Gamma Process Research Articles

Degradation modeling for reliability estimation of DC film capacitors subject to humidity acceleration

To investigate the lifetime of DC film capacitors subject to humidity environment, this paper proposes a degradation model to extract reliability characteristics from performance degradation measures. The percentage of capacitance loss of DC film capacitors is driven by a non-stationary gamma process and the component heterogeneity is characterized by random effects. Based on the model, the method of maximum likelihood for the model parameter estimation and the reliability characteristics including the probability density function of failure time and its confidence interval are developed. The effectiveness and superiority of the proposed method are illustrated and verified by comparing with the conventional failure time analysis method. The results indicate that the proposed method is capable of obtaining comparable lifetime prediction with reduced accelerated testing time.

ON SPREAD OPTION PRICING USING TWO-DIMENSIONAL FOURIER TRANSFORM

Spread options are multi-asset options with payoffs dependent on the difference of two underlying financial variables. In most cases, analytically closed form solutions for pricing such payoffs are not available, and the application of numerical pricing methods turns out to be nontrivial. We consider several such nontrivial cases and explore the performance of the highly efficient numerical technique of Hurd & Zhou[(2010) A Fourier transform method for spread option pricing, SIAM J. Financial Math. 1(1), 142–157], comparing this with Monte Carlo simulation and the lower bound approximation formula of Caldana & Fusai[(2013) A general closed-form spread option pricing formula, Journal of Banking & Finance 37, 4893–4906]. We show that the former is in essence an application of the two-dimensional Parseval’s Identity. As application examples, we price spread options in a model where asset prices are driven by a multivariate normal inverse Gaussian (NIG) process, in a three-factor stochastic volatility model, as well as in examples of models driven by other popular multivariate Lévy processes such as the variance Gamma process, and discuss the price sensitivity with respect to volatility. We also consider examples in the fixed-income market, specifically, on cross-currency interest rate spreads and on LIBOR/OIS spreads.

On Bayesian Tracking and Prediction of Radar Cross Section

We consider the problem of Bayesian tracking of radar cross section. The adopted observation model employs the gamma family, which covers all Swerling cases in a unified framework. State dynamics are modeled using a nonstationary autoregressive gamma process. The principal component of the proposed solution is a nontrivial gamma approximation, applied during the time update recursion. The superior performance of the proposed approach is confirmed using simulations and a real-world dataset.

The disorder problem for purely jump Lévy processes with completely monotone jumps

We study the problem of detecting as quickly as possible the disorder time at which a purely jump Lévy process changes its probabilistic features. Assuming that its jumps are completely monotone, the monitored process is approximated by a sequence of hyperexponential processes. Then, the solution to the disorder problem for a hyperexponential process is used to approximate the one of the original problem. The efficiency of the proposed approximation scheme is investigated for some popular Lévy processes, such as the gamma, inverse Gaussian, variance-gamma and CGMY processes.

Minimal relative entropy for equivalent martingale measures by low-discrepancy sequence in Lévy process

In this paper, we focused on computing the minimal relative entropy between the original probability and all of the equivalent martin gale measure for the Lévy process. For this purpose, the quasiMonte Carlo method is used. The probability with minimal relative entropy has many suitable properties. This probability has the minimal Kullback-Leibler distance to the original probability. Also, by using the minimal relative entropy the exponential utility indifference price can be found. In this paper, the Monte Carlo and quasi-Monte Carlo methods have been applied. In the quasi-Monte Carlo method, two types of widely used lowdiscrepancy sequences, Halton sequence and Sobol sequence, are used. These methods have been used for exponential Lévy process such as variance gamma and CGMY process. In these two processes, the minimal relative entropy has been computed by Monte Carlo and quasi-Monte Carlo, and compared their results. The results show that quasi-Monte Carlo with Sobol sequence performs better in terms of fast convergence and less error. Finally, this method by fitting the variance gamma model and parameters estimation for the model has been implemented for financial data and the corresponding minimal relative entropy has been computed.

The first and second order moment structure of an inhomogeneous gamma point process

The nonhomogeneous Poisson process (NHPP) and the Gamma renewal process (GRP) are two stochastic models which are commonly used to describe the pattern of repeated occurrence data. An inhomogeneous Gamma process (IGP) is a point process which generalizes both the NHPP and GRP. This paper focuses on establishing the first and second order moment structure for an IGP.

Prediction of Expected Residual Useful Life of Rubble-Mound Breakwaters Using Stochastic Gamma Process

Prediction of Expected Residual Useful Life of Rubble-Mound Breakwaters Using Stochastic Gamma Process

Maintenance policy for a system with a weighted linear combination of degradation processes

This paper develops maintenance policies for a system under condition monitoring. We assume that a number of defects may develop and the degradation process of each defect follows a gamma process. The system is said failed if a linear combination of the degradation processes exceeds a pre-specified threshold. Preventive maintenance is performed. The system is renewed after several preventive maintenance activities have been performed. The main objective of this paper is to optimise the time between preventive maintenance actions and the number of the preventive maintenance. Numerical examples are given to illustrate the results.

