Realization Of Stochastic Process Research Articles

Problem-Driven Scenario Generation for Stochastic Programming Problems: A Survey

Stochastic Programming is a powerful framework that addresses decision-making under uncertainties, which is a frequent occurrence in real-world problems. To effectively solve Stochastic Programming problems, scenario generation is one of the common practices that organizes realizations of stochastic processes with finite discrete distributions, which enables the use of mathematical programming models of the original problem. The quality of solutions is significantly influenced by the scenarios employed, necessitating a delicate balance between incorporating informative scenarios and preventing overfitting. Distributions-based scenario generation methodologies have been extensively studied over time, while a relatively recent concept of problem-driven scenario generation has emerged, aiming to incorporate the underlying problem’s structure during the scenario generation process. This survey explores recent literature on problem-driven scenario generation algorithms and methodologies. The investigation aims to identify circumstances under which this approach is effective and efficient. The work provides a comprehensive categorization of existing literature, supplemented by illustrative examples. Additionally, the survey examines potential applications and discusses avenues for its integration with machine learning technologies. By shedding light on the effectiveness of problem-driven scenario generation and its potential for synergistic integration with machine learning, this survey contributes to enhanced decision-making strategies in the context of uncertainties.

Beyond expected regularity of aggregate urban mobility: A case study of ridesourcing service

The regularity of urban mobility is a tacit understanding in public transportation planning. Many studies investigated regularity in the context of individual travel while fewer were found on aggregate mobility. This work proposes a generic measure of regularity to quantify the degree of repetition of a time or spatial data series. In addition, it discusses how the properties of a data series itself may dictate the regularity and adopts parametric bootstrapping to estimate the residual regularity that deviates from realizations of stochastic processes. The proposed measure can examine not only the periodicity of trip generation in locations but also the stationarity of travel flows. The experiments were conducted on a ridesourcing dataset, including more than six million trips in Chengdu, China. Both the temporal and spatial regularities were investigated at distinct time intervals. We discussed how the total travel volume and the peakedness of probability distribution could affect the regularity. It was found that regularity was positively associated with the total volume and the peakedness. We further examined the impact of the built environment on regularity and the effects of spatial and temporal scales. The results show both point-of-interest density and diversity and tourist attractions contribute to regularity. Regularity grows linearly as the spatial and temporal scales increase exponentially. Lastly, different specifications of regularities were compared, and the outcome was generally consistent across different similarity measures including coefficient of determination, Manhattan distance, Euclidean distance, cross-entropy and cosine similarity.

Optimal Stochastic Process Optimizer: A New Metaheuristic Algorithm With Adaptive Exploration-Exploitation Property

Metaheuristic algorithms are constructed to solve optimization problems, but they cannot solve all the problems with best solutions. This work proposes a novel self-adaptive metaheuristic optimization algorithm, named Optimal Stochastic Process Optimizer (OSPO), which can solve different kinds of optimization problems with promising performance. Specifically, OSPO regards the procedure of optimization as a realization of stochastic process, and with the help of Subjective Probability Distribution Function (SPDF) and Receding Sampling Strategy proposed in this paper, OSPO can control the exploration-exploitation property online by the adaptive modification of the parameters in SPDF. This adaptive exploration-exploitation property of OSPO contributes to dealing with different kinds of problems; thus, it makes OSPO have the potential to solve at least a vast majority of optimization problems. The proposed algorithm is first benchmarked on uni-modal, multi-modal and composite test functions both in low and high dimensions. The results are verified by comparative studies with seven well-performed metaheuristic algorithms. Then, 21 real-world optimization problems are used to further investigate the effectiveness of OSPO. The winners of CEC2020 Competition on Real-World Single Objective Constrained Optimization, SASS algorithm, sCMAgES algorithm, EnMODE algorithm and COLSHADE algorithm are used as four comparative algorithms in real-world optimization problems. The analysis of simulations demonstrates that OSPO is able to provide very competitive performance compared to the comparative meta-heuristics both in benchmark functions and in real-world optimization problems; thus, the potential of OSPO to solve at least a vast majority of optimization problems is verified. A corresponding MATLAB APP demo is available on https://github.com/JiahongXu123/OSPO-algorithm.git.

