Non-homogeneous Process Research Articles

A bootstrap test of independence between three temporal nonhomogeneous Poisson processes and its application to heat wave modeling

The main contribution of this work is a bootstrap test to check the independence between temporal nonhomogeneous Poisson processes. The test statistic is based on the close point relation, which adapts the crossed nearest neighbour distance ideas of spatial point processes to the case of nonhomogeneous time point processes. Since it is complicated to obtain the probability distribution of the test statistic under the null hypothesis and the parameters of the processes are usually unknown, the \(p\) value is obtained using a parametric bootstrap approach. A simulation study shows that the size of the test is close to the nominal one. The power is analyzed considering three approaches for generating dependent nonhomogeneous processes and different levels of dependence, and satisfactory results are obtained in all cases. Although the test was initially intended for Poisson processes, it can be applied to any type of point process in one dimension which can be simulated. This test is a valuable tool in the validation analysis of common Poisson shock models. For the bivariate case, the process can be decomposed into three independent Poisson processes, and the assumption of independence between them has to be checked. As an application, the joint modeling of the occurrence process of extreme heat events in daily maximum and minimum temperatures is described.

Dynamic Response of Bridge Subjected to Eccentrically Moving Flexible Vehicle: A Semianalytical Approach

Dynamic response of a single span bridge subjected to moving flexible vehicles has been studied using a semianalytical approach. The eccentricity of vehicle path giving rise to torsional motion of the bridge has been incorporated in the approach. The bridge surface irregularity has been considered as the nonhomogeneous process in spatial domain. A closed form expression has been derived to generate response samples corresponding to each input of roughness profile to form an ensemble. Thereafter, averaging across the ensemble has been carried out at each time step to determine mean and standard deviation of bridge and vehicle response. Further, dynamic amplification factor (DAF) of the bridge response has been obtained for several combinations of bridge-vehicle parameters. The study reveals that structural bending modes of vehicle can significantly reduce dynamic response of the bridge. The eccentricity of vehicle path and flexural/torsional rigidity ratios plays a significant role in dynamic amplification of bridge response.

Temporal and Spatial Evolution of Bursts in a Fiber Bundle Model of Creep Rupture

We present a theoretical study of the creep rupture of heterogeneous materials based on a fiber bundle model which provides a direct connection between the microscopic fracture mechanisms and the macroscopic time evolution. In the model, material elements fail either due to immediate breaking or undergo a damage accumulating ageing process. We found that on the micro-level the competition of the two failure modes gives rise to bursts of breakings with power law distributed size and waiting time between events. We demonstrate that approaching macroscopic failure the system accelerates which can be fully described as a non-homogeneous Poissonian process for long range load sharing, however, when localization occurs breaking events get clustered. Bursts are composed of sub-avalanches which lead to a non-trivial temporal shape comparable to measurements. The pulse shape proved to be sensitive to the range of load sharing.

The Distributions and Moments of the First Entrance Time for Nonhomogeneous (H,Q) Process

In this paper, we discuss one kind of Markov Skeleton processes, nonhomogeneous (H,Q) processes. We mainly study their distributions and moments of the first entrance time, and we obtain their recursive formula and some related properties.

The Distributions and Moments of the First Entrance Time for Nonhomogeneous Semi-Markov Process

In this paper we discuss one kind of nonhomogeneous (H,Q) processes, Nonhomogeneous Semi-Markov Process. We get the distributions and moments of the first entrance time for this process. And furthermore, we obtain their recursive formula.

