Nonstationary Stochastic Processes Research Articles

An improved analytical solution to outcrossing rate for scalar nonstationary and non-gaussian processes

For time-dependent reliability methods, there is a lack of analytical solutions to outcrossing rates of nonstationary and non-Gaussian processes. This paper aims to propose an analytical method to determine this type of outcrossing rate with improved accuracy and efficiency. The novelty of this proposed method is that it can consider the probabilistic properties of non-Gaussian processes by combining a two-component parallel system model with higher-order moments-based reliability indexes. The main contributions of this method are: (1) compared with PHI2+ method, it is more accurate for the calculation of the outcrossing rate of nonstationary and non-Gaussian processes with nonlinear limit state functions; (2) compared with MPHI2 method, it is analytical and not only applicable for the nonstationary stochastic processes but also insensitive to time increments; and (3) it is not only more accurate than the existing methods for nonstationary and non-Gaussian processes but also more computation efficient than the Monte Carlo simulation method. The proposed method has shown its advantages for practical structures with neither Gaussian processes nor linear limit state functions, which are beneficial for both researchers and asset managers to evaluate the time-dependent reliability of structures accurately and to develop risk-informed maintenance schemes with a view to prolonging their service life.

Sequential simulated annealing for life-cycle optimization of nonlinear stochastic systems via arbitrary polynomial chaos expansion

Quantifying the uncertainties of engineering systems modeled as nonlinear oscillators subject to random excitation is a theoretically complex and computationally demanding task. Consequently, finding an optimal design of such systems considering their life-cycle performance is prohibitive with Monte Carlo simulation methods. In this paper, an efficient performance-based design optimization approach is developed for finding the optimal parameters of engineering systems modeled as nonlinear/hysteretic oscillators subject to stationary and non-stationary excitation. This novel approach utilizes an arbitrary polynomial chaos expansion to estimate the expected cost of failure without performing a computationally expensive integration over the hazard levels. Moreover, we introduce a novel sequential heuristic optimization scheme based on simulated annealing to minimize the total expected cost over the structure life-cycle. Three examples are included in the paper to assess the developed optimization scheme. First, we use the developed framework to optimize a linear single-degree-of-freedom oscillator subject to broadband excitation. Second, a multi-degree-of-freedom oscillator with cubic nonlinearity in damping and stiffness, subject to stationary broadband excitation, is optimized to show the influence of the problem dimensionality in the optimization process. In the last example, a multi-story reinforced concrete shear building modeled as a multi-degree-of-freedom Bouc-Wen oscillator with stiffness and strength degradation and subject to multi-hazards modeled as stationary (wind excitation) and non-stationary (earthquake) stochastic processes, is optimized.

Generalized spectrum analysis of Chirp Cyclostationary signals associate with linear canonical transform

A chirp cyclostationary (CCS) process is a nonstationary stochastic process, whose statistics have been defined and utilized in radar and communication signal processing. However, spectrum analysis of CCS signals, which provides a spectral view to characterize a signal, has not been performed. Classical spectrum analysis is based on the Fourier transform, which is not suitable for CCS signals because of the multiplicative chirp signals in a CCS model. In this study, we propose a generalized spectral representation of chirp cyclostationary signals associated with a linear canonical transform. The proposed representation shows that a CCS signal can be represented by a linear combination of a series of jointly stationary and equal-bandwidth processes, which bridge the nonstationary and stationary processes. Furthermore, the relationships between the second-order statistics of the CCS process and its stationary representations are introduced. Finally, based on the above observations, a linear time-varying Wiener filter was designed for CCS signal denoising and time-varying channel estimation. Finally, the relationships between the nonstationarity of the input and output and time-varying properties of the systems are discussed, which are helpful for determining the nonstationarity of the outputs or the time-varying property of a system.

