NADARAYA-WATSON ESTIMATORS FOR REFLECTED STOCHASTIC PROCESSES

We study the Nadaraya-Watson estimators for the drift function of two-sided reflected stochastic differential equations.The estimates,based on either the continuously observed process or the discretely observed process,are considered.Under certain conditions,we prove the strong consistency and the asymptotic normality of the two estimators.Our method is also suitable for one-sided reflected stochastic differential equations.Simulation results demonstrate that the performance of our estimator is superior to that of the estimator proposed by Cholaquidis et al.(Stat Sin,2021,31:29-51).Several real data sets of the currency exchange rate are used to illustrate our proposed methodology.

Applications of Dynamic-Equilibrium Continuous Markov Stochastic Processes to Elements of Survival Analysis

In this article, we summarize some results on invariant non-homogeneous and dynamic-equilibrium (DE) continuous Markov stochastic processes. Moreover, we discuss a few examples and consider a new application of DE processes to elements of survival analysis. These elements concern the stochastic quadratic-hazard-rate model, for which our work 1) generalizes the reading of its It? stochastic ordinary differential equation (ISODE) for the hazard-rate-driving independent (HRDI) variables, 2) specifies key properties of the hazard-rate function, and in particular, reveals that the baseline value of the HRDI variables is the expectation of the DE solution of the ISODE, 3) suggests practical settings for obtaining multi-dimensional probability densities necessary for consistent and systematic reconstruction of missing data by Gibbs sampling and 4) further develops the corresponding line of modeling. The resulting advantages are emphasized in connection with the framework of clinical trials of chronic obstructive pulmonary disease (COPD) where we propose the use of an endpoint reflecting the narrowing of airways. This endpoint is based on a fairly compact geometric model that quantifies the course of the obstruction, shows how it is associated with the hazard rate, and clarifies why it is life-threatening. The work also suggests a few directions for future research.

Hierarchical multihead self-attention for time-series-based fault diagnosis

Fault diagnosis is important for maintaining the safety and effectiveness of chemical process.Considering the multivariate,nonlinear,and dynamic characteristic of chemical process,many time-series-based data-driven fault diagnosis methods have been developed in recent years.However,the existing methods have the problem of long-term dependency and are difficult to train due to the sequential way of training.To overcome these problems,a novel fault diagnosis method based on time-series and the hierarchical multihead self-attention(HMSAN)is proposed for chemical process.First,a sliding window strategy is adopted to construct the normalized time-series dataset.Second,the HMSAN is developed to extract the time-relevant features from the time-series process data.It improves the basic self-attention model in both width and depth.With the multihead structure,the HMSAN can pay attention to different aspects of the complicated chemical process and obtain the global dynamic features.However,the multiple heads in parallel lead to redundant information,which cannot improve the diagnosis performance.With the hierarchical structure,the redundant information is reduced and the deep local time-related features are further extracted.Besides,a novel many-to-one training strategy is introduced for HMSAN to simplify the training procedure and capture the long-term dependency.Finally,the effectiveness of the proposed method is demonstrated by two chemical cases.The experimental results show that the proposed method achieves a great performance on time-series industrial data and outperforms the state-of-the-art approaches.

THE SOLUTION OF RANDOM EIGENVALUE PROBLEM WITH SMALL STOCHASTIC PROCESSES

This paper considers an eigenvalue problem containing small stochastic processes. For every fixed is, we can use the Prufer substitution to prove the existence of the random solutions lambda(n) and u(n) in the meaning of large probability. These solutions can be expanded in epsilon regularly, and their correction terms can be obtained by solving some random linear differential equations.

Research on Production Process Control Method Combined Stochastic Process Algebra and Stochastic Petri Nets

A hierarchical closed-loop production control scheme integrating scheduling,control and performance evaluation is discussed.Firstly,the production process is divided into two main hierarchies:the lower level is the physical operation level and the upper one is the management level.Secondly,the schedule template for the management level and the activity template for the physical operation level are constructed separately,the tasks in the schedule have the ability to make partial decisions,and the per- formance parameters are introduced into activity template.Thirdly,the two levels use different model representations:stochastic process algebra for the management level whose output is the control commands and stochastic Petri net for the physical operation lev- el which is the execution of the control commands.Then,the integration of the two levels is the control commands mapping into the lower physical operations and the responses feeding back to the upper decision-making that are defined by some transition functions. Under the proposed scheme,the production process control of a flexible assembly is exemplified.It is concluded that the process con- trol model has partial ability to make decision on-line for uncertain and dynamic environments and facilitates reasoning about the be- haviors of the process control,and performance evaluation can be done online for real-time scheduling to ensure the global optimiza- tion.