Addressing Gene Tree Discordance and Non-Stationarity to Resolve a Multi-Locus Phylogeny of the Flatfishes (Teleostei: Pleuronectiformes)

Non-homogeneous processes and, in particular, base compositional non-stationarity have long been recognized as a critical source of systematic error. But only a small fraction of current molecular systematic studies methodically examine and effectively account for the potentially confounding effect of non-stationarity. The problem is especially overlooked in multi-locus or phylogenomic scale analyses, in part because no efficient tools exist to accommodate base composition heterogeneity in large data sets. We present a detailed analysis of a data set with 20 genes and 214 taxa to study the phylogeny of flatfishes (Pleuronectiformes) and their position among percomorphs. Most genes vary significantly in base composition among taxa and fail to resolve flatfish monophyly and other emblematic groups, suggesting that non-stationarity may be causing systematic error. We show a strong association between base compositional bias and topological discordance among individual gene partitions and their inferred trees. Phylogenetic methods applying non-homogeneous models to accommodate non-stationarity have relatively minor effect to reduce gene tree discordance, suggesting that available computer programs applying these methods do not scale up efficiently to the data set of modest size analysed in this study. By comparing phylogenetic trees obtained with species tree (STAR) and concatenation approaches, we show that gene tree discordance in our data set is most likely due to base compositional biases than to incomplete lineage sorting. Multi-locus analyses suggest that the combined phylogenetic signal from all loci in a concatenated data set overcomes systematic biases induced by non-stationarity at each partition. Finally, relationships among flatfishes and their relatives are discussed in the light of these results. We find support for the monophyly of flatfishes and confirm findings from previous molecular phylogenetic studies suggesting their close affinity with several carangimorph groups (i.e., jack and allies, barracuda, archerfish, billfish and swordfish, threadfin, moonfish, beach salmon, and snook and barramundi).

Neyman, Markov processes and survival analysis

J. Neyman used stochastic processes extensively in his applied work. One example is the Fix and Neyman (F-N) competing risks model (1951) that uses finite homogeneous Markov processes to analyse clinical trials with breast cancer patients. We revisit the F-N model, and compare it with the Kaplan-Meier (K-M) formulation for right censored data. The comparison offers a way to generalize the K-M formulation to include risks of recovery and relapses in the calculation of a patient's survival probability. The generalization is to extend the F-N model to a nonhomogeneous Markov process. Closed-form solutions of the survival probability are available in special cases of the nonhomogeneous processes, like the popular multiple decrement model (including the K-M model) and Chiang's staging model, but these models do not consider recovery and relapses while the F-N model does. An analysis of sero-epidemiology current status data with recurrent events is illustrated. Fix and Neyman used Neyman's RBAN (regular best asymptotic normal) estimates for the risks, and provided a numerical example showing the importance of considering both the survival probability and the length of time of a patient living a normal life in the evaluation of clinical trials. The said extension would result in a complicated model and it is unlikely to find analytical closed-form solutions for survival analysis. With ever increasing computing power, numerical methods offer a viable way of investigating the problem.

A Bayesian Phylogeographical Analysis of Type 1 Porcine Reproductive and Respiratory Syndrome Virus (PRRSV)

Understanding viral transmission is an important factor for the effective prevention one of the most devastating swine diseases, porcine reproductive and respiratory syndrome. Focusing on molecular epidemiology of type 1 PRRSV, this study analysed a large ORF5 dataset collected worldwide from 1991 to 2012 using a coalescent-based Bayesian Markov chain Monte Carlo approach. The results suggested that the virus diversified into unique subpopulations in Russia & Belarus and Italy approximately 100years ago. Previously unreported consecutive diffusions of the virus were identified, which showed that some countries, such as Spain and Germany, acted as distribution sources to some extent. This study also provided statistical evidence for the existence of an ORF5-based phylogeographical structure of type 1 PRRSV, in which the virus tended to cluster by geographical locations more tightly than expected by chance. In contrast to this tight geographical structure, the evolution of the ORF5 gene, based on mapping of non-synonymous/synonymous substitutions, was best described by a non-homogeneous process that could be implicated as a mechanism for viral immune evasion.

Extended optimal replacement policy for a system subject to non-homogeneous pure birth shocks

This article studies the optimal replacement policy with general repairs for an operating system subject to shocks occurring to a non-homogeneous pure birth process (NHPBP). A shock causes that the system experiences one of two types of failures: type-I failure (minor failure) is rectified by a general repair, or type-II failure (catastrophic failure) is removed by an unplanned replacement. The probabilities of these two types of failures depend on the number of shocks since the last replacement. We consider a bivariate replacement policy (n,T) under which the system is replaced at planned life age T, or at the nth type-I failure, or at any type-II failure, whichever occurs first. The optimal replacement schedule which minimizes the expected cost rate model is derived analytically and discussed numerically.

