Multiplicative Noise and Non-Gaussianity: A Paradigm for Atmospheric Regimes?

2005 ◽  
Vol 62 (5) ◽  
pp. 1391-1409 ◽  
Author(s):  
Philip Sura ◽  
Matthew Newman ◽  
Cécile Penland ◽  
Prashant Sardeshmukh
Keyword(s):  
Large Scale ◽  
Multiplicative Noise ◽  
Probability Distributions ◽  
Joint Probability ◽  
Joint Probability Distribution ◽  
Stochastic Perturbations ◽  
The Past ◽  
State Dependent ◽  
Non Gaussian ◽  
Noise Statistics

Abstract Atmospheric circulation statistics are not strictly Gaussian. Small bumps and other deviations from Gaussian probability distributions are often interpreted as implying the existence of distinct and persistent nonlinear circulation regimes associated with higher-than-average levels of predictability. In this paper it is shown that such deviations from Gaussianity can, however, also result from linear stochastically perturbed dynamics with multiplicative noise statistics. Such systems can be associated with much lower levels of predictability. Multiplicative noise is often identified with state-dependent variations of stochastic feedbacks from unresolved system components, and may be treated as stochastic perturbations of system parameters. It is shown that including such perturbations in the damping of large-scale linear Rossby waves can lead to deviations from Gaussianity very similar to those observed in the joint probability distribution of the first two principal components (PCs) of weekly averaged 750-hPa streamfunction data for the past 52 winters. A closer examination of the Fokker–Planck probability budget in the plane spanned by these two PCs shows that the observed deviations from Gaussianity can be modeled with multiplicative noise, and are unlikely the results of slow nonlinear interactions of the two PCs. It is concluded that the observed non-Gaussian probability distributions do not necessarily imply the existence of persistent nonlinear circulation regimes, and are consistent with those expected for a linear system perturbed by multiplicative noise.

scholarly journals Bayesian computational methods

1991 ◽  
Vol 337 (1647) ◽  
pp. 369-386 ◽  
Keyword(s):  
Probability Distributions ◽  
Joint Probability ◽  
Statistical Modelling ◽  
Joint Probability Distribution ◽  
Adaptive Quadrature ◽  
Adaptive Monte Carlo ◽  
Integrated Likelihood ◽  
The Past ◽  
Likelihood Approach ◽  
Probability Description

The bayesian (or integrated likelihood) approach to statistical modelling and analysis proceeds by representing all uncertainties in the form of probability distributions. Learning from new data is accomplished by application of Bayes’s Theorem, the latter providing a joint probability description of uncertainty for all model unknowns. To pass from this joint probability distribution to a collection of marginal summary inferences for specified interesting individual (or subsets of) unknowns, requires appropriate integration of the joint distribution. In all but simple stylized problems, these (typically high-dimensional) integrations will have to be performed numerically. This need for efficient simultaneous calculation of potentially many numerical integrals poses novel computational problems. Developments over the past decade are reviewed, including adaptive quadrature, adaptive Monte Carlo, and a variant of a Markov chain simulation procedure known as the Gibbs sampler.

scholarly journals Optimal compromise between incompatible conditional probability distributions, with application to Objective Bayesian Kriging

2019 ◽  
Vol 23 ◽  
pp. 271-309
Author(s):  
Joseph Muré
Keyword(s):  
Probability Distribution ◽  
Gibbs Sampler ◽  
Prediction Interval ◽  
Probability Distributions ◽  
Joint Probability ◽  
Joint Probability Distribution ◽  
Conditional Distributions ◽  
Stationary Probability Distribution ◽  
Gibbs Sampling Algorithm ◽  
Correlation Parameters

Models are often defined through conditional rather than joint distributions, but it can be difficult to check whether the conditional distributions are compatible, i.e. whether there exists a joint probability distribution which generates them. When they are compatible, a Gibbs sampler can be used to sample from this joint distribution. When they are not, the Gibbs sampling algorithm may still be applied, resulting in a “pseudo-Gibbs sampler”. We show its stationary probability distribution to be the optimal compromise between the conditional distributions, in the sense that it minimizes a mean squared misfit between them and its own conditional distributions. This allows us to perform Objective Bayesian analysis of correlation parameters in Kriging models by using univariate conditional Jeffreys-rule posterior distributions instead of the widely used multivariate Jeffreys-rule posterior. This strategy makes the full-Bayesian procedure tractable. Numerical examples show it has near-optimal frequentist performance in terms of prediction interval coverage.

