Nonlinear Stochastic Processes Research Articles

Intelligent spare ordering and replacement optimisation leveraging adaptive prediction information

Predicting system health using inspection technologies is crucial for efficiently managing the maintenance of various industrial products. This study introduced an innovative policy for intelligently ordering and replacing spare parts. It utilises real-time prediction data to make sequential decisions on whether to schedule spares and when to conduct non-immediate maintenance. A generalised non-linear stochastic process was established to capture the underlying degradation path, with a lifetime coefficient updated through Bayesian inference. Conditional reliability assessed during regular inspections determines when spare preparations and delayed replacements are warranted. Predictive replacements can also be postponed based on the expected remaining lifetime adjusted for safety factors and spare lead times. The model dynamically optimises operational costs by iteratively optimising spare-ordering times, postponement intervals, and adjustment coefficients. Numerical experiments on high-speed train gearboxes validated its superior cost-effectiveness compared to conventional approaches.

NIRK-based mixed-type accurate continuous–discrete Gaussian filters with deterministically sampled expectation and covariance for state estimation in continuous-time stochastic process models with discrete measurements

This paper advances further the idea of overall continuous–discrete Gaussian filtering with deterministically sampled expectation and covariance towards efficient mixed-type accurate Kalman-like schemes based on self-regulated Ordinary Differential Equation (ODE) solvers with automatic local and global error controls intended for treating continuous-time nonlinear stochastic process models with discrete measurements. The universal state estimation algorithms of such sort designed recently by Kulikov and Kulikova (2022) possess three potential limitations, which might affect their successful applications in practice. First, these deal with complicated Moment Differential Equations (MDEs) and, hence, can be rather time-consuming in estimating stochastic process models of large size, that compromises their efficiency for online computations. Second, the aforementioned filters do not employ any kind of global error regulation and, hence, can be insufficiently accurate in solving difficult MDEs arisen. Third, these are hardly effective in treating stiff continuous–discrete stochastic systems. Below, all the listed issues are addressed and solved via utilization of the much simpler but more robust MDEs arisen in the usual continuous–discrete Extended Kalman Filter (EKF) within the mixed-type accurate filtering approach initiated by Kulikova and Kulikov (2015), which is rooted in the variable-stepsize Nested Implicit Runge–Kutta (NIRK) methods. For the sake of stability of our novel methods to the numerical integration and round-off errors, we devise square-root versions of the mixed-type Gaussian filters, which are based either on hyperbolic QR factorizations or on hyperbolic Singular Value Decompositions (SVDs). Performances of these new mixed-type square-root Gaussian filters are assessed and compared to their original (non-mixed-type) counterparts as well as to their non-square-root versions in a simulated non-stiff ill-conditioned radar tracking scenario and on a stiff stochastic Oregonator reaction.

Simple stochastic model with safe speed using Exchange approach to recognizing synchronized flow as speed-synchronized phase

In order to unravel the physical and mathematical mystery of synchronized-flow mechanism and to reveal the fundamental mechanism and origin of synchronized flow produced by nonlinear stochastic processes, we have produced simple stochastic traffic flow models (the gradual-NaSch, Phoenix and mPhoenix models) with nonlinear safe speeds. In the mNaSch model and our gradual-NaSch model, the [Formula: see text]th vehicle’s speed is the same as or different from the [Formula: see text]th vehicle’s speed because they are discrete values. In the mNaSch and gradual-NaSch models, the same discrete values of the [Formula: see text]th and [Formula: see text]th states make it easy to identify synchronized flow with speed-synchronized phase of the [Formula: see text]th and [Formula: see text]th states. On the other hand, when we deal with synchronized flow in continuous traffic flow models, we face a problem. Continuous values cause the difficulty in identification of synchronization. In order to definitely clarify whether or not synchronized flow occurs in continuous models, we have established a novel idea of Exchange approach to recognizing synchronized flow as speed-synchronized phase. Our idea is generated from the analogical image that the whole system of synchronized metronomes does not change even if the [Formula: see text]th and [Formula: see text]th synchronized metronomes are exchanged. The Exchange approach (exchanging the [Formula: see text]th vehicle’s state for any [Formula: see text]th vehicle’s state in the whole system of traffic flow), which causes distortion such as a collision if non-synchronized vehicle’s states are exchanged and makes it possible to definitely clarify whether or not synchronized flow occurs in continuous models, is applied to our simple stochastic continuous model (the mPhoenix model). On the basis of the Exchange approach, we can recognize that the mPhoenix model surely reproduces synchronized flow. In addition, we have proposed mathematical approach to deriving nonlinear safe speeds which guarantee collision free driving and reproduce synchronized flow.