A Least-Squares Temporal Difference based method for solving resource allocation problems

Value function approximation has a central role in Approximate Dynamic Programming (ADP) to overcome the so-called curse of dimensionality associated to real stochastic processes. In this regard, we propose a novel Least-Squares Temporal Difference (LSTD) based method: the “Multi-trajectory Greedy LSTD” (MG-LSTD). It is an exploration-enhanced recursive LSTD algorithm with the policy improvement embedded within the LSTD iterations. It makes use of multi-trajectories Monte Carlo simulations in order to enhance the system state space exploration.This method is applied for solving resource allocation problems modeled via a constrained Stochastic Dynamic Programming (SDP) based framework. In particular, such problems are formulated as a set of parallel Birth–Death Processes (BDPs). Some operational scenarios are defined and solved to show the effectiveness of the proposed approach. Finally, we provide some experimental evidence on the MG-LSTD algorithm convergence properties in function of its key-parameters.

Time-variant reliability-based design optimization using sequential kriging modeling

This paper presents a sequential Kriging modeling approach (SKM) for time-variant reliability-based design optimization (tRBDO) involving stochastic processes. To handle the temporal uncertainty, time-variant limit state functions are transformed into time-independent domain by converting the stochastic processes and time parameter to random variables. Kriging surrogate models are then built and enhanced by a design-driven adaptive sampling scheme to accurately identify potential instantaneous failure events. By generating random realizations of stochastic processes, the time-variant probability of failure is evaluated by the surrogate models in Monte Carlo simulation (MCS). In tRBDO, the first-order score function is employed to estimate the sensitivity of time-variant reliability with respect to design variables. Three case studies are introduced to demonstrate the efficiency and accuracy of the proposed approach.

Simulation of Spatially Varying Non-Gaussian and Nonstationary Seismic Ground Motions by the Spectral Representation Method

AbstractSimulation of sample realizations of stochastic processes is the bedrock of the Monte Carlo method, and the accurate modeling of stochastic processes is crucial to determine realistic struc...

Experimental Study of Stochastic Processes, Determining their Characteristics and Conducting Real Experiments on the Operated Bridge

Abstract The paper describes the process of actual experiment for practical confirmation of the effectiveness of new technologies. New software is proposed for investigating a real stochastic process. Technology and software are designed for fast calculation, analysis of adequate estimates of statistical characteristics, confirming the effectiveness of suggested technology, with high accuracy.

Stabilization of Input-Disturbed Stochastic Port-Hamiltonian Systems Via Passivity

It is a universal phenomenon that the inputs of a real stochastic process are disturbed. However, this phenomenon is often ignored when stochastic port-Hamiltonian system (SPHS) is used for modeling, in which only the disturbance on the states is considered. For this reason, an extension of SPHS, named input-disturbed SPHS (Id-SPHS), is proposed to describe the inputs disturbance as well as the states one. The definition and some properties of Id-SPHS, including structure invariance and stochastic passivity, are presented one by one to construct the basic framework of Id-SPHS. Based on this framework, a passivity-based controller is further developed to stabilize Id-SPHS at its steady state. The control strategy is demonstrated in a thermodynamic heat conduction process with the results supporting strongly the proposed Id-SPHS theory.

A wavelet threshold denoising procedure for multimodel predictions: An application to economic time series

Noise‐affected economic time series, realizations of stochastic processes exhibiting complex and possibly nonlinear dynamics, are dealt with. This is often the case of time series found in economics, which notoriously suffer from problems such as low signal‐to‐noise ratios, asymmetric cycles and multiregimes patterns. In such a framework, even sophisticated statistical models might generate suboptimal predictions, whose quality can further deteriorate unless time consuming updating or deeper model revision procedures are carried out on a regular basis. However, when the models' outcomes are expected to be disseminated in timeliness manner (as in the case of Central Banks or national statistical offices), their modification might not be a viable solution, due to time constraints. On the other hand, if the application of simpler linear models usually entails relatively easier tuning‐up procedures, this would come at the expenses of the quality of the predictions yielded. A mixed, self‐tuning forecasting method is therefore proposed. This is an automatic, 2‐stage procedure, able to generate predictions by exploiting the denoising capabilities provided by the wavelet theory in conjunction with a compounded forecasting generator. Its out‐of‐sample performances are evaluated through an empirical study carried out on macroeconomic time series.

