scholarly journals Multilevel Regression Modeling of Nonlinear Processes: Derivation and Applications to Climatic Variability

2005 ◽  
Vol 18 (21) ◽  
pp. 4404-4424 ◽  
Author(s):  
S. Kravtsov ◽  
D. Kondrashov ◽  
M. Ghil
Keyword(s):  
Nonlinear Dynamics ◽  
Polynomial Regression ◽  
Climatic Variability ◽  
Low Frequency ◽  
Inverse Model ◽  
Empirical Orthogonal Functions ◽  
Multilevel Regression ◽  
Coherent Noise ◽  
Orthogonal Functions ◽  
Constant Matrix

Abstract Predictive models are constructed to best describe an observed field’s statistics within a given class of nonlinear dynamics driven by a spatially coherent noise that is white in time. For linear dynamics, such inverse stochastic models are obtained by multiple linear regression (MLR). Nonlinear dynamics, when more appropriate, is accommodated by applying multiple polynomial regression (MPR) instead; the resulting model uses polynomial predictors, but the dependence on the regression parameters is linear in both MPR and MLR. The basic concepts are illustrated using the Lorenz convection model, the classical double-well problem, and a three-well problem in two space dimensions. Given a data sample that is long enough, MPR successfully reconstructs the model coefficients in the former two cases, while the resulting inverse model captures the three-regime structure of the system’s probability density function (PDF) in the latter case. A novel multilevel generalization of the classic regression procedure is introduced next. In this generalization, the residual stochastic forcing at a given level is subsequently modeled as a function of variables at this level and all the preceding ones. The number of levels is determined so that the lag-0 covariance of the residual forcing converges to a constant matrix, while its lag-1 covariance vanishes. This method has been applied to the output of a three-layer, quasigeostrophic model and to the analysis of Northern Hemisphere wintertime geopotential height anomalies. In both cases, the inverse model simulations reproduce well the multiregime structure of the PDF constructed in the subspace spanned by the dataset’s leading empirical orthogonal functions, as well as the detailed spectrum of the dataset’s temporal evolution. These encouraging results are interpreted in terms of the modeled low-frequency flow’s feedback on the statistics of the subgrid-scale processes.

scholarly journals A Hierarchy of Data-Based ENSO Models

2005 ◽  
Vol 18 (21) ◽  
pp. 4425-4444 ◽  
Author(s):  
D. Kondrashov ◽  
S. Kravtsov ◽  
A. W. Robertson ◽  
M. Ghil
Keyword(s):  
El Niño ◽  
Additive Noise ◽  
Polynomial Regression ◽  
El Nino ◽  
Southern Oscillation ◽  
Model Performance ◽  
Empirical Orthogonal Functions ◽  
Quadratic Model ◽  
Orthogonal Functions ◽  
Optimizing Model

Abstract Global sea surface temperature (SST) evolution is analyzed by constructing predictive models that best describe the dataset’s statistics. These inverse models assume that the system’s variability is driven by spatially coherent, additive noise that is white in time and are constructed in the phase space of the dataset’s leading empirical orthogonal functions. Multiple linear regression has been widely used to obtain inverse stochastic models; it is generalized here in two ways. First, the dynamics is allowed to be nonlinear by using polynomial regression. Second, a multilevel extension of classic regression allows the additive noise to be correlated in time; to do so, the residual stochastic forcing at a given level is modeled as a function of variables at this level and the preceding ones. The number of variables, as well as the order of nonlinearity, is determined by optimizing model performance. The two-level linear and quadratic models have a better El Niño–Southern Oscillation (ENSO) hindcast skill than their one-level counterparts. Estimates of skewness and kurtosis of the models’ simulated Niño-3 index reveal that the quadratic model reproduces better the observed asymmetry between the positive El Niño and negative La Niña events. The benefits of the quadratic model are less clear in terms of its overall, cross-validated hindcast skill; this model outperforms, however, the linear one in predicting the magnitude of extreme SST anomalies. Seasonal ENSO dependence is captured by incorporating additive, as well as multiplicative forcing with a 12-month period into the first level of each model. The quasi-quadrennial ENSO oscillatory mode is robustly simulated by all models. The “spring barrier” of ENSO forecast skill is explained by Floquet and singular vector analysis, which show that the leading ENSO mode becomes strongly damped in summer, while nonnormal optimum growth has a strong peak in December.

