Theory Of Stochastic Processes Research Articles

FIXED POINT THEOREM FOR AN INFINITE TOEPLITZ MATRIX

AbstractFor an infinite Toeplitz matrix T with nonnegative real entries we find the conditions under which the equation $\boldsymbol {x}=T\boldsymbol {x}$ , where $\boldsymbol {x}$ is an infinite vector column, has a nontrivial bounded positive solution. The problem studied in this paper is associated with the asymptotic behaviour of convolution-type recurrence relations and can be applied to problems arising in the theory of stochastic processes and other areas.

Comfort evaluation of bridge based on stochastic process theory

Vehicle or pedestrian loads can cause bridge vibration. The pedestrian load is essentially a stochastic process, and the vibration response of pedestrian bridges is also a stochastic process. In order to evaluate the human comfort of a pedestrian bridge comprehensively, this paper presents a comfort evaluation method based on stochastic process theory. The autocorrelation function of the displacement response of a pedestrian bridge under random pedestrian load is derived in this paper. On this basis, the root mean square (RMS) of acceleration is proposed as the comfort evaluation index. Taking an asymmetric single-tower cable-stayed bridge as an example, the modal analysis of the bridge is proceeded by finite element method (FEM). Then the acceleration time histories of four measuring points are obtained by fi eld experiment. FEM results and fi eld experiment results are used to analyse random vibration of the bridge. Through a fi eld experiment and random vibration analysis of the bridge, the acceleration peak value evaluation index and the RMS of acceleration value evaluation index are compared. The results show that the RMS of acceleration index is more comprehensive and reasonable because it considers the whole-time history of the vibration response. The RMS of acceleration index is based on stochastic process theory, so it is more rigorous in mathematical expression and clearer in physical concept. It provides a reference for the evaluation of pedestrian bridge comfort.

Quantile Regression under Local Misspecification

The frequentist model averaging (FMA) and the focus information criterion (FIC) under a local framework have been extensively studied in the likelihood and regression setting since the seminal work of Hjort and Claeskens in 2003. One inconvenience, however, of the existing works is that they usually require the involved criterion function to be twice differentiable which thus prevents a direct application to the case of quantile regression (QR). This as well as some other intrinsic merits of QR motivate us to study the FIC and FMA in a locally misspecified linear QR model. Specifically, we derive in this paper the explicit asymptotic risk expression for a general submodel-based QR estimator of a focus parameter. Then based on this asymptotic result, we develop the FIC and FMA in the current setting. Our theoretical development depends crucially on the convexity of the objective function, which makes possible to establish the asymptotics based on the existing convex stochastic process theory. Simulation studies are presented to illustrate the finite sample performance of the proposed method. The low birth weight data set is analyzed.

Dynamical Phase Transitions in Dissipative Quantum Dynamics with Quantum Optical Realization.

We study dynamical phase transitions (DPT) in the driven and damped Dicke model, realizable for example by a driven atomic ensemble collectively coupled to a damped cavity mode. These DPTs are characterized by nonanalyticities of certain observables, primarily the overlap of time evolved and initial state. Even though the dynamics is dissipative, this phenomenon occurs for a wide range of parameters and no fine-tuning is required. Focusing on the state of the "atoms" in the limit of a bad cavity, we are able to asymptotically evaluate an exact path integral representation of the relevant overlaps. The DPTs then arise by minimization of a certain action function, which is related to the large deviation theory of a classical stochastic process. Finally, we present a scheme which allows a measurement of the DPT in a cavity-QED setup.

Generalized Description of Intermittency in Turbulence via Stochastic Methods

We present a generalized picture of intermittency in turbulence that is based on the theory of stochastic processes. To this end, we rely on the experimentally and numerically verified finding by R. Friedrich and J. Peinke [Phys. Rev. Lett. 78, 863 (1997)] that allows for an interpretation of the turbulent energy cascade as a Markov process of velocity increments in scale. It is explicitly shown that phenomenological models of turbulence, which are characterized by scaling exponents ζn of velocity increment structure functions, can be reproduced by the Kramers–Moyal expansion of the velocity increment probability density function that is associated with a Markov process. We compare the different sets of Kramers–Moyal coefficients of each phenomenology and deduce that an accurate description of intermittency should take into account an infinite number of coefficients. This is demonstrated in more detail for the case of Burgers turbulence that exhibits pronounced intermittency effects. Moreover, the influence of nonlocality on Kramers–Moyal coefficients is investigated by direct numerical simulations of a generalized Burgers equation. Depending on the balance between nonlinearity and nonlocality, we encounter different intermittency behavior that ranges from self-similarity (purely nonlocal case) to intermittent behavior (intermediate case that agrees with Yakhot’s mean field theory [Phys. Rev. E 63 026307 (2001)]) to shock-like behavior (purely nonlinear Burgers case).

