Time-dependent Probability Research Articles

Poisson distributions in stochastic dynamics of gene expression: What events do they count?

The Poisson distribution is the probability distribution of the number of independent events in a given period. Although the Poisson distribution appears ubiquitously in various stochastic dynamics of gene expression, both as time-dependent distributions and stationary distributions, underlying independent events that give rise to such distributions have not been clear, especially in the presence of the degradation of gene products, which is not a Poisson process. I show that the variable following the Poisson distribution is the number of independent events where biomolecules are created, destined to survive until the end of a given time duration. This new viewpoint enables me to rederive the Poisson distribution as a time-dependent probability distribution for molecule numbers in various monomolecular reaction models of stochastic gene dynamics. Additionally, it allows me to derive an analytic form of the time-dependent probability distribution for multispecies monomolecular reaction models with species whose lifetimes follow nonexponential distributions, which is the convolution of the Poisson distribution with the multinomial distribution. This distribution is then utilized for deriving a novel series expansion form of a time-dependent distribution for a model with a stochastic production rate.

Joint Air-Ground Distributed Federated Learning for Intelligent Transportation Systems

Supported by some of the major revolutionary technologies, such as Internet of Vehicles (IoVs), Edge Computing, and Machine Learning (ML), the traditional Vehicular Networks (VNs) are changing drastically and converging rapidly into one of the most complex, highly intelligent, and advanced networking systems, mostly known as Intelligent Transportation System (ITS). Recently, distributed ML techniques, such as Federated Learning (FL) have gained huge popularity mainly for their advantages in terms of intelligence sharing and privacy concerns. VNs are a natural contender for exploiting FL for solving challenging problems; however, their limited resources, dynamic nature, high speed, and reduced latency requirements often become the bottleneck. V2X communication technologies allow vehicular terminals (VTs) to share their valuable local environment parameters and become aware of their surroundings. Such information can be utilized to build a more sustainable and affordable FL platform for serving VTs. Gaining from recently introduced 3D architectures, integrating terrestrial and aerial edge computing layers, we present here a distributed FL platform able to distribute the FL process on a 3D fashion while reducing the overall communication cost for providing vehicular services. The framework is defined as a constrained optimization problem for reducing the overall FL process cost through a proper network selection between various nodes. We have modeled the FL network selection problem as a sequential decision-making process through a Markov Decision Process (MDP) with time-dependent state transition probabilities. A computation-efficient value iteration algorithm is adapted for solving the MDP. Comparison with various benchmark methods shows the overall improvement in terms of latency, energy, and FL performance.

Noise induced dynamics of two-qubit entangled Bell’s states

The stability of different types two-qubit entangled states was analyzed by a direct calculations of the time dependent probability to find the system in its initial state in the presence of Gaussian noise. The entangled states time evolution was studied in the presence of both classical and quantum noise. It was demonstrated that one pair of Bell’s states is robust to the Gaussian fluctuations, while the another one decays. The decay law of unstable entangled states is determined by the noise correlation function. The developed general formalism was applied for analysis the dynamics of entangled states in the presence of fluctuations produced by the electron–phonon interaction. It was revealed that unstable entangled states interacting with phonon bath can demonstrate the crossover between short time exponential decay law and long time power law decay. Obtained results can be effectively exploited for quantum computation, in which it is essential to be able to retrieve information about the initial quantum state even in the presence of interaction with the environment.

Minimum structure of high-order harmonic spectrum from molecular multi-orbital effects involving inner-shell orbitals.

The spectral features of high-order harmonic spectra can provide rich information for probing the structure and dynamics of molecules in intense laser fields. We theoretically study the high harmonic spectrum with the laser polarization direction perpendicular to the N2O molecule and find a minimum structure in the plateau region of the harmonic spectrum. Through analyzing the time-dependent survival probability of different electronic orbitals and the time-dependent wave packet evolution, it is found that this minimum position is caused by the harmonic interference of HOMO a, HOMO-1, and HOMO-3 a orbitals. Moreover, this interference minimum is discovered over a wide frequency range of 0.087a.u. to 0.093a.u., as well as a range of driving laser intensities with peak amplitudes between 0.056 a.u. and 0.059 a.u.. This study sheds light on the multi-electron effects and ultrafast dynamics of inner-shell electrons in intense laser pulses, which are crucial for understanding and controlling chemical reactions in molecules.

