Non-stationary Research Articles

Spatiotemporal variability identification and analysis for non-stationary climatic trends for a tropical river basin of India

The non-stationary behavior of climatic variables has been increasingly recognized as a challenge that disrupts the equilibrium of human-defined climate-based stationary processes, including hydrological and agricultural practices, and irrigation systems. This study aims to investigate long-term trends and non-stationarity in climatic variables across 23 stations of the Krishna River basin, India. Prominent trends in rainfall, temperature, and their extreme indices were identified using the Modified Mann-Kendall (MMK), Bootstrapped Mann-Kendall (BMK), and Sen's Slope Estimator tests, while the Innovative Trend Analysis (ITA) test uncovered hidden trends and potential shifts in climatic patterns. This study addresses a critical research gap by exploring both significant and hidden trends in climatic variables, providing a better understanding of future dynamics. Traditional methods like MMK and Sen's Slope were insufficient to reveal these hidden trends, but ITA offered a more comprehensive analysis. The findings revealed an increase in total annual rainfall for almost 50% of the basin, which aligns with rising maximum temperatures, suggesting enhanced evaporation rates and subsequent fluctuations in rainfall patterns. Seasonal analysis indicated a shift towards decreased rainfall during winter and pre-monsoon seasons, contrasted by increased precipitation during the monsoon and post-monsoon periods, highlighting a clear alteration in rainfall distribution. The Simple Daily Intensity Index (SDII) and other indices suggest intensified rainfall events despite a decrease in the number of rainy days, indicating fewer but more intense events. Temperature analysis showed an overall increase in maximum temperatures, with the Diurnal Temperature Range (DTR) significantly increasing across all stations, implying greater daily temperature variations and potential for intensified water cycles and extreme climatic events. Furthermore, the study simplifies these trends by classifying them into two attributes: intensity and frequency, aiding policymakers in site-specific management of water resources and planning for future climatic scenarios. The presence of non-stationarity in extreme rainfall was confirmed by the Augmented Dickey-Fuller (ADF), Phillips-Perron (PP), and Kwiatkowski-Phillips-Schmidt-Shin (KPSS) tests. These findings are significant as they conclude how climate change is altering hydrological patterns at each station. The study emphasizes the necessity for adaptive management strategies to mitigate the adverse impacts on agriculture, infrastructure, and human safety.

Debiasing Welch’s method for spectral density estimation

Abstract Welch’s method provides an estimator of the power spectral density that is statistically consistent. This is achieved by averaging over periodograms calculated from overlapping segments of a time series. For a finite-length time series, while the variance of the estimator decreases as the number of segments increases, the magnitude of the estimator’s bias increases: a bias-variance trade-off ensues when setting the segment number. We address this issue by providing a novel method for debiasing Welch’s method that maintains the computational complexity and asymptotic consistency, and leads to improved finite-sample performance. Theoretical results are given for fourth-order stationary processes with finite fourth-order moments and an absolutely convergent fourth-order cumulant function. The significant bias reduction is demonstrated with numerical simulation and an application to real-world data. Our estimator also permits irregular spacing over frequency and we demonstrate how this may be employed for signal compression and further variance reduction. The code accompanying this work is available in R and python.

On Asymptotic Equipartition Property for Stationary Process of Moving Averages

Let {Xn}n∈Z be a stationary process with values in a finite set. In this paper, we present a moving average version of the Shannon–McMillan–Breiman theorem; this generalize the corresponding classical results. A sandwich argument reduced the proof to direct applications of the moving strong law of large numbers. The result generalizes the work by Algoet et. al., while relying on a similar sandwich method. It is worth noting that, in some kind of significance, the indices an and ϕ(n) are symmetrical , i.e., for any integer n, if the growth rate of (an)n∈Z is slow enough, all conclusions in this article still hold true.

Solid Oxide Cell Reactor Model for Transient and Stationary Electrochemical H2O and CO2 Conversion Process Studies

The ability of high-temperature solid oxide cell (SOC) electrochemical reactors to efficiently convert atmospheric carbon to high value chemicals for industrial and energy storage applications via CO2 and co-electrolysis makes them a key technology for active carbon utilisation. However, due to additional operational risks from thermochemical reactions on thermal management, limited experimental capacity, and relative novelty, CO2 and co-electrolysis lag behind steam electrolysis in large-scale adoption. Here, a 1D+1D SOC model based on fundamental first principles considering three-dimensional heat transfer was improved via a unique method for representing co-electrolysis electrochemistry, solving with low computational effort. Validation against experimental data for two compositions and pressures, showed high levels of accuracy with respect to characteristic cell voltages, temperatures, and outlet compositions. The model also showed CO2 reduction during co-electrolysis mainly occurred via reverse water gas shift, while CO2 electrolysis still accounted for up to 35% of the total share. Pressurised co-electrolysis operation promotes exothermic methanation, causing pronounced heating of the reactor, consequently reducing the isothermal current density. Therefore, low to moderate pressurisation is likely most suited for coupling with downstream synthesis processes to avoid the installation of unnecessarily large systems and associated high costs.

