Continuous Time Discrete State Research Articles

State‐of‐Charge Prediction of Lithium‐Ion Batteries Based on Sparse Autoencoder and Gated Recurrent Unit Neural Network

Lithium‐ion batteries are widely used in daily life because of their fast charging and high energy density. The accurate prediction of state‐of‐charge (SOC), critical for the quality evaluation and long‐term planning of lithium‐ion batteries, has still been challenging owing to sophisticated battery dynamics and ever‐changing ambient conditions. Herein, a joint sparse autoencoder (SAE) and gated recurrent unit (GRU) model are developed to improve SOC prediction performance. The sliding time window method is first utilized for data reconstruction to explore the continuous time trend and discrete state characteristics in multi‐cycle data of the lithium‐ion batteries. The SAE is then used to extract features from the reconstruction data. The GRU is finally applied to realize an accurate SOC prediction. The performance of proposed method is tested on a benchmark lithium‐ion battery degradation dataset and actual battery degradation experiments. Experimental tests show that the present model has the lowest root mean square error of 0.176 and 0.271, the lowest mean absolute error of 0.1396 and 0.2412, and the lowest mean square error of 0.03098 and 0.07355 on two datasets, respectively, demonstrating that the proposed method outperforms the state‐of‐the‐art methods in terms of accuracy and robustness.

Time-discretization approximation enriches continuous-time discrete-space models for animal movement

Continuous time discrete state models are a valuable tool for explaining animal movement. However, data collection to fit such models over a specified window of time can be misaligned with the actual realization of the movement process. This necessitates approximate model fitting, at present, through approximate imputation distributions (AIDs). Here, we propose a direct time-discretization approximation to the likelihood. The approach employs familiar ideas from hidden Markov modeling. Computation is implemented through the induced infinitesimal generator matrix. Linearization of this matrix expedites computation time. Through simulation and a real data application involving whale movement, we demonstrate that this model fitting strategy can outperform AID approaches.

Deterministic approximation for the nucleation-growth type model of nanoparticle formation: A validation against stochastic simulations

A deterministic approximation method is introduced here for a complicated kinetic model that describes nanoparticle formation with nucleation and growth steps. The model starts from monomer units, some of which combine in a nucleation reaction, and the number of monomer units (n) in a kinetically effective nucleus is an unbounded integer parameter in the model. The growth steps are second-order processes between a particle and a monomer unit, the rate constant of which depends on the size of the growing nanoparticle in a way that is given in a kernel function. Four different kernels are considered: diffusion kernel (size independence), Brownian kernel (direct proportionality to the linear size), surface kernel (direct proportionality to the surface), mass kernel (direct proportionality to the volume or mass). Aggregation or coagulation are not part of the model. The approximation method works well for all four kernels and all integer values of n, and yields analytical formulas for the average and the most important moments (zeroth, first, and the one that is relevant for the kernel function) of the nanoparticle size distribution at every stage of the process. The approximation is validated against stochastic simulations using the Gillespie algorithm within the framework of the continuous time discrete state approach to stochastic kinetics. The comparison shows that the obtained analytical formulas are applicable for all practically important cases of the model and provide a way to interpret the dependence of average nanoparticle size on the ratio of the rate constants of nucleation and particle growth.

Continuous-Time Discrete-State Modeling for Deep Whale Dives

Understanding unexposed/baseline behavior of marine mammals is required to assess the effects of increasing levels of anthropogenic noise exposure in the marine environment. However, quantifying variation in the baseline behavior of whales is challenging due to the fact that they spend much of their time at depth, and therefore, their diving behavior is not directly observable. Data collection employs tags as measurement devices to record vertical movement. We focus here on satellite tags, which have the advantage of collection over a time window of weeks. The type of data we analyze here suffers the disadvantage of being in the form of depths attached to an arbitrarily created set of depth bins and being sparse in time. We provide a multi-stage generative model for deep dives using a continuous-time discrete-space Markov chain. Then, we build a likelihood, incorporating dive-specific random effects, in order to fit this model to a set of satellite tag records, each consisting of a temporally misaligned collection of deep dives with sparse binned depths for each dive. Through simulation, we demonstrate the ability to recover true model parameters. With real satellite tag records, we validate the model out of sample and also provide inference regarding stage behavior, inter-tag record behavior, dive duration, and maximum dive depth. Supplementary materials accompanying this paper appear online.