A Two-Step Estimation Method for a Time-Varying INAR Model

This paper proposes a new time-varying integer-valued autoregressive (TV-INAR) model with a state vector following a logistic regression structure. Since the autoregressive coefficient in the model is time-dependent, the Kalman-smoothed method is applicable. Some statistical properties of the model are established. To estimate the parameters of the model, a two-step estimation method is proposed. In the first step, the Kalman-smoothed estimation method, which is suitable for handling time-dependent systems and nonstationary stochastic processes, is utilized to estimate the time-varying parameters. In the second step, conditional least squares is used to estimate the parameter in the error term. This proposed method allows estimating the parameters in the nonlinear model and deriving the analytical solutions. The performance of the estimation method is evaluated through simulation studies. The model is then validated using actual time series data.

Non-equilibrium Onsager–Machlup theory

This paper proposes a simple mathematical model of non-stationary and non-linear stochastic dynamics, which approximates a (globally) non-stationary and non-linear stochastic process by its locally (or ‘piecewise’) stationary version. Profiting from the elegance and simplicity of both, the exact mathematical model referred to as the Ornstein–Uhlenbeck stochastic process (which is globally stationary, Markov and Gaussian) and of the Lyapunov criterion associated with the stability of stationarity, we show that the proposed non-linear non-stationary model provides a natural extension of the Onsager–Machlup theory of equilibrium thermal fluctuations, to the realm of non-stationary, non-linear and non-equilibrium processes. As an illustrative application, we then apply the extended non-equilibrium Onsager–Machlup theory, to the description of thermal fluctuations and irreversible relaxation processes in liquids, leading to the main exact equations employed to construct the non-equilibrium self-consistent generalised Langevin equation (NE-SCGLE) theory of irreversible processes in liquids. This generic theory has demonstrated that the most intriguing and long-unsolved questions of the glass and gel transitions are understood as a natural consequence of the second law of thermodynamics, enunciated in terms of the proposed piecewise stationary stochastic mathematical model.

A time variant uncertainty propagation method for high-dimensional dynamic structural system via K-L expansion and Bayesian deep neural network.

In this paper, a time variant uncertainty propagation (TUP) method for dynamic structural system with high-dimensional input variables is proposed. Firstly, an arbitrary stochastic process simulation (ASPS) method based on Karhunen-Loève (K-L) expansion and numerical integration is developed, expressing the stochastic process as the combination of its marginal distributions and eigen functions at several discrete time points. Secondly, the iterative sorting method is implemented to the statistic samples of marginal distributions for matching the constraints of covariance function. Since marginal distributions are directly used to express the stochastic process, the proposed ASPS is suitable for stationary or non-stationary stochastic processes with arbitrary marginal distributions. Thirdly, the high-dimensional TUP problem is converted into several high-dimensional static uncertainty propagation (UP) problems after implementing ASPS. Then, the Bayesian deep neural network based UP method is used to compute the marginal distributions as well as the eigen functions of dynamic system response, the high-dimensional TUP problem can thus be solved. Finally, several numerical examples are used to validate the effectiveness of the proposed method. This article is part of the theme issue 'Physics-informed machine learning and its structural integrity applications (Part 1)'.

Simulation of non-stationary and non-Gaussian stochastic processes by the AFD-Type Sparse Representations

Simulation of a stochastic process to simultaneously meet a given marginal distribution condition and be compatible with a given covariance function is a well-known problem called the two-match problem in the sequel. We propose the adaptive Fourier decomposition (AFD) type methods as constructive steps to solve the two-match problem. AFD-Type methods used in the physical domain (e.g., time or frequency) are replacements of the Karhunen–Loève expansion, or spectral decomposition, in general, while, when being used in the probability space are replacements of the polynomial chaos or other types of chaos methods. The AFD-Type methods in both circumstances play the role of an approximation. Being compared with the Karhunen–Loève expansion, the proposed AFD-Type methods do not need to compute out the eigenvalues and eigenfunctions of the kernel integral operator defined by the target covariance function, while compared with the polynomial chaos or other chaos methods the AFD-Type methods offer a great deal of flexibility and efficiency. The proposed methods can be applied to stationary and non-stationary, weakly, and strongly non-Gaussian stochastic processes. We provide several examples to show effectiveness and efficiency of the AFD-Type methods.

