Techniques of linear prediction, with application to oceanic and atmospheric fields in the tropical Pacific

The problem of constructing optimal linear prediction models by multivariance regression methods is reviewed. It is well known that as the number of predictors in a model is increased, the skill of the prediction grows, but the statistical significance generally decreases. For predictions using a large number of candidate predictors, strategies are therefore needed to determine optimal prediction models which properly balance the competing requirements of skill and significance. The popular methods of coefficient screening or stepwise regression represent a posteriori predictor selection methods and therefore cannot be used to recover statistically significant models by truncation if the complete model, including all predictors, is statistically insignificant. Higher significance can be achieved only by a priori reduction of the predictor set. To determine the maximum number of predictors which may be meaningfully incorporated in a model, a model hierarchy can be used in which a series of best fit prediction models is constructed for a (prior defined) nested sequence of predictor sets, the sequence being terminated when the significance level either falls below a prescribed limit or reaches a maximum value. The method requires a reliable assessment of model significance. This is characterized by a quadratic statistic which is defined independently of the model skill or artificial skill. As an example, the method is applied to the prediction of sea surface temperature anomalies at Christmas Island (representative of sea surface temperatures in the central equatorial Pacific) and variations of the central and east Pacific Hadley circulation (characterized by the second empirical orthogonal function (EOF) of the meridional component of the trade wind anomaly field) using a general multiple‐time‐lag prediction matrix. The ordering of the predictors is based on an EOF sequence, defined formally as orthogonal variables in the composite space of all (normalized) predictors, irrespective of their different physical dimensions, time lag, and geographic position. The choice of a large set of 20 predictors at 12 time lags yields significant predictability only for forecast periods of 3 to 5 months. However, a prior reduction of the predictor set to 4 predictors at 10 time lags leads to 95% significant predictions with skill values of the order of 0.4 to 0.7 up to 6 or 8 months. For infinitely long time series the construction of optimal prediction models reduces essentially to the problem of linear system identification. However, the model hierarchies normally considered for the simulation of general linear systems differ in structure from the model hierarchies which appear to be most suitable for constructing pure prediction models. Thus the truncation imposed by statistical significance requirements can result in rather different models for the two cases. The relation between optimal prediction models and linear dynamical models is illustrated by the prediction of east‐west sea level changes in the equatorial Pacific from wind field anomalies. It is shown that the optimal empirical prediction is statistically consistent in this case with both the first‐order relaxation and damped oscillator models recently proposed by McWilliams and Gent (but with somewhat different model parameters than suggested by the authors). Thus the data do not allow a distinction between the two physical models; the simplest acceptable model is the first‐order damped response. Finally, the problem of estimating forecast skill is discussed. It is usually stated that the forecast skill is smaller than the true skill, which in turn is smaller than the hindcast skill, by an amount which in both cases is approximately equal to the artificial skill. However, this result applies to the mean skills averaged over the ensemble of all possible hindcast data sets, given the true model. Under the more appropriate side condition of a given hindcast data set and an unknown true model, the estimation of the forecast skill represents a problem of statistical inference and is dependent on the assumed prior probability distribution of true models. The Bayesian hypothesis of a uniform prior distribution yields an average forecast skill equal to the hindcast skill, but other (equally acceptable) assumptions yield lower forecast skills more compatible with the usual hindcast‐averaged expressio

Computations and parameterizations of the nonlinear energy transfer in a gravity-wave spectrum. Part II: Parameterizations of the nonlinear energy transfer for application in wave models

Four different parameterizations of the nonlinear energy transfer Snl in a surface wave spectrum are in investigated. Two parameterizations are based on a relatively small number of parameters and are useful primarily for application in parametrical or hybrid wave models. In the first parameterization, shape-distortion parameters are introduced to relate the distribution Snl for different values of the peak-enhancement parameter γ. The second parameterization is based on an EOF expansion of a set of Snl computed for a number of different spectral distributions. The remaining two parameterizations represent operator forms that contain the same number of free parameters as used to describe he wave spectrum. Such parameterizations with a matched number of input and output parameters are required for numerical stability in high-resolution discrete spectral models. A cubic, fourth-order diffusion-operator expression derived by a local-interaction expansion is found to be useful for understanding many of the properties of Snl, but is regarded as too inaccurate in detail for application in most wave models. The best results are achieved with a discrete-interaction operator parameterization, in which a single interaction configuration, together with its mirror image (representing a two-dimensional continuum of interactions with respect to a variable reference wavenumber scale and direction) is used to simulate the net effect of the full five-dimensional interaction continuum

