Statistical Stationarity Research Articles

AN ASTEROID BELT INTERPRETATION FOR THE TIMING VARIATIONS OF THE MILLISECOND PULSAR B1937+21

Pulsar timing observations have revealed companions to neutron stars that include other neutron stars, white dwarfs, main-sequence stars, and planets. We demonstrate that the correlated and apparently stochastic residual times of arrival from the millisecond pulsar B1937+21 are consistent with the signature of an asteroid belt having a total mass less than approximately 0.05 Earth masses. Unlike the solar system's asteroid belt, the best fit pulsar asteroid belt extends over a wide range of radii, consistent with the absence of any shepherding companions. We suggest that any pulsar that has undergone accretion-driven spin-up and subsequently evaporated its companion may harbor orbiting asteroid mass objects. The resulting timing variations may fundamentally limit the timing precision of some of the other millisecond pulsars. Observational tests of the asteroid belt model include identifying periodicities from individual asteroids, which are difficult; testing for statistical stationarity that become possible when observations are conducted over a longer observing span; and searching for reflected radio emission.

Visualizing lateral anisotropy in modern carbonates

Statistical stationarity is a key assumption for the many modeling techniques based on variograms and transiograms used for geostatistical reconstruction of the subsurface. Stationarity expresses the property that the rules of geometry and neighborhood in the model are translation invariant, that is, no directional change in either mean or variance is observed. These criteria are met when the lateral arrangement of lithologic elements into a facies mosaic is isotropic. The balance between isotropy and anisotropy is a defining statistic in the configuration of both real and modeled carbonate landscapes. Even a cursory look at a satellite image of a modern carbonate platform shows that gradients in environment and hydrodynamics cause radical departures from isotropy. Although reef-forming organisms have changed through time, we do not expect that ancient reef systems behaved any differently than today. Hence, significant anisotropy should also be anticipated in the vertical and lateral arrangements of lithologies in the subsurface. To maintain sufficient geologic realism, it is paramount that process-imitating and pattern-replicating models alike be capable of honoring an expected degree of nonstationarity. Despite this need, few studies exist that provide quantitative information to the reach and location of zones of geometric isotropy and anisotropy in carbonate systems, let alone methods with which this property can be assessed. In an effort to close this disjoint, we develop a method for evaluating a modern Pacific depositional system, the Saipan Lagoon, for which we have created a geographic information system stack consisting of mapped facies distributions and a seabed topographic model, both at meter-scale resolution. By developing a lagged spatial metric based on the Markov property of facies transitions, we demonstrate that the degree of anisotropy is influenced by water depth; the shallowest areas (5 m [16 ft]) of the platform interior tend to be anisotropic whereas areas at greater depth are isotropic. This behavior suggests a possible extension to a genetic rule set that could be imparted to subsurface models based on the environment of deposition. This marks an advance in the understanding and, ultimately, handling of geometric nonstationarity in models of carbonate depositional systems.

A novel transient turbulent jet flame for studying turbulent combustion

This paper introduces a new class of transient turbulent flames as a new benchmark case for studying turbulent combustion. It also brings together the latest high-speed laser imaging experiments with large-eddy simulation (LES)/probability density function (PDF) to advance the calculations of transient processes in the newly introduced test cases. The otherwise steady jet flames are subjected to a pulse of inflow velocity such that the flow experiences sequentially increased local extinction, the formation of upper and lower burning regions separated by a blown-off section, re-ignition and reconnection of the separated flame regions, and then the returning of the flame to statistical stationarity. The base test cases considered here are Sydney University’s turbulent methane piloted flames L, B and M. The LES/PDF model predicts the flame’s dynamic response to the velocity pulse qualitatively, with the extinction limit predicted accurately compared to the measurement. This paper reports the first LES/PDF calculations of these flames which have narrow reaction zones compared to Sandia’s partially premixed flames, D, E and F. The statistically-transient jet flame introduced here provides a new benchmark for studying the effects of turbulence-chemistry interaction.