A novel shock life model for BGA solder joints based on evolutionary power spectral density

Multiple shock damage of Ball grid array (BGA) solder joint is one of the important reasons for space mission failure. A novel shock life model is proposed for BGA solder joints to evaluate its life under shock loads based on the evolutionary power spectral density (EPSD). Considering the bandwidth and non-Gaussian features of the stress signals during different stages, the shock stress response was treated as a non-stationary and non-Gaussian stochastic process. EPSD of the shock stress response is derived in matrix form based on wavelet transforms. The weight coefficient and non-Gaussian damage factor are proposed to describe the influence of the bandwidth and non-Gaussian characteristics of shock response on fatigue damage, respectively. The damage values of different stress stages were calculated by weight coefficient and corrected using non-Gaussian damage factor. Then the shock life model is proposed based on damage accumulation theory. Tests for the BGA solder joints shock life were conducted, and the result shows that the proposed model is feasible. Therefore, the developed method in the paper is expected to provide reference for the prediction of BGA solder joint shock life in aerospace engineering.

Design sensitivity analysis of structural systems with damping devices subjected to fully non-stationary stochastic seismic excitations

In the framework of dynamic excitations to be considered during the design phase of structures, the most crucial one is the ground motion acceleration. To increase the structural performances against seismic actions, one of the most effective design criteria is to introduce damping devices.An efficient approach to define the optimal parameters of the damping system is based on the design sensitivity analysis, which provides a quantitative estimate of desirable design change, by relating the available design variables.In this paper a method to evaluate the sensitivities of stochastic response characteristics of structural systems with damping devices subjected to seismic excitations, modelled as fully non-stationary Gaussian stochastic processes, is proposed. The main steps are: i) to define the time–frequency varying response (TFR) function for non-classically damped systems; ii) to evaluate closed form solutions for the first-order derivatives of the TFR function as well as of the one-sided evolutionary power spectral density function of the structural response, with respect to damping parameters of devices; iii) to perform a design sensitivity analysis selecting as performance measure function the non-geometric spectral moments of nodal displacements.A numerical application demonstrates how the proposed approach is suitable to cope with practical problems of engineering interest.

A harmonic-wavelet-based representation for non-stationary stochastic excitations

Representation of nonstationary stochastic excitations is crucial for stochastic response analyses of (time-varying) linear and nonlinear structural systems. This paper proposes a new representation method of non-stationary stochastic excitations based on the generalized harmonic wavelet (GHW) that takes the phase angles and frequencies as basic random variables. The orthogonal properties of the discrete-form spectral process increments describing non-stationary stochastic processes are formulated. Then the GHW-based representation is derived by using the orthogonal properties. This method can be used to accurately reproduce non-stationary stochastic excitations with the target asymptotic Gaussianity and evolutionary power spectrum density. The effectiveness and accuracy of the proposed method have been validated via numerical examples. This study provides a novel way for the representation of non-stationary processes and deserves to be applied in the stochastic response analyses of structures.

Stochastic Optimization of Dissipation Structures Based on Lyapunov Differential Equations and the Full Stress Design Method

This article presents a Lyapunov precise integral-based analysis method for seismic structures with added viscous fluid dampers. This study uses the full stress algorithm as the optimization method, considering the mean square of interstory drifts as the optimization objective, the position of the damper as the optimization object, and the random vibration analysis method as the calculation method to optimize seismic frame structures with viscous dampers. A precise integral solution is derived for the Lyapunov equation based on the general expression of the Lyapunov differential equation for the damping system under the excitation of a nonstationary stochastic process using two types of modulation functions: g(t)=1 and g(t)=t. Finally, the optimal damping arrangement is achieved using this method with a six-layer non-eccentric planar frame. In addition, the optimization results of this study are verified with those in the literature using time-history analysis, which verifies the feasibility and effectiveness of the proposed method. This study provides a method for the optimal configuration of dampers for seismic response of structures, which is beneficial for engineering applications and the protection of seismic structures.