Techniques of linear prediction for systems with periodic statistics

Many parameters that measure climatic variability have nonstationary statistics, that is, they depend strongly on the phase of the annual cycle. In this case normal statistical analysis techniques based on time-invariant models are inappropriate. Generalized methods accounting for seasonal nonstationarity (phase averaged or cyclostationary models) have been developed to treat such data. The methods are applied to the problem of predicting El Niño off South America. It is shown that El Niños may be predicted up to a year in advance with considerably more confidence and accuracy using phase-averaged models than with time-invariant models. In a second application surface air temperature anomalies are predicted over North America from Pacific Ocean sea surface temperatures. Again, the phase-averaged models consistently outperform models based on standard statistical procedures

Statistical prediction of seasonal air temperature over Eurasia

Statistical models for the prediction of seasonal surface air temperature anomalies over Eurasia were constructed. The models were designed to test the relative predictive skill of Atlantic sea surface temperatures (SST), sea level pressure (SLP) and persistence in a cyclostationary setting. Significant forecast skill was found for the spring season in central and eastern Europe. The main predictors were persistence and SLPs (the north Atlantic oscillation). SSTs had little predictive value. All results were confirmed with independent forecast experiments. The statistical results were attributed to (a) a positive feedback between given winter atmospheric circulation regimes, the snow cover they produce and the snow-induced enhancement/retardation of normal season warming and (b) the persistence of large-scale circulation patterns over the Atlantic Ocean

Functional renormalization group in the broken symmetry phase: momentum dependence and two-parameter scaling of the self-energy

We include spontaneous symmetry breaking into the functional renormalization group (RG) equations for the irreducible vertices of Ginzburg-Landau theories by augmenting these equations by a flow equation for the order parameter, which is determined from the requirement that at each RG step the vertex with one external leg vanishes identically. Using this strategy, we propose a simple truncation of the coupled RG flow equations for the vertices in the broken symmetry phase of the Ising universality class in D dimensions. Our truncation yields the full momentum dependence of the self-energy Sigma (k) and interpolates between lowest order perturbation theory at large momenta k and the critical scaling regime for small k. Close to the critical point, our method yields the self-energy in the scaling form Sigma (k) = k_c^2 sigma^{-} (k | xi, k / k_c), where xi is the order parameter correlation length, k_c is the Ginzburg scale, and sigma^{-} (x, y) is a dimensionless two-parameter scaling function for the broken symmetry phase which we explicitly calculate within our truncation.Comment: 9 pages, 4 figures, puplished versio

Topological Defects and the Spin Glass Phase of Cuprates

We propose that the spin glass phase of cuprates is due to the proliferation of topological defects of a spiral distortion of the antiferromagnet order. Our theory explains straightforwardly the simultaneous existence of short range incommensurate magnetic correlations and complete a-b symmetry breaking in this phase. We show via a renormalization group calculation that the collinear O(3)/O(2) symmetry is unstable towards the formation of local non-collinear correlations. A critical disorder strength is identified beyond which topological defects proliferate already at zero temperature.Comment: 7 pages, 2 figures. Final version with some changes and one replaced figur

Climate system modeling in the framework of the tolerable windows approach: The ICLIPS climate model

The computational burden associated with applications of theTolerable Windows Approach (TWA) considerably exceeds that oftraditional integrated assessments of global climate change. Aspart of the ICLIPS (Integrated Assessment of Climate ProtectionStrategies) project, a computationally efficient climate model hasbeen developed that can be included in integrated assessmentmodels of any kind. The ICLIPS climate model (ICM) is implementedin GAMS. It is driven by anthropogenic emissions of CO2,CH4, N2O, halocarbons, SF6, andSO2. Theoutput includes transient patterns of near-surface airtemperature, total column-integrated cloud cover fraction,precipitation, humidity, and global mean sea-level rise. Thecarbon cycle module explicitly treats the nonlinear sea watercarbon chemistry and the nonlinear CO2 fertilized biosphereuptake. Patterns of the impact-relevant climate variables arederived form empirical orthogonal function (EOF) analysis andscaled by the principal component of temperature change. Theevolution of the latter is derived from a box-model-typedifferential analogue to its impulse response function convolutionintegral. We present a description of the ICM components and someresults to demonstrate the model's applicability in the TWA setting