Adaptive Green-Kubo estimates of transport coefficients from molecular dynamics based on robust error analysis

We present a rigorous Green-Kubo methodology for calculating transport coefficients based on on-the-fly estimates of: (a) statistical stationarity of the relevant process, and (b) error in the resulting coefficient. The methodology uses time samples efficiently across an ensemble of parallel replicas to yield accurate estimates, which is particularly useful for estimating the thermal conductivity of semi-conductors near their Debye temperatures where the characteristic decay times of the heat flux correlation functions are large. Employing and extending the error analysis of Zwanzig and Ailawadi [Phys. Rev. 182, 280 (1969)] and Frenkel [in Proceedings of the International School of Physics "Enrico Fermi", Course LXXV (North-Holland Publishing Company, Amsterdam, 1980)] to the integral of correlation, we are able to provide tight theoretical bounds for the error in the estimate of the transport coefficient. To demonstrate the performance of the method, four test cases of increasing computational cost and complexity are presented: the viscosity of Ar and water, and the thermal conductivity of Si and GaN. In addition to producing accurate estimates of the transport coefficients for these materials, this work demonstrates precise agreement of the computed variances in the estimates of the correlation and the transport coefficient with the extended theory based on the assumption that fluctuations follow a Gaussian process. The proposed algorithm in conjunction with the extended theory enables the calculation of transport coefficients with the Green-Kubo method accurately and efficiently.

Low-dimensional model of turbulent Rayleigh-Bénard convection in a Cartesian cell with square domain

A low-dimensional model (LDM) for turbulent Rayleigh-Bénard convection in a Cartesian cell with square domain, based on the Galerkin projection of the Boussinesq equations onto a finite set of empirical eigenfunctions, is presented. The empirical eigenfunctions are obtained from a joint proper orthogonal decomposition (POD) of the velocity and temperature fields using the snapshot method on the basis of direct numerical simulation (DNS). The resulting LDM is a quadratic inhomogeneous system of coupled ordinary differential equations which we use to describe the long-time temporal evolution of the large-scale mode amplitudes for a Rayleigh number of 105 and a Prandtl number of 0.7. The truncation to a finite number of degrees of freedom, which does not exceed a number of 310 for the present, requires the additional implementation of an eddy viscosity-diffusivity to capture the missing dissipation of the small-scale modes. The magnitude of this additional dissipation mechanism is determined by taking statistical stationarity and a total dissipation that corresponds with the original DNS data. We compare the performance of two models, a constant so-called Heisenberg viscosity-diffusivity and a mode-dependent or modal one. The latter viscosity-diffusivity model turns out to reproduce the large-scale properties of the turbulent convection qualitatively well, even for a model with only a few hundred POD modes.

Scale-wise coherent vorticity extraction for conditional statistical modeling of homogeneous isotropic two-dimensional turbulence

Classical statistical theories of turbulence have shown their limitations, in that they cannot predict much more than the energy spectrum in an idealized setting of statistical homogeneity and stationarity. We explore the applicability of a conditional statistical modeling approach: can we sort out what part of the information should be kept, and what part should be modeled statistically, or, in other words, “dissipated”? Our mathematical framework is the initial value problem for the two-dimensional (2D) Euler equations, which we approximate numerically by solving the 2D Navier–Stokes equations in the vanishing viscosity limit. In order to obtain a good approximation of the inviscid dynamics, we use a spectral method and a resolution going up to 8192 2. We introduce a macroscopic concept of dissipation, relying on a split of the flow between coherent and incoherent contributions: the coherent flow is constructed from the large wavelet coefficients of the vorticity field, and the incoherent flow from the small ones. In previous work, a unique threshold was applied to all wavelet coefficients, while here we also consider the effect of a scale by scale thresholding algorithm, called scale-wise coherent vorticity extraction. We study the statistical properties of the coherent and incoherent vorticity fields, and the transfers of enstrophy between them, and then use these results to propose, within a maximum entropy framework, a simple model for the incoherent vorticity. In the framework of this model, we show that the flow velocity can be predicted accurately in the L 2 norm for about 10 eddy turnover times.

How much trust should risk managers place on “Brownian motions” of financial markets?

PurposeThe purpose of this paper is to investigate the recent global economic downturn. Particularly, the study explores the utilization of the concept of Brownian motion in financial risk management in organizations in the USA.Design/methodology/approachThe three assumptions, namely, independence, stationarity, and normal distribution that underlie the concept of Brownian motion are examined.FindingsIt is concluded that the widely used risk management strategies predicated on Brownian motion fail to provide a rational understanding of financial turmoil. Consequently, prescriptive insights are offered to aid the industry in developing an apposite mechanism for risk management.Research limitations/implicationsThis paper offers new and improved risk management strategies that need to be undertaken to augment our understanding and prediction of financial scenarios.Practical implicationsThe paper is useful for managers in all financial organizations, which employ computer models using Brownian motions. Specifically, this study contends that static models are unsuitable and dynamic models are more useful for risk assessment.Originality/valueThe paper reveals the weaknesses of the key assumptions of the risk management models used in financial organizations, namely, normal distribution of stock market price fluctuations, statistical stationarity, and efficient market assumption. Valuable guidelines are provided for financial managers who either do not have the inclination or time to sift through the voluminous literature related to the risk management models and computer software designed on these models.