Statistical description of the East Australian Current low-frequency variability from the WOCE PCM3 array

2006 ◽  
Vol 57 (3) ◽  
pp. 273 ◽  
Author(s):  
Mauricio M. Mata ◽  
Susan Wijffels ◽  
John A. Church ◽  
Matthias Tomczak
Keyword(s):  
Ocean Circulation ◽  
World Ocean ◽  
Low Frequency ◽  
General Rule ◽  
Empirical Orthogonal Functions ◽  
East Australian Current ◽  
Orthogonal Functions ◽  
The Pacific ◽  
Strong Surface ◽  
The Impact

The in situ dataset used in the current study consists of the Pacific Current Meter 3 (PCM3) array, which was a significant part of the Australian contribution to the World Ocean Circulation Experiment to study the variability of the East Australian Current (EAC), and was operational between September 1991 and March 1994. Area-preserving spectral analysis has been used to investigate the typical time scales observed by the current meters. As a general rule, the spectra from the top layers of the shallow (1, 2 and 3) and the deep (4, 5 and 6) moorings have a distinct peak in the temporal mesoscale band (periods between 70 and 170 days), with a general redistribution of energy towards the higher-frequencies near the ocean floor. This peak has been linked with eddy variability of the EAC system, which influences the fluctuations of the current main jet. The vertical modes of the velocity profile show that the strong surface-intensified baroclinic signal of the EAC dominated the variability at mooring 4 location. Further offshore the predominant configuration resembles more closely the barotropic mode. Ultimately, spatial empirical orthogonal functions (EOF) analysis point out the impact of the presence/absence of the EAC jet in the array.

scholarly journals Multiple Regimes and Low-Frequency Oscillations in the Northern Hemisphere’s Zonal-Mean Flow

2006 ◽  
Vol 63 (3) ◽  
pp. 840-860 ◽  
Author(s):  
S. Kravtsov ◽  
A. W. Robertson ◽  
M. Ghil
Keyword(s):  
Singular Spectrum Analysis ◽  
Low Frequency ◽  
Zonal Flow ◽  
Empirical Orthogonal Functions ◽  
Mean Flow ◽  
Orthogonal Functions ◽  
Atlantic Sector ◽  
Low Frequency Oscillations ◽  
Multiple Regimes ◽  
Flow States

Abstract This paper studies multiple regimes and low-frequency oscillations in the Northern Hemisphere zonal-mean zonal flow in winter, using 55 yr of daily observational data. The probability density function estimated in the phase space spanned by the two leading empirical orthogonal functions exhibits two distinct, statistically significant maxima. The two regimes associated with these maxima describe persistent zonal-flow states that are characterized by meridional displacements of the midlatitude jet, poleward and equatorward of its time-mean position. The geopotential height anomalies of either regime have a pronounced zonally symmetric component, but largest-amplitude anomalies are located over the Atlantic and Pacific Oceans. High-frequency synoptic transients participate in the maintenance of and transitions between these regimes. Significant oscillatory components with periods of 147 and 72 days are identified by spectral analysis of the zonal-flow time series. These oscillations are described by singular spectrum analysis and the multitaper method. The 147-day oscillation involves zonal-flow anomalies that propagate poleward, while the 72-day oscillation only manifests northward propagation in the Atlantic sector. Both modes mainly describe changes in the midlatitude jet position and intensity. In the horizontal plane though, the two modes exhibit synchronous centers of action located over the Atlantic and Pacific Oceans. The two persistent flow regimes are associated with slow phases of either oscillation.

scholarly journals Climate Regime Variability for Past and Present Time Slices Simulated by the Fast Ocean Atmosphere Model

2009 ◽  
Vol 22 (1) ◽  
pp. 58-70 ◽  
Author(s):  
Dörthe Handorf ◽  
Klaus Dethloff ◽  
Andrew G. Marshall ◽  
Amanda Lynch
Keyword(s):  
Climate Model ◽  
Low Frequency ◽  
Empirical Orthogonal Functions ◽  
Climate Regime ◽  
Orthogonal Functions ◽  
Medium Resolution ◽  
Ocean Atmosphere ◽  
Dimensional State Space ◽  
Atmosphere Model ◽  
Low Dimensional