Determination of the characteristics of drill string vibrations during the drilling process

The object of research is vibration processes of a certain origin in the drill string with typical design deviations depending on the mode parameters of drilling. A drill string is an oscillating system with an infinite number of degrees of freedom of a multifactor system. An exhaustive study of oscillatory processes in the drill string is impossible neither analytically nor experimentally, due to the specifics of the hole deepening in various rocks, the design of the well, its shape, etc. Therefore, in practice, they try to solve the problems of the dynamics of the drill string for an idealized system and, while preserving the main oscillatory properties, solve some problems of the rod system. The work carried out was aimed at experimental studies of vibrations of the drill string during the drilling process. It is shown that the effectiveness of the use of hydrodynamic cavitation requires the development of methods and devices for intensifying the well drilling process. It is proven that the design of the cavitation generator organically fits into the existing well drilling equipment and allows for the intensification of technological processes with lower specific energy consumption. It is found that all oscillatory processes that occur in the drill string are random in nature and must be considered using the mathematical apparatus of the theory of random oscillations. The study of vibrations during well drilling shows that vibrations can be considered as random stationary processes, since transient modes have a sufficiently short duration for homogeneous rocks with fixed drilling modes. The analysis of the vibrations of the drill string elements based on random oscillations in a number of cases allows to increase the reliability of determining the vibration reliability of the drill string elements. It has been proven that the response of drill string elements to broadband random vibration can be defined as the combined effect of several narrowband random vibrations.

TESTING MEAN REVERSION OF STOCK PRICES IN OECD COUNTRIES: EVIDENCE FROM FOURIER THRESHOLD UNIT ROOT TEST

Researchers focus on whether stock prices have a unit root, that is, whether they contain a random walk process. If stock prices have a stationary process, that is, if they return to the mean, the effects of shocks are temporary, and it is interpreted that they will return to the trend path over time. If stock prices have transitory shocks, it allows for the prediction of future movements based on past behavior in terms of investment. This study investigates whether stock prices revert to the mean and thus have a random walk process. For this purpose, the Fourier Threshold Unit Root (FTUR) test based on the test methodology of Caner and Hansen (2001) for the period January 1990–January 2021 for 26 OECD countries is applied. The FTUR test takes into account both structural breaks and nonlinearities. The purpose of using Fourier functions to account for structural changes is that they are not affected by the number, location, or shape of breaks. Thus, the power of the test increases. According to the results of this test, stock prices in Austria, Canada, Germany, Italy, New Zealand, Spain, and the UK are linear. Therefore, Fourier Augmented Dickey-Fuller (FADF) unit root analysis was performed for these countries. The FTUR test was performed in other countries. According to the results of FTUR and FADF unit root tests, stock prices are found to contain unit roots in some countries except Italy. In some countries, stock prices have a partial unit root structure. In other words, the effects of shocks are permanent, and it is concluded that future returns cannot be predicted in these countries with the random walk process.

Generalized kinetic theory of coarse-grained systems. I. Partial equilibrium and Markov approximations

The general kinetic theory of coarse-grained systems is presented in the abstract formalism of communication theory developed by Shannon and Weaver, Khinchin and Kolmogorov. The martingale theory shows that, under reasonable, general hypotheses, coarse-grained systems can be approximated by generalized Markov systems. For mixing systems, the Kolmogorov entropy production can be defined for nonstationary processes as Kolmogorov defined it for stationary processes.