Second-order structure function in fully developed turbulence

We relate the second-order structure function of a time series with the power spectrum of the original variable, taking an assumption of statistical stationarity. With this approach, we find that the structure function is strongly influenced by the large scales. The large-scale contribution and the contribution range are, respectively, 79% and 1.4 decades for a Kolmogorov -5/3 power spectrum. We show numerically that a single scale influence range, over smaller scales is about 2 decades. We argue that the structure function is not a good method to extract the scaling exponents when the data possess large energetic scales. An alternative methodology, the arbitrary order Hilbert spectral analysis which may constrain this influence within 0.3 decade, is proposed to characterize the scaling property directly in an amplitude-frequency space. An analysis of passive scalar (temperature) turbulence time series is presented to show the influence of large-scale structures in real turbulence and the efficiency of the Hilbert-based methodology. The corresponding scaling exponents ζ(θ)(q) provided by the Hilbert-based approach indicate that the passive scalar turbulence field may be less intermittent than what was previously believed.

The Role of Climate and Human Influences in the Dry‐Up of the Jinci Springs, China1

Abstract: One of the largest karst springs in North China, the Jinci Springs, dried up and has remained dry since 1994. We develop a correlation analysis with time‐lag and a regression analysis with time‐lag to study the relation between spring flow and precipitation. This allows us to obtain a better understanding of karst hydrological processes by differentiating the contribution of variation in precipitation from anthropogenic impacts on the dry‐up of Jinci Springs. We divided the karstic hydrological processes into two phases: pre‐1961 and post‐1961. In the first phase (i.e., 1954‐1960) the groundwater recharge was affected by precipitation alone, and in the second phase (i.e., 1961‐1994) the groundwater recharge was influenced by both precipitation and human activities. Using precipitation and groundwater recharge data in the first phase, we set up a groundwater recharge model with time‐lags. By running the time‐lags model, we acquired the groundwater recharge likely to occur under the sole effect of precipitation in the second phase. Using a water‐balance calculation, we conclude that the groundwater recharge exhibited statistical stationarity, and the Jinci Springs dry‐up was the result of anthropogenic activities. At least three specific types of anthropogenic activities contributed to the drying‐up of Jinci Springs: (1) groundwater pumping accounts for 51%, (2) the dewatering from coal mining accounts for 33%, (3) and dam‐building 14%. The drying‐up of Jinci Springs meant that the groundwater drained from the aquifer’s fractures, and subsequently changed the structure of the karst aquifer. Although groundwater exploitation has been reduced, the flow at Jinci Springs has not reoccurred.

Application of stochastic averaging to non-linear ecosystems

Stochastic models for the prey–predator type ecosystems in a random environment are proposed and investigated. They are the stochastic versions of the classical Lotka–Volterra model, the predator-saturation model, and the time-delay model. Variations in the birth rate of the preys and the death rate of the predators are modeled as random processes. For the three stochastic models, the stochastic averaging procedure of Stratonovich and Khasminskii is applied to obtain the probability distributions of the prey and predator populations at the state of statistical stationarity. Effects of different system parameters on the ecosystem behaviors are evaluated. The analytical results are substantiated by results obtained from Monte Carlo simulations.

A combined sea clutter and noise model appropriate to the operation of high‐frequency pulsed Doppler radar in regions constrained by external noise

In recent years, bistatic pulsed high‐frequency ground wave radar models of the ocean clutter have been developed. Several new features, distinct from earlier monostatic developments, appear as products of those analyses. One question that needs to be addressed is, “What characteristics of the theoretical clutter models are likely to be visible in experimental data collected from the ocean surface?” A major consideration in answering this question is the development of an appropriate noise model. Such a model along with an analogous clutter model is derived. This allows a simulation of time series data for both clutter and noise which may be treated using standard Fourier transform techniques to provide a periodogram for the typical combined noise/clutter spectrum of scattering from the ocean surface. The analysis proceeds on the assumption of an externally noise‐limited system, with the noise being characterized as a white Gaussian zero‐mean process. The aliasing due to noise undersampling is seen to be an integral part of the model. Statistical stationarity is assumed throughout. Both infinite and finite pulse trains are considered.