Convection of a passive scalar by a quasi-uniform random straining field

1974 ◽  
Vol 64 (4) ◽  
pp. 737-762 ◽  
Author(s):  
Robert H. Kraichnan
Keyword(s):  
Velocity Field ◽  
Probability Distributions ◽  
Joint Probability ◽  
Passive Scalar ◽  
Joint Probability Distribution ◽  
Molecular Diffusivity ◽  
Velocity Reversal ◽  
Amplitude Profile ◽  
Line Elements ◽  
Rates Of Growth

The stretching of line elements, surface elements and wave vectors by a random, isotropic, solenoidal velocity field in D dimensions is studied. The rates of growth of line elements and (D – 1)-dimensional surface elements are found to be equal if the statistics are invariant to velocity reversal. The analysis is applied to convection of a sparse distribution of sheets of passive scalar in a random straining field whose correlation scale is large compared with the sheet size. This is Batchelor's (1959) κ−1 spectral regime. Some exact analytical solutions are found when the velocity field varies rapidly in time. These include the dissipation spectrum and a joint probability distribution that describes the simultaneous effect of Stretching and molecular diffusivity κ on the amplitude profile of a sheet. The latter leads to probability distributions of the scalar field and its space derivatives. For a growing κ−1 range at zero κ, these derivatives have essentially lognormal statistics. In the steady-state κ−1 regime at κ > 0, intermittencies measured by moment ratios are much smaller than for lognormal statistics, and they increase less rapidly with the order of the derivative than in the κ = 0 case. The κ > 0 distributions have singularities a t zero amplitude, due to a background of highly diffused sheets. The results do not depend strongly on D. But as D → ∞, temporal fluctuations in the stretching rates become negligible and Batchelor's (1959) constant-strain dissipation spectrum is recovered.

scholarly journals Need for Caution in Interpreting Extreme Weather Statistics

2015 ◽  
Vol 28 (23) ◽  
pp. 9166-9187 ◽  
Author(s):  
Prashant D. Sardeshmukh ◽  
Gilbert P. Compo ◽  
Cécile Penland
Keyword(s):  
Process Model ◽  
Large Scale ◽  
Probability Distributions ◽  
Extreme Weather ◽  
Recent Change ◽  
Atmospheric Variability ◽  
Tail Probabilities ◽  
The North ◽  
Special Cases ◽  
Non Gaussian

Abstract Given the reality of anthropogenic global warming, it is tempting to seek an anthropogenic component in any recent change in the statistics of extreme weather. This paper cautions that such efforts may, however, lead to wrong conclusions if the distinctively skewed and heavy-tailed aspects of the probability distributions of daily weather anomalies are ignored or misrepresented. Departures of several standard deviations from the mean, although rare, are far more common in such a distinctively non-Gaussian world than they are in a Gaussian world. This further complicates the problem of detecting changes in tail probabilities from historical records of limited length and accuracy. A possible solution is to exploit the fact that the salient non-Gaussian features of the observed distributions are captured by so-called stochastically generated skewed (SGS) distributions that include Gaussian distributions as special cases. SGS distributions are associated with damped linear Markov processes perturbed by asymmetric stochastic noise and as such represent the simplest physically based prototypes of the observed distributions. The tails of SGS distributions can also be directly linked to generalized extreme value (GEV) and generalized Pareto (GP) distributions. The Markov process model can be used to provide rigorous confidence intervals and to investigate temporal persistence statistics. The procedure is illustrated for assessing changes in the observed distributions of daily wintertime indices of large-scale atmospheric variability in the North Atlantic and North Pacific sectors over the period 1872–2011. No significant changes in these indices are found from the first to the second half of the period.

scholarly journals Linking Nonlinearity and Non-Gaussianity of Planetary Wave Behavior by the Fokker–Planck Equation

2005 ◽  
Vol 62 (7) ◽  
pp. 2098-2117 ◽  
Author(s):  
Judith Berner
Keyword(s):  
Stochastic Model ◽  
Probability Density ◽  
Planetary Wave ◽  
Multiplicative Noise ◽  
Planck Equation ◽  
Fokker Planck Equation ◽  
State Dependent ◽  
Nonlinear Stochastic Model ◽  
Non Gaussian ◽  
Fokker Planck