Precision-dissipation trade-off for driven stochastic systems

Over the last few decades, stochastic thermodynamics has emerged as a framework to study the thermodynamics of small-scaled systems. The relation between entropy production and precision is one of the most prominent research topics in this field. In this paper, I answer the question how much dissipation is needed to follow a pre-determined trajectory. This will be done by deriving a trade-off relation between how precisely a mesoscopic system can follow a pre-defined trajectory and how much the system dissipates. In the high-precision limit, the minimal amount of dissipation is inversely proportional to the expected deviation from the pre-defined trajectory. Furthermore, I will derive the protocol that maximizes the precision for a given amount of dissipation. The optimal time-dependent force field is a conservative energy landscape which combines a shifted version of the initial energy landscape and a quadratic energy landscape. The associated time-dependent probability distribution conserves its shape throughout the optimal protocol. Potential applications are discussed in the context of bit erasure and electronic circuits.

Time-dependent probability density function for partial resetting dynamics

Stochastic resetting is a rapidly developing topic in the field of stochastic processes and their applications. It denotes the occasional reset of a diffusing particle to its starting point and effects, inter alia, optimal first-passage times to a target. Recently the concept of partial resetting, in which the particle is reset to a given fraction of the current value of the process, has been established and the associated search behaviour analysed. Here we go one step further and we develop a general technique to determine the time-dependent probability density function (PDF) for Markov processes with partial resetting. We obtain an exact representation of the PDF in the case of general symmetric Lévy flights with stable index . For Cauchy and Brownian motions (i.e. ), this PDF can be expressed in terms of elementary functions in position space. We also determine the stationary PDF. Our numerical analysis of the PDF demonstrates intricate crossover behaviours as function of time.

Random Dynamic Analysis and Running Safety Assessment of Long-Span Cable-Stayed Bridge Under High-Speed Trains Subjected to Earthquake Excitations and Track Irregularities

Running trains on a long-span bridge under earthquake is an inevitable problem for the high-speed railway in seismic regions. This paper aims to investigate the seismically random dynamic behavior of train–cable stayed bridges interaction system using the probability density evolution method (PDEM). The system simultaneously involves the multi-effect of random multi-point earthquake excitations with traveling waves and random track irregularity. The multi-point earthquake excitations, composed of lateral and vertical positions, are generated by stochastic harmful functions (SHFs) and uniformly modulated into nonstationary random processes. The motion equation of such a system is established by coupling the 38-degree vehicles and bridges through the refined random wheel/rail contact relationship and accounting for the phase-lags of seismic travelling waves between pier excitations. PDEM is proven to be applicable to such time-dependent systems and is then used to transform the random excitations into a series of deterministic representative excitations with initial probability. By solving for the corresponding deterministic probability responses, various nonstationary random responses, including the time-dependent probability density evolution functions of random responses, mean values curves and standard deviations, can be obtained easily. A case study based on the cable-stayed bridge is then presented, and the results show the effectiveness and accuracy of the proposed method by comparison with Monte Carlo Simulation. Additionally, the influence of train speed and seismic intensity on the safety and reliability of trains is discussed.

AN \(M^{[X]}/G/1\) QUEUE WITH OPTIONAL SERVICE AND WORKING BREAKDOWN

In this study, a batch arrival single service queue with two stages of service (second stage is optional) and working breakdown is investigated. When the system is in operation, it may breakdown at any time. During breakdown period, instead of terminating the service totally, it continues at a slower rate. We find the time-dependent probability generating functions in terms of their Laplace transforms and derive explicitly the corresponding steady state results. Furthermore, numerous measures indicating system performances, such as the average queue size and the average queue waiting time, has been obtained. Some of the numerical results and graphical representations were also presented.

