Nonstationary Stochastic Processes Research Articles

Measuring Financial Impacts of the Renewable Energy Based Fiscal Policy in Colombia under Electricity Price Uncertainty

This paper analyses the financial implications, from the point of view of an investor in renewable energy, which sells the energy for an uncertain price of electricity and decides to take advantage of the Colombian tax policy over the renewable energy. The policy, known as Investment Tax Allowance (ITA), encourages installation of renewable projects in a country traditionally dominated by hydro power. Price is modeled as a non-stationary autoregressive stochastic process with normally distributed error terms. Costs, and uncertain revenue and taxes are considered to assess the financial impact on a solar project when the policy is implemented. Since impact varies according to project ownership, two cases are evaluated: a generation company (GENCO-1) that only owns the solar project; and, an existent generation company (GENCO-2) that owns a portfolio of projects. Results indicate that if ITA is applied, it is likely that the GENCO-1 cannot take the full advantage of the incentive, as opposed to the GENCO-2. Although this policy might not satisfy planner objectives since it does not guarantee the construction of significantly high capacity of new renewable energy projects, it definitely represents an attractive mechanism to decrease tax obligations at the GENCO-2 level. Finally, a theoretical analysis shows that investment cost affects the mean of the present value; whereas tax rates impacts both its mean and standard deviation.

Effect of Epidemic Interference on the Performance of M-ASK, M-PSK and M-QAM Modulation Schemes

This article presents a study on the performance of different digital modulation systems in the presence of epidemic interference. This type of interference is caused by the fast increase in the number of users in a system at a given time. The epidemic interference can be modeled as a non-stationary stochastic process, which presents random power levels over time. In a previous work, the authors assessed the effect of the epidemic interference on the performance of BPSK, QPSK, and M-QAM systems. The present paper extends the previous results for MASK and M-PSK systems. Furthermore, numerical results from Monte Carlo simulations are presented for all evaluated digital modulation systems.

Denoising, Outlier/Dropout Correction, and Sensor Selection in Range-Based Positioning

Range-based target localization combines the range measurements and appropriate algorithms [such as time of arrival (TOA)] to determine the real-time position of a moving target not only in a satellite-denied environment but also for high-precision applications in an open (i.e., satellite-available) environment. However, the measurements from sensors always suffer from uncertainties, such as noises, outliers, dropouts, and biases, which make measurements not reliable enough to directly utilize in positioning. This article is, therefore, concerned with two kinds of sensor correction problems: 1) attenuating the noises, removing the outliers, and completing the dropouts and 2) identifying the discredited sensors and forbidding them from use. Specifically, we model a range measurement time series from a ranging sensor as a nonstationary stochastic process and then use a local polynomial to regress the mean function in an online manner. The derivatives of the mean function are identified as the state variables in the state space, and the Kalman filter is used to estimate the states. We will show that: 1) the proposed method is simple yet effective in denoising the measurements and correcting the outliers/dropouts with very high accuracy and precision and 2) the defined and estimated states can clearly reflect the pattern of one type of nonspecific anomaly contained in the range time series, which enables the recognition of the dysfunctional sensors.

Regularized Local Basis Function Approach to Identification of Nonstationary Processes

The problem of identification of nonstationary stochastic processes (systems or signals) is considered and a new class of identification algorithms, combining the basis functions approach with local estimation technique, is described. Unlike the classical basis function estimation schemes, the proposed regularized local basis function estimators are not used to obtain interval approximations of the parameter trajectory, but provide a sequence of point estimates corresponding to consecutive instants of time. Based on the results of theoretical analysis, the paper addresses and solves all major problems associated with implementation of the new class of estimators, such as optimization of the regularization matrix, adaptive selection of the number of basis functions and the width of the local analysis interval, and reduction of complexity of the computational algorithms.

A sample-based iterative scheme for simulating non-stationary non-Gaussian stochastic processes

This paper presents a new numerical scheme for simulating stochastic processes specified by their marginal distribution functions and covariance functions. Stochastic samples are first generated to satisfy target marginal distribution functions. An iterative algorithm is proposed to match the simulated covariance function of stochastic samples to the target covariance function, and only a few iterations can converge to a required accuracy. Several explicit representations, based on Karhunen-Loève expansion and Polynomial Chaos expansion, are further developed to represent the obtained stochastic samples in series forms. Proposed methods can be applied to non-stationary non-Gaussian stochastic processes, and three examples illustrate their accuracies and efficiencies.