Erratum to: Analyzing the effect of dynamically assigned treatments using duration models, binary treatment models, and panel data models

Often, the moment of a treatment and the moment at which the outcome of interest occurs are realizations of stochastic processes with dependent unobserved determinants. Notably, both treatment and outcome are characterized by the moment they occur. In this paper, we compare different methods of inference of the treatment effect. We argue that the timing of the treatment relative to the outcome conveys useful information on the treatment effect, which is discarded in binary treatment frameworks.

Forecasting stochastic processes using singular spectrum analysis: Aspects of the theory and application

This paper presents theoretical results on the properties of forecasts obtained by using singular spectrum analysis to forecast time series that are realizations of stochastic processes. The mean squared forecast errors are derived under broad regularity conditions, and it is shown that, in practice, the forecasts obtained will converge to their population ensemble counterparts. The theoretical results are illustrated by examining the performances of singular spectrum analysis forecasts when applied to autoregressive processes and a random walk process. Simulation experiments suggest that the asymptotic properties developed are reflected in the behaviour of observed finite samples. Empirical applications using real world data sets indicate that forecasts based on singular spectrum analysis are competitive with other methods currently in vogue.

The Vibration Monitoring of the Gascompressor Units with Application of Hurst's Statistics

The gascompressor units work in the unsteady regime which depends on many internal and external factors. For the solution of problems of control, management and forecasting the statistical assessment of a condition of object in real time according to noisy observations of one realization of stochastic process is carried out. The traditional spectral analysis on the basis of Fast Fourier Transformation in this case is inefficient if time scale is much less than duration, which is subject to the analysis. Therefore for monitoring of a condition of the equipment it is offered to use Hurst's statistics by means of which it is possible to investigate quantitatively law of development of a vibration signal and unambiguously to analyze a condition of the gascompressor units. Calculation of economic efficiency from introduction of such technology of monitoring will provide decrease on average by 2.5% of a consumption of gas and specific costs of capital repairs of the gascompressor units at the expense of increase in a total operating time by 20%.

Stochastic Long-term Hydrothermal Scheduling with Parameter Uncertainty in Autoregressive Streamflow Models

The optimal scheduling of hydrothermal systems requires the representation of uncertainties in future streamflows to devise a cost-effective operations policy. Stochastic optimization has been widely used as a powerful tool to solve this problem but results will necessarily depend on the stochastic model used to generate future scenarios for streamflows. Periodic autoregressive (PAR) models have been widely used in this task. However, its parameters are typically unknown and must be estimated from historical data, incorporating a natural estimation error. Furthermore, the model is just a linear approximation of the real stochastic process. The consequence is that the operator will be uncertain about the correct linear model that should be used at each period. The objective of this work is to assess the impacts of incorporating the uncertainty of the parameters of the PAR models into a stochastic hydrothermal scheduling model. The proposed methodology is tested with case studies based on data from the Brazilian hydroelectric system. It is shown that when the uncertainty of the parameters is ignored, the policies given by the stochastic optimization tend to be too optimistic.

Eigenvalue Attraction

We prove that the complex conjugate (c.c.) eigenvalues of a smoothly varying real matrix attract (Eq. 15). We offer a dynamical perspective on the motion and interaction of the eigenvalues in the complex plane, derive their governing equations and discuss applications. C.c. pairs closest to the real axis, or those that are ill-conditioned, attract most strongly and can collide to become exactly real. As an application we consider random perturbations of a fixed matrix M. If M is Normal, the total expected force on any eigenvalue is shown to be only the attraction of its c.c. (Eq. 24) and when M is circulant the strength of interaction can be related to the power spectrum of white noise. We extend this by calculating the expected force (Eq. 41) for real stochastic processes with zero-mean and independent intervals. To quantify the dominance of the c.c. attraction, we calculate the variance of other forces. We apply the results to the Hatano-Nelson model and provide other numerical illustrations. It is our hope that the simple dynamical perspective herein might help better understanding of the aggregation and low density of the eigenvalues of real random matrices on and near the real line respectively. In the appendix we provide a Matlab code for plotting the trajectories of the eigenvalues.

