Stochastic process

About: Stochastic process is a research topic. Over the lifetime, 31227 publications have been published within this topic receiving 898736 citations. The topic is also known as: random process & stochastic processes.

Large Deviations and Ensembles of Trajectories in Stochastic Models

Abstract: We consider ensembles of trajectories associated with large deviations of time-integrated quantities in stochastic models. Motivated by proposals that these ensembles are relevant for physical processes such as shearing and glassy relaxation, we show how they can be generated directly using auxiliary stochastic processes. We illustrate our results using the GlauberIsing chain, for which biased ensembles of trajectories can exhibit ferromagnetic ordering. We discuss the relation between such biased ensembles and quantum phase transitions.

A stochastic process approach to the analysis of temporal dynamics in transportation networks

Abstract: Equilibrium analyses of transportation networks are by their nature “static,” with equilibrium configuration defined as “fixed” or “autoreflexive” points, i.e. flow patterns reproducing themselves on the basis of the assumptions made on users' behavior once reached by the system. In this paper it is argued that no transportation system remains in the same state over successive periods because of the action of several causes (e.g. temporal fluctuation of level and composition of demand, users' choices, and travel costs). This implies that the sequence of states occupied by the system over successive epochs or times of similar characteristics (e.g. am peak hour of working days) is the realization of a stochastic process, the type of which depends on, among other things, the choice mechanism followed by travelers. Stationarity of the stochastic process within fixed potential demand and network structures is considered to be a desirable property because it allows a flow pattern distribution to be associated to each demand-network system independently of its starting configuration and elapsed time. Furthermore, this stationarity makes it possible to define expected path and link flows and compare them with those of stochastic user equilibrium (SUE). In this paper rather general sufficient conditions for the process stationarity are given, essentially calling for a “stable” choice mechanism of potential users. In the following a particular model of temporal dynamics (STODYN), based upon a number of simplifying assumptions on users' behavior common to most assignment models, is described. Exact and approximate relationships between STODYN steady-state expected flows and SUE average flows are also analyzed both in the case of unique and multiple equilibria. The possible use of STODYN as an assignment model giving unique average flows along with their variances and covariances is then discussed. The model takes into account stochastic fluctuations of demand and can be easily extended to other “dimensions” such as distribution and modal choice. Some results of an empirical analysis comparing STODYN average flows with SUE and observed flows on two urban car networks are also reported.

Particle RRT for Path Planning with Uncertainty

Abstract: This paper describes a new extension to the rapidly-exploring random tree (RRT) path planning algorithm. The particle RRT algorithm explicitly considers uncertainty in its domain, similar to the operation of a particle filter. Each extension to the search tree is treated as a stochastic process and is simulated multiple times. The behavior of the robot can be characterized based on the specified uncertainty in the environment, and guarantees can be made as to the performance under this uncertainty. Extensions to the search tree, and therefore entire paths, may be chosen based on the expected probability of successful execution. The benefit of this algorithm is demonstrated in the simulation of a rover operating in rough terrain with unknown coefficients of friction

A statistical approach to learning and generalization in layered neural networks

Abstract: A general statistical description of the problem of learning from examples is presented. Learning in layered networks is posed as a search in the network parameter space for a network that minimizes an additive error function of a statistically independent examples. By imposing the equivalence of the minimum error and the maximum likelihood criteria for training the network, the Gibbs distribution on the ensemble of networks with a fixed architecture is derived. The probability of correct prediction of a novel example can be expressed using the ensemble, serving as a measure to the network's generalization ability. The entropy of the prediction distribution is shown to be a consistent measure of the network's performance. The proposed formalism is applied to the problems of selecting an optimal architecture and the prediction of learning curves. >