A weighted sieve estimator for nonparametric time series models with nonstationary variables

We study a class of nonparametric regression models that includes deterministic time trends and both stationary and nonstationary stochastic processes (whose shocks are allowed to be mutually correlated). We propose a unified approach to estimation based on the weighted sieve method to tackle the issue of unbounded support of the covariates. This approach improves on the existing technology in terms of some key regularity conditions such as moment conditions and the α-mixing coefficients for the stationary process. We establish self-normalized central limit theorems for the sieve estimator and other related quantities. Monte Carlo simulation confirms the theoretical results. We use our methodology to study the effect of CO2 and solar irradiance on global sea level rise.

Time–Frequency Spectral Representation Models to Simulate Nonstationary Processes and Their Use to Generate Ground Motions

In the present study, two new models are proposed to model nonstationary stochastic processes with time-frequency-dependent spectra and coherence. The models are developed based on the discrete orthonormal S-transform and are enhanced by using the time-frequency representation from the S-transform. One of the models is shown to be usable in carrying out the conditional simulation. A parallel is drawn between the proposed models and those based on the spectral representation method using the ordinary Fourier analysis to facilitate an understanding of the proposed models. Unlike the use of the evolutionary spectral theory, no amplitude modulation or frequency modulation functions need to be assigned and identified for the proposed models because they are included explicitly in the time-frequency spectra. Moreover, the use of the S-transform and discrete orthonormal S-transform provides a familiar time-frequency spectral representation rather than the timescale representation obtained from the wavelet transform. The exposition and use of the models are focused on the simulation of ground motions at single and multiple support, although the models can be equally applicable to nonstationary processes, such as the winds and waves. The adequacy of the simulated ground motion records is assessed in terms of the time-frequency spectral representation and response spectrum. Numerical examples, including conditionally simulate ground motions, are also given.

Kinetic equation for particle transport in turbulent flows

With a new approach based on the ensemble average over particle state transition paths in phase space, a kinetic equation for particles transported in turbulent flows is derived. The probability density function (PDF) for particles is defined as an ensemble average of a special fine-grained PDF, referred to as the local path density operator. The kinetic equation is derived from a Taylor series expansion of the PDF in terms of the cumulants with respect to particle paths in phase space and leads to a closed expression for its diffusion terms. It shows that the random forcing of eddy fluctuations, non-stationarity of turbulence, and inertia of particles are explicitly presented in the diffusion coefficient, which could help us to understand how particles are diffused by these underlying mechanisms. The kinetic equation is applicable to non-Markovian, non-Gaussian, and non-stationary stochastic processes, while for Markovian processes, it recovers the classical Fokker–Planck equation. The macroscopic equations for particle phase are derived based on the kinetic equation and compared with the direct numerical simulation of particles transported in turbulent flows.

Time-dependent system kinematic reliability analysis for planar parallel manipulators

Time-dependent system kinematic reliability of the planar parallel manipulator refers to the probability of the pose error falling into the allowable safe region over the whole specified trajectory, which is essential for its work performance. However, works regarding this issue are quite limited. Consequently, this study conducts time-dependent kinematic reliability analysis for planar parallel manipulators considering joint clearance, input uncertainty, and manufacturing imperfection based on the first-passage method. First, the limit-state pose error function is established by means of the Baker–Campbell–Hausdorff formula and the theory of the Lie group and Lie algebra. Then, the analytical solution to the outcrossing rate of the non-stationary vector stochastic kinematic process is derived by employing the expectation propagation and closed-form properties of conditional and marginal statistical moments (mean and covariance) of the multivariate Gaussian. On this basis, the time-dependent system kinematic reliability is calculated upon the assumption that the outcrossing events are independent. Finally, the planar 3-RRR parallel manipulator is used to demonstrate the proposed method, and its effectiveness is validated by comparison with the Monte Carlo simulation method.

Discrepancy-Based Theory and Algorithms for Forecasting Non-Stationary Time Series

We present data-dependent learning bounds for the general scenario of non-stationary non-mixing stochastic processes. Our learning guarantees are expressed in terms of a data-dependent measure of sequential complexity and a discrepancy measure that can be estimated from data under some mild assumptions. Our learning bounds guide the design of new algorithms for non-stationary time series forecasting for which we report several favorable experimental results.

Time-dependent reliability prediction using transfer learning

One of the key challenges in reliability estimation is the acquisition of failure information especially under real-life scenarios or computationally expensive long-run simulations. Due to the lack of failure information, it becomes a challenging task to estimate time-dependent reliability of a structure or component. In this paper, a preliminary study of time-dependent reliability prediction using transfer learning is given by utilizing only current time performance function information. Transfer learning, specifically domain adaptation, is used in this work in order to find a representation in latent space where the statistical properties of nonstationary stochastic processes such as variance are preserved and their distribution parameters at different times are made close to each other. Correlated random samples of the stochastic processes are generated in present and future time intervals in order to find common latent space using domain adaptation via transfer component analysis (TCA). Due to the excellent computational efficiency, a Kriging surrogate model is built in the present interval only using an adaptive sampling strategy. The reliability in the future is predicted using the same surrogate model without retraining it using future performance function information. Monte Carlo simulation (MCS) is used to validate the results of the proposed method. The results show that the proposed method can predict the failure probability of time-dependent problems efficiently with significant accuracy.

