Random Instants Research Articles

Stochastic resetting with refractory periods: pathway formulation and exact results

We look into the problem of stochastic resetting with refractory periods. The model dynamics comprises diffusive and motionless phases. The diffusive phase ends at random time instants, at which the system is reset to a given position—where the system remains at rest for a random time interval, termed the refractory period. A pathway formulation is introduced to derive exact analytical results for the relevant observables in a broad framework, with the resetting time and the refractory period following arbitrary distributions. For the paradigmatic case of Poissonian distributions of the resetting and refractory times, in general with different characteristic rates, closed-form expressions are obtained that successfully describe the relaxation to the steady state. Finally, we focus on the single-target search problem, in which the survival probability and the mean first passage time to the target can be exactly computed. Therein, we also discuss optimal strategies, which show a non-trivial dependence on the refractory period.

Tight-binding model subject to conditional resets at random times.

We investigate the dynamics of a quantum system subjected to a time-dependent and conditional resetting protocol. Namely, we ask what happens when the unitary evolution of the system is repeatedly interrupted at random time instants with an instantaneous reset to a specified set of reset configurations taking place with a probability that depends on the current configuration of the system at the instant of reset? Analyzing the protocol in the framework of the so-called tight-binding model describing the hopping of a quantum particle to nearest-neighbor sites in a one-dimensional open lattice, we obtain analytical results for the probability of finding the particle on the different sites of the lattice. We explore a variety of dynamical scenarios, including the one in which the resetting time intervals are sampled from an exponential as well as from a power-law distribution, and a setup that includes a Floquet-type Hamiltonian involving an external periodic forcing. Under exponential resetting, and in both the presence and absence of the external forcing, the system relaxes to a stationary state characterized by localization of the particle around the reset sites. The choice of the reset sites plays a defining role in dictating the relative probability of finding the particle at the reset sites as well as in determining the overall spatial profile of the site-occupation probability. Indeed, a simple choice can be engineered that makes the spatial profile highly asymmetric even when the bare dynamics does not involve the effect of any bias. Furthermore, analyzing the case of power-law resetting serves to demonstrate that the attainment of the stationary state in this quantum problem is not always evident and depends crucially on whether the distribution of reset time intervals has a finite or an infinite mean.

CasODE: Modeling Irregular Information Cascade via Neural Ordinary Differential Equations (Student Abstract)

Predicting information cascade popularity is a fundamental problem for understanding the nature of information propagation on social media. However, existing works fail to capture an essential aspect of information propagation: the temporal irregularity of cascade event -- i.e., users' re-tweetings at random and non-periodic time instants. In this work, we present a novel framework CasODE for information cascade prediction with neural ordinary differential equations (ODEs). CasODE generalizes the discrete state transitions in RNNs to continuous-time dynamics for modeling the irregular-sampled events in information cascades. Experimental evaluations on real-world datasets demonstrate the advantages of the CasODE over baseline approaches.

Suboptimal Filtering Over Sensor Networks With Random Communication

The problem of filter design is considered for continuous-time linear stochastic systems using distributed sensors. Each sensor unit, represented by a node in an undirected and connected graph, collects some information about the state and communicates its own estimate with the neighbors. It is stipulated that this communication between sensor nodes connected by an edge is time-sampled randomly and for each edge, the sampling process is an independent Poisson counter. Our proposed filtering algorithm for each sensor node is a stochastic hybrid system: It comprises a continuous-time differential equation, and at random time instants when communication takes place, each sensor node updates its state estimate based on the information received by its neighbors. In this setting, we compute the expectation of the error covariance matrix for each unit which is governed by a matrix differential equation. To study the asymptotic behavior of these covariance matrices, we show that if the gain matrices are appropriately chosen and the mean sampling rate is large enough, then the error covariances practically converge to a constant matrix.

A Time-Inhomogeneous Prendiville Model with Failures and Repairs

We consider a time-inhomogeneous Markov chain with a finite state-space which models a system in which failures and repairs can occur at random time instants. The system starts from any state j (operating, F, R). Due to a failure, a transition from an operating state to F occurs after which a repair is required, so that a transition leads to the state R. Subsequently, there is a restore phase, after which the system restarts from one of the operating states. In particular, we assume that the intensity functions of failures, repairs and restores are proportional and that the birth-death process that models the system is a time-inhomogeneous Prendiville process.