Non-equilibrium Onsager–Machlup theory

This paper proposes a simple mathematical model of non-stationary and non-linear stochastic dynamics, which approximates a (globally) non-stationary and non-linear stochastic process by its locally (or ‘piecewise’) stationary version. Profiting from the elegance and simplicity of both, the exact mathematical model referred to as the Ornstein–Uhlenbeck stochastic process (which is globally stationary, Markov and Gaussian) and of the Lyapunov criterion associated with the stability of stationarity, we show that the proposed non-linear non-stationary model provides a natural extension of the Onsager–Machlup theory of equilibrium thermal fluctuations, to the realm of non-stationary, non-linear and non-equilibrium processes. As an illustrative application, we then apply the extended non-equilibrium Onsager–Machlup theory, to the description of thermal fluctuations and irreversible relaxation processes in liquids, leading to the main exact equations employed to construct the non-equilibrium self-consistent generalised Langevin equation (NE-SCGLE) theory of irreversible processes in liquids. This generic theory has demonstrated that the most intriguing and long-unsolved questions of the glass and gel transitions are understood as a natural consequence of the second law of thermodynamics, enunciated in terms of the proposed piecewise stationary stochastic mathematical model.

A Bimodal Model for Oil Prices

Oil price behaviour over the last 10 years has shown to be bimodal in character, displaying a strong tendency to congregate around one range of high oil prices and one range of low prices, indicating two distinct peaks in its frequency distribution. In this paper, we propose a new, single nonlinear stochastic process to model the bimodal behaviour, namely, dp=α(p1−p)(p2−p(p3−p)dt+σpγdZ, γ=0,0.5. Further, we find analytic approximations of oil price futures under this model in the cases where the stable fixed points of the corresponding deterministic model are (a) evenly spaced about the unstable fixed point and (b) are spaced in the ratio 1:2 about the unstable fixed point. The solutions are shown to produce accurate prices when compared to numerical solutions.

Time reversal of spinal processes for linear and non-linear branching processes near stationarity

We consider a stochastic individual-based population model with competition, trait-structure affecting reproduction and survival, and changing environment. The changes of traits are described by jump processes, and the dynamics can be approximated in large population by a non-linear PDE with a non-local mutation operator. Using the fact that this PDE admits a non-trivial stationary solution, we can approximate the non-linear stochastic population process by a linear birth-death process where the interactions are frozen, as long as the population remains close to this equilibrium. This allows us to derive, when the population is large, the equation satisfied by the ancestral lineage of an individual uniformly sampled at a fixed time T, which is the path constituted of the traits of the ancestors of this individual in past times t≤T. This process is a time inhomogeneous Markov process, but we show that the time reversal of this process possesses a very simple structure (e.g. time-homogeneous and independent of T). This extends recent results where the authors studied a similar model with a Laplacian operator but where the methods essentially relied on the Gaussian nature of the mutations.