Multifractional Brownian motion and quantum-behaved particle swarm optimization for short term power load forecasting: An integrated approach

Power load fluctuation is generally agreed to be a non-stationary stochastic process. The Fractional Brownian Motion (FBM) model is proposed to forecast a non-stationary time series with high accuracy. Computation of the Hurst exponent (H) for the power load data series using the Rescaled Range Analysis (R/S) in this study. This method is used to verify the Long-Range Dependent (LRD) characteristics of non-stationary power load data. For the real power load, however, H exponent takes on the self-similarity characteristics in a certain finite range of intervals, the global self-similarity is very rare to exist. The H exponent of the self-similarity usually has more than one value. We generalize multifractional H(t) to replace constant H. To improve the forecasting accuracy, the H(t) is optimized by the Quantum-Behaved Particle Swarm Optimization (QPSO). Once the optimal H(t) is obtained, then the optimal and parameters in the multi-Fractional Brownian Motion (mFBM) model can be deduced to forecast next power load data series with a higher accuracy.

A Discrete Increment Model for Electricity Price Forecasting Based on Fractional Brownian Motion

Electricity price forecasting is essential for all participants of the power consumption market. However, electricity price series has complex properties such as high volatility and non-stationarity that make forecasting turn out to be very difficult. In this study, we aim to forecast electricity price series by the discrete increment model of fractional Brownian motion (fBm). A specific feature of the fBm is that it represents a typical non-stationary stochastic process with long-range dependent (LRD) characteristics. Analysis of electricity price series has LRD characteristics, and the Hurst exponent can be calculated. The Hurst exponent is the key parameter of fBm, and it is a measure of self-similarity. The stochastic differential equation driven by fBm is discretized into the discrete increment model for electricity price forecasting. Other parameters of the discrete increment model can be evaluated by the maximum likelihood estimation (MLE). The performance of the proposed method is demonstrated by using the data from the U.S. Energy Information Administration. The validity of the proposed model was compared with other methods.

Dynamic predictive probabilities to monitor rapid cystic fibrosis disease progression.

Cystic fibrosis (CF) is a progressive, genetic disease characterized by frequent, prolonged drops in lung function. Accurately predicting rapid underlying lung‐function decline is essential for clinical decision support and timely intervention. Determining whether an individual is experiencing a period of rapid decline is complicated due to its heterogeneous timing and extent, and error component of the measured lung function. We construct individualized predictive probabilities for “nowcasting” rapid decline. We assume each patient's true longitudinal lung function, , follows a nonlinear, nonstationary stochastic process, and accommodate between‐patient heterogeneity through random effects. Corresponding lung‐function decline at time t is defined as the rate of change, . We predict conditional on observed covariate and measurement history by modeling a measured lung function as a noisy version of . The method is applied to data on 30 879 US CF Registry patients. Results are contrasted with a currently employed decision rule using single‐center data on 212 individuals. Rapid decline is identified earlier using predictive probabilities than the center's currently employed decision rule (mean difference: 0.65 years; 95% confidence interval (CI): 0.41, 0.89). We constructed a bootstrapping algorithm to obtain CIs for predictive probabilities. We illustrate real‐time implementation with R Shiny. Predictive accuracy is investigated using empirical simulations, which suggest this approach more accurately detects peak decline, compared with a uniform threshold of rapid decline. Median area under the ROC curve estimates (Q1‐Q3) were 0.817 (0.814‐0.822) and 0.745 (0.741‐0.747), respectively, implying reasonable accuracy for both. This article demonstrates how individualized rate of change estimates can be coupled with probabilistic predictive inference and implementation for a useful medical‐monitoring approach.

Spatio-Temporal Segmentation in 3-D Echocardiographic Sequences Using Fractional Brownian Motion.

An important aspect for an improved cardiac functional analysis is the accurate segmentation of the left ventricle (LV). A novel approach for fully-automated segmentation of the LV endocardium and epicardium contours is presented. This is mainly based on the natural physical characteristics of the LV shape structure. Both sides of the LV boundaries exhibit natural elliptical curvatures by having details on various scales, i.e. exhibiting fractal-like characteristics. The fractional Brownian motion (fBm), which is a non-stationary stochastic process, integrates well with the stochastic nature of ultrasound echoes. It has the advantage of representing a wide range of non-stationary signals and can quantify statistical local self-similarity throughout the time-sequence ultrasound images. The locally characterized boundaries of the fBm segmented LV were further iteratively refined using global information by means of second-order moments. The method is benchmarked using synthetic 3D+time echocardiographic sequences for normal and different ischemic cardiomyopathy, and results compared with state-of-the-art LV segmentation. Furthermore, the framework was validated against real data from canine cases with expert-defined segmentations and demonstrated improved accuracy. The fBm-based segmentation algorithm is fully automatic and has the potential to be used clinically together with 3D echocardiography for improved cardiovascular disease diagnosis.