Abstract This paper presents an analysis of Northern Hemisphere climate regime variability for three different time slices, simulated by the Fast Ocean Atmosphere Model (FOAM). The three time slices are composed of present-day conditions, the mid-Holocene, and the Last Glacial Maximum (LGM). Climate regimes have been determined by analyzing the structure of a spherical probability density function in a low-dimensional state space spanned by the three leading empirical orthogonal functions. This study confirms the ability of the FOAM medium-resolution climate model to reproduce low-frequency climate variability in the form of regime-like behavior. Three to four regimes have been detected for each time slice. Compared with present-day conditions, new climate regimes appeared for the LGM. For the mid-Holocene, which had slightly different boundary conditions and external forcings than the present-day simulation, the frequency of occurrence of the regimes was altered while only slight changes were found in the structure of some regimes.

scholarly journals Signatures of Nonlinear Dynamics in an Idealized Atmospheric Model

2011 ◽  
Vol 68 (1) ◽  
pp. 3-12 ◽  
Author(s):  
Dmitri Kondrashov ◽  
Sergey Kravtsov ◽  
Michael Ghil
Keyword(s):  
Nonlinear Dynamics ◽  
Phase Space ◽  
Model Reduction ◽  
Empirical Model ◽  
Multiplicative Noise ◽  
Atmospheric Model ◽  
Empirical Orthogonal Functions ◽  
Orthogonal Functions ◽  
Low Order

Abstract Signatures of nonlinear dynamics are analyzed by studying the phase-space tendencies of a global baroclinic, quasigeostrophic, three-level (QG3) model with topography. Nonlinear, stochastic, low-order prototypes of the full QG3 model are constructed in the phase space of this model’s empirical orthogonal functions using the empirical model reduction (EMR) approach. The phase-space tendencies of the EMR models closely match the full QG3 model’s tendencies. The component of these tendencies that is not linearly parameterizable is shown to be dominated by the interactions between “resolved” modes rather than by multiplicative “noise” associated with unresolved modes. The method of defining the leading resolved modes and the interactions between them plays a key role in understanding the nature of the QG3 model’s dynamics, whether linear or nonlinear, deterministic or stochastic.

scholarly journals Linear Contributions of Different Time Scales to Teleconnectivity

2009 ◽  
Vol 22 (13) ◽  
pp. 3720-3728 ◽  
Author(s):  
Panos J. Athanasiadis ◽  
Maarten H. P. Ambaum
Keyword(s):  
Time Scales ◽  
Low Frequency ◽  
Reanalysis Data ◽  
Empirical Orthogonal Functions ◽  
Orthogonal Functions ◽  
Low Frequencies ◽  
Teleconnection Patterns ◽  
The North ◽  
The North Atlantic Oscillation ◽  
Different Time Scales

Abstract The contributions of different time scales to extratropical teleconnections are examined. By applying empirical orthogonal functions and correlation analyses to reanalysis data, it is shown that eddies with periods shorter than 10 days have no linear contribution to teleconnectivity. Instead, synoptic variability follows wavelike patterns along the storm tracks, interpreted as propagating baroclinic disturbances. In agreement with preceding studies, it is found that teleconnections such as the North Atlantic Oscillation (NAO) and the Pacific–North America (PNA) pattern occur only at low frequencies, typically for periods more than 20 days. Low-frequency potential vorticity variability is shown to follow patterns analogous to known teleconnections but with shapes that differ considerably from them. It is concluded that the role, if any, of synoptic eddies in determining and forcing teleconnections needs to be sought in nonlinear interactions with the slower transients. The present results demonstrate that daily variability of teleconnection indices cannot be interpreted in terms of the teleconnection patterns, only the slow part of the variability.

scholarly journals Common metrics of calibration for continuous Gaussian data and exceedance probabilities

2018 ◽  
Author(s):  
Rita Glowienka-Hense ◽  
Andreas Hense ◽  
Thomas Spangehl ◽  
Marc Schröder
Keyword(s):  
Gaussian Processes ◽  
Low Frequency ◽  
Empirical Orthogonal Functions ◽  
Brier Score ◽  
Data Sets ◽  
Orthogonal Functions ◽  
Leibler Divergence ◽  
The Mean ◽  
Decadal Climate ◽  
Concept Of Information