Information quantity evaluation of nonlinear time series processes and applications

Among several models proposed in the time series literature, the Self-Exciting Threshold Autoregressive (SETAR) model is non-linear and considers threshold values to model time series affected by regimes. Beyond the linear models, the computation of information and dependence metrics in non-linear time series is of great interest to compare processes and test non-linearity. This paper considers the stationary marginal density of a SETAR(2;1,1) process to compute explicit differential entropy and Kullback–Leibler and Jeffrey’s divergences. In addition, an asymptotic homogeneity test for statistical significance of the disparity between two SETAR(2;1,1) processes was built. Numerical illustrations and applications to fish condition factor and Chilean economic perception time series are presented to illustrate the proposed methodology. Besides, a numerical algorithm based on Riemann–Stieltjes integral definition is implemented to calculate information measures based on stationary cumulative density functions of other types of nonlinear stationary stochastic processes, such as the first-order double autoregressive bilinear threshold moving average and threshold autoregressive moving average processes.

Ambiguity tube MPC

This paper is about a class of distributionally robust model predictive controllers (MPC) for nonlinear stochastic processes, which evaluate risk and control performance measures by propagating ambiguity sets in the space of state probability measures. A framework for formulating such ambiguity tube MPC controllers is presented using methods from the field of optimal transport theory. Moreover, an analysis technique based on supermartingales is proposed, leading to stochastic stability results for a large class of distributionally robust controllers. In this context, we also discuss how to construct terminal cost functions for stochastic and distributionally robust MPC that ensure closed-loop stability and asymptotic convergence to robust invariant sets. The corresponding theoretical developments are illustrated by tutorial-style examples and a numerical case study.

The two-particle irreducible effective action for classical stochastic processes

By combining the two-particle-irreducible (2PI) effective action common in non-equilibrium quantum field theory with the classical Martin–Siggia–Rose formalism, self-consistent equations of motion for the first and second cumulants of non-linear classical stochastic processes are constructed. Such dynamical equations for correlation and response functions are important in describing non-equilibrium systems, where equilibrium fluctuation–dissipation relations are unavailable. The method allows to evolve stochastic systems from arbitrary Gaussian initial conditions. In the non-linear case, it is found that the resulting integro-differential equations can be solved with considerably reduced computational effort compared to state-of-the-art stochastic Runge–Kutta methods. The details of the method are illustrated by several physical examples.

Analysis of growth in population through nonlinear stochastic process

Probability density evolution of an insect population is discussed by taking non linear stochastic differential equations for the growth and also using the related Fokker-Plank equations for the probability densities which are time dependent. It can be seen that the variance behavior depends on initial conditions whereas the effect of initial conditions disappears rapidly in mean case. We can observe the difference in the response of mean and variance as the growth of insects proceeds. It is found that variance of the process may increase monotonically to a level above the steady state variance before getting back to steady state variance.

Quasi-likelihood analysis for nonlinear stochastic processes

A brief overview of the theory of quasi-likelihood analysis (QLA) is given and its usefulness is demonstrated with applications to estimation for a volatility parameter of a semimartingale. A simplified version of the QLA is recalled. The role of non-degeneracy of a key index reflecting identifiability is highlighted. In an application of the QLA, the concept of global jump filters is introduced for precise estimation of the volatility parameter from the data contaminated with jumps.

Quasi-likelihood analysis and its applications

The Ibragimov–Khasminskii theory established a scheme that gives asymptotic properties of the likelihood estimators through the convergence of the likelihood ratio random field. This scheme is extending to various nonlinear stochastic processes, combined with a polynomial type large deviation inequality proved for a general locally asymptotically quadratic quasi-likelihood random field. We give an overview of the quasi-likelihood analysis and its applications to ergodic/non-ergodic statistics for stochastic processes.

