Showing papers on "Stochastic process published in 2009"

Level sets and extrema of random processes and fields

Abstract: Introduction. Reading diagram. Chapter 1: Classical results on the regularity of the paths. 1. Kolmogorov's Extension Theorem. 2. Reminder on the Normal Distribution. 3. 0-1 law for Gaussian processes. 4. Regularity of the paths. Exercises. Chapter 2: Basic Inequalities for Gaussian Processes. 1. Slepian type inequalities. 2. Ehrhard's inequality. 3. Gaussian isoperimetric inequality. 4. Inequalities for the tails of the distribution of the supremum. 5. Dudley's inequality. Exercises. Chapter 3: Crossings and Rice formulas for 1-dimensional parameter processes. 1. Rice Formulas. 2. Variants and Examples. Exercises. Chapter 4: Some Statistical Applications. 1. Elementary bounds for P{M > u}. 2. More detailed computation of the first two moments. 3. Maximum of the absolute value. 4. Application to quantitative gene detection. 5. Mixtures of Gaussian distributions. Exercises. Chapter 5: The Rice Series. 1. The Rice Series. 2. Computation of Moments. 3. Numerical aspects of Rice Series. 4. Processes with Continuous Paths. Chapter 6: Rice formulas for random fields. 1. Random fields from Rd to Rd. 2. Random fields from Rd to Rd!, d> d!. Exercises. Chapter 7: Regularity of the Distribution of the Maximum. 1. The implicit formula for the density of the maximum. 2. One parameter processes. 3. Continuity of the density of the maximum of random fields. Exercises. Chapter 8: The tail of the distribution of the maximum. 1. One-dimensional parameter: asymptotic behavior of the derivatives of FM. 2. An Application to Unbounded Processes. 3. A general bound for pM. 4. Computing p(x) for stationary isotropic Gaussian fields. 5. Asymptotics as x! +". 6. Examples. Exercises. Chapter 9: The record method. 1. Smooth processes with one dimensional parameter. 2. Non-smooth Gaussian processes. 3. Two-parameter Gaussian processes. Exercises. Chapter 10: Asymptotic methods for infinite time horizon. 1. Poisson character of "high" up-crossings. 2. Central limit theorem for non-linear functionals. Exercises. Chapter 11: Geometric characteristics of random sea-waves. 1. Gaussian model for infinitely deep sea. 2. Some geometric characteristics of waves. 3. Level curves, crests and velocities for space waves. 4. Real Data. 5. Generalizations of the Gaussian model. Exercises. Chapter 12: Systems of random equations. 1. The Shub-Smale model. 2. More general models. 3. Non-centered systems (smoothed analysis). 4. Systems having a law invariant under orthogonal transformations and translations. Chapter 13: Random fields and condition numbers of random matrices. 1. Condition numbers of non-Gaussian matrices. 2. Condition numbers of centered Gaussian matrices. 3. Non-centered Gaussian matrices. Notations. References.

Stochastic Dynamics of Structures

Abstract: In Stochastic Dynamics of Structures, Li and Chen present a unified view of the theory and techniques for stochastic dynamics analysis, prediction of reliability, and system control of structures within the innovative theoretical framework of physical stochastic systems. The authors outline the fundamental concepts of random variables, stochastic process and random field, and orthogonal expansion of random functions. Readers will gain insight into core concepts such as stochastic process models for typical dynamic excitations of structures, stochastic finite element, and random vibration analysis. Li and Chen also cover advanced topics, including the theory of and elaborate numerical methods for probability density evolution analysis of stochastic dynamical systems, reliabilitybased design, and performance control of structures.