Monounireducible Nonhomogeneous Continuous Time Semi-Markov Processes Applied to Rating Migration Models

Monounireducible nonhomogeneous semi- Markov processes are defined and investigated. The mono- unireducible topological structure is a sufficient condition that guarantees the absorption of the semi-Markov process in a state of the process. This situation is of fundamental importance in the modelling of credit rating migrations because permits the derivation of the distribution function of the time of default. An application in credit rating modelling is given in order to illustrate the results.

A Block Replacement Policy for Systems Subject to Non-homogeneous Pure Birth Shocks

This note studies the block replacement policy with general repairs for an operating system subject to shocks occurring according to a non-homogeneous pure birth process. A shock causes the system to fail. There are two types of failures: a type-I failure (minor failure) is fixed by a general repair, whereas a type-II failure (catastrophic failure) is removed by an unplanned (or unscheduled) replacement. The failure type probabilities depend on the number of type-I failure shocks that occurred since the last replacement. Under the block replacement policy, the operating system is replaced every time units to reduce the chances of more expensive unplanned replacements due to type-II failures. The aim of this note is to determine the optimal block interval T*, which minimizes the expected cost rate and the expected total discounted cost rate of the proposed policy. As the shock process is a more general non-homogeneous pure birth process, several previous models become the special cases of our model.

Discrete Time Non-Homogeneous Semi-Markov Processes Applied to Models for Disability Insurance

In this paper, we present a stochastic model for disability insurance contracts. The model is based on a discrete time non-homogeneous semi-Markov process (DTNHSMP) to which the backward recurrence time process is introduced. This permits a more exhaustive study of disability evolution and a more efficient approach to the duration problem. The use of semi-Markov reward processes facilitates the possibility of deriving equations of the prospective and retrospective mathematical reserves. The model is applied to a sample of contracts drawn at random from a mutual insurance company.

Mean and variance of first passage time of non-homogeneous random walk

The prime concern of this paper is the first passage time of a non-homogeneous random walk, which is nearest neighbor but able to stay at its position. It is revealed that the branching structure of the walk corresponds to a 2-type non-homogeneous branching process and the first passage time of the walk can be expressed by that branching process. Therefore, one can calculate the mean and variance of the first passage time, though its exact distribution is unknown.

The Delay-Time Maintenance Optimization Model with Two Failure Modes

The concept of delay time and its maintenance theory are introduced, considering failure modes of system have different probability of occurrence and repair time, two revised model are established with the defects follow homogeneous processes and non-homogeneous process, The revised model could improve the expected unit downtime and optimize inspection interval.

A Necessary Characteristic Equation of Diffusion Processes Having Gaussian Marginals

The aim of this work is to characterize one‐dimensional homogeneous diffusion process, under the assumption that marginal density of the process is Gaussian. The method considers the forward Kolmogorov equation and Fourier transform operator approach. The result establishes the necessary characteristic equation between drift and diffusion coefficients for homogeneous and nonhomogeneous diffusion processes. The equation for homogeneous diffusion process leads to characterize the possible diffusion processes that can exist. Two well‐known examples using the necessary characteristic equation are also given.

Maximum Likelihood Estimators for a Supercritical Branching Diffusion Process

The log-likelihood of a nonhomogeneous Branching Diffusion Process under several conditions assuring existence and uniqueness of the diffusion part and nonexplosion of the branching process. Expressions for different Fisher information measures are provided. Using the semimartingale structure of the process and its local characteristics, a Girsanov-type result is applied. Finally, an Ornstein-Uhlenbeck process with finite reproduction mean is studied. Simulation results are discussed showing consistency and asymptotic normality.