Shotgun Weddings in Control Engineering and Postwar Economics, 1940–72

During World War II there was a “shotgun wedding” forged in the processes of designing feedback mechanisms used to control gunfire targeting fast-moving enemy aircraft. Hendrik Bode perceived the shotgun marriage as a union of the classic regulator approach and long-distance telephone communications engineering. Norbert Weiner took the wedding description a step further by nesting communication engineering into a statistical mechanics framework in which the information engineer had to filter the useful signal from the useless noise. A main goal in designing weapon-fire control systems was to use negative feedback loops to ensure the stability of a system that was always in a transient mode. In the 1950s, control engineers such as Arnold Tustin, Bill Phillips, and Charles Holt offered their new analytic tools of block diagrams of systems with feedback loops, stability criteria, and physical analogues to model the economy, realize the promise of stability of employment and prices, and avoid another depression similar to that of the 1930s. By the early 1970s Paul Samuelson and the rational expectations theorists were primed to be receptive to a second, Cold War shotgun wedding in control engineering. Richard Bellman described it as a marriage of classical optimization theory and the probabilistic theory of stochastic processes.

Nonhomogeneus Generalisations of Poisson Process in the Modeling of Random Processes Related to Road Accidents

Abstract The stochastic processes theory provides concepts, and theorems, which allow to build the probabilistic models concerning accidents. “Counting process” can be applied for modelling the number of road, sea, and railway accidents in the given time intervals. A crucial role in construction of the models plays a Poisson process and its generalizations. The nonhomogeneous Poisson process, and the corresponding nonhomogeneous compound Poisson process are applied for modelling the road accidents number, and number of people injured and killed in Polish roads. To estimate model parameters were used data coming from the annual reports of the Polish police.

Joint probability analysis of marine environmental elements

Reasonably determining the fortification criterion is one of the important links in the design of breakwaters and flood walls, and it is directly related to the construction cost and protection efficiency of these engineering works. Wave height is one of the most important element of the marine environment determined by the breakwater and flood protection standards. Based on the stochastic process theory, this paper gives a mathematical description of the stochastic process that can reflect the deeper level of statistical characteristics regarding the wave heights. By regarding the number of typhoons in a certain sea area as a Poisson process, this paper establishes a new model for the joint probability analysis of marine environmental elements: the nested random composite distribution model, and it gives the display expression of the new model. We also take typhoon data, wave height, and water increase data under typhoon influence from Weizhou Hydrological Station in western Guangdong, China from 1990 to 2016 as samples, and use the Gumbel-Pareto distribution as the edge distribution to apply the nested random composite distribution model to typhoon-affected sea areas. The joint recurrence period under different time combinations is calculated, and the joint statistical characteristics of the same element or different elements in different periods are also given.

The evaluation of a weighted sum of Gauss hypergeometric functions and its connection with Galton–Watson processes

We evaluate a weighted sum of Gauss hypergeometric functions for certain ranges of the argument, weights and parameters. We establish the domain of absolute convergence of this series by determining the growth of the hypergeometric function for large summation index. We present an application to Galton–Watson branching processes arising in the theory of stochastic processes. We introduce a new class of positive integer-valued distributions with power tails.

Neuro-fuzzy based identification of Hammerstein OEAR systems

This paper considers the parameter identification of neuro-fuzzy based Hammerstein output error auto-regressive (OEAR) systems by combining multiple signal source separation principle and auxiliary model identification idea. The unmeasurable internal variable is replaced by the correlation function of input and output data, then correlation analysis method is adopted to identify the parameters of linear part. In order to solve the parameter identification of the nonlinear part and the noise model, this paper presents a recursive generalized least squares algorithm based on auxiliary model. The convergence analysis in stochastic process theory shows that the parameter estimation error converges to zero under the persistent excitation condition. Examples results indicate that the proposed algorithm has significant advantages of good recognition accuracy to noise disturbance.