Probability, Markov Chains, Queues, and Simulation: The Mathematical Basis of Performance Modeling

Abstract: Probability, Markov Chains, Queues, and Simulation provides a modern and authoritative treatment of the mathematical processes that underlie performance modeling. The detailed explanations of mathematical derivations and numerous illustrative examples make this textbook readily accessible to graduate and advanced undergraduate students taking courses in which stochastic processes play a fundamental role. The textbook is relevant to a wide variety of fields, including computer science, engineering, operations research, statistics, and mathematics. The textbook looks at the fundamentals of probability theory, from the basic concepts of set-based probability, through probability distributions, to bounds, limit theorems, and the laws of large numbers. Discrete and continuous-time Markov chains are analyzed from a theoretical and computational point of view. Topics include the Chapman-Kolmogorov equations; irreducibility; the potential, fundamental, and reachability matrices; random walk problems; reversibility; renewal processes; and the numerical computation of stationary and transient distributions. The M/M/1 queue and its extensions to more general birth-death processes are analyzed in detail, as are queues with phase-type arrival and service processes. The M/G/1 and G/M/1 queues are solved using embedded Markov chains; the busy period, residual service time, and priority scheduling are treated. Open and closed queueing networks are analyzed. The final part of the book addresses the mathematical basis of simulation. Each chapter of the textbook concludes with an extensive set of exercises. An instructor's solution manual, in which all exercises are completely worked out, is also available (to professors only).Numerous examples illuminate the mathematical theories Carefully detailed explanations of mathematical derivations guarantee a valuable pedagogical approach Each chapter concludes with an extensive set of exercises Professors: A supplementary Solutions Manual is available for this book. It is restricted to teachers using the text in courses. For information on how to obtain a copy, refer to: http://press.princeton.edu/class_use/solutions.html

Interference and Outage in Clustered Wireless Ad Hoc Networks

Abstract: In the analysis of large random wireless networks, the underlying node distribution is almost ubiquitously assumed to be the homogeneous Poisson point process. In this paper, the node locations are assumed to form a Poisson cluster process on the plane. We derive the distributional properties of the interference and provide upper and lower bounds for its distribution. We consider the probability of successful transmission in an interference-limited channel when fading is modeled as Rayleigh. We provide a numerically integrable expression for the outage probability and closed-form upper and lower bounds. We show that when the transmitter-receiver distance is large, the success probability is greater than that of a Poisson arrangement. These results characterize the performance of the system under geographical or MAC-induced clustering. We obtain the maximum intensity of transmitting nodes for a given outage constraint, i.e., the transmission capacity (of this spatial arrangement) and show that it is equal to that of a Poisson arrangement of nodes. For the analysis, techniques from stochastic geometry are used, in particular the probability generating functional of Poisson cluster processes, the Palm characterization of Poisson cluster processes, and the Campbell-Mecke theorem.

Non-parametric Threshold Estimation for Models with Stochastic Diffusion Coefficient and Jumps

Abstract: We consider a stochastic process driven by diffusions and jumps Given a discrete record of observations, we devise a technique for identifying the times when jumps larger than a suitably defined threshold occurred This allows us to determine a consistent non-parametric estimator of the integrated volatility when the infinite activity jump component is Levy Jump size estimation and central limit results are proved in the case of finite activity jumps Some simulations illustrate the applicability of the methodology in finite samples and its superiority on the multipower variations especially when it is not possible to use high frequency data

Elucidating the origin of anomalous diffusion in crowded fluids.

Abstract: Anomalous diffusion in crowded fluids, e.g., in the cytoplasm of living cells, is a frequent phenomenon. So far, however, the associated stochastic process, i.e., the propagator of the random walk, has not been uncovered. Here we show by means of fluorescence correlation spectroscopy and simulations that the properties of crowding-induced subdiffusion are consistent with the predictions for fractional Brownian motion or obstructed (percolationlike) diffusion, both of which have stationary increments. In contrast, our experimental results cannot be explained by a continuous time random walk with its distinct non-Gaussian propagator.