A General Analytic Approachto Analysing Nonstationary Stochastic Processes in Thermodynamic Systems.Part III: The Practical Application of the Approach

A General Analytic Approachto Analysing Nonstationary Stochastic Processes in Thermodynamic Systems.Part III: The Practical Application of the Approach

Functional time series forecasting: Functional singular spectrum analysis approaches

AbstractWe introduce two novel nonparametric forecasting methods designed for functional time series (FTS), namely, functional singular spectrum analysis (FSSA) recurrent and vector forecasting. Our algorithms rely on extracted signals obtained from the FSSA method and innovative recurrence relations to make predictions. These techniques are model‐free, capable of predicting nonstationary FTS and utilize a computational approach for parameter selection. We also employ a bootstrap algorithm to assess the goodness‐of‐prediction. Through comprehensive evaluations on both simulated and real‐world climate data, we showcase the effectiveness of our techniques compared to various parametric and nonparametric approaches for forecasting nonstationary stochastic processes. Furthermore, we have implemented these methods in the Rfssa R package and developed a shiny web application for interactive exploration of the results.

Statistical Analysis of Silicon Photomultiplier Output Signals

Silicon photomultipliers are relatively new devices designed as a matrix of single-photon avalanche detectors, which have become popular for their miniature dimensions and low operating voltage. Their superior sensitivity allows detecting low-photon-count optical pulses, e.g., in ranging and LIDAR applications. The output signal of the photomultiplier is a non-stationary stochastic process, from which a weak periodic pulse can be extracted by means of statistical processing. Using the double-exponential approximation of output avalanche pulses the paper presents a simple analytical solution to the mean and variance of the stochastic process. It is shown that even for an ideal square optical pulse the rising edge of the statistically detected signal is longer than the edge of individual avalanche pulses. The knowledge of the detected waveform can be used to design an optimum laser pulse waveform or algorithms for estimating the time of arrival. The experimental section demonstrates the proposed procedure.

A General Analytic Approachto Analysing Nonstationary Stochastic Processes in Thermodynamic Systems. Part II: Basic Featuresof the Constitutive Equation Analytic Solutions

A General Analytic Approachto Analysing Nonstationary Stochastic Processes in Thermodynamic Systems. Part II: Basic Featuresof the Constitutive Equation Analytic Solutions

Analysis of the Impact of Energy Factor Allocation Efficiency on Green Total Point Productivity from a Low-Carbon Perspective

The high consumption of energy resources but low utilization, low factor allocation, and unreasonable economic structure affect the sustainability of economic development. Based on the energy factor theory from a low-carbon perspective, this paper constructs an impact model of green total point productivity and conducts a detailed analysis of the resource mismatch in the energy industry. Under the premise of efficient resource allocation, the factor input of an enterprise is positively correlated with total point productivity. Second, the model adds environmental constraints, considers the undesired output of carbon dioxide, and considers the input of energy factors, and reduces the undesired output as much as possible while increasing the expected output. The model realizes the update estimation of each component of the stationary part of a class of nonstationary stochastic processes and can use the information at a certain time and its previous time in a cycle to forecast all subsequent times. In the simulation process, in order to make the results more objective and accurate, this paper calculates the index and its decomposition results under two conditions, and uses MaxDEA 6.0 professional software to measure the regional industry total point productivity index. The experimental results show that the effective factor of low-carbon data is 0.91, and the relative error is less than 3.11%, which meets the test threshold and effectively improves the robustness of the model.

A Novel Efficient Method for Simulating Non-Stationary Random Processes Combining Generalized Harmonic Wavelet and Stochastic Harmonic Function

Integrating the general expression of the generalized harmonic wavelet (GHW)-based spectral representation method (SRM) and the idea of stochastic harmonic function (SHF), a novel stochastic generalized harmonic wavelet (SGHW) method for fully nonstationary stochastic processes, is established. The advantages of the proposed method are (1) a stochastic process with accurate probability information can be obtained by retaining fewer components. A smaller number of components greatly reduces the number of random variables. Fewer random variables also reduce the difficulty of random process analysis. The superior fitting effect between evolutionary power spectral density (EPSD) obtained from the samples using the SGHW method and the EPSD model proves the validity and effectiveness of the SGHW method. (2) The proposed method contains more accurate probability information and has a higher computational efficiency. The comparisons of relative errors and computational time between the SGHW method and the SRM demonstrate the accuracy and efficiency of the proposed method.

