White Noise Stochastic Process Research Articles

Improved performance of BDS-3 time and frequency transfer based on an epoch differenced model with receiver clock estimation

The carrier phase (CP) technique based on the BeiDou Global Satellite Navigation System (BDS-3) has proven to be a crucial spatial tool for remote time and frequency transfer. The current CP technique models the receiver clock offset as a white noise stochastic process and easily absorbs some unmodeled errors, thus compromising the time and frequency transfer performance. To further improve the performance of time and frequency transfer, a new BDS-3 receiver clock estimation algorithm based on the epoch difference (ED) model is presented, and the mathematical principles and applied modes are discussed. The algorithm makes full use of both observations of the current epoch and practical variations of the receiver clock offset, further improving the performance of time and frequency transfer. Five Multi-Global Navigation Satellite System Experiment network stations equipped with various types of receivers and antennas with dual-frequency BDS-3 signals were used to establish four time transfer links (i.e., AMC4–PTBB, BRUX–PTBB, OP71–PTBB, and WTZS–PTBB) to evaluate their effectiveness. The ED model improves all the four time links in terms of noise level, with improvements of 17.0%, 18.3%, 20.3%, and 5.9%, respectively, when compared with the results from a non-ED model. The ED model outputs were better than the raw solutions in terms of frequency stability at all time links, particularly for average time intervals (tau) < 1000 s. The mean improvement was 8.1% for AMC4–PTBB, 16.1% for BRUX–PTBB, 10.0% for OP71–PTBB, and 18.6% for WTZS–PTBB when the average time (tau) was less than 1000 s.

Seismic multi-objective stochastic parameters optimization of multiple tuned mass damper system for a large podium twin towers structure

The lateral-torsional coupled dynamic response under earthquake excitation is a critical problem for eccentric building structures. To control the hazardous coupled seismic response, a tuned mass damper (TMD) is an attractive vibration control device. A multiple TMD (MTMD) system, which is composed by several TMDs with different frequencies and damping ratios, has a better structural control effect under stochastic excitation with a wide frequency range such as earthquake. To improve the lateral-torsional coupled seismic mitigation performance of MTMD for eccentric structures, a multi-objective stochastic parameters optimization method is proposed, and a building with a large podium and twin towers is proposed as the case study. Considering the randomness of earthquakes, a stationary white noise stochastic process based on the Kanai-Tajimi spectrum is used as the input in the optimization. According to structural modal shapes, three TMDs are distributed in the podium, left tower and right tower respectively in the X direction, and two TMDs are in the left tower and right tower respectively in the Y direction, while both the maximum and root mean square (RMS) displacements of top stories of the podium, left tower and right tower are all considered in the optimization. Artificial fish swarm algorithm (AFSA) is used to search optimized parameters of MTMD in this study. For comparison, a MTMD system designed based on the traditional method is presented. Numerical results show that under bi-directional earthquake excitations, the proposed MTMD system has the best seismic mitigation effect generally, in both displacement and acceleration responses. Though the proposed MTMD system is aimed at X- and Y- directional translational responses, it can control torsional responses effectively as well.

Multi-objective stochastic optimization of tuned mass dampers under earthquake excitation considering soil–structure interaction

ABSTRACT Tuned mass dampers (TMDs) are attractive vibration control devices, and their seismic attenuation depends on their frequency and damping coefficient. Structural characteristics are changed by the soil-structure interaction (SSI); therefore, SSI should be considered in the optimization of TMD. A high-rise building with 40 stories, including four different soil conditions, is proposed as a case study. For the objective function, considering the randomness of earthquakes, a stationary white noise stochastic process based on the Kanai-Tajimi spectrum was used as the input. To improve the performance of a TMD in controlling multimodal and overall dynamic responses, three objectives were considered in the objective function: the maximum and root mean square (RMS) displacements of the top story and the maximum displacement of all stories, respectively. The weight coefficient of each objective function was obtained according to the modal mass participation coefficient. To clearly observe the damping effect of the TMD with different parameter combinations, an ergodic parameter calculation is used. The TMD optimization parameters for each benchmark model with different soil conditions were proposed. Far-field ground motions were considered, and four optimized TMDs from reference were compared. The results show that the proposed TMDs can significantly mitigate earthquake oscillations in a great degree. They performed better than contrastively optimized TMDs under several earthquakes.