Quantifying cross-correlations using local and global detrending approaches

Abstract: In order to quantify the long-range cross-correlations between two time series qualitatively, we introduce a new cross-correlations test QCC(m), where m is the number of degrees of freedom. If there are no cross-correlations between two time series, the cross-correlation test agrees well with the χ2(m) distribution. If the cross-correlations test exceeds the critical value of the χ2(m) distribution, then we say that the cross-correlations are significant. We show that if a Fourier phase-randomization procedure is carried out on a power-law cross-correlated time series, the cross-correlations test is substantially reduced compared to the case before Fourier phase randomization. We also study the effect of periodic trends on systems with power-law cross-correlations. We find that periodic trends can severely affect the quantitative analysis of long-range correlations, leading to crossovers and other spurious deviations from power laws, implying both local and global detrending approaches should be applied to properly uncover long-range power-law auto-correlations and cross-correlations in the random part of the underlying stochastic process.

Coordination and Consensus of Networked Agents with Noisy Measurements: Stochastic Algorithms and Asymptotic Behavior

Abstract: This paper considers the coordination and consensus of networked agents where each agent has noisy measurements of its neighbors' states. For consensus seeking, we propose stochastic approximation-type algorithms with a decreasing step size, and introduce the notions of mean square and strong consensus. Although the decreasing step size reduces the detrimental effect of the noise, it also reduces the ability of the algorithm to drive the individual states towards each other. The key technique is to ensure a trade-off for the decreasing rate of the step size. By following this strategy, we first develop a stochastic double array analysis in a two-agent model, which leads to both mean square and strong consensus, and extend the analysis to a class of well-studied symmetric models. Subsequently, we consider a general network topology, and introduce stochastic Lyapunov functions together with the so-called direction of invariance to establish mean square consensus. Finally, we apply the stochastic Lyapunov analysis to a leader following scenario.

Using Copulas for Modeling Stochastic Dependence in Power System Uncertainty Analysis

Abstract: The increasing penetration of renewable generation in power systems necessitates the modeling of this stochastic system infeed in operation and planning studies. The system analysis leads to multivariate uncertainty analysis problems, involving non-Normal correlated random variables. In this context, the modeling of stochastic dependence is paramount for obtaining accurate results; it corresponds to the concurrent behavior of the random variables, having a major impact to the aggregate uncertainty (in problems where the random variables correspond to spatially spread stochastic infeeds) or their evolution in time (in problems where the random variables correspond to infeeds over specific time-periods). In order to investigate, measure and model stochastic dependence, one should transform all different random variables to a common domain, the rank/uniform domain, by applying the cumulative distribution function transformation. In this domain, special functions, copulae, can be used for modeling dependence. In this contribution the basic theory concerning the use of these functions for dependence modeling is presented and focus is given on a basic function, the Normal copula. The case study shows the application of the technique for the study of the large-scale integration of wind power in the Netherlands.

A Stochastic Model for the Optimal Operation of a Wind-Thermal Power System

Abstract: This paper presents a stochastic cost model and a solution technique for optimal scheduling of the generators in a wind integrated power system considering the demand and wind generation uncertainties. The proposed robust unit commitment solution methodology will help the power system operators in optimal day-ahead planning even with indeterminate information about the wind generation. A particle swarm optimization based scenario generation and reduction algorithm is used for modeling the uncertainties. The stochastic unit commitment problem is solved using a new parameter free self adaptive particle swarm optimization algorithm. The numerical results indicate the low risk involved in day-ahead power system planning when the stochastic model is used instead of the deterministic model.

Distributed Sensor Localization in Random Environments Using Minimal Number of Anchor Nodes

Abstract: The paper introduces DILOC, a distributed, iterative algorithm to locate M sensors (with unknown locations) in Rm, m ges 1, with respect to a minimal number of m + 1 anchors with known locations. The sensors and anchors, nodes in the network, exchange data with their neighbors only; no centralized data processing or communication occurs, nor is there a centralized fusion center to compute the sensors' locations. DILOC uses the barycentric coordinates of a node with respect to its neighbors; these coordinates are computed using the Cayley-Menger determinants, i.e., the determinants of matrices of internode distances. We show convergence of DILOC by associating with it an absorbing Markov chain whose absorbing states are the states of the anchors. We introduce a stochastic approximation version extending DILOC to random environments, i.e., when the communications among nodes is noisy, the communication links among neighbors may fail at random times, and the internodes distances are subject to errors. We show a.s. convergence of the modified DILOC and characterize the error between the true values of the sensors' locations and their final estimates given by DILOC. Numerical studies illustrate DILOC under a variety of deterministic and random operating conditions.