A Markov reward process-based framework for resilience analysis of multistate energy systems under the threat of extreme events

Energy systems are increasingly exposed to the threats of extreme events like floods, earthquakes and hurricanes. In practice, the behaviors of the systems affected by these extreme events are often modeled by multistate models to facilitate the analysis. In this paper, we develop a generic framework for resilience modeling and analysis of multistate energy systems. A multistate resilience model is developed based on a Markov reward process model, where the degradation and recovery of system performance are characterized by a continuous time discrete state Markov chain and the losses caused by the extreme event is modeled by the reward rates associated with the sojourns in the degradation states and the transitions among the states. Four numerical metrics are defined to describe different aspects of system resilience, i.e. the resistant, absorption, recovery and overall resilience. A simulation-based algorithm is proposed for resilience analysis of multistate energy systems. The developed methods are applied for resilience modeling and analysis of a Nuclear Power Plant (NPP) under the threat of earthquakes. The Markov reward process model is developed following a probabilistic seismic hazard analysis, a fragility analysis and an event tree modeling of accident evolutions. Both a time-static and time-dependent resilience analysis are conducted and the results show that the developed model is able to comprehensively describe the resilience of multistate energy systems under the threats of extreme events.

Reliability-Based Design Optimization of Load Sharing Systems Using SSI-Markov Models

This paper presents a novel single loop approach to design the components of the load sharing systems by optimally allocating the failure probabilities to each component, thereby satisfying the overall system reliability requirement. The Reliability–Based Design Optimization (RBDO) of load sharing systems is computationally intensive due to the dynamic nature of component failure probabilities, since the failure of one component will vary the failure probabilities of other working components. Many RBDO methods have been successfully utilized to design individual components, however using these methods for handling system level reliability constraints is still a challenging task. This is because of a drop in accuracy and computational efficiency, especially when considering a load sharing system, where there is dependency in failure probabilities of components. The key idea is to integrate Stress–Strength Interference (SSI) theory with discrete (or) continuous time-discrete state Markov model for the reliability assessment of system, with the states being the condition of components (working/failed). This method takes advantage of the state transition probability matrix to represent the dynamic nature of the system performance. A numerical example of a simple load sharing system with two I-Beams is presented to illustrate and evaluate the performance of the proposed methodology.

Slow-scale split-step tau-leap method for stiff stochastic chemical systems

Tau-leaping is a family of algorithms for the approximate simulation of the discrete state continuous time Markov chains. Motivation for the development of such methods can be found, for instance, in the fields of chemical kinetics and systems biology. It is known that the dynamical behavior of biochemical systems is often intrinsically stiff representing a serious challenge for their numerical approximation. The naive extension of stiff deterministic solvers to stochastic integration often yields numerical solutions with either impractically large relaxation times or incorrectly resolved covariance. In this paper, we propose a splitting heuristic which helps to resolve some of these issues. The proposed integrator contains a number of unknown parameters which are estimated for each particular problem from the moment equations of the corresponding linearized system. We show that this method is able to reproduce the exact mean and variance of the linear scalar test equation and demonstrates a good accuracy for the arbitrarily stiff systems at least in the linear case. The numerical examples for both linear and nonlinear systems are also provided and the obtained results confirm the efficiency of the considered splitting approach.

Representation of electrical mode in arc furnaces by a state change model and determination of probabilities of those states

It is the first paper where the electrical mode of an arc furnace (AF) is proposed to be considered as a state change. This work also proposes a methodology for calculating the time values of the probabilities of these states. The methodology is based on the representation of state-change processes by the Markov model of continuous time, discrete state (CDS) stochastic processes. The state of electrical mode in each phase of an arc furnace is identified by the value of arc current that can be set for a given melting period, may be in the range of permitted deviations, or may get to the range of large operational or emergency deviations. Assuming that the system goes from state to state under the action of the Poisson flows of events, the concept of intensity of disturbance flows, and the intensity of flows of control actions are introduced. This makes it possible to form a system of Kolmogorov differential equations to change the state probabilities of the AF electrical mode. The solution of the system results in obtaining time dependencies of change in state probabilities. When analyzing graphs of changes over time in the probabilities of AF electrical mode states, it is possible to choose the desired intensity of the flow of control actions, which ensures that the electrical mode is in a given state under the action of the corresponding disturbance flow.

The Liquidity Regimes and the Prepayment Option of a Corporate Loan in the Finite Horizon Case

We investigate the prepayment option related to a corporate loan. The default intensity of the firm is supposed to follow a Cox-Ingersoll-Ross (CIR) process and the short interest rate is assumed constant. A liquidity term that represents the funding costs of the bank is introduced and modeled as a continuous time discrete state Markov jump process. The prepayment option is an American option with the payoff being an implicit function of the parameters of the problem. We give a verification result that allows to compute the price of the option. Numerical results are completely consistent with the theory; it is seen that the exercise domain may entirely disappear during such a liquidity crisis meaning that it is not optimal for the borrower to prepay. The method allows to quantify and interpret these findings.