On Stochastic ISS of Time-Varying Switched Systems With Generic Lévy Switching Signals

Switched systems in which switching among subsystems occurs at random time instants find numerous applications in engineering. Stability analysis of such systems, however, is quite challenging. This letter investigates the stochastic input-to-state stability of this class of switched systems. The random switching instants are modeled by a non-decreasing, positive, and real-valued Lévy process, which, at every time instant, selects the active subsystem from a family of deterministic systems. No assumption on the stability of subsystems is presumed; they can be stable or unstable. Stochastic properties of the switching signal are coupled with a family of Lyapunov-like functions to obtain a sufficient condition for stochastic input-to-state stability.

Multiclass classification of dephasing channels

We address the use of neural networks (NNs) in classifying the environmental parameters of single-qubit dephasing channels. In particular, we investigate the performance of linear perceptrons and of two non-linear NN architectures. At variance with time-series-based approaches, our goal is to learn a discretized probability distribution over the parameters using tomographic data at just two random instants of time. We consider dephasing channels originating either from classical 1/f{\alpha} noise or from the interaction with a bath of quantum oscillators. The parameters to be classified are the color {\alpha} of the classical noise or the Ohmicity parameter s of the quantum environment. In both cases, we found that NNs are able to exactly classify parameters into 16 classes using noiseless data (a linear NN is enough for the color, whereas a single-layer NN is needed for the Ohmicity). In the presence of noisy data (e.g. coming from noisy tomographic measurements), the network is able to classify the color of the 1/f{\alpha} noise into 16 classes with about 70% accuracy, whereas classification of Ohmicity turns out to be challenging. We also consider a more coarse-grained task, and train the network to discriminate between two macro-classes corresponding to {\alpha} \lessgtr 1 and s \lessgtr 1, obtaining up to 96% and 79% accuracy using single-layer NNs.

Motility and energetics of randomly flashing ratchets

We consider randomly flashing ratchets, where the potential acting can be switched to another at random time instants with Poisson statistics. Using coupled Fokker–Planck equations, we formulate explicit expressions of mean velocity, dispersion and quantities measuring thermodynamics. How potential landscapes and transitions affect the motility and energetics is exemplified by numerical calculations on random on-off ratchets. Randomly flashing ratchets with shifted sawtooth potentials are further discussed. We find that the dynamics and output power of such system present symmetry w.r.t. the shift between the two potentials Δmax + Δmin, which is the sum of the shift between the two peaks (Δmax) and the shift between the two bottoms (Δmin). The mean velocity and output power both reach the optimal performance at Δmax + Δmin = 1, provided that the asymmetry α i of potential U i implies a positive flux respectively, i.e., α i > 0.5 for i = 1, 2.

Measurement of 1\u201310\xa0Hz 3D vibration modes with a CT-scanner

A new in situ vibration mode measurement method within a tomograph is proposed based on Projection-based Digital Volume Correlation techniques. Several projection angles are selected and a large number of radiographs of the vibrating sample are acquired at random instants with a small exposure time in order to ‘freeze out’ the displacement and avoid motion blurring. Based on an initial reconstruction acquired in a static configuration, the displacement field measurement is performed using a Proper Generalized Decomposition technique. All projections are analyzed as being due to a few vibration modes deforming the known reference volume. The different projection directions are related to each other assuming that the modal amplitude probability distribution functions are statistically similar. A synthetic test case, mock-up of a liver, is used to illustrate and validate the approach. In this case, 5 projection angles were chosen, 300 radiographs per angle, and the first three vibration modes could be recovered with a good accuracy.