Industrial Time Series Modeling With Causal Precursors and Separable Temporal Convolutions

When applied to industrial processes, predictive models (PMs) easily fail to depict the overall complexity of a production plant. Multiple factors can interfere during the production process and the requirements in accuracy, safety and efficiency are very high. To fully achieve the potential of PMs, it is useful to integrate typical characteristics of the manufacturing process and build procedures that account for the domain peculiarities. Cause-effect relationships that entail the flow of an industrial process are rarely considered in PMs for manufacturing despite the relevance that these relationships have in practice. In this work we present a two-step procedure that uses a causal discovery method for time series, eventually validated by domain experts, together with a new neural network architecture named Separable Temporal Convolutional Network (S-TCN) to create a PM for industrial multivariate time series. Causal precursors exploit the sequentiality of the process flow, grouping process machines by their effective temporal activation. The S-TCN architecture is based on separable temporal convolution and enables the efficient forecast of more distant temporal connections than common temporal models. A numerical validation is presented on a large class of synergetic nonlinear stochastic processes. We apply the proposed procedure in an ultra-processed food plant collecting more than 100 days of active production. The presented procedure achieves better results when compared with state-of-the-art algorithms in both conditions.

Microscopic dynamics modeling unravels the role of asymptomatic virus carriers in SARS-CoV-2 epidemics at the interplay between biological and social factors

The recent experience of SARS-CoV-2 epidemics spreading revealed the importance of passive forms of infection transmissions. Apart from the virus survival outside the host, the latent infection transmissions caused by asymptomatic and presymptomatic hosts represent major challenges for controlling the epidemics. In this regard, social mixing and various biological factors play their subtle, but often critical, role. For example, a life-threatening condition may result in the infection contracted from an asymptomatic virus carrier. Here, we use a new recently developed microscopic agent-based modelling framework to shed light on the role of asymptomatic hosts and unravel the interplay between the biological and social factors of these nonlinear stochastic processes at high temporal resolution. The model accounts for each human actor's susceptibility and the virus survival time, as well as traceability along the infection path. These properties enable an efficient dissection of the infection events caused by asymptomatic carriers from those which involve symptomatic hosts before they develop symptoms and become removed to a controlled environment. Consequently, we assess how their relative proportions in the overall infection curve vary with changing model parameters. Our results reveal that these proportions largely depend on biological factors in the process, specifically, the virus transmissibility and the critical threshold for developing symptoms, which can be affected by the virus pathogenicity. Meanwhile, social participation activity is crucial for the overall infection level, further modulated by the virus transmissibility.

Comparison of Classical and Robust Estimates of Threshold Auto-regression Parameters

The exponential auto-regression model is a discrete analog of the second-order nonlinear differential equations of the type of Duffing and van der Pol oscillators. It is used to describe nonlinear stochastic processes with discrete time, such as vehicle vibrations, ship roll, electrical signals in the cerebral cortex. When applying the model in practice, one of the important tasks is its identification, in particular, an estimate of the model parameters from observations of the stochastic process it described. A traditional technique to estimate autoregressive parameters is the nonlinear least squares method. Its disadvantage is high sensitivity to the measurement errors of the process observed. The M-estimate method largely has no such a drawback. The M-estimates are based on the minimization procedure of a non-convex function of several variables. The paper studies the effectiveness of several well-known minimization methods to find the M-estimates of the parameters of an exponential autoregressive model. The paper demonstrates that the sequential quadratic programming algorithm, the active set algorithm, and the interior-point algorithm have shown the best and approximately the same accuracy. The quasi-Newton algorithm is inferior to them in accuracy a little bit, but is not inferior in time. These algorithms had approximately the same speed and were one and a half times faster than the Nelder-Mead algorithm and 14 times faster than the genetic algorithm. The Nelder-Mead algorithm and the genetic algorithm have shown the worst accuracy. It was found that all the algorithms are sensitive to initial conditions. The estimate of parameters, on which the autoregressive equation linearly depends, is by an order of magnitude more accurate than that of the parameter on which the auto-regression equation depends in a nonlinear way.