Abstract To link prominent nonlinearities in the dynamics of 500-hPa geopotential heights to non-Gaussian features in their probability density, a nonlinear stochastic model of atmospheric planetary wave behavior is developed. An analysis of geopotential heights generated by extended integrations of a GCM suggests that a stochastic model and its associated Fokker–Planck equation call for a nonlinear drift and multiplicative noise. All calculations are carried out in the reduced phase space spanned by the leading EOFs. It is demonstrated that this nonlinear stochastic model of planetary wave behavior captures the non-Gaussian features in the probability density function of atmospheric states to a remarkable degree. Moreover, it not only predicts global temporal characteristics, but also the nonlinear, state-dependent divergence of state trajectories. In the context of this empirical modeling, it is discussed on which time scale a stochastic model is expected to approximate the behavior of a continuous deterministic process. The reduced model is then used to determine the importance of the nonlinearities in the drift and the role of the multiplicative noise. While the nonlinearities in the drift are crucial for a good representation of planetary wave behavior, multiplicative (i.e., state dependent) noise is not absolutely essential. It is found that a major contributor to the stochastic component is the Branstator–Kushnir oscillation, which acts as a fluctuating force for physical processes with even longer time scales, like those that project on the Arctic Oscillation pattern. In this model, the oscillation is represented by strongly correlated noise.

scholarly journals Vertical fluxes conditioned on vorticity and strain reveal submesoscale ventilation

2021 ◽  
Author(s):  
Dhruv Balwada ◽  
Qiyu Xiao ◽  
Shafer Smith ◽  
Ryan Abernathey ◽  
Alison R. Gray
Keyword(s):  
Large Scale ◽  
Joint Probability ◽  
Horizontal Resolution ◽  
Surface Velocity ◽  
Small Scale ◽  
Joint Probability Distribution ◽  
Tracer Transport ◽  
Multi Scale ◽  
Flow Features ◽  
Circumpolar Current

AbstractIt has been hypothesized that submesoscale flows play an important role in the vertical transport of climatically important tracers, due to their strong associated vertical velocities. However, the multi-scale, non-linear, and Lagrangian nature of transport makes it challenging to attribute proportions of the tracer fluxes to certain processes, scales, regions, or features. Here we show that criteria based on the surface vorticity and strain joint probability distribution function (JPDF) effectively decomposes the surface velocity field into distinguishable flow regions, and different flow features, like fronts or eddies, are contained in different flow regions. The JPDF has a distinct shape and approximately parses the flow into different scales, as stronger velocity gradients are usually associated with smaller scales. Conditioning the vertical tracer transport on the vorticity-strain JPDF can therefore help to attribute the transport to different types of flows and scales. Applied to a set of idealized Antarctic Circumpolar Current simulations that vary only in horizontal resolution, this diagnostic approach demonstrates that small-scale strain dominated regions that are generally associated with submesoscale fronts, despite their minuscule spatial footprint, play an outsized role in exchanging tracers across the mixed layer base and are an important contributor to the large-scale tracer budgets. Resolving these flows not only adds extra flux at the small scales, but also enhances the flux due to the larger-scale flows.

scholarly journals Tests for quantum contextuality in terms of Q-entropies

2014 ◽  
Vol 14 (11&12) ◽  
pp. 996-1013
Author(s):  
Alexey E. Rastegin
Keyword(s):  
Probability Distribution ◽  
Probability Distributions ◽  
Joint Probability ◽  
Joint Probability Distribution ◽  
Theoretic Approach ◽  
Entropic Inequalities ◽  
Information Theoretic ◽  
The Family ◽  
Information Theoretic Approach ◽  
New Inequalities

The information-theoretic approach to Bell's theorem is developed with use of the conditional $q$-entropies. The $q$-entropic measures fulfill many similar properties to the standard Shannon entropy. In general, both the locality and noncontextuality notions are usually treated with use of the so-called marginal scenarios. These hypotheses lead to the existence of a joint probability distribution, which marginalizes to all particular ones. Assuming the existence of such a joint probability distribution, we derive the family of inequalities of Bell's type in terms of conditional $q$-entropies for all $q\geq1$. Quantum violations of the new inequalities are exemplified within the Clauser--Horne--Shimony--Holt (CHSH) and Klyachko--Can--Binicio\v{g}lu--Shumovsky (KCBS) scenarios. An extension to the case of $n$-cycle scenario is briefly mentioned. The new inequalities with conditional $q$-entropies allow to expand a class of probability distributions, for which the nonlocality or contextuality can be detected within entropic formulation. The $q$-entropic inequalities can also be useful in analyzing cases with detection inefficiencies. Using two models of such a kind, we consider some potential advantages of the $q$-entropic formulation.