Stochastic modeling of stationary scalar Gaussian processes in continuous time from autocorrelation data

We consider the problem of constructing a vector-valued linear Markov process in continuous time, such that its first coordinate is in good agreement with given samples of the scalar autocorrelation function of an otherwise unknown stationary Gaussian process. This problem has intimate connections to the computation of a passive reduced model of a deterministic time-invariant linear system from given output data in the time domain. We construct the stochastic model in two steps. First, we employ the AAA algorithm to determine a rational function which interpolates the z-transform of the discrete data on the unit circle and use this function to assign the poles of the transfer function of the reduced model. Second, we choose the associated residues as the minimizers of a linear inequality constrained least squares problem which ensures the positivity of the transfer function’s real part for large frequencies. We apply this method to compute extended Markov models for stochastic processes obtained from generalized Langevin dynamics in statistical physics. Numerical examples demonstrate that the algorithm succeeds in determining passive reduced models and that the associated Markov processes provide an excellent match of the given data.

The [formula omitted] stochastic volatility model

This article introduces a new class of stochastic volatility models called logGARCH Stochastic Volatility models (logGARCH-SV). We establish the strict stationarity and second-order stationarity properties of this model class. Additionally, we provide conditions for the existence of higher-order moments. To estimate the parameters of the proposed model, we utilize a sequential Monte Carlo method. Finally, we assess the performance of the suggested estimation method through a simulation study.

An Approximate Algorithm for Simulating Stationary Discrete Random Processes with Bivariate Distributions of Their Consecutive Components in the Form of Mixtures of Gaussian Distributions

An Approximate Algorithm for Simulating Stationary Discrete Random Processes with Bivariate Distributions of Their Consecutive Components in the Form of Mixtures of Gaussian Distributions

Optimality conditions for bilevel programmes via Moreau envelope reformulation*

For bilevel programmes with a convex lower-level programme, the classical approach replaces the lower-level programme with its Karush-Kuhn-Tucker condition and solve the resulting mathematical programme with complementarity constraint (MPCC). It is known that when the set of lower-level multipliers is not unique, MPCC may not be equivalent to the original bilevel problem, and many MPCC-tailored constraint qualifications do not hold. In this paper, we study bilevel programmes where the lower level is generalized convex. Applying the equivalent reformulation via Moreau envelope, we derive new directional optimality conditions. Even in the nondirectional case, the new optimality condition is stronger than the strong stationarity for the corresponding MPCC.

Towards autonomous learning and optimisation in textile production: data-driven simulation approach for optimiser validation

AbstractThe textile industry is a traditional industry branch that remains highly relevant in Europe. The industry is under pressure to remain profitable in this high-wage region. As one promising approach, data-driven methods can be used for process optimisation in order to reduce waste, increase profitability and relieve mental burden on staff members. However, approaches from research rarely get adopted into practice. We identify the high dimensionality of textile production processes leading to high model uncertainty as well as an incomplete problem formulation as the two main problems. We argue that some form of an autonomous learning agent can address this challenge, when it safely explores advantageous, unknown new settings by interacting with the process. Our main goal is to facilitate the adoption of promising research into practical applications. The main contributions of this paper include the derivation and formulation of a probabilistic optimisation problem for high-dimensional, stationary production processes. We also create a highly adaptable simulation of the textile carded nonwovens production process in Python that implements the optimisation problem. Economic and technical behavior of the process is approximated using both Gaussian Process Regression (GPR) models trained with industrial data as well as physics-motivated explicit models. This ’simulation first’-approach makes the development of autonomous learning agents for practical applications feasible because it allows for cheap testing and validation before physical trials. Future work will include the comparison of the performance of different agent approaches.

Nonstationary frequency analysis of extreme precipitation: Embracing trends in observations

Knowledge of the recurrence intervals of precipitation extremes is vital for infrastructure design, risk assessment, and insurance planning. However, trends and shifts in rainfall patterns globally pose challenges to the application of extreme value analysis (EVA) which relies critically on the assumption of stationarity. In this paper, we explore: (1) the suitability of nonstationary (NS) models in the presence of statistically significant trends, and (2) their potential in modeling out-of-sample data to improve frequency analysis of extreme precipitation. We analyze the benefits of using a nonstationary Generalized Extreme Value (GEV) model for annual extreme precipitation records from Southern Brazil. The location of the GEV distribution is allowed to change with time. The unknown GEV model parameters are estimated using Bayesian techniques coupled with Markov chain Monte Carlo simulation. Next, we use GAME sampling to compute the evidence (and their ratios, the so-called Bayes factors) for stationary and nonstationary models of annual maximum precipitation. Our results show that the presence of a statistically significant trend in annual maximum precipitation alone does not justify the use of a NS model. The location parameter of the GEV distribution must also be well defined, otherwise, stationary models of annual maximum precipitation receive more support by the data. These findings reiterate the importance of accounting for GEV model parameters and predictive uncertainty in frequency analysis and hypothesis testing of annual maximum precipitation data records. Furthermore, a meaningful EVA demands detailed knowledge about the origin and persistence of observed changes.