Safety Lifetime Analysis Using Two-Phase Subset Simulation Combined with Kriging Model

The safety lifetime analysis under the required time-dependent failure probability (R-FP) constraint is essential to ensure structure service safety, but existing methods are unaffordable for solving the safety lifetime under a small R-FP constraint. For this issue, a novel method is proposed by combining a two-phase subset simulation with an adaptive kriging model (TP-SS-AK). The innovations of the TP-SS-AK include three aspects. First, it creatively transforms the safety lifetime under a small R-FP constraint into that under a large conditional R-FP constraint by subset simulation, which can greatly improve efficiency because the large conditional R-FP constraint only needs a small candidate sample pool for searching the safety lifetime. Second, by deducing the misclassification probability of the time-dependent structure state in the first phase, the first new learning strategy is established to train the kriging model of the time-dependent performance function and further determine a reasonable lifetime search interval efficiently. Third, by deriving the misclassification probability of the first failure instant, the second new learning strategy is constructed to efficiently update the kriging model in the second phase and further search the safety lifetime. Because the proposed TP-SS-AK combines the advantages of the two-phase subset simulation and two new learning strategies, it is superior to the existing methods in efficiency, and its superiority is verified by the examples.

Variational deep learning of equilibrium transition path ensembles.

We present a time-dependent variational method to learn the mechanisms of equilibrium reactive processes and efficiently evaluate their rates within a transition path ensemble. This approach builds off of the variational path sampling methodology by approximating the time-dependent commitment probability within a neural network ansatz. The reaction mechanisms inferred through this approach are elucidated by a novel decomposition of the rate in terms of the components of a stochastic path action conditioned on a transition. This decomposition affords an ability to resolve the typical contribution of each reactive mode and their couplings to the rare event. The associated rate evaluation is variational and systematically improvable through the development of a cumulant expansion. We demonstrate this method in both over- and under-damped stochastic equations of motion, in low-dimensional model systems, and in the isomerization of a solvated alanine dipeptide. In all examples, we find that we can obtain quantitatively accurate estimates of the rates of the reactive events with minimal trajectory statistics and gain unique insights into transitions through the analysis of their commitment probability.

Modeling Regime Shifts in Multiple Time Series

We investigate the problem of discovering and modeling regime shifts in an ecosystem comprising multiple time series known as co-evolving time series. Regime shifts refer to the changing behaviors exhibited by series at different time intervals. Learning these changing behaviors is a key step toward time series forecasting. While advances have been made, existing methods suffer from one or more of the following shortcomings: (1) failure to take relationships between time series into consideration for discovering regimes in multiple time series; (2) lack of an effective approach that models time-dependent behaviors exhibited by series; (3) difficulties in handling data discontinuities which may be informative. Most of the existing methods are unable to handle all of these three issues in a unified framework. This, therefore, motivates our effort to devise a principled approach for modeling interactions and time-dependency in co-evolving time series. Specifically, we model an ecosystem of multiple time series by summarizing the heavy ensemble of time series into a lighter and more meaningful structure called a mapping grid . By using the mapping grid, our model first learns time series behavioral dependencies through a dynamic network representation, then learns the regime transition mechanism via a full time-dependent Cox regression model. The originality of our approach lies in modeling interactions between time series in regime identification and in modeling time-dependent regime transition probabilities, usually assumed to be static in existing work.

A Reliability-Based Framework for Damage Accumulation Due to Multiple Earthquakes: A Case Study on Bridges

Damage accumulation due to multiple seismic impacts over time has a significant effect on the residual service life of the bridge. A reliability-based framework was developed to make decisions in bridge maintenance activities. The feature of the framework enables quantifying the time-dependent probability of failure of bridges due to the impact of multiple earthquakes and progressive deterioration. To estimate the reliability of the bridge systems, the probability of failure of the bridge was used. Two case studies were utilised to demonstrate how the method can be applied to the real world. Results show that the accumulated damage caused by multiple earthquakes and progressive deterioration significantly impact the remaining useful life of the bridge. Furthermore, the soil conditions predominantly influence the progressive deterioration and reduce the service life of the bridge. Overall, the proposed framework enables the sustainable decision-making process for bridge maintenance activities. The results reveal the necessity of including the combined impact in the bridge maintenance system and that there is a more than 40% increase in the probability of failure, due to the combined effect of progressive deterioration and earthquake impacts, compared to the impact only due to seismic loads for the considered case study bridge.