The Probabilistic Method of Failure Analysis to Transmission Facilities under Ice Storms

2010 ◽  
Vol 37-38 ◽  
pp. 1525-1528
Author(s):  
Wen Jun Xu ◽  
Hong Ming Yang ◽  
Ming Yong Lai ◽  
Shuang Wang
Keyword(s):  
Transmission Lines ◽  
Power Transmission ◽  
Probability Distributions ◽  
Probabilistic Method ◽  
Joint Probability ◽  
Copula Function ◽  
Value Theory ◽  
Joint Probability Distribution ◽  
Ice Storms ◽  
Power Transmission Systems

Based on Extreme Value Theory (EVT), the Generalized Pareto Distributions (GPDs) of meteorological variables wind speed and freezing precipitation is simulated. Considering the dependence of them, a joint probability distribution is calculated by the Copula function. Further more, the probability distributions of ice loads and wind loads on transmission lines are analyzed, and the failure probability of broken lines and collapsed towers under ice storms is calculated. The accuracy and validity of this analytical method is demonstrated with comparison between numerical results and the historical datas of Chen Zhou power transmission systems.

Joint probability of storm surge and wave along Hainan Island based on bi-variate copulas analysis

2020 ◽  
Author(s):  
Meng Cheng ◽  
Weihua Fang
Keyword(s):  
Probability Distribution ◽  
Storm Surge ◽  
Wave Height ◽  
Significant Wave Height ◽  
Probability Distributions ◽  
Joint Probability ◽  
Hainan Island ◽  
Joint Probability Distribution ◽  
Significant Wave ◽  
Surge Height

<p>Tropical cyclones (TCs) often bring multiple hazards to offshore and onshore areas, including wind, rainfall, riverine flood, wave and storm surge. These hazards usually interact with each other and cause greater amplified hazard intensity. In the coastal areas, wave may damage coastal defense system like sea walls and dykes, and overtopping storm surge could hence become severe flooding due to the breach of the dykes. The probability distributions of wave and surge, as univariate respectively, have been studies and used in the design in various research. However, far less investigations on their joint probability distribution have been carried out in the past.</p><p>In this study, the dataset of hourly surge height, and significant wave height of 89 TC events impacting along Hainan Island during 1949~2013 was obtained, which are simulated numerically with ADCIRC and SWAN respectively. Following that, 4 types of probability distributions for univariate were used to fit the marginal distribution of storm surge and wave. Secondly, Frank, Clayton and Gumbel Copula were tried to construct the joint probability distribution of wave and surge, and the optimal Copula was determined by K-S test and AIC, BIC criteria. Based on the optimal Copula selected for each area of interest, the joint return period of wave and surge was estimated.</p><p>The results show that, 1) the annual maximum value of the storm surge height and significant wave height of Hainan Island has a relatively obvious geographical distribution regularity. 2) GEV and Gumbel are the most optimal distribution for storm surge height and significant wave height respectively. 3) Clayton Copula is the best model for fitting joint probability of storm surge and wave. The estimated joining probability distribution can help the determination of design standard, and typical TC disaster scenario development.</p>

The joint probability distribution function of structure factors with rational indices. IV. The P1 case

1999 ◽  
Vol 55 (3) ◽  
pp. 512-524
Author(s):  
Carmelo Giacovazzo ◽  
Dritan Siliqi ◽  
Cristina Fernández-Castaño
Keyword(s):  
Probability Distribution ◽  
Probability Distributions ◽  
Probabilistic Approach ◽  
Joint Probability ◽  
Distribution Functions ◽  
Accurate Estimation ◽  
Joint Probability Distribution ◽  
Structure Factors ◽  
Joint Probability Distributions ◽  
The Hilbert Transform

The method of the joint probability distribution functions of structure factors has been extended to reflections with rational indices. The most general case, space group P1, has been considered. The positional parameters are the primitive random variables of our probabilistic approach, while the reflection indices are kept fixed. Quite general joint probability distributions have been considered from which conditional distributions have been derived: these proved applicable to the accurate estimation of the real and imaginary parts of a structure factor, given prior information on other structure factors. The method is also discussed in relation to the Hilbert-transform techniques.

Sign in / Sign up

Export Citation Format

Share Document