برآورد احتمالاتی توزیع بار در واحدهای فرآوری مواد معدنی با روش زنجیره مارکف

در روش‌های مرسوم موازنه جرم، برای تعیین عیار فلز، درصد جامد و توزیع دانه‌بندی نمونه‌برداری در مسیرها (شاخه‌ها) انجام می‌شود و در نهایت خطاهای مربوط به آنها سرشکن می‌شود. در این روش‌ها، برای هر واحد فرآوری (تجهیزات کارخانه)، در خصوص مقدار بار و عیار یا محتوای فلز اطلاعات لحظه‌ای وجود ندارد. در این مقاله با بهره‌گیری از نظریه فرآیند‌های تصادفی، مقادیر کمی و کیفی بار ورودی به هر یک از واحدهای عملیاتی در مسیر فرآوری به شکل احتمالات در نظر گرفته و توزیع بار در واحدهای مذکور برآورد شده است. به این منظور، ابتدا کارخانه فرآوری به صورت یک گراف جهت‌دار و موزون مدل‌سازی شده است. در این گراف، هرگره بیانگر یک واحد عملیاتی و هر یال، نشان‌دهنده مسیر واقع بین دو واحد است. وزن هر یال معرف وزن مواد عبوری در مسیر از گره‌ای به گره دیگر بوده و در یک بازه زمانی مشخص به‌عنوان یک متغیر تصادفی منظور شده است. پس از مدل‌سازی فرآیند، مدل گرافی به‌دست آمده که در برگیرنده شاخص‌های تصادفی است با استفاده از نظریهزنجیرهای مارکف تحلیل شده است. با روش ارایه شده در این مقاله نه تنها توزیع بار، بلکه احتمال حضور هر ذره یا کانی، در هر یک از واحدهای عملیاتی کارخانه فرآوری نیز قابل پیش‌بینی است. همچنین در صورت توقف کارخانه، می‌توان تناژ موجود در هر یک از واحدها را تخمین زد و اقدامات لازم برای تخلیه آنها را فراهم کرد. در این تحقیق، خط فرآوری سنگ آهن چادرملو با روش بالا، مدل‌سازی و نتایج حاصل تحلیل و سپس با استفاده از شاخص‌های احتمالات گزارش شده است.

Research on the statistical characteristics of typhoon frequency

Extreme meteorological events are becoming more frequent as a consequence of global warming. Typhoon as one of the natural disasters has caused significant damages globally. Therefore, developing theoretical models to effectively predict typhoon probability is urgent. Based on the stochastic process theory, this paper studies the local statistics of typhoon frequency and suggests a stochastic process model that can describe the number of typhoons occurring locally in time domain. Mathematical evidence has been provided to prove that the number of typhoons during a certain period can be described by using a stochastic process model. Furthermore, the dependency of typhoon occurrences in different time periods and the probability distribution of typhoon occurrence intervals are explored. The suggested model is employed to discuss the probability of typhoon on Naozhou Island in South China Sea during May to September from year 2000–2016 subject to the absence of the event in May. The results reveal that the probability distribution of typhoon events predicted by the suggested model is more reliable in comparison with conventional approach as it considered the typhoon occurrence in time domain, which can provides useful information for predicting the probability of typhoons occurrence and intervals in engineering practice.

Evaluation of scenario reduction algorithms with nested distance

Multistage stochastic optimization is used to solve many real-life problems where decisions are taken at multiple times. Such problems need the representation of stochastic processes, which are usually approximated by scenario trees. In this article, we implement seven scenario reduction algorithms: three based on random extraction, named Random, and four based on specific distance measures, named Distance-based. Three of the latter are well known in literature while the fourth is a new approach, namely nodal clustering. We compare all the algorithms in terms of computational cost and information cost. The computational cost is measured by the time needed for the reduction, while the information cost is measured by the nested distance between the original and the reduced tree. Moreover, we also formulate and solve a multistage stochastic portfolio selection problem to measure the distance between the optimal solutions and between the optimal objective values of the original and the reduced tree.

Traffic accident prediction based on Markov chain cloud model

A dynamic prediction model based on Markov chain and cloud model is established to predict the volume of road traffic accidents which is under the guidance of stochastic process and cloud theory referring to the characters of the road traffic accidents and time series data. First of all, we establish the evaluation model based on cloud model, and get the relative error range of the observed value and the predicted value. Then we can use Markov chain to correct the relative error. The result shows that this method balances both randomicity and fuzziness and has higher accuracy. Therefore, it can be used for analysing the trend of road traffic accidents in different traffic conditions and provide evidence for safety early warning and corresponding accident prevention countermeasures formulation in relative road segments.

Short-term supply reliability assessment of a gas pipeline system under demand variations

It is the basic responsibility for pipeline companies to satisfy the gas demand of customers. A methodology is proposed to assess the short-term supply reliability of a gas pipeline system under demand variations in this paper. Based on the failure rate and repair rate of the pipe segments and compressor units obtained through historical operational data, the probability under different structural failure scenarios of a gas pipeline system is computed through the stochastic process and probability theory. Indices from aspects of adequacy, availability, and shortage rate are developed to reflect the supply reliability. To reflect the assessment results intuitively, the cumulative probability function is adopted to reflect the supply adequacy and availability, and the probability distribution is adopted to reflect the supply shortage rate during the assessment period. For all scenarios, supply reliability is calculated through batch transient hydraulic and thermal simulations of a gas pipeline system by Stoner Pipeline Simulator considering practical operational processes and legal constraints. The proposed methodology is performed based on the Trans-Asian gas pipeline A-B system, and the functions of cross-over valves on supply reliability of the Trans-Asian gas pipeline A-B system is studied.