A Dynamic Analysis of Stock Price Ratios

Stock price ratios have long been used by finance practitioners as a relative value metric. A popular argument for this widespread use is that stock price ratios would tend to revert to their long-run mean so that substantial deviations from historical averages could successfully be arbitraged away. In this work, we lay out the theoretical conditions for the ratio of stock prices to be a trend stationary process. In particular, we establish that, in the context of our model, market completeness entails stationary price ratios. We also theoretically relate statistical price ratio stationarity to economic mean reversion in profitability (as measured by dividends or earnings price ratios) across securities. We further test our theoretical predictions using standard unit root tests and cointegration analysis on a popular example of stocks. To illustrate the implications of the theoretical work, we provide a simple empirical exercise where we analyze the time series behavior of the Coca Cola and Pepsi stock price ratio. These two stocks provide us with a straightforward example of relative pricing between close substitutes. Our results have important implications for practitioners who seek to apply pairs-trading investment strategies in the stock market as they gives clear economic intuition to this popular practice. Indeed, as long as theoretical requirements are met, an investment strategy that exploits short-term error deviations of stock prices of close firms apart from their long run (cointegrated) relation, e.g., matching stocks by minimizing the sum of squared deviations between normalized stock prices as in Gatev, Goetzmann, and Rouwenhorst (2006) should produce significant risk-adjusted returns.

Statistical characteristics and stationarity properties of low-frequency seismic signals

Statistical properties of microseismic signals are studied in the frequency range 0.03–15 Hz at various points of the Earth near and far from sources of microseisms. It is found that various frequency ranges differ in the property of stationarity. The minimum interval of stationarity of microseisms in the range 0.12–1.1 Hz is estimated at 1–1.5 h. A conclusion is drawn that the measurement accuracy of the spectral density of microseisms cannot be improved above a certain limit by increasing the time of signal stacking.

Stochastic analysis of time-delayed ecosystems

The predator-prey-type ecosystem is investigated, taking into account the time-delay effect of the prey population on the predator population, as well as random variations in the birth rate of the prey and the death rate of the predators. The stochastic averaging procedure is applied to obtain the probability distributions of the predator and prey populations at the state of statistical stationarity. It is found that two system parameters, quantifying the effects of prey self-competition and the time delay, respectively, play the most important roles. Results are also obtained from Monte Carlo simulations to compare with the analytical results.

Analysis of correlations between time series with the use of the phase spectrum

In connection with the problem of revealing cause-effect relations among different climatic characteristics, methods for determining the dependence between time series on the basis of selected auto-and cross-covariance functions and periodograms expressed in terms of the Fourier coefficients of the initial series are discussed. Real time series do not always satisfy the conditions of statistical stationarity, so that their analysis requires a combination of statistical and deterministic approaches. The possibility of using the above numerical characteristics in relation to such series is considered. Characteristic features of describing finite segments of time series with the use of their Fourier coefficients is studied in detail. The main emphasis is on the determination of the time shifts (delays) at which the covariance between the series is maximal. The problems that appear during a practical implementation of the periodogram method for shift estimation are discussed. A previously unknown formula that is necessary for further studies and relates the Fourier transform of a selected correlation function to the periodogram of the series is derived.

Stochastic analysis of predator–prey type ecosystems

A stochastic model for the predator–prey type ecosystems in a random environment is proposed and investigated. The model is a variation of the Lotka–Volterra type with an additional self-competition mechanism within the prey population. Two different situations are considered: (1) saturation of predators, and (2) competition among predators. Changes in the birth rate of the preys and the death rate of the predators are modeled as random processes. The stochastic averaging procedure of Stratonovich and Khasminskii is applied to obtain the probability distribution of the system state variables at the state of statistical stationarity. Asymptotic behaviors of the system are also investigated. Effects on the ecosystem behaviors are evaluated of (1) prey self-competition, (2) predator saturation and predator competition, (3) random variation in the prey birth rate, and (4) random variation in the predator death rate.

Probability density function of non-reactive solute concentration in heterogeneous porous formations

Available models of solute transport in heterogeneous formations lack in providing complete characterization of the predicted concentration. This is a serious drawback especially in risk analysis where confidence intervals and probability of exceeding threshold values are required. Our contribution to fill this gap of knowledge is a probability distribution model for the local concentration of conservative tracers migrating in heterogeneous aquifers. Our model accounts for dilution, mechanical mixing within the sampling volume and spreading due to formation heterogeneity. It is developed by modeling local concentration dynamics with an Ito Stochastic Differential Equation (SDE) that under the hypothesis of statistical stationarity leads to the Beta probability distribution function (pdf) for the solute concentration. This model shows large flexibility in capturing the smoothing effect of the sampling volume and the associated reduction of the probability of exceeding large concentrations. Furthermore, it is fully characterized by the first two moments of the solute concentration, and these are the same pieces of information required for standard geostatistical techniques employing Normal or Log-Normal distributions. Additionally, we show that in the absence of pore-scale dispersion and for point concentrations the pdf model converges to the binary distribution of [Dagan, G., 1982. Stochastic modeling of groundwater flow by unconditional and conditional probabilities, 2, The solute transport. Water Resour. Res. 18 (4), 835–848.], while it approaches the Normal distribution for sampling volumes much larger than the characteristic scale of the aquifer heterogeneity. Furthermore, we demonstrate that the same model with the spatial moments replacing the statistical moments can be applied to estimate the proportion of the plume volume where solute concentrations are above or below critical thresholds. Application of this model to point and vertically averaged bromide concentrations from the first Cape Cod tracer test and to a set of numerical simulations confirms the above findings and for the first time it shows the superiority of the Beta model to both Normal and Log-Normal models in interpreting field data. Furthermore, we show that assuming a-priori that local concentrations are normally or log-normally distributed may result in a severe underestimate of the probability of exceeding large concentrations.