A RESEARCH ON SEGMENTATION OF NONSTATIONARY STOCHASTIC PROCESS INTO PIECEWISE STATIONARY STOCHASTIC PROCESS

P. M. Djuric, etc.(1992) researched on the segmentation of nonstationary stochastic process into piecewise stationary stochastic process by Bayesian criterion ,and gave a dynamic equation about the number of segments, their boundaries and AR model orders for each segment, but did not give detailed solution for the equation. Because the solution for the equation is very complex, this paper investigates the solution, derives some recursive relations, simplifies the problem ,saves computation time and goes further into the segmentation of nonstationary stochastic process into piecewise stationary stochastic process.

Modeling and Analysis for Supply Chain Using Stochastic Process Algebra

In order to improve the influence of the uncertain and dynamic of node enterprise behavior on the performance of supply chain,the method based on stochastic process algebra for description,analysis,validation and evaluation of supply chain business process model is proposed.Firstly,the description of the uncertainty of node enterprise behavior is given using the extended Unified Modeling Language sequence diagram,and mapping rule is defined from the extended Unified Modeling Language sequence diagram to stochastic process algebra.Secondly,on the basis of the acquired stochastic process algebra model,the supply chain business process model is verified with Mobility Workbench.Finally,according to the operational semantics of stochastic process algebra,the continuous-time Markov chain,isomorphic with stochastic process algebra model,is built; and the system performance evaluation of transient status and stable status is respectively conducted in accordance with Markov transfer relations and the current state of system,obtaining the predicted performance value and average performance index value for a specific period of time.The simulation experiments show that the proposed method can accurately describe the stochastic behaviors of supply chain system and interactions among nodes,effectively verify the validity of the model,and objectively and exactly evaluate design of the supply chain.

Lebesgues-Stieltjes Integrals of Fuzzy Stochastic Processes with Respect to Finite Variation Processes

Let be a fuzzy stochastic process and be a real valued finite variation process. We define the Lebesgue-Stieltjes integral denoted by for each by using the selection method, which is direct, nature and different from the indirect definition appearing in some references. We shall show that this kind of integral is also measurable, continuous in time t and bounded a.s. under the Hausdorff metric.

Numerical Approximation of Fractal Dimension of Gaussian Stochastic Processes

In this paper we propose a numerical method to estimate the fractal dimension of stationary Gaussian stochastic processes using the random Euler numerical scheme and based on an analytical formulation of the fractal dimension for filtered stochastic signals. The discretization of continuous time processes through this random scheme allows us to find, numerically, the expected value, variance and correlation functions at any point of time. This alternative method for estimating the fractal dimension is easy to implement and requires no sophisticated routines. We use simulated data sets for stationary processes of the type Random Ornstein Uhlenbeck to graphically illustrate the results and compare them with those obtained whit the box counting theorem.

Understanding Molecular Dynamics with Stochastic Processes via Real or Virtual Dynamics

Molecular dynamics with the stochastic process provides a convenient way to compute structural and thermodynamic properties of chemical, biological, and materials systems. It is demonstrated that the virtual dynamics case that we proposed for the Langevin equation [J. Chem. Phys. 147, 184104 （2017）] in principle exists in other types of stochastic thermostats as well. The recommended ＂middle＂ scheme [J. Chem. Phys. 147, 034109 （2017）] of the Andersen thermostat is investigated as an example. As shown by both analytic and numerical results, while the real and virtual dynamics cases approach the same plateau of the characteristic correlation time in the high collision frequency limit, the accuracy and efficiency of sampling are relatively insensitive to the value of the collision frequency in a broad range. After we compare the behaviors of the Andersen thermostat to those of Langevin dynamics, a heuristic schematic representation thermostatting processes with molecular is proposed for understanding efficient stochastic dynamics.

