: the monsoon rainfall in two Indian regions, the Southern Oscillation Index for the Tropical Pacific, and the NAO index for the North Atlantic. The individual records exhibit highly significant oscillatory modes with spectral peaks at 7–8 yr and in the quasi-biennial and quasi-quadrennial bands. The interactions between the three regions were investigated in the light of the synchronization theory of chaotic oscillators. The theory was applied here by combining multichannel singular-spectrum analysis (M-SSA) with a recently introduced varimax rotation of the M-SSA eigenvectors. A key result is that the 7–8-yr and 2.7-yr oscillatory modes in all three regions are synchronized, at least in part. The energy-ratio analysis, as well as time-lag results, suggest that the NAO plays a leading role in the 7–8-yr mode. It was found therewith that the South Asian monsoon is not slaved to forcing from the equatorial Pacific, although it does interact strongly with it. The time-lag analysis pinpointed this to be the case in particular for the quasi-biennial oscillatory modes. Overall, these results confirm that the approach of synchronized oscillators, combined with varimax-rotated M-SSA, is a powerful tool in studying teleconnections between regional climate modes and that it helps identify the mechanisms that operate in various frequency bands. This approach should be readily applicable to ocean modes of variability and to the problems of air-sea interaction as well.