Non-Linear Response of Cable-Mass-Spring System in High-Rise Buildings under Stochastic Seismic Excitation.

In high-rise buildings earthquake ground motions induce bending deformation of the host structure. Large dynamic displacements at the top of the building can be observed which in turn lead to the excitation of the cables/ropes within lift installations. In this paper, the stochastic dynamics of a cable with a spring-damper and a mass system deployed in a tall cantilever structure under earthquake excitation is considered. The non-linear system is developed to describe lateral displacements of a vertical cable with a concentrated mass attached at its lower end. The system is moving slowly in the vertical direction. The horizontal displacements of the main mass are constrained by a spring-viscous damping element. The earthquake ground motions are modelled as a filtered Gaussian white noise stochastic process. The equivalent linearization technique is then used to replace the original non-linear system with a linear one with the coefficients determined by utilising the minimization of the mean-square error between both systems. Mean values, variances and covariances of particular random state variables have been obtained by using the numerical calculation. The received results were compared with the deterministic response of the system to the harmonic process and were verified against results obtained by Monte Carlo simulation.

Constraining two climate field reconstruction methodologies over the North Atlantic realm using pseudo-proxy experiments

This study presents pseudo-proxy experiments to quantify the reconstruction skill of two climate field reconstruction methodologies for a marine proxy network subject to age uncertainties. The BARCAST methodology (Bayesian Algorithm for Reconstructing Climate Anomalies in Space and Time) is tested for sea surface temperature (SST) reconstruction for the first time over the northern North Atlantic region, and compared with a classic analogue reconstruction methodology. The reconstruction experiments are performed at annual and decadal resolution. We implement chronological uncertainties inherent to marine proxies as a novelty, using a simulated age-model ensemble covering the past millennium. Our experiments comprise different scenarios for the input data network, with the noise levels added to the target variable extending from ideal to realistic. Results show that both methodologies are able to reconstruct the Summer mean SST skillfully when the proxy network is considered absolutely dated, but the skill of the analogue method is superior to BARCAST. Only the analogue method provides skillful correlations with the true target variable in the case of a realistic noisy and age-uncertain proxy network. The spatiotemporal properties of the input target data are partly contrasting with the BARCAST model formulations, resulting in an inferior reconstruction ensemble that is similar to a white-noise stochastic process in time. The analogue method is also successful in reconstructing decadal temperatures, while BARCAST fails. The results contribute to constraining uncertainties in CFR for ocean dynamics which are highly important for climate across the globe.

Stochastic Multi-objective Optimisation of Exoskeleton Structures

In this study, a structural optimisation problem, addressed through a stochastic multi-objective approach, is formulated and solved. The problem deals with the optimal design of exoskeleton structures, conceived as vibration control systems under seismic loading. The exoskeleton structure is assumed to be coupled to an existing primary inner structure for seismic retrofit: the aim is to limit the dynamic response of the primary structure to prevent structural damage. A non-stationary filtered Gaussian white noise stochastic process is taken as the seismic input. Design variables pertain to the mechanical properties (stiffness, damping) of the exoskeleton structure. Two concurrent and competing objective functions are introduced, in order to take into account not only safety performance but also economic cost considerations. The resulting trade-off is solved searching the Pareto front by way of a controlled elitist genetic algorithm, derived from the Non-dominated Sorting Genetic Algorithm-II. Sensitivities of Pareto fronts and Pareto optimal sets to different system parameters are finally investigated by way of a numerical application.