Study of Volatility Stochastic Processes in the Context of Solvency Forecasting for Sri Lankan Life Insurers

The main business of Life Insurers is Long Term contractual obligations with a typical lifetime of 20 - 40 years. Therefore, the Solvency metric is defined by the adequacy of capital to service the cash flow requirements arising from the said obligations. The main component inducing volatility in Capital is market sensitive Assets, such as Bonds and Equity. Bond and Equity prices in Sri Lanka are highly sensitive to macro-economic elements such as investor sentiment, political stability, policy environment, economic growth, fiscal stimulus, utility environment and in the case of Equity, societal sentiment on certain companies and industries. Therefore, if an entity is to accurately forecast the impact on solvency through asset valuation, the impact of macro-economic variables on asset pricing must be modelled mathematically. This paper explores mathematical, actuarial and statistical concepts such as Brownian motion, Markov Processes, Derivation and Integration as well as Probability theorems such as the Probability Density Function in determining the optimum mathematical model which depicts the accurate relationship between macro-economic variables and asset pricing.

Importance Sampling Strategy for Oscillatory Stochastic Processes

This paper contributes to the structural reliability problem by presenting a novel approach that enables for identification of stochastic oscillatory processes as a critical input for given mechanical models. Identification development follows a transparent image processing paradigm completely independent of state-of-the-art structural dynamics, aiming at delivering a simple and wide purpose method. Validation of the proposed importance sampling strategy is based on multi-scale clusters of realizations of digitally generated non-stationary stochastic processes. Good agreement with the reference pure Monte Carlo results indicates a significant potential in reducing the computational task of first passage probabilities estimation, an important feature in the field of e.g., probabilistic seismic design or risk assessment generally.

GPS time-series and its response to M_S=8.1 Kunlunshan earthquake

In this paper, observation data in 25 GPS reference stations of China have been analyzed by calculating GPS position coordinate time-series with GIPSY. Result shows there is an obvious trend variation in such time-series. The trend variations of time series along the longitude and latitude coordinate reflect the motion of each position in the global-plate, in which the trend variation in the vertical direction reveals some large-scale construction information or reflects the local movement around the positions. The analysis also shows that such time-series have a variation cycle of nearly 1.02 a, but the reason still remains to be further studied. At the end of this paper, response of the time-series to MS=8.1 Kunlunshan earthquake was analyzed, and the seismogenic process of MS=8.1 Kunlunshan earthquake, according to the time proceeding and the feature of anomaly, was divided into 3 phases-changes in blocks with forces, strain accumulation, quick accumulation and slow release of energy. At the initial stage of seismogenic process of MS=8.1 earthquake and at the imminent earthquake, coseismic process as well as during the post earthquake recovery, anomaly in vertical direction is always in a majority. The anomalous movement in vertical direction at the initial stage resulted in a blocking between faults, while at the middle stage of seismogenic process, the differential movement between blocks are in a majority, which is the major reason causing energy accumulating at the blocking stage of faults.

Nitrogen deposition mediates more stochastic processes in structuring plant community than soil microbial community in the Eurasian steppe

Anthropogenic environmental changes may affect community assembly through mediating both deterministic(e.g.,competitive exclusion and environmental filtering)and stochastic processes(e.g.,birth/death and dispersal/colonization).It is traditionally thought that environmental changes have a larger mediation effect on stochastic processes in structuring soil microbial community than aboveground plant community;however,this hypothesis remains largely untested.Here we report an unexpected pattern that nitrogen(N)deposition has a larger mediation effect on stochastic processes in structuring plant community than soil microbial community(those<2 mm in diameter,including archaea,bacteria,fungi,and protists)in the Eurasian steppe.We performed a ten-year nitrogen deposition experiment in a semiarid grassland ecosystem in Inner Mongolia,manipulating nine rates(0–50 g N m^(-2)per year)at two frequencies(nitrogen added twice or 12 times per year)under two grassland management strategies(fencing or mowing).We separated the compositional variation of plant and soil microbial communities caused by each treatment into the deterministic and stochastic components with a recently-developed method.As nitrogen addition rate increased,the relative importance of stochastic component of plant community first increased and then decreased,while that of soil microbial community first decreased and then increased.On the whole,the relative importance of stochastic component was significantly larger in plant community(0.552±0.035;mean±standard error)than in microbial community(0.427±0.035).Consistently,the proportion of compositional variation explained by the deterministic soil and community indices was smaller for plant community(0.172–0.186)than microbial community(0.240–0.767).Meanwhile,as nitrogen addition rate increased,the linkage between plant and microbial community composition first became weaker and then became stronger.The larger stochasticity in plant community relative to microbial community assembly suggested that more stochastic strategies(e.g.,seeds addition)should be adopted to maintain above-than below-ground biodiversity under the pressure of nitrogen deposition.