Asymptotical Stability of Logic Dynamical Systems With Random Impulsive Disturbances

This article presents an investigation of the asymptotical set stability (ASS) of logic dynamical systems (LDSs) with impulsive disturbances at random instants. Unlike ordinary probabilistic Boolean networks, such an impulsive LDS is not Markovian in general. In this article, an impulsive LDS is described by using a hybrid index model and the sequence of intervals between adjacent impulsive instants is assumed to be independent and identically distributed. Under this assumption, the subsequence of a solution obtained by sampling at impulsive instants is a Markov chain. The initial distribution, the transition probability matrix, and the reachable set of the Markovian subsequence are calculated. Calculations of different invariant subsets are discussed and a necessary and sufficient condition for the convergence of finite Markov chains to subsets is obtained. Based on these results, necessary conditions for the ASS of impulsive LDSs in the hybrid and time domains, respectively, are obtained. In addition, we prove that the necessary conditions become sufficient if the impulsive intervals and the initial impulsive instant are with bounded expectations and variances. An example is provided to show the application of the proposed results to node synchronization of logic dynamical networks.

ImdLMS: An Imputation Based LMS Algorithm for Linear System Identification With Missing Input Data

In many practical systems concerning linear system identification with adaptive filters, the performance of the system is intrinsically limited by the problem of missing data, which might arise either because the input or the output data is missing at random time instants or subsets of input or output data are missing. In this paper the problem of linear system identification is studied with only input data missing at random time instants while output data is obtained correctly at all time instants. To model the input data missing process, the available input data sequence is modeled as the product of the actual (unknown) input data sequence with an i.i.d. sequence of Bernoulli random variables with known mean. A new data sequence, called the imputed data sequence is created from the available data sequence where, at any time instant, the imputed data is either the available data at that instant, if it is nonzero, or is set to a constant factor times the imputed data at the previous time instance. This imputed sequence is then used to propose an LMS-type algorithm called Imputation based missing data LMS (ImdLMS). Performance analysis of the algorithm is carried out where the extreme complications and possible intractability vis-a-vis an exact mean square performance analysis is avoided by finding an upper bound of the mean square deviation of the algorithm. Simulation results demonstrate the efficacy of the proposed ImdLMS algorithm, especially when the actual (unknown) input data sequence is correlated.

Transient temperature distribution and thermal stresses in a thick plate subjected to random pulses

This paper deals with the determination of second-order statistics (the mean of each component and correlations between components) of the quasi-static temperature distribution and thermal stresses on a thick plate under thermal load. The heated thick circular plate with random internal heat generation acting at random instants of time, in addition to sectional heat supply in the form of random impulses on the upper face, while the lower face is kept at zero temperature, and the fixed circular edge thermally insulated in considered for investigation. A type of Laplace transform method is employed to obtain the analytical solutions for the temperature and stresses statistics. Numerical results demonstrate the effects of the magnitude with the statistics of the temperature and thermal stresses and represented graphically.

Coincidence functions and Bartlett spectra of point processes

The second order statistical properties of time point processes (PP) are described by the time coincidence function (CF) and the frequency Bartlett spectrum (BS). For PPs recorded by pulses appearing at random time instants, as in photodetection experiments, the CF can be measured by various physical devices showing in particular the famous bunching effect of photons. On the other hand, for PPs recorded by the intervals between successive points (lifetimes), especially for renewal processes, there is no usual procedure for the estimation of the CF, and the aim of this paper is to describe an approach of this problem. The starting point is a mathematical relation between the CF and the set of probabilities density functions of the lifetimes of any order of the PP. As a consequence, the CF can be obtained by processing the results of the multiple normalized histograms of these lifetimes. In the cases, relatively rare, where the mathematical expression of the CF is known in closed form, the correct behavior of the procedure is verified by an experimental analysis of simulated data. The method is extended in order to verify the relationship between the CF and the BS.

Deterministic and randomized motions in single-well potentials

Newtonian, undamped motion in single-well potentials belong to a class of well-studied conservative systems. Here, we investigate and compare long-time properties of fully deterministic motions in single-well potentials with analogous randomized systems. We consider a special type of energy-conserving randomization process: the deterministic motion is interrupted by hard velocity reversals at random time instants ti. In the 1D case, for fixed initial conditions, the differences in probability distributions disappear in the long-time limit making asymptotic densities insensitive to the selection of random time instants when velocity is reversed. Substantially different probability distributions can be obtained, for instance, through the additional randomization of initial conditions. Analogously, in 2D setups, the probability distributions asymptotically are insensitive to a protocol of velocity reversals.