Abstract. A framework of ensemble forecast verification tools is discussed which is founded on the concept of information entropy. It can be based on a common yardstick namely that of "correlation". With these measures calibration is deduced from the balance between ensemble sharpness and resolution. With the same units these features can be put into one diagram for continuous time series from Gaussian processes and exceedance probabilities, the latter usually tested with the reliability term from the Brier score. The sharpness and resolution terms allow to use the same vocabulary of over- and underdispersion which is established for frequency histograms. The concept is based on the fact that mutual information (MI) of two Gaussian processes is directly related to Pearson's anomaly correlation. Further MI can be written as the Kullback-Leibler divergence of the conditional probability of observations given the model forecasts and the unconditioned observations. Thus the MI is a measure of resolution. The mean of the UTILITY defined by (Kleeman, 2002) is the corresponding measure of sharpness. For Gaussian processes the mean UTILITY is very close to the ratio of ensemble mean variance to mean ensemble variance (ANOVA) which is the analysis of variance factor when time is taken as treatment. The ensemble spread score (ESS) (Palmer et al., 2006) is shown to be a measure of calibration if model and observed data are scaled with their respective means and standard deviations. For exceedance probabilities the resolution term of the divergence score (Weijs et al., 2010) is already defined as a MI term and it is here complemented with a mean UTILITY formed similarly to the resolution term but with forecasts only. The entropy terms are then rescaled to the "correlation" yardstick. The concept is applied to temperature data from the German project on decadal climate prediction, Mittelfristige Klimaprognose (MiKlip). It is shown that both over – and underdispersion can be found for the 2m temperature forecasts. Increasing ensemble sharpness of surface ocean temperature with lead year in the southern ocean hints at model-data inconsistencies at some locations in the ocean. Finally empirical orthogonal functions (EOF) of northern hemisphere annual mean surface temperature for ERA-40/ERA-Interim and MiKlip retrospective hindcasts are determined. For both data sets the respective first EOF represents the low frequency temperature development. The time coefficients of the EOF are used to compare resolution and sharpness of continuous data and exceedance probabilities in one diagram.

scholarly journals Bimodal Behavior in the Zonal Mean Flow of a Baroclinic β-Channel Model

2005 ◽  
Vol 62 (6) ◽  
pp. 1746-1769 ◽  
Author(s):  
S. Kravtsov ◽  
A. W. Robertson ◽  
M. Ghil
Keyword(s):  
Channel Model ◽  
Low Frequency ◽  
Zonal Flow ◽  
Relative Energy ◽  
Empirical Orthogonal Functions ◽  
Stationary Mode ◽  
Nonlinear Interactions ◽  
Mean Flow ◽  
Orthogonal Functions ◽  
Synoptic Eddies

Abstract The dynamical origin of midlatitude zonal-jet variability is examined in a thermally forced, quasigeostrophic, two-layer channel model on a β plane. The model’s behavior is studied as a function of the bottom-friction strength. Two distinct zonal-flow states exist at realistic, low, and intermediate values of the bottom drag; these two states are maintained by the eddies and differ mainly in terms of the meridional position of their climatological jets. The system’s low-frequency evolution is characterized by irregular transitions between the two states. For a given branch of model solutions, the leading stationary and propagating empirical orthogonal functions are related to eigenmodes of the model’s dynamical operator, linearized about the climatological state on this branch. Nonlinear interactions between these modes are instrumental in determining their relative energy level. In particular, the stationary modes’ self-interaction is shown to vanish. Thus, these modes do not exchange energy with the mean flow and, consequently, dominate the lowest-frequency behavior in the model. The leading stationary mode resembles the observed annular mode in the Southern Hemisphere. The bimodality is due to nonlinear interactions between nearly equivalent barotropic, stationary, and propagating modes, while the synoptic eddies play a modest role in determining the relative persistence of the two states. The role of synoptic eddies is very substantial only at unrealistically high values of the bottom drag, where they give rise to ultralow frequency variability by modifying the jet in a way that reinforces generation of the eddy field. This type of behavior is related to the presence of a homoclinic orbit in the model’s phase space and is not apparent for more realistic, lower values of the bottom drag.

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