Time-dependent reliability analysis under random and interval uncertainties based on Kriging modeling and saddlepoint approximation

Time-dependent reliability analysis is capable of evaluating the reliability of the system over its full life cycle, which is of highly concerned for designers. In practice, some uncertainties cannot be simply described by a deterministic distribution due to lack of knowledge or information, and generally, their upper and lower bounds can be obtained. Thus, this paper develops a novel computational method for time-dependent reliability under random and interval uncertainties. The first step of the proposed method is to construct the Kriging surrogating model of the most probable point trajectory, with which the bound of the limit-state function can be efficiently recast as an equivalent Gaussian process. In this case, the computation of the bound of time-dependent failure probability (TDFP) is converted to a high-dimensional Gaussian integral problem. To solve this problem, the expansion optimal linear estimation, sparse-grid Gaussian quadrature technique and saddlepoint approximation method are integrated so that the above integral can be efficiently computed. Finally, four engineering examples are studied to demonstrate the effectiveness of the proposed method by comparison with previous methods. The results show this novel method can provide accurate and efficient time-dependent reliability analysis in terms of random and interval uncertainties.

Levonorgestrel intrauterine device use and incident idiopathic intracranial hypertension among commercially insured women

ObjectivesTo estimate the hazard of incident idiopathic intracranial hypertension, a potentially blinding condition, among women using levonorgestrel intrauterine devices (LNG-IUD) compared to copper IUD, as conflicting associations have been reported. Study designThis retrospective, longitudinal cohort study identified women ages 18–45 years in a large care network (January 1, 2001, to December 31, 2015) using LNG-IUD, subcutaneous etonogestrel implant, copper IUD, tubal device/surgery, or hysterectomy. Incident idiopathic intracranial hypertension was defined as the first diagnosis code for after 1 year without any codes and following brain imaging or lumbar puncture. Kaplan-Meier analysis estimated time-dependent probabilities of idiopathic intracranial hypertension at 1 and 5 years after incident contraception use, stratified by type. Cox regression estimated the hazard of idiopathic intracranial hypertension associated with LNG-IUD use compared to copper IUD (primary comparison) after adjusting for sociodemographics and factors associated with idiopathic intracranial hypertension (e.g., obesity) or contraception selection. A sensitivity analysis with propensity score-adjusted models was performed. ResultsOf 268,280 women, 78,175 (29%) used LNG-IUD, 8715 (3%) etonogestrel implant, 20,275 (8%) copper IUD, 108,216 (40%) hysterectomy, 52,899 (20%) tubal device/surgery, and 208 (0.08%) developed idiopathic intracranial hypertension over a mean follow-up of 2.4 ± 2.4 years. Also, 1-/5-year Kaplan-Meier idiopathic intracranial hypertension probabilities were 0.0004/0.0021 for LNG-IUD and 0.0005/0.0006 for copper IUD users. LNG-IUD use did not show significantly different hazard of idiopathic intracranial hypertension compared to copper IUD (adjusted hazard ratio 1.84 [95% CI 0.88, 3.85]). Sensitivity analyses were similar. ConclusionsWe did not observe a significantly increased hazard of idiopathic intracranial hypertension among women using LNG-IUD compared to copper IUDs. ImplicationsThe lack of an association between LNG-IUD use and idiopathic intracranial hypertension in this large observational study provides reassurance to women considering initiation or continued use of this highly effective contraceptive method.

Simulation to investigate site-based monitoring of pest insect species for trade.

Pest insect surveillance using lures is widely used to support market access requirements for traded articles that are hosts or carriers of quarantine pests. Modeling has been used extensively to guide the design of surveillance to support pest free area claims but is less commonly applied to provide confidence in pest freedom or low pest prevalence within sites registered for trade. Site-based surveillance typically needs to detect pests that are already present in the site or that may be entering the site from surrounding areas. We assessed the ability of site-based surveillance strategies to detect pests originating from within or outside the registered site using a probabilistic trapping network simulation model with random-walk insect movement and biologically realistic parameters. For a given release size, time-dependent detection probability was primarily determined by trap density and lure attractiveness, whereas mean step size (daily dispersal) had limited effect. Results were robust to site shape and size. For pests already within the site, detection was most sensitive using regularly spaced traps. Perimeter traps performed best for detecting pests moving into the site, although the importance of trap arrangement decreased with time from release, and random trap placement performed relatively well compared to regularly spaced traps. High detection probabilities were achievable within 7 days using realistic values for lure attractiveness and trap density. These findings, together with the modeling approach, can guide the development of internationally agreed principles for designing site-based surveillance of lure-attractant pests that is calibrated against the risk of non-detection.