The stochastic gravitational wave background from turbulence and magnetic fields generated by a first-order phase transition

Abstract: We analytically derive the spectrum of gravitational waves due to magneto-hydrodynamical turbulence generated by bubble collisions in a first-order phase transition. In contrast to previous studies, we take into account the fact that turbulence and magnetic fields act as sources of gravitational waves for many Hubble times after the phase transition is completed. This modifies the gravitational wave spectrum at large scales. We also model the initial stirring phase preceding the Kolmogorov cascade, while earlier works assume that the Kolmogorov spectrum sets in instantaneously. The continuity in time of the source is relevant for a correct determination of the peak position of the gravitational wave spectrum. We discuss how the results depend on assumptions about the unequal-time correlation of the source and motivate a realistic choice for it. Our treatment gives a similar peak frequency as previous analyses but the amplitude of the signal is reduced due to the use of a more realistic power spectrum for the magneto-hydrodynamical turbulence. For a strongly first-order electroweak phase transition, the signal is observable with the space interferometer LISA.

Shortest path problem considering on-time arrival probability

Abstract: This paper studies the problem of finding a priori shortest paths to guarantee a given likelihood of arriving on-time in a stochastic network. Such “reliable” paths help travelers better plan their trips to prepare for the risk of running late in the face of stochastic travel times. Optimal solutions to the problem can be obtained from local-reliable paths, which are a set of non-dominated paths under first-order stochastic dominance. We show that Bellman’s principle of optimality can be applied to construct local-reliable paths. Acyclicity of local-reliable paths is established and used for proving finite convergence of solution procedures. The connection between the a priori path problem and the corresponding adaptive routing problem is also revealed. A label-correcting algorithm is proposed and its complexity is analyzed. A pseudo-polynomial approximation is proposed based on extreme-dominance. An extension that allows travel time distribution functions to vary over time is also discussed. We show that the time-dependent problem is decomposable with respect to arrival times and therefore can be solved as easily as its static counterpart. Numerical results are provided using typical transportation networks.

Fractional brownian motion versus the continuous-time random walk: a simple test for subdiffusive dynamics.

Abstract: Fractional Brownian motion with Hurst index less then $1/2$ and continuous-time random walk with heavy tailed waiting times (and the corresponding fractional Fokker-Planck equation) are two different processes that lead to a subdiffusive behavior widespread in complex systems We propose a simple test, based on the analysis of the so-called $p$ variations, which allows distinguishing between the two models on the basis of one realization of the unknown process We apply the test to the data of Golding and Cox [Phys Rev Lett 96, 098102 (2006)], describing the motion of individual fluorescently labeled mRNA molecules inside live E coli cells It is found that the data does not follow heavy tailed continuous-time random walk The test shows that it is likely that fractional Brownian motion is the underlying process

Random trees : an interplay between combinatorics and probability

Abstract: Trees are a fundamental object in graph theory and combinatorics as well as a basic object for data structures and algorithms in computer science. During the last years research related to (random) trees has been constantly increasing and several asymptotic and probabilistic techniques have been developed in order to describe characteristics of interest of large trees in different settings. The aim of this book is to provide a thorough introduction into various aspects of trees in random settings and a systematic treatment of the involved mathematical techniques. It should serve as a reference book as well as a basis for future research. One major conceptual aspect is to bridge combinatorial and probabilistic methods that range from counting techniques (generating functions, bijections) over asymptotic methods (saddle point techniques, singularity analysis) to various sophisticated techniques in asymptotic probability (martingales, convergence of stochastic processes, concentration inequalities).

The Stochastic Process π

Abstract: The principal aim of this chapter is to familiarize the reader with the fact that the conditional distribution of the signal can be viewed as a stochastic process with values in the space of probability measures. While it is true that this chapter sets the scene for the subsequent chapters, it can be skipped by those readers whose interests are biased towards the applied aspects of the subject. The gist of the chapter can be summarized by the following.