Spatial covariation of Azotobacter abundance and soil properties: A case study using the wavelet transform

The soil microflora is very heterogeneous in its spatial distribution. The origins of this heterogeneity and its significance for soil function are not well understood. A problem for understanding spatial variation better is the assumption of statistical stationarity that is made in most of the statistical methods used to assess it. These assumptions are made explicit in geostatistical methods that have been increasingly used by soil biologists in recent years. Geostatistical methods are powerful, particularly for local prediction, but they require the assumption that the variability of a property of interest is spatially uniform, which is not always plausible given what is known about the complexity of the soil microflora and the soil environment. We have used the wavelet transform, a relatively new innovation in mathematical analysis, to investigate the spatial variation of abundance of Azotobacter in the soil of a typical agricultural landscape. The wavelet transform entails no assumptions of stationarity and is well suited to the analysis of variables that show intermittent or transient features at different spatial scales. In this study, we computed cross-variograms of Azotobacter abundance with the pH, water content and loss on ignition of the soil. These revealed scale-dependent covariation in all cases. The wavelet transform also showed that the correlation of Azotobacter abundance with all three soil properties depended on spatial scale, the correlation generally increased with spatial scale and was only significantly different from zero at some scales. However, the wavelet analysis also allowed us to show how the correlation changed across the landscape. For example, at one scale Azotobacter abundance was strongly correlated with pH in part of the transect, and not with soil water content, but this was reversed elsewhere on the transect. The results show how scale-dependent variation of potentially limiting environmental factors can induce a complex spatial pattern of abundance in a soil organism. The geostatistical methods that we used here make assumptions that are not consistent with the spatial changes in the covariation of these properties that our wavelet analysis has shown. This suggests that the wavelet transform is a powerful tool for future investigation of the spatial structure and function of soil biota.

Structure of a spatially developing turbulent lean methane–air Bunsen flame

Direct numerical simulation of a three-dimensional spatially developing turbulent slot-burner Bunsen flame has been performed with a new reduced methane–air mechanism. The mechanism, derived from sequential application of directed relation graph theory, sensitivity analysis and computational singular perturbation over the GRI-1.2 detailed mechanism is non-stiff and tailored to the lean conditions of the DNS. The simulation is performed for three flow through times, long enough to achieve statistical stationarity. The turbulence parameters have been chosen such that the combustion occurs in the thin reaction zones regime of premixed combustion. The data is analyzed to study possible influences of turbulence on the structure of the preheat and reaction zones. The results show that the mean thickness of the turbulent flame, based on progress variable gradient, is greater than the corresponding laminar flame. The effects of flow straining and flame front curvature on the mean flame thickness are quantified through conditional means of the thickness and by examining the balance equation for the evolution of the flame thickness. Finally, conditional mean reaction rate of key species compared to the laminar reaction rate profiles show that there is no significant perturbation of the heat release layer.

The texture of rain: Exploring stochastic micro-structure at small scales

Summary The main theme of this review is the importance of a discrete approach to describing rain at spatial scales comparable to inter-drop separation. We propose that the pair correlation function should be used to define and measure the texture of rain. To that end, we discuss the pivotal role of the Poisson process for examining this micro-structure of rain. The importance of statistical stationarity and the essential distinction between a Poisson distribution and a Poisson process are emphasized. It is argued that the correlation-fluctuation theorem (which relates drop count variance to the pair-correlation function) is ideally suited for scale-dependent exploration of rain micro-structure in the discrete “shot noise” limit. The likelihood of spurious negative correlations at fine spatial scales is pointed out as instruments are pushed to their resolution limits. One of the consequences is that possibly spurious Poisson statistics at a given spatial scale may result from a cancellation on sub-scales. We then proceed to examine implications of stochastic microstructure and show that the notion of spatially variable and random concentration (or size distribution) does not always provide an adequate description of rain texture.