Study on the time-varying extreme value characteristic of the transient loads on a 5:1 rectangular cylinder subjected to a thunderstorm-like wind

A reasonable extreme probability model to estimate the transient loads is essential for structural wind resistance design. The transient aerodynamic characteristics of a 5:1 rectangular cylinder under thunderstorm-like wind are experimentally studied based on the multiple-fan active wind tunnel technique. In this paper, the non-Gaussian characteristics of the transient pressures are analyzed with emphasis on the time-varying skewness and time-varying kurtosis. A generalized time-varying extreme value model for uniformly modulated nonstationary stochastic processes is proposed, which is adopted to study the extreme value of transient loads on a 5:1 rectangular cylinder under a thunderstorm-like wind. Compared with the stationary flow condition, the results show that the non-Gaussian features of the transient pressures is noticeable, which is determined by the time-varying turbulence intensity. The greater the time-varying turbulence intensity, the stronger the non-Gaussian characteristic of the transient loads is, which is consistent with the stationary wind case. Notably, the presented work confirms that the conventional mean extreme value model may seriously underestimate the extreme value of transient loads. Moreover, the extreme values under stationary flow are slightly higher than those under transient wind pressures. The results show that the peak factors in the suction region are significantly greater than that in the reattachment region, and the time-varying peak factors under transient wind pressures are slightly less than in the stationary flow case.

The decomposition-based nonstationary flood frequency analysis

• The performance-based NS-FFA is challenged theoretically and practically. • A new decomposition procedure for the decomposition-based NS-FFA is proposed. • The proposed decomposition procedure preserves the stationary component. • The proposed decomposition-based NS-FFA captures the nonstationary process better. • This approach overcomes the previous issues of the NS-FFA and performs competitively. The nonstationary flood frequency analysis (NS-FFA) is conducted when the assumption of stationarity in hydrologic extremes is violated. The commonly used approach is the performance-based NS-FFA, which jointly determines the distribution and the nonstationary structure based upon the model performance. However, this approach is challenged from both theoretical and practical perspectives. An alternative is the herein named decomposition-based NS-FFA, which determines the two NS-FFA model components separately and explicitly uses the available knowledge of the nonstationarity. However, it has been barely implemented in practice. This paper proposed a novel decomposition procedure that strictly follows the theoretical decomposition of nonstationary stochastic processes to advance the decomposition-based NS-FFA. The proposed decomposition procedure was compared with a previously reported method in both an analytical deduction and a simulation study. The proposed decomposition-based NS-FFA was further compared to the performance-based NS-FFA using both synthetic and real datasets from North America, which exhibit different patterns of nonstationarity. The Particle Filter was adopted for uncertainty quantification and parameter estimation. The results revealed that the proposed decomposition-based approach was advantageous in preserving the moments of underlying stochastic component, particularly the higher-order moment (i.e., skewness). In addition, the comparison of the two NS-FFA approaches demonstrated the superiority of the proposed decomposition-based approach in capturing the underlying (known) nonstationary stochastic process and being competitive with the performance-based approach from the performance perspective in real applications. The results from the simulation study and the real application also revealed several caveats of the performance-based approach, including the potential overfitting and equifinality problems as well as the selection of distinct models when adopting different performance metrics. In addition, differing from the performance-based approach, the decomposition-based NS-FFA avoided/alleviated the ergodicity violation. All these results demonstrated the advancements of the proposed decomposition-based NS-FFA and advocated its application in the NS-FFA.

A Tripolar Model of Gas Price Formation in Germany. Does the Shale Revolution in the US Matter?

Abstract Presented analysis of gas price formation mechanism in Germany was prompted by changes brought about by technological advancements and the liberalization and harmonization of natural gas markets in the European Union after the year 2000. Because the data used in the study is generated by nonstationary stochastic processes, the cointegrated vector autoregressive model was applied as the most appropriate. The analysis pointed out that the price of natural gas, oil and the USD/EUR exchange rate influence each other in the long run and thus should be modelled together. Gas price in Germany is driven by both fundamental and financial factors, and so it rises with economic expansion, oil price increases, and the depreciation of the USD. It also reacts to changes in short-term interest rates and the volume of gas production in the US, which confirms that the shale revolution in this country has been consequential for gas prices in Europe, like any other supply shock would have been.