Multi-Stage Algorithms of Localization of a Random Pulsed Source with a Two-Step Probability Density Function in the Search Interval

Time-optimal algorithms of spatial localization of random pulsed-point sources are proposed, and their parameters are calculated. These sources have a two-step probability density function in the search interval and are detected due to generation of instantaneous pulses at random time instants (delta functions). The advantage of the developed optimal localization algorithms over the search procedures that ignore specific features of the spatial distribution of the detected pulsed-point objects is demonstrated.

Statistical properties of intermittent bursts in the Texas Helimak

We present investigations of intermittent turbulence in the Texas Helimak, a simple toroidal plasma device in which the turbulence properties are modified by applying a bias voltage. The analyzed turbulence presents high density bursts, detected by Langmuir probes measuring ion saturation current fluctuations. The turbulent time series is reproduced by a synthetic signal model that has bursts with the same temporal profile and random amplitudes, plus a low amplitude fluctuating stochastic background. Using this model, we identify two burst regimes, observed according to the radial position and external bias: in the first regime, the bursts occur in random instants, leading to a Poisson distribution of the time interval between bursts, while in the second regime, the time interval between large bursts is correlated and modeled by a Gamma distribution. Furthermore, we use the shape parameter k that measures the correlation between occurrence times of successive bursts to characterize the burst regime in most of the low field side. We find that in the region described by the second regime, the k values increase with positive applied bias.

Randomized Transmissions for Networked Control Under High-Frequency Jamming

We explore the networked control problem of a linear continuous-time plant under jamming attacks. We introduce a state-feedback control framework where the sensor transmits state information packets to the controller at random time instants. We show that our randomized transmission approach can handle jamming attacks with high occurrence frequencies and closed-loop stability can be achieved as long as the average jamming duration is sufficiently small. We illustrate the efficacy of our approach with a numerical example.

Power Spectral Density Response of Bridge‐Like Structures Loaded by Stochastic Moving Forces

In this study, simple and manageable closed form expressions are obtained for the mean value, the spectral density function, and the standard deviation of the deflection induced by stochastic moving loads on bridge‐like structures. As a basic case, a simply supported beam is considered, loaded by a sequence of concentrated forces moving in the same direction, with random instants of arrival, constant random crossing speeds, and constant random amplitudes. The loads are described by three stochastic processes, representing an idealization of vehicular traffic on a bridge in case of negligible inertial coupling effects between moving masses and structure. System’s responses are analytically determined in terms of mean values and power spectral density functions, yielding standard deviations, with the possibility to easily extend the results to more refined models of single span bridge‐like structures. Potential applications regard structural analysis, vibration control, and condition monitoring of traffic excited bridges.

Networked control of a power system: A non-uniform sampling approach

This article deals with the damping of low-frequency power systems oscillations using a networked controller on a smart grid. We assume that the network channel is band-limited and has random communication delay whose probability distribution can be found. To mitigate the effect of delay on control system performance, the measurements from the system are sampled at a random and non-uniform period but the inputs are updated at a faster and constant rate. This makes the closed-loop control system a stochastic switching system. A framework to analyze the mean square stability of the closed-loop system is developed and a switching observer-based controller is designed to ensure the stability of the system at random sampling instants. A case study has been performed using the Western-System-Coordinating-Council (WSCC) 3-machines, 9-bus system for verification of the technique. The contribution of the paper is a bandwidth efficient delay compensation technique and its application to power systems stability enhancement in a smart grid.

Event-Driven Cooperative Receding Horizon Control for Multi-Agent Systems in Uncertain Environments

We propose an event-driven Cooperative Receding Horizon (CRH) controller to solve maximum reward collection problems (MRCP) where multiple agents cooperate to maximize the total reward collected from a set of targets in a given mission space. In previous work, a CRH controller was developed and in this paper, we overcome several limitations of this controller, including potential instabilities in the agent trajectories and poor performance due to inaccurate estimation of a reward-to-go function. Rewards are non-increasing functions of time and the environment is uncertain with new targets detected by agents at random time instants. The controller sequentially solves optimization problems over a planning horizon and executes the control for a shorter action horizon, where both are defined by certain events associated with new information becoming available. In contrast to the earlier CRH controller, we reduce the originally infinite-dimensional feasible control set to a finite set at each control update event. We prove some properties of this new controller and include simulation results showing its improved performance.