A numerical technique for the identification of discrete-state continuous-time Markov models

A numerical technique for the identification of discrete-state continuoustime Markov models is presented. The technique is characterized by utilizing both initial approximations derived from observed data and estimates of minimized criterion sensitivity to small variations of identified parameters. A new feature of the proposed approach is improved running time of the applied combinatorial optimization algorithm achieved through replacing, at each iteration, of enumeration of various combinations of parameter values in the neighborhood of their current estimates by enumeration of values of only those parameters to which the minimized criterion is highly sensitive.

Classification and clustering with continuous time Bayesian network models

Classification and clustering of streaming data are relevant in finance, computer science, and engineering while they are becoming increasingly important in medicine and biology. Streaming data are analyzed with algorithms and models capable to represent dynamics, sequences and time. Dynamic Bayesian networks and hidden Markov models are commonly used to analyze streaming data. However, they are concerned with evenly spaced time series data and thus suffer from several limitations. Indeed, it is not clear how timestamps should be discretized even if some approaches to mitigate this problem have been recently made available. In this paper we describe the class of continuous time Bayesian networks classifiers and develop algorithms for their parametric and structural learning to solve classification and clustering of multivariate discrete state continuous time trajectories. Numerical experiments on synthetic and real world data are used to compare the performance of continuous time Bayesian network models to that achieved by dynamic Bayesian networks. In particular, post-stroke rehabilitation data is used for the classification task while urban traffic data from continuous time loop is used for the clusteirng task. The achieved results confirm the effectiveness of the proposed approaches.

PREPAYMENT OPTION OF A PERPETUAL CORPORATE LOAN: THE IMPACT OF THE FUNDING COSTS

We investigate in this paper a perpetual prepayment option related to a corporate loan. The short interest rate and default intensity of the firm are supposed to follow Cox–Ingersoll–Ross (CIR) processes. A liquidity term that represents the funding costs of the bank is introduced and modeled as a continuous time discrete state Markov chain. The prepayment option needs specific attention as the payoff itself is a derivative product and thus an implicit function of the parameters of the problem and of the dynamics. We prove verification results that allows to certify the geometry of the exercise region and compute the price of the option. We show moreover that the price is the solution of a constrained minimization problem and propose a numerical algorithm building on this result. The algorithm is implemented in a two-dimensional code and several examples are considered. It is found that the impact of the prepayment option on the loan value is not to be neglected and should be used to assess the risks related to client prepayment. Moreover, the Markov chain liquidity model is seen to describe more accurately clients' prepayment behavior than a model with constant liquidity.

Modelling and analysis of a multi-stage system involving batch processors with incompatible job families

Batch processors can concurrently process more than one job, and are commonly used in complex manufacturing systems such as wafer fabrication facilities (wafer fabs). Previous work has shown that there can be a significant reduction in cycle time when the control of the upstream serial processor is dependent on the batch processor. We wish to determine whether this is also true when the upstream processor is also a batch processor, with smaller capacity. We model a two-stage system with an upstream batch processor and a downstream batch processor under differing simple control policies as a continuous time-discrete state Markov chain. In doing so, we show that the concept of constraining the production of the upstream processor in accordance to the anticipated needs of the downstream batch processor can similarly reduce mean system cycle time when the upstream processor is a smaller batch processor. We also perform simulation experiments, using parameters close to those found in wafer fabs, and confirm that the strategy of prioritising the formation of larger batches for the downstream batch processor typically results in substantial cycle time reductions for the two-stage system.

Possibility of Application of the Theory of Semi-Markov Processes to Determine Reliability of Diagnosing Systems / MOŻLIWOŚĆ ZASTOSOWANIA TEORII PROCESÓW SEMI-MARKOWA DO OKREŚLENIA NIEZAWODNOŚCI SYSTEMÓW DIAGNOZUJĄCYCH

Abstract The paper provides justification for the necessity to define reliability of diagnosing systems (SDG) in order to develop a diagnosis on state of any technical mechanism being a diagnosed system (SDN). It has been shown that the knowledge of SDG reliability enables defining diagnosis reliability. It has been assumed that the diagnosis reliability can be defined as a diagnosis property which specifies the degree of recognizing by a diagnosing system (SDG) the actual state of the diagnosed system (SDN) which may be any mechanism, and the conditional probability p(S*/K*) of occurrence (existence) of state S* of the mechanism (SDN) as a diagnosis measure provided that at a specified reliability of SDG, the vector K* of values of diagnostic parameters implied by the state, is observed. The probability that SDG is in the state of ability during diagnostic tests and the following diagnostic inferences leading to development of a diagnosis about the SDN state, has been accepted as a measure of SDG reliability. The theory of semi-Markov processes has been used for defining the SDG reliability, that enabled to develop a SDG reliability model in the form of a seven-state (continuous-time discrete-state) semi-Markov process of changes of SDG states.