Study on Nonlinear Stochastic Process of Deck Slamming on Floating Offshore Platform

In this paper, a semi-analytic method is introduced to predict the deck-slamming probability and corresponding loads. This method is based on a nonlinear statistical approach that takes into account the linear and second-order components of the relative wave elevation up to the second order. The linear and second-order wave elevation is assumed to be a two-term Volterra series. The joint probability density function of the relative wave elevation and velocity are formulated using the Hermite-moment method, and the probability distributions of the wave crest and relative wave velocity are calculated. These probability distributions are verified using the data sampled from the linear and second-order relative wave elevation. Based on this formulation, the probabilities of deck slamming and slamming-induced loads are estimated. This method is applied to a tension leg platform (TLP) model, and the effects of the second-order component of the relative wave elevation on the deck slamming are investigated.

First passage time of nonlinear diffusion processes with singular boundary behavior

New theorems for moments of the first passage time of one dimensional nonlinear stochastic processes with an entrance boundary xe are formulated. This important class of one dimensional stochastic processes results among others from approximations of the energy or amplitude of second order nonlinear stochastic differential equations. Since the diffusion of a stochastic process vanishes at an entrance boundary, xe is called a singular point of the stochastic process. The theorems for the moments of the first passage times are validated based on existing analytical results. In addition, the first passage times of a forced and damped Mathieu oscillator, as well as a nonlinear stochastic differential equation, which is important for the determination of dangerous ship roll dynamics, are calculated. The proposed analytical expressions for the moments of the first passage times can be calculated very fast using standard quadrature formulas.

A Langevin dynamics approach to the distribution of animal move lengths

Movement is fundamental to the animal ecology, determining how, when, and where an individual interacts with the environment. The animal dynamics is usually inferred from trajectory data described as a combination of moves and turns, which are generally influenced by the vast range of complex stochastic stimuli received by the individual as it moves. Here we consider a statistical physics approach to study the probability distribution of animal move lengths based on stochastic differential Langevin equations and the superstatistics formalism. We address the stochastic influence on the move lengths as a Wiener process. Two main cases are considered: one in which the statistical properties of the noise do not change along the animal’s path and another with heterogeneous noise statistics. The latter is treated in a compounding statistics framework and may be related to heterogeneous landscapes. We study Langevin dynamics processes with different types of nonlinearity in the deterministic component of movement and both linear and nonlinear multiplicative stochastic processes. The move length distributions derived here comprise the possibility of movement multiscales, diffusive and superdiffusive (Lévy-like) dynamics, and include most of the distributions currently considered in the literature of animal movement, as well as some new proposals.

Dynamic performance enhancement for nonlinear stochastic systems using RBF driven nonlinear compensation with extended Kalman filter

In this paper, a novel hybrid control method is proposed to enhance the tracking performance of the Proportional–Integral (PI) based control system for a class of nonlinear and non-Gaussian stochastic dynamic processes with unmeasurable states. The system performance is presented by tracking error entropy as the system is nonlinear and subjected to non-Gaussian noises. The well-known kernel density estimation (KDE) technique is employed to estimate the entropy because the precise statistical property of noises is not available for many industrial processes. Since in many industrial cases gains of PI controllers are fixed, a compensative controller is designed without changing the existing closed loop PI controller. Moreover, the compensative signal is formed using the estimated states from the extended Kalman filter (EKF) and a nonlinear compensation realized by the radial basis function (RBF) neural network. The weights of RBF are trained to minimize the entropy of the closed loop tracking error. The convergence of RBF network is discussed and the stability of the resulting closed-loop control system is analysed in mean square sense. Finally, two numerical examples and a practical system simulation are given to illustrate the effectiveness of the proposed control method.

The Marginal Density of a TMA(1) Process

This note reconsiders the marginal density of a threshold moving average process and proposes a simple yet effective numerical algorithm to implement that by solving an associated integral equation. This algorithm can also be applied to calculate stationary probability density or distribution functions of a few other types of nonlinear stationary stochastic processes numerically.