Diffusionmodels for time-series applications: a survey

Diffusion models, a family of generative models based on deep learning, have become increasinglyprominent in cutting-edge machine learning research. With distinguished performance in generating samples thatresemble the observed data, diffusion models are widely used in image, video, and text synthesis nowadays. Inrecent years, the concept of diffusion has been extended to time-series applications, and many powerful models havebeen developed. Considering the deficiency of a methodical summary and discourse on these models, we providethis survey as an elementary resource for new researchers in this area and to provide inspiration to motivate futureresearch. For better understanding, we include an introduction about the basics of diffusion models. Except forthis, we primarily focus on diffusion-based methods for time-series forecasting, imputation, and generation, andpresent them, separately, in three individual sections. We also compare different methods for the same applicationand highlight their connections if applicable. Finally, we conclude with the common limitation of diffusion-basedmethods and highlight potential future research directions.

Stochastic Analysis of Interconnect Delay in the Presence of Process Variations

Process variations can reduce the accuracy in estimation of interconnect performance. This work presents a process variation based stochastic model and proposes an effective analytical method to estimate interconnect delay. The technique decouples the stochastic interconnect segments by an improved decoupling method. Combined with a polynomial chaos expression （PCE）, this paper applies the stochastic Galerkin method （SGM） to analyze the system response. A finite representation of interconnect delay is then obtained with the complex approximation method and the bisection method. Results from the analysis match well with those from SPICE. Moreover, the method shows good computational efficiency, as the running time is much less than the SPICE simulation＇s.

Approximation of weak sense stationary stochastic processes from local averages

We show that a weak sense stationary stochastic process can be approximated by local averages. Explicit error bounds are given. Our result improves an early one from Splettst?sser.

A novel imprecise stochastic process model for time-variant or dynamic uncertainty quantification

This paper proposes a novel model named as “imprecise stochastic process model” to handle the dynamic uncertainty with insufficient sample information in real-world problems. In the imprecise stochastic process model, the imprecise probabilistic model rather than a precise probability distribution function is employed to characterize the uncertainty at each time point for a time-variant parameter, which provides an effective tool for problems with limited experimental samples. The linear correlation between variables at different time points for imprecise stochastic processes is described by defining the auto-correlation coefficient function and the crosscorrelation coefficient function. For the convenience of analysis, this paper gives the definition of the P-box-based imprecise stochastic process and categorizes it into two classes: parameterized and non-parameterized P-box-based imprecise stochastic processes. Besides, a time-variant reliability analysis approach is developed based on the P-box-based imprecise stochastic process model,through which the interval of dynamic reliability for a structure under uncertain dynamic excitations or time-variant factors can be obtained. Finally, the effectiveness of the proposed method is verified by investigating three numerical examples.

An Extended Birth-Death Processes with Catastrophes——Stochastically Monotone, Feller and Symmetric Properties

A new structure with the special property that catastrophes is imposed to ordinary Birth_Death processes is considered. The necessary and sufficient conditions of stochastically monotone, Feller and symmetric properties for the extended birth_death processes with catastrophes are obtained.

A Novel Formal Analysis Method of Network Survivability Based on Stochastic Process Algebra

Stochastic process algebras have been proposed as compositional specification formalisms for performance models. A formal analysis method of survivable network was proposed based on stochastic process algebra, which incorporates formal modeling into performance analysis perfectly, and then various performance parameters of survivable network can be simultaneously obtained after formal modeling. The formal description with process expression to the survivable network system was carried out based on the simply introduced syntax and operational semantics of stochastic process algebra. Then PEPA workbench tool was used to obtain the probability of system’s steady state availability and transient state availability. Simulation experiments show the effectiveness and feasibility of the developed method.