Atomic clock modelling augmenting time and frequency transfer using carrier phase observation

The carrier phase (CP) technique based on the global positioning system is a widely applied spatial tool for precise time and frequency transfer. However, the traditional CP technique considers the receiver clock offset to be a white noise stochastic process, which introduces unknown errors into the model and thereby compromises time and frequency transfer performance. To improve the performance of the traditional CP technique, an atomic clock modelling CP time transfer (AMCP) method is proposed. A first-order, one variable grey model is employed to establish the receiver clock offset forecast model for the formation of prior information. The results show that the AMCP approach, which uses an atomic clock correlation between nearby epochs, outperformed the traditional CP technique. The noise level of the AMCP solution was improved by 7.7% for the NTSC–PT11 link, and the improvement for the other three time links were all ∼2.0%. The average improvement of the absolute jumps in the day-boundary discontinuities was ∼13.1%. Additionally, the frequency stability of the AMCP solution was improved, and the mean improvement was 10.2, 12.0, 10.7, and 8.3% for TLT4–PT11, USN9–PT11, WAB2–PT11, and NTSC–PT11, respectively.

First-passage time study of a stochastic growth model

In this paper, a stochastic nonlinear growth model is proposed, which can be considered a generalization of the stochastic logistic model. It can be applied to the growth of a generic population as well as to the propagation of the fracture in engineering materials. The excitation is assumed to be a stationary Gaussian white noise stochastic process, which affects the system parametrically. A preliminary study of the stochastic differential equation governing the model reveals that there is a phase transition when the nonlinearity parameter c reaches one: when $$c<1$$ , the system tends to the stationary state, while it is never stationary when $$c\ge 1$$ . Then, attention is focused on the first-passage time problem, which is of crucial importance for dynamical systems. The first-order stochastic differential equation (SDE) that describes the model is transformed into an Ito’s SDE by adding the Wong–Zakai–Stratonovich corrective term. For the last equation, the backward Kolmogorov equation is formulated. By solving it with appropriate initial and boundary conditions, the probability of survival is obtained, that is, the probability of not exceeding a given threshold. The solution is looked for three cases $$c 1$$ . In any case, the numerical analyses show that the survival probability decays fast.

Stochastic stability of the inverted pendulum subjected to delta- correlated base excitation

This paper is concerned with the stochastic stability of an inverted pendulum with a point mass at the top and a spring at the base; the bar is massless. The base is subjected at the base to a vertical acceleration A(t) that is supposed to be a white noise (delta-correlated) stochastic process. Both Gaussian and Poissonian white noises are considered. A line-like structure excited by a vertical ground motion can be idealized in this way. It is assumed that during the motion the angle of rotation ϑ remains small so that sin ϑ≅ϑ. In this way, the motion equation assumes the classical form of the second order oscillator, but the excitation is parametric so that there is a possibility of stochastic instability. The almost sure (sample) stability and the stability in the second moments are considered herein. It is found that the two stability criteria lead to notable differences in the stability boundaries and the almost sure stability is not conservative. The mean square stability under the Poisson white noise is determined only by the arrival rate of underlying Poisson counting process and by the mean square amplitude of the pulses: the cumulants beyond the second order do not affect it.

Dynamical symmetries of Markov processes with multiplicative white noise

We analyse various properties of stochastic Markov processes with multiplicative white noise. We take a single-variable problem as a simple example, and we later extend the analysis to the Landau–Lifshitz–Gilbert equation for the stochastic dynamics of a magnetic moment. In particular, we focus on the non-equilibrium transfer of angular momentum to the magnetization from a spin-polarised current of electrons, a technique which is widely used in the context of spintronics to manipulate magnetic moments. We unveil two hidden dynamical symmetries of the generating functionals of these Markovian multiplicative white-noise processes. One symmetry only holds in equilibrium and we use it to prove generic relations such as the fluctuation-dissipation theorems. Out of equilibrium, we take profit of the symmetry-breaking terms to prove fluctuation theorems. The other symmetry yields strong dynamical relations between correlation and response functions which can notably simplify the numerical analysis of these problems. Our construction allows us to clarify some misconceptions on multiplicative white-noise stochastic processes that can be found in the literature. In particular, we show that a first-order differential equation with multiplicative white noise can be transformed into an additive-noise equation, but that the latter keeps a non-trivial memory of the discretisation prescription used to define the former.