Kolmogorov extension theorem for (quantum) causal modelling and general probabilistic theories

In classical physics, the Kolmogorov extension theorem lays the foundation for the theory of stochastic processes. It has been known for a long time that, in its original form, this theorem does not hold in quantum mechanics. More generally, it does not hold in any theory of stochastic processes -- classical, quantum or beyond -- that does not just describe passive observations, but allows for active interventions. Such processes form the basis of the study of causal modelling across the sciences, including in the quantum domain. To date, these frameworks have lacked a conceptual underpinning similar to that provided by Kolmogorov’s theorem for classical stochastic processes. We prove a generalized extension theorem that applies to all theories of stochastic processes, putting them on equally firm mathematical ground as their classical counterpart. Additionally, we show that quantum causal modelling and quantum stochastic processes are equivalent. This provides the correct framework for the description of experiments involving continuous control, which play a crucial role in the development of quantum technologies. Furthermore, we show that the original extension theorem follows from the generalized one in the correct limit, and elucidate how a comprehensive understanding of general stochastic processes allows one to unambiguously define the distinction between those that are classical and those that are quantum.

Vehicle Cooperative Network Model Based on Hypergraph in Vehicular Fog Computing.

In this paper, we propose an optimization framework of vehicular fog computing and a cooperation vehicular network model. We aim to improve the performance of vehicular fog computing and solve the problem that the data of the vehicle collaborative network is difficult to obtain. This paper applies the hypergraph theory to study the underlying structure, considering the social characteristics of the vehicles and vehicle communication. Since the vehicles join the network in accordance with the Poisson process law, the model is analyzed by using Poisson stochastic process and mean field theory. This paper uses MATLAB to simulate the evolution process of cooperative networks. The results show that the vehicle’s super-degree in vehicular fog computing has scale-free characteristics. Through this model, the vehicle cooperation situation can be analyzed, and the vehicle dynamics can be accurately predicted to further improve the performance of vehicular fog computing. The model can be transformed into some complex network models by adjusting the parameters. It has strong universality and has certain reference significance for the research on the related characteristics of VANETs and the theoretical research of the cooperative network.

Semi-parametric resampling with extremes

Nonparametric resampling methods such as Direct Sampling are powerful tools to simulate new datasets preserving important data features such as spatial patterns from observed datasets while using only minimal assumptions. However, such methods cannot generate extreme events beyond the observed range of data values. We here propose using tools from extreme value theory for stochastic processes to extrapolate observed data towards yet unobserved high quantiles. Original data are first enriched with new values in the tail region, and then classical resampling algorithms are applied to enriched data. In a first approach to enrichment that we label “naive resampling”, we generate an independent sample of the marginal distribution while keeping the rank order of the observed data. We point out inaccuracies of this approach around the most extreme values, and therefore develop a second approach that works for datasets with many replicates. It is based on the asymptotic representation of extreme events through two stochastically independent components: a magnitude variable, and a profile field describing spatial variation. To generate enriched data, we fix a target range of return levels of the magnitude variable, and we resample magnitudes constrained to this range. We then use the second approach to generate heatwave scenarios of yet unobserved magnitude over France, based on daily temperature reanalysis training data for the years 2010 to 2016.

Maximum likelihood identification of dual‐rate Hammerstein output‐error moving average system

This study discusses the parameter estimation of the Hammerstein output-error moving average system using the dual-rate sampled data. The polynomial transformation technique is used to obtain the identification model of the discussed dual-rate sampled systems. The stochastic gradient optimisation method is an effective optimisation method. Compared with the Newton optimisation, it only needs to calculate the first derivative during the optimisation and the amount of calculation is relatively small. It is a good choice to use the stochastic gradient algorithm for the identification of Hammerstein dual-rate model after using the polynomial transformation technique. In order to improve the convergence speed, a maximum likelihood forgetting factor stochastic gradient identification algorithm is proposed by combining the maximum likelihood principle and the gradient search method. The convergence of the algorithm is analysed by using the stochastic process theory. Furthermore, in order to improve the estimation accuracy of the identification algorithm, a maximum likelihood multi-innovation forgetting factor stochastic gradient algorithm is proposed by using the multi-innovation identification theory. The effectiveness of the proposed algorithms is illustrated by a numerical simulation example and a water tank system.

Reduction of random variables in the Stochastic Harmonic Function representation via spectrum-relative dependent random frequencies

Two significant developments pertaining to the application of the Stochastic Harmonic Function representation of stochastic processes are presented. Together, they allow for Gaussian records to be simulated within the bounds of the representation with the fewest number of random variables. Specifically, independent random frequencies that form a staple component of the Stochastic Harmonic Function are replaced by dependent random frequencies, along with a specific scheme for choosing frequency interval widths. Numerical examples demonstrating spectrum reconstruction accuracy and estimated PDF convergence to the Gaussian are presented to support the work.