A comparison of non-homogeneous Markov regression models with application to Alzheimer's disease progression

Markov regression models are useful tools for estimating risk factor effects on transition rates between multiple disease states. Alzheimer's disease (AD) is an example of a multi-state disease process where great interest lies in identifying risk factors for transition. In this context, non-homogeneous models are required because transition rates change as subjects age. In this report we propose a non-homogeneous Markov regression model that allows for reversible and recurrent states, transitions among multiple states between observations, and unequally spaced observation times. We conducted simulation studies to compare performance of estimators for covariate effects from this model and alternative models when the underlying non-homogeneous process was correctly specified and under model misspecification. In simulation studies, we found that covariate effects were biased if non-homogeneity of the disease process was not accounted for. However, estimates from non-homogeneous models were robust to misspecification of the form of the non-homogeneity. We used our model to estimate risk factors for transition to mild cognitive impairment (MCI) and AD in a longitudinal study of subjects included in the National Alzheimer's Coordinating Center's Uniform Data Set. We found that subjects with MCI affecting multiple cognitive domains were significantly less likely to revert to normal cognition.

Marine traffic risk modelling – an innovative approach and a case study

This paper presents a model to analyse the risk of two common marine accidents: collision and grounding. Attention is focused on oil tankers since they pose the highest environmental risks. A case study in selected areas of the Gulf of Finland in ice-free conditions is presented. The model utilizes a formula for risk calculation that considers both the probability of an unwanted event and its consequences. The model can be decomposed into a block representation in which blocks for the probability of a collision, probability of a grounding event, and the consequences of an accident are linked. The probability of vessel colliding is assessed in terms of a minimum-distance-to-collision-based model. The model defines the collision zone using a mathematical ship motion model and considers the traffic flow to be a non-homogeneous process. Calculations are performed using data for traffic flows in the Gulf of Finland with particular attention being paid to the crossing of the channel used by scheduled ferries between Helsinki and Tallinn, and the main shipping channel. For the assessment of a grounding probability, a new approach is proposed, which utilizes a gravity-like model, where a ship and navigational obstructions are perceived as interacting objects and their repulsion is modelled by a formulation inspired by gravitational force. The considered situation in this case is the movement of oil tankers in the approach channel to an oil terminal at Sköldvik, near Helsinki. The consequences of an accident are expressed in monetary terms, and concern the costs of cleaning up an oil spill, based on the statistics of compensation levels claimed from the International Oil Pollution Compensation Fund.

Occurrence analysis of daily rainfalls through non-homogeneous Poissonian processes

Abstract. A stochastic model based on a non-homogeneous Poisson process, characterised by a time-dependent intensity of rainfall occurrence, is employed to explain seasonal effects of daily rainfalls exceeding prefixed threshold values. The data modelling has been performed with a partition of observed daily rainfall data into a calibration period for parameter estimation and a validation period for checking on occurrence process changes. The model has been applied to a set of rain gauges located in different geographical areas of Southern Italy. The results show a good fit for time-varying intensity of rainfall occurrence process by 2-harmonic Fourier law and no statistically significant evidence of changes in the validation period for different threshold values.

Monte carlo model in metal recrystallization simulation

Some new advices are proposed for Monte Carlo model of recrystallization annealing simulation, including an additional recovery process, a new stored energy distribution, a more advanced nucleation model and a conversion probability of site orientation. Then a Monte Carlo model is established absorbing the above improvements, and used to isothermal annealing simulation of 1060 pure aluminum cold-rolled sheet. For contrast, a recrystallization annealing experiment of 1060 pure aluminum is carried out at the same time. The results show that the 1060 aluminum alloy in the annealing experiment produces obvious recrystallization, and the grain morphology of recrystallization is similar to isometric. The new model can effectively simulate the nonhomogeneous nucleation process and the microstructure evolution of 1060 industrial pure aluminum in annealing, and the simulated recrystallization kinetics curve is similar to the theoretical curve. The Avrami exponent coincides with the experimental result but is lower than the theoretical value of 2. As the Monte Carlo model does not consider the preferential growth orientation, some simulated results are not completely consistent with that from experiments.