Robust stability criteria for uncertain neutral type stochastic system with Takagi–Sugeno fuzzy model and Markovian jumping parameters

In this paper, the robust stability for uncertain neutral stochastic system with Takagi–Sugeno (T–S) fuzzy model and Markovian jumping parameters (MJPs) are investigated. The jumping parameters considered here are generated from a continuous-time discrete-state homogeneous Markov process, which are governed by a Markov process with discrete and finite-state space. Some novel sufficient conditions are derived to guarantee the asymptotic stability of the equilibrium point in the mean square. By utilizing the Lyapunov–Krasovskii functional, stochastic analysis theory, some free weighting matrices and linear matrix inequality (LMI) technique, the upper bound of time-varying delay is obtained by using Matlab® control toolbox. Finally, some numerical examples are given to show the effectiveness of the obtained results.

Robust μ-stability analysis of Markovian switching uncertain stochastic genetic regulatory networks with unbounded time-varying delays

This paper investigates the global robust stability problem of Markovian switching uncertain stochastic genetic regulatory networks with unbounded time-varying delays and norm bounded parameter uncertainties. The structure variations at discrete time instances during the process of gene regulations known as hybrid genetic regulatory networks based on Markov process is proposed. The jumping parameters considered here are generated from a continuous-time discrete-state homogeneous Markov process, which are governed by a Markov process with discrete and finite state space. The concept of global robust μ-stability in the mean square for genetic regulatory networks is given. Based on Lyapunov function, stochastic theory and Itô’s differential formula, the stability criteria are presented in the form of linear matrix inequalities (LMIs). Numerical examples are presented to demonstrate the effectiveness of the main result.

Evaluating the performance of look-ahead policies for upstream serial processor with downstream batch processor serving incompatible job families and finite buffer sizes

Batch processors can concurrently process more than one job. In wafer fabrication, the processing time of a batch is independent of batch size, and only jobs from the same job family can be batched together. We consider a two-stage subsystem of a wafer fabrication facility (wafer fab), comprised of the diffusion furnace (a batch processor) and its upstream serial processor, with random job arrivals. We hypothesise that allowing the serial processor to anticipate the job family preference of the batch processor will reduce the overall cycle time of jobs passing through this system. To evaluate this hypothesis, we model the performance of the two-stage system under different system parameters and processor control policies as discrete state continuous time Markov chains. We characterise the system performance and show that the concept of constraining the production of the upstream processor according to the anticipated needs of the batch processor can reduce the mean cycle time of jobs being processed. We also perform simulation experiments to show that a simple heuristic based on this insight can translate to substantial cycle time reductions for systems with assumptions closer to those found in wafer fabs.

Stochastic interpretation of the asymmetry of enantiomeric distribution observed in the absolute asymmetric Soai reaction

The asymmetry of the experimentally observed distribution of enantiomeric excesses in the Soai reaction [Kawasaki, T.; Suzuki, K.; Shimizu, M.; Ishikawa, K.; Soai, K. Chirality 2006, 18, 479–482] is interpreted in this work using the continuous time discrete state stochastic kinetic approach assuming the presence of minor amounts of a chiral initiator at the beginning of the reaction. Schemes with first order and second order enantioselective autocatalysis are considered with the latter giving a better interpretation of the published results. For the number of initiator molecules, Poisson and geometric distributions are considered and give equally good fits. It is shown that the macroscopically observed asymmetry of the distribution of enantiomeric excesses could be the consequence of a few molecules acting as asymmetric inductors.

Modeling of Combined Heat and Power Generating Power Plant Units Using Markov Processes with Continuous Time Parameter and Discrete State Space

Power plant units with extraction and back-pressure turbines cogenerating heat and power play a decisive role in the heat and power generation of the former centrally planned Eastern European countries. Cogeneration requires that for a certain portion of the operating period, extraction-condensing and back-pressure steam turbine power plant units make less than their nominal capacity available to the power system. Consequently, the two-state reliability description is not suitable for differentiated reliability modeling of cogeneration power plants. This article briefly sets out a three-or-more-state reliability description of these power plant units using Markov processes with continuous time parameter and discrete state space.

Performance evaluation for tandem multi-factory supply chains: an approximate solution

Predicting the performance of multistage manufacturing systems is usually viewed as challenging because of unexpected machine breakdowns, random processing times, uncertain inter-factory transportation times, etc. In this paper, the authors formulate an approximate model for the tandem manufacturing systems, where the inventory in each buffer is monitored based on the (s, Q) discipline. This model divides a multistage system into a series of primitive line segments, each of which is characterised by a continuous time discrete state Markov process. The model may be applied in two types of systems: (1) tandem flow lines with batch processing and (2) multi-factory manufacturing supply chain, where inter-factory material transportation is required. Based on the model, a number of commonly used performance measures, including throughput, inventory, transportation frequency, etc., can be estimated. These estimates may enable manufacturers to evaluate the performance of the systems, and hence improve the management of production and inventory.