Decadal predictability of soil water, vegetation, and wildfire frequency over North America

The potential decadal predictability of land hydrological and biogeochemical variables in North America is examined using a 900-year-long pre-industrial control simulation, conducted with the NCAR Community Earth System Model (CESM) version 1.0.3. The leading modes of simulated North American precipitation and soil water storage are characterized essentially by qualitatively similar meridional seesaw patterns associated with the activity of the westerly jet. Whereas the corresponding precipitation variability can be described as a white noise stochastic process, power spectra of vertically integrated soil water exhibit significant redness on timescales of years to decades, since the predictability of soil water storage arises mostly from the integration of precipitation variability. As a result, damped persistence hindcasts following a 1st order Markov process are skillful with lead times of up to several years. This potential multi-year skill estimate is consistent with ensemble hindcasts conducted with the CESM for various initial conditions. Our control simulation further suggests that decadal variations in soil water storage also affect vegetation and wildfire occurrences. The long-term potential predictability of soil water variations in combination with the slow regrowth of vegetation after major disruptions leads to enhanced predictability on decadal timescales for vegetation, terrestrial carbon stock, and fire frequency, in particular in the Southern United States (US)/Mexico region. By contrast, the prediction skill of fire frequency in the Northern US is limited to 1 year. Our results demonstrate that skillful decadal predictions of soil water storage, carbon stock, and fire frequency are feasible with proper initialization of soil conditions. Although the potential predictability in our idealized modeling framework would overestimate the real predictability of the coupled climate-land-vegetation system, the decadal climate prediction may become beneficial for water resource management, forestry, and agriculture.

The Role of Modulation Function in Nonstationary Stochastic Earthquake Model

In structural engineering earthquakes are often represented as random phenomena. Frequently, filtered white noise stochastic processes are adopted to properly model their frequency content. In order to model the time variation of earthquake intensity, these processes are assumed nonstationary, and time modulation functions (MFs) are used. For these, different shapes and formulas have been proposed in literature till now, but only few works have dealt with their comparison in terms of structural response. This paper focuses on this topic: at this aim, a simple linear single degree of freedom (SDoF) system, which represents a structure vibrating in its fundamental mode, is considered subject to a time modulated filtered stochastic process. Different shapes of the MF are considered and the influence on two structural response indices, i.e. the maximum displacement standard deviation and the failure probability, is investigated. A sensitivity analysis is finally performed by varying peak ground acceleration (PGA), Arias intensity and structural period.

Mean square stability of a second-order parametric linear system excited by a colored Gaussian noise

The stochastic stability of a second-order linear parametric oscillator whose stiffness is perturbed by a stationary zero-mean colored Gaussian process is investigated in this paper referring to the stability in the second response moments (mean square stability). The stochastic perturbation is the output of a linear filter excited by a stationary Gaussian white noise stochastic process, and the hypothesis of weak excitation is not formulated. The first assumption allows using Itô׳s stochastic differential calculus. The moment equations of the response are written till the second order by means of Itô׳s differential rule. They form an infinite hierarchy, that is the equations for the moments of order r contain moments of order larger than r. In order to close the hierarchy, the cumulant neglect closure method is applied herein. The moment equation set is closed by neglecting the cumulants of third order. Nonlinear moment equations result: they are linearized, and the limit of stability is searched by studying the eigenvalues of the matrix of the coefficients of the linearized moment equations. Several numerical analyses are performed to evaluate the critical mean square value of the excitation when a system parameter is varied for the cases of both first and second-order parametric excitation. A first-order excitation, an Ornstein–Uhlenbeck process, does not cause stochastic resonance, while a second-order excitation does. The first result was not known previously.

Una medida de eficiencia de mercado. Un enfoque de teoría de la información

In this paper, based on the concept of Shannon entropy, we propose a measure of market efficiency by using the empirical density function of returns. Under certain conditions of ergodicity and stationarity, it is shown that the sample entropy converges to the entropy of the dominant state. It is also shown that the proposed measure is consistent with some of the axioms from Artzner et al. (1999) of a coherent risk measure. Bounds on the behavior of entropy as a measure of efficiency on the basis of extreme cases are also established; going from deterministic processes to pure white noise stochastic processes. Finally, for illustrative purposes, we carry out several applications of the proposed efficiency measure of capital to different markets: DJIA, S&P500, FTSE100 and IPC.