Research on the contact fatigue failure of thermal sprayed coating based on infrared thermography

ABSTRACTAiming to improve the accuracy of the contact life prediction of sprayed coatings, a series of contact fatigue tests are conducted on RM-1 tester. Five kinds of plasma sprayed Al2O3–40 wt%TiO2 composite ceramic coatings (AT40 ceramic coatings) of different quality and surface performance parameters are tested on the principle of central composite design. The passive infrared thermography is used to monitor the temperature change of the coating during the process of contact fatigue failure. The findings reveal that the infrared signal which is characteristic of non-stationary stochastic process shows a pre-warning against the failure of coatings. Both singularity detection and k-means clustering analysis come to a conclusion that the invalidating process of coating caused by contact fatigue is an interwoven result of crack initiation, crack propagation and crack instability fracture, etc. The nonstationarity of the infrared signal is further verified by non-parametric hypothesis test. Based on the results of correlation analysis between the contact point temperature range of the coating surface and the contact fatigue life, a reliable nonlinear prediction model is established.

Subdifferentials of Value Functions in Nonconvex Dynamic Programming for Nonstationary Stochastic Processes

The main goal of this paper is to apply the machinery of variational analysis and generalized differentiation to study infinite horizon stochastic dynamic programming (DP) with discrete time in the Banach space setting without convexity assumptions. Unlike to standard stochastic DP with stationary Markov processes, we investigate here stochastic DP in $L^p$ spaces to deal with nonstationary stochastic processes, which describe a more flexible learning procedure for the decision-maker. Our main concern is to calculate generalized subgradients of the corresponding value function and to derive necessary conditions for optimality in terms of the stochastic Euler inclusion under appropriate Lipschitzian assumptions. The usage of the subdifferential formula for integral functionals on $L^p$ spaces allows us, in particular, to find verifiable conditions to ensure smoothness of the value function without any convexity and/or interiority assumptions.

Dynamics of a Duffing oscillator with the stiffness modeled as a stochastic process

The Duffing equation has been extensively studied in the last decades, especially in systems of vibrating micro-beams subjected to electro-mechanical fluctuating fields. However, mechanical systems are often subjected to uncertainties coming from the excitation and/or the design parameters. These uncertainties can then be considered as stationary or non-stationary stochastic processes. In many engineering applications, we consider stationary stochastic processes as they are easier to model and simplify our problems. However, some of them do not present stationarity properties and it is desirable to know what are the consequences of leaving the non-stationarity aside and consider a stationary case. For this purpose, two stochastic processes are applied to a Duffing oscillator and the results are analyzed when its stiffness modeling is based on (i) a stationary stochastic process generated with Langevin’s equation; and when it is based on (ii) a non-stationary stochastic process known as Brownian Bridge. A methodology is proposed to modify these stochastic processes in order to establish a lower limit for their support. In fact, this methodology can be applied to stationary and non-stationary stochastic processes. The stochastic results showed that the case with uncertainties in the stiffness coefficient of the non-linear term presented the highest variation on system’s response. In addition, the non-stationary case presented a result much closer to the deterministic one for the set of parameters chosen. The smooth variation of the second moment with time might explain it. There are cases in which the assumption of a stationary process might be not appropriate and non-stationarity must be assessed.

Detecting dynamic spatial correlation patterns with generalized wavelet coherence and non-stationary surrogate data

Time series measured from real-world systems are generally noisy, complex and display statistical properties that evolve continuously over time. Here, we present a method that combines wavelet analysis and non-stationary surrogates to detect short-lived spatial coherent patterns from multivariate time-series. In contrast with standard methods, the surrogate data proposed here are realisations of a non-stationary stochastic process, preserving both the amplitude and time-frequency distributions of original data. We evaluate this framework on synthetic and real-world time series, and we show that it can provide useful insights into the time-resolved structure of spatially extended systems.

Disorder indicator for nonstationary stochastic processes

The properties of a statistic called a self-consistent stationarity level of nonstationary time series are examined in this work. It is shown that a change in this statistic can be treated as a disorder in the nonstationarity properties of the series. The significance level of decision making is estimated, characteristic periods in a non-stationary stochastic process are detected, and an optimal sample size for constructing indicators in stochastic control problems is determined. A disorder indicator for electroencephalogram data from epilepsy patients is studied as a practical application.