Isothermal-isobaric molecular dynamics using stochastic velocity rescaling.

Abstract: The authors present a new molecular dynamics algorithm for sampling the isothermal-isobaric ensemble. In this approach the velocities of all particles and volume degrees of freedom are rescaled by a properly chosen random factor. The technical aspects concerning the derivation of the integration scheme and the conservation laws are discussed in detail. The efficiency of the barostat is examined in Lennard-Jones solid and liquid near the triple point and compared to the deterministic Nose–Hoover and the stochastic Langevin methods. In particular, the dependence of the sampling efficiency on the choice of the thermostat and barostat relaxation times is systematically analyzed.

Scenario tree modeling for multistage stochastic programs

Abstract: An important issue for solving multistage stochastic programs consists in the approximate representation of the (multivariate) stochastic input process in the form of a scenario tree. In this paper, we develop (stability) theory-based heuristics for generating scenario trees out of an initial set of scenarios. They are based on forward or backward algorithms for tree generation consisting of recursive scenario reduction and bundling steps. Conditions are established implying closeness of optimal values of the original process and its tree approximation, respectively, by relying on a recent stability result in Heitsch, Romisch and Strugarek (SIAM J Optim 17:511–525, 2006) for multistage stochastic programs. Numerical experience is reported for constructing multivariate scenario trees in electricity portfolio management.

Marked point processes for crowd counting

Abstract: A Bayesian marked point process (MPP) model is developed to detect and count people in crowded scenes. The model couples a spatial stochastic process governing number and placement of individuals with a conditional mark process for selecting body shape. We automatically learn the mark (shape) process from training video by estimating a mixture of Bernoulli shape prototypes along with an extrinsic shape distribution describing the orientation and scaling of these shapes for any given image location. The reversible jump Markov Chain Monte Carlo framework is used to efficiently search for the maximum a posteriori configuration of shapes, leading to an estimate of the count, location and pose of each person in the scene. Quantitative results of crowd counting are presented for two publicly available datasets with known ground truth.

Basics of Applied Stochastic Processes

Abstract: Markov Chains.- Renewal and Regenerative Processes.- Poisson Processes.- Continuous-Time Markov Chains.- Brownian Motion.

Stochastic dynamics and non-equilibrium thermodynamics of a bistable chemical system: the Schlögl model revisited

Abstract: Schlogl's model is the canonical example of a chemical reaction system that exhibits bistability. Because the biological examples of bistability and switching behaviour are increasingly numerous, this paper presents an integrated deterministic, stochastic and thermodynamic analysis of the model. After a brief review of the deterministic and stochastic modelling frameworks, the concepts of chemical and mathematical detailed balances are discussed and non-equilibrium conditions are shown to be necessary for bistability. Thermodynamic quantities such as the flux, chemical potential and entropy production rate are defined and compared across the two models. In the bistable region, the stochastic model exhibits an exchange of the global stability between the two stable states under changes in the pump parameters and volume size. The stochastic entropy production rate shows a sharp transition that mirrors this exchange. A new hybrid model that includes continuous diffusion and discrete jumps is suggested to deal with the multiscale dynamics of the bistable system. Accurate approximations of the exponentially small eigenvalue associated with the time scale of this switching and the full time-dependent solution are calculated using Matlab. A breakdown of previously known asymptotic approximations on small volume scales is observed through comparison with these and Monte Carlo results. Finally, in the appendix section is an illustration of how the diffusion approximation of the chemical master equation can fail to represent correctly the mesoscopically interesting steady-state behaviour of the system.