Truth, models, model sets, AIC, and multimodel inference: A Bayesian perspective

Statistical inference begins with viewing data as realizations of stochastic processes. Mathematical models provide partial descriptions of these processes; inference is the process of using the data to obtain a more complete description of the stochastic processes. Wildlife and ecological scientists have become increasingly concerned with the conditional nature of model-based inference: what if the model is wrong? Over the last 2 decades, Akaike's Information Criterion (AIC) has been widely and increasingly used in wildlife statistics for 2 related purposes, first for model choice and second to quantify model uncertainty. We argue that for the second of these purposes, the Bayesian paradigm provides the natural framework for describing uncertainty associated with model choice and provides the most easily communicated basis for model weighting. Moreover, Bayesian arguments provide the sole justification for interpreting model weights (including AIC weights) as coherent (mathematically self consistent) model probabilities. This interpretation requires treating the model as an exact description of the data-generating mechanism. We discuss the implications of this assumption, and conclude that more emphasis is needed on model checking to provide confidence in the quality of inference. Published 2015. This article is a U.S. Government work and is in the public domain in the USA.

Models Associated with Extended Exponential Smoothing

We study an extended form of exponential smoothing which is more flexible than the original one: let be a real stochastic process, observed until time n, consider the probabilistic predictor of Xn + 1 defined aswhere the series is supposed to be convergent in mean square. We look for stochastic models such that X*n + 1 is the best linear predictor of Xn + 1, given Xt, t ⩽ n. We obtain various ARIMA models depending on (α, β). In this context we study estimation of (α, β) and give some indications concerning hypotheses testing. Finally, extension to functional stochastic processes is considered.

The identification of the distribution density in the realization of stochastic processes by the regularization method

The identification of the distribution density in the realization of stochastic processes by the regularization method

Identification of critical samples of stochastic processes towards feasible structural reliability applications

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. The proposed method is based on a graphical representation of such processes utilizing state of the art image processing and pattern recognition techniques, leading to a set of finite rules that consistently identifies those realizations of stochastic processes that would lead to a critical response of a given mechanical model. To examine the validity of the suggested method, large sets of realizations of artificial non-stationary processes were generated from known models, several criteria for critical response were formulated and the results were statistically evaluated. The promising results suggest important applications that would dramatically decrease computational costs e.g. in the field of probabilistic seismic design. Further examination may lead to a formulation of a new class of importance sampling techniques.

Bayesian nonparametric modeling for functional analysis of variance

Analysis of variance is a standard statistical modeling approach for comparing populations. The functional analysis setting envisions that mean functions are associated with the populations, customarily modeled using basis representations, and seeks to compare them. Here, we adopt the modeling approach of functions as realizations of stochastic processes. We extend the Gaussian process version to allow nonparametric specifications using Dirichlet process mixing. Several metrics are introduced for comparison of populations. Then we introduce a hierarchical Dirichlet process model which enables comparison of the population distributions, either directly or through functionals of interest using the foregoing metrics. The modeling is extended to allow us to switch the sampling scheme. There are still population level distributions but now we sample at levels of the functions, obtaining observations from potentially different individuals at different levels. We illustrate with both simulated data and a dataset of temperature versus depth measurements at different locations in the Atlantic Ocean.

Band-limited component estimation in time-limited economic series

This paper tackles the issue of economic time-series modeling from a joint time and frequency-domain standpoint, with the objective of estimating the latent trend-cycle component. Since time-series records are data strings over a finite time span, they read as samples of contiguous data drawn from realizations of stochastic processes aligned with the time arrow. This accounts for the interpretation of time series as time-limited signals. Economic time series (up to a disturbance term) result from latent components known as trend, cycle, and seasonality, whose generating stochastic processes are harmonizable on a finite average-power argument. In addition, since trend is associated with long-run regular movements, and cycle with medium-term economic fluctuation, both of these turn out to be band-limited components. Recognizing such a frequency-domain location permits a filter-based approach to component estimation. This is accomplished through a Toeplitz matrix operator with sinc functions as entries, mirroring the ideal low-pass filter impulse response. The notion of virtual transfer function is developed and its closed-form expression derived in order to evaluate the filter features. The paper is completed by applying this filter to quarterly data from Italian industrial production, thus shedding light on the performance of the estimation procedure.