Time-dependent probability density function analysis of H-mode transitions

The first application of time-dependent probability density function (PDF) analysis to the L-H transition in fusion plasmas is presented. PDFs are constructed using Doppler Backscattering data of perpendicular fluctuation velocity, , and turbulence from the edge region of the DIII-D tokamak. These raw time-series data are sliced into millisecond-long sliding time-windows to create PDFs. During the transition, the PDFs develop strong right tails, indicative of turbulence-suppressing localised flows in the plasma edge; such features and other subtle behaviours are explored using novel information geometry techniques. This letter examines the applicability of these techniques to predict L-H transitions and investigate predator-prey self-regulation theories between turbulence and perpendicular velocity.

Improvement of burst capacity model for pipelines containing surface cracks and its implication for reliability analysis

This paper presents the improvement of a widely used burst capacity model for steel oil and gas pipelines that contain longitudinal external surface cracks, namely the CorLAS model, through the addition of a correction factor that is quantified by the Gaussian process regression (GPR). The correction factor is assumed to depend on four non-dimensional input features that characterize both the crack geometry and pipe material properties. A database consisting of 212 full-scale burst tests of pipe specimens that contain longitudinal surface cracks is established based on the open literature, which is employed to train the GPR model and evaluate its performance. It is shown that GPR is highly effective in improving the accuracy of the CorLAS model predictions. The improvement is further shown to have a marked effect on the time-dependent probability of burst of pipelines containing growing surface cracks through two hypothetical pipeline examples: when employing the CorLAS model, the probabilities of burst are significantly higher, exceeding those obtained using the improved CorLAS model by more than one order of magnitude.

Coherent dynamics of a photon-dressed qubit

We consider the dynamics and stationary regime of a capacitively-shunted transmon-type qubit in front of a mirror, affected by two signals: probe and dressing signals. By varying the parameters of these signals and then analyzing the probe signal (reflected by the atom-mirror system), it is possible to explore the system dynamics, which can be described by the Bloch equation. The obtained time-dependent occupation probabilities are related to the experimentally measured reflection coefficient. The study of this type of dynamics opens up new horizons for better understanding of the system properties and underlying physical processes, such as Landau-Zener-Stuckelberg-Majorana transitions.

The Cramér-Lundberg model with a fluctuating number of clients

This paper considers the Cramér-Lundberg model, with the additional feature that the number of clients can fluctuate over time. Clients arrive according to a Poisson process, where the times they spend in the system form a sequence of independent and identically distributed non-negative random variables. While in the system, every client generates claims and pays premiums. In order to describe the model's rare-event behaviour, we establish a sample-path large-deviation principle. This describes the joint rare-event behaviour of the reserve-level process and the client-population size process. The large-deviation principle can be used to determine the decay rate of the time-dependent ruin probability as well as the most likely path to ruin. Our results allow us to determine whether the chance of ruin is greater with more or with fewer clients and, more generally, to determine to what extent a large deviation in the reserve-level process can be attributed to an unusual outcome of the client-population size process.

An Entropy Dynamics Approach for Deriving and Applying Fractal and Fractional Order Viscoelasticity to Elastomers

Abstract Entropy dynamics is a Bayesian inference methodology that can be used to quantify time-dependent posterior probability densities that guide the development of complex material models using information theory. Here, we expand its application to non-Gaussian processes to evaluate how fractal structure can influence fractional hyperelasticity and viscoelasticity in elastomers. We investigate how kinematic constraints on fractal polymer network deformation influences the form of hyperelastic constitutive behavior and viscoelasticity in soft materials such as dielectric elastomers, which have applications in the development of adaptive structures. The modeling framework is validated on two dielectric elastomers, VHB 4910 and 4949, over a broad range of stretch rates. It is shown that local fractal time derivatives are equally effective at predicting viscoelasticity in these materials in comparison to nonlocal fractional time derivatives under constant stretch rates. We describe the origin of this accuracy that has implications for simulating large-scale problems such as finite element analysis given the differences in computational efficiency of nonlocal fractional derivatives versus local fractal derivatives.