Rainfall Potential and Consequences on Structural Soil Degradation of the Most Important Agricultural Region of Mexico

This study investigates the historical variability in annual average precipitation in the northwest region of Mexico, aiming to evaluate the cumulative impact of precipitation on soil degradation and associated risks posed by rainfall. Despite being known as “The Agricultural Heart of Mexico”, the region’s soil has experienced significant damage to its granulometric structure due to unpredictable rainfall patterns attributed to climate change. Sixteen historical series of average annual rainfall were analyzed as stationary stochastic processes for spectral analysis. The results revealed exponential decay curves in each radial spectrum, indicating a linear relationship between frequency and amplitude. These curves identified initial impulses correlated with moments of severity for structural damages caused by rainfall-induced degradation. The degradation process, exacerbated by water stress, accelerates, as evidenced by maps illustrating approximately 75% soil damage. In the context of climate change and the uncertainty surrounding soil responses to extreme meteorological events, understanding this phenomenon becomes crucial. Recognizing the dynamic nature of soil responses to environmental stressors is essential for effective soil management. Emphasizing the need to employ numerical processes tailored to new environmental considerations related to observed soil damages is crucial for sustainable soil management practices in any region.

MGF-based SNC for stationary independent Markovian processes with localized application of martingales

MGF-based SNC for stationary independent Markovian processes with localized application of martingales

Smallest gaps between zeros of stationary Gaussian processes

In this paper, we study the smallest gaps between successive zeros of nondegenerate smooth stationary centered Gaussian processes on the real line with the assumption that the covariance kernel κ(x) and its derivatives decay to 0 as |x|→∞. We prove that, after rescaling, the smallest gaps converge to a Poisson point process with a specific rate. Moreover, the positions where these smallest gaps occur tend to a uniform distribution. Consequently, we can derive the limiting density for the k-th smallest gap.

Randomized limit theorems for stationary ergodic random processes and fields

Using the randomization approach, introduced by A. Tempelman in Randomized multivariate central limit theorems for ergodic homogeneous random fields, Stochastic Processes and their Applications. 143 (2022), 89–105, we prove: (a) a randomized version of the invariance principle (the functional CLT); (b) a version the Glivenko–Cantelli theorem; (c) a randomized theorem about convergence of empirical processes to the Brownian bridge. We also weaken the moment condition in the randomized CLTs, proved in the mentioned article. The main point of our work is that most of our theorems are valid for all ergodic homogeneous random fields on Zm and Rm,m≥1.

Adaptive Neural Control for a Class of Random Fractional-Order Multi-Agent Systems with Markov Jump Parameters and Full State Constraints

Based on an adaptive neural control scheme, this paper investigates the consensus problem of random Markov jump multi-agent systems with full state constraints. Each agent is described by the fractional-order random nonlinear uncertain system driven by random differential equations, where the random noise is the second-order stationary stochastic process. First, in order to deal with the unknown functions with Markov jump parameters, a radial basis function neural network (RBFNN) structure is introduced to achieve approximation. Second, for the purpose of keeping the agents’ states from violating the constraint boundary, the tan-type barrier Lyapunov function is employed. By using the stochastic stability theory and adopting the backstepping technique, a novel adaptive neural control design method is presented. Furthermore, to cope with the differential explosion problem in the design course, the extended state observer (ESO) is developed instead of neural network (NN) approximation or command filtering techniques. Finally, the exponentially noise-to-state stability in the mean square is analyzed rigorously by the Lyapunov method, which guarantees the consensus of the considered multi-agent systems and all the agents’ outputs are bounded in probability. Two simulation examples are provided to verify the effectiveness of the suggested control strategy.

A new shape of extremal clusters for certain stationary semi-exponential processes with moderate long range dependence

A new shape of extremal clusters for certain stationary semi-exponential processes with moderate long range dependence

Recursive generalized gamma kernel density estimation for nonnegative dependent data

In this article, we consider the recursive generalized gamma (GG) kernel density estimator (KDE) for nonnegative dependent data from stationary α-mixing process. Asymptotic bias, variance and mean integrated squared error are provided. Simulation experiments from an autoregressive conditional duration model and a stochastic volatility model are conducted to compare the recursive GG KDE with non-recursive GG KDE. Two applications are provided for nonnegative time series data.