Hidden symmetries and equilibrium properties of multiplicative white-noise stochastic processes

Multiplicative white-noise stochastic processes continue to attract attention in a wide area of scientific research. The variety of prescriptions available for defining them makes the development of general tools for their characterization difficult. In this work, we study equilibrium properties of Markovian multiplicative white-noise processes. For this, we define the time reversal transformation for such processes, taking into account that the asymptotic stationary probability distribution depends on the prescription. Representing the stochastic process in a functional Grassmann formalism, we avoid the necessity of fixing a particular prescription. In this framework, we analyze equilibrium properties and study hidden symmetries of the process. We show that, using a careful definition of the equilibrium distribution and taking into account the appropriate time reversal transformation, usual equilibrium properties are satisfied for any prescription. Finally, we present a detailed deduction of a covariant supersymmetric formulation of a multiplicative Markovian white-noise process and study some of the constraints that it imposes on correlation functions using Ward–Takahashi identities.

Supersymmetric formulation of multiplicative white-noise stochastic processes

We present a supersymmetric formulation of Markov processes, represented by a family of Langevin equations with multiplicative white noise. The hidden symmetry encodes equilibrium properties such as fluctuation-dissipation relations. The formulation does not depend on the particular prescription to define the Wiener integral. In this way, different equilibrium distributions, reached at long times for each prescription, can be formally treated on the same footing.

Research on the control and simulation of vehicle suspension systems

A four-degree-of-freedom half-body dynamic model of vehicle active suspension system is established. Road roughness height is modelled as a filtered white noise stochastic process. Based on control theory, the fuzzy control system of the active suspension is proposed. Using the Matlab/Simulink software, the simulation model is obtained. Simulation results indicate that the proposed active suspension system proves to be effective in the vibration isolation of the suspension system. What is more, a mechanical model of four-degree-of-freedom half-body of vehicle passive suspension system is constituted with the aid of software ADAMS. The combined simulation results demonstrate the effectiveness of the proposed mathematical model.

Moment Lyapunov exponents of the stochastic parametrical Hill’s equation

The Lyapunov exponent and moment Lyapunov exponents of Hill’s equation with frequency and damping coefficient fluctuated by white noise stochastic process are investigated. A perturbation approach is used to obtain explicit expressions for these exponents in the presence of small intensity noises. The results are applied to the study of the almost-sure and the moment stability of the stationary solutions of the thin simply supported beam subjected to axial compressions and time-varying damping which are small intensity stochastic excitations.

Stretching of Polymers in Isotropic Turbulence: A Statistical Closure

We present a new closure for the mean rate of stretching of a dissolved polymer by homogeneous isotropic turbulence. The polymer is modeled by a bead-spring-type model (e.g., Oldroyd B, FENE-P, Giesekus) and the analytical closure is obtained assuming the Lagrangian velocity gradient can be modeled as a Gaussian, white-noise stochastic process. The resulting closure for the mean stretching depends upon the ratio of the correlation time for strain and rotation. Additionally, we derived a second-order expression for circumstances when strain and rotation have a finite correlation time. Finally, the base level closure is shown to reproduce results from direct numerical simulations by simply modifying the coefficients.

Stochastic optimal preview control of a vehicle suspension

Stochastic optimal control of a vehicle suspension on a random road is studied. The road roughness height is modelled as a filtered white noise stochastic process and a four-degree-of-freedom half-car model is used in the analysis. It is assumed that a sensor is mounted in the front bumper that measures the road irregularity at some distances in the front of the vehicle. Two other sensors also measure relative velocities of the vehicle body with respect to the unsprung masses in the vehicle suspension spaces. All measurements are assumed to be conducted in a noisy environment. The state variables of the vehicle system are estimated using a method similar to the Kalman filter. The suspension system is optimized by minimizing the performance index containing the mean-square values of body accelerations (including effects of heave and pitch), tire deflections and front and rear suspension rattle spaces. The effect of delay between front and rear wheels is included in the analysis. For stochastic active control with and without preview, the suspension performance and the power demand are evaluated and compared with those of the passive system. The results show that the inclusion of time delay between the front and rear axles and the preview information measured by the sensor mounted on the vehicle improves all aspects of the suspension performance, while reducing the energy consumption.