Fault Detection for Fuzzy Systems With Intermittent Measurements

Abstract: This paper investigates the problem of fault detection for Takagi-Sugeno (T-S) fuzzy systems with intermittent measurements. The communication links between the plant and the fault detection filter are assumed to be imperfect (i.e., data packet dropouts occur intermittently, which appear typically in a network environment), and a stochastic variable satisfying the Bernoulli random binary distribution is utilized to model the unreliable communication links. The aim is to design a fuzzy fault detection filter such that, for all data missing conditions, the residual system is stochastically stable and preserves a guaranteed performance. The problem is solved through a basis-dependent Lyapunov function method, which is less conservative than the quadratic approach. The results are also extended to T--S fuzzy systems with time-varying parameter uncertainties. All the results are formulated in the form of linear matrix inequalities, which can be readily solved via standard numerical software. Two examples are provided to illustrate the usefulness and applicability of the developed theoretical results.

Stabilization of Systems With Probabilistic Interval Input Delays and Its Applications to Networked Control Systems

Abstract: Motivated by the study of a class of networked control systems, this correspondence paper is concerned with the design problem of stabilization controllers for linear systems with stochastic input delays Different from the common assumptions on time delays, it is assumed here that the probability distribution of the delay taking values in some intervals is known a priori By making full use of the information concerning the probability distribution of the delays, criteria for the stochastic stability and stabilization controller design are derived Traditionally, in the case that the variation range of the time delay is available, the maximum allowable bound of time delays can be calculated to ensure the stability of the time-delay system It is shown, via numerical examples, that such a maximum allowable bound could be made larger in the case that the probability distribution of the time delay is known

State Estimation for Coupled Uncertain Stochastic Networks With Missing Measurements and Time-Varying Delays: The Discrete-Time Case

Abstract: This paper is concerned with the problem of state estimation for a class of discrete-time coupled uncertain stochastic complex networks with missing measurements and time-varying delay. The parameter uncertainties are assumed to be norm-bounded and enter into both the network state and the network output. The stochastic Brownian motions affect not only the coupling term of the network but also the overall network dynamics. The nonlinear terms that satisfy the usual Lipschitz conditions exist in both the state and measurement equations. Through available output measurements described by a binary switching sequence that obeys a conditional probability distribution, we aim to design a state estimator to estimate the network states such that, for all admissible parameter uncertainties and time-varying delays, the dynamics of the estimation error is guaranteed to be globally exponentially stable in the mean square. By employing the Lyapunov functional method combined with the stochastic analysis approach, several delay-dependent criteria are established that ensure the existence of the desired estimator gains, and then the explicit expression of such estimator gains is characterized in terms of the solution to certain linear matrix inequalities (LMIs). Two numerical examples are exploited to illustrate the effectiveness of the proposed estimator design schemes.

On unbounded path-loss models: effects of singularity on wireless network performance

Abstract: This paper addresses the following question: how reliable is it to use the unbounded path-loss model G(d) = d-alpha, where alpha is the path-loss exponent, to model the decay of transmitted signal power in wireless networks? G(d) is a good approximation for the path-loss in wireless communications for large values of d but is not valid for small values of d due to the singularity at 0. This model is often used along with a random uniform node distribution, even though in a group of uniformly distributed nodes some may be arbitrarily close to one another. The unbounded path-loss model is compared to a more realistic bounded path-loss model, and it is shown that the effect of the singularity on the total network interference level is significant and cannot be disregarded when nodes are uniformly distributed. A phase transition phenomenon occurring in the interference behavior is analyzed in detail. Several performance metrics are also examined by using the computed interference distributions. In particular, the effects of the singularity at 0 on bit error rate, packet success probability and wireless channel capacity are analyzed.

Using copulas for modeling stochastic dependence in power system uncertainty analysis

Abstract: The increasing penetration of renewable generation in power systems necessitates the modeling of this stochastic system infeed in operation and planning studies. The system analysis leads to multivariate uncertainty analysis problems, involving non-Normal correlated random variables. In this context, the modeling of stochastic dependence is paramount for obtaining accurate results; it corresponds to the concurrent behavior of the random variables, having a major impact to the aggregate uncertainty (in problems where the random variables correspond to spatially spread stochastic infeeds) or their evolution in time (in problems where the random variables correspond to infeeds over specific time-periods).