Hierarchical Space-Time Modeling of Asymptotically Independent Exceedances With an Application to Precipitation Data

The statistical modeling of space-time extremes in environmental applications is key to understanding complex dependence structures in original event data and to generating realistic scenarios for impact models. In this context of high-dimensional data, we propose a novel hierarchical model for high threshold exceedances defined over continuous space and time by embedding a space-time Gamma process convolution for the rate of an exponential variable, leading to asymptotic independence in space and time. Its physically motivated anisotropic dependence structure is based on geometric objects moving through space-time according to a velocity vector. We demonstrate that inference based on weighted pairwise likelihood is fast and accurate. The usefulness of our model is illustrated by an application to hourly precipitation data from a study region in Southern France, where it clearly improves on an alternative censored Gaussian space-time random field model. While classical limit models based on threshold-stability fail to appropriately capture relatively fast joint tail decay rates between asymptotic dependence and classical independence, strong empirical evidence from our application and other recent case studies motivates the use of more realistic asymptotic independence models such as ours. Supplementary materials for this article, including a standardized description of the materials available for reproducing the work, are available as an online supplement.

A new class of survival distribution for degradation processes subject to shocks

Many systems experience gradual degradation while simultaneously being exposed to a stream of random shocks of varying magnitudes that eventually cause failure when a shock exceeds the residual strength of the system. In this paper, we present a family of stochastic processes, called shock-degradation processes, that describe this failure mechanism. In our failure model, system strength follows a geometric degradation process. The degradation process itself is any Lévy process, that is, any stochastic process with stationary independent increments. The shock stream is a Fréchet stochastic process, a process derived from the Fréchet extreme-value distribution. Finally, the shock-degradation process is a convolution of the Fréchet shock process and any one of the candidate degradation processes. The system fails at the first occasion when a shock takes system strength across a threshold at zero. The paper presents results for Wiener diffusion processes and gamma processes as examples of Lévy degradation processes. The paper develops key statistical properties of the process model and its survival distribution, including several that are important for its practical application. As the failure mechanism is a first hitting time event, applications that require regression structures fall within the domain of threshold regression methodology.

Exponential Dispersion Process for Degradation Analysis

The methods for analyzing degradation data are usually based on a specific degradation model. However, this may result in a large bias to estimate the reliability of the product when the assumed model is wrong. In this paper, a new stochastic process, called exponential dispersion process, is developed to describe the degradation path of the product's physical or chemical characteristics. Exponential dispersion process includes the most used degradation models (Wiener process, gamma process, inverse Gaussian process, and compound Poisson process) as special cases. Thus, it is useful for suggesting an appropriate degradation model for a specific dataset. We investigate the exponential dispersion model with nonlinear degradation path, and use approximated maximum likelihood method and bootstrap approach to obtain the point and confidence interval estimates of the unknown model parameters. Furthermore, exponential dispersion process models with acceleration factors and random effects are also explored. The developed methodologies are then applied to real data analysis.

The Duration of Load Effect in Lumber as Stochastic Degradation

This paper proposes a Bayesian framework for applying a gamma process model for reliability analysis in structural engineering under time–varying stochastic loads. We focus on an important application to engineered wood products and thereby demonstrate how the proposed methodology can be implemented in practice. We use a gamma process to characterize the degradation over time in each piece of lumber, allowing for variability between pieces when external forces from loads are applied to a population of lumber. We propose a model for the shape parameter to accommodate time–varying loads. Our approach is compared with traditional accumulated damage models. Our statistical analysis of experimental data highlight the limitations of using accelerated testing to assess long–term reliability, as seen in the wide posterior intervals. This suggests the need for more comprehensive testing in future applications, or to encode appropriate expert knowledge in the priors used for Bayesian analysis.

Time-variant reliability analysis of a continuous system with strength deterioration based on subset simulation

To conduct a reliability analysis for mechanical components, it is necessary to consider the combined influence of strength deterioration and dynamic loads. An efficient method based on subset simulation is proposed in this paper to analyze time-variant reliability by considering the strength deterioration of mechanical components in a continuous system. A gamma process is used to describe the deterioration of system strength. A model for time-variant reliability considering strength deterioration is constructed for a continuous system. A representative example and tubular cantilever structure are assessed to demonstrate the efficiency and accuracy of the proposed method. The reliability probability examples were analyzed using a first-order reliability method and benchmark results for the proposed method were derived using direct Monte Carlo simulation (MCS). The results of the proposed method and MCS are consistent, indicating that the proposed method is an effective reliability analysis method for evaluating small failure probabilities in a continuous system subjected to strength deterioration and dynamic loads.

Linnik Lévy process and some extensions

In the literature, the Linnik, Mittag-Leffler, Laplace and asymmetric Laplace distributions are the most known examples of geometric stable distributions. The geometric stable distributions are especially useful in the modeling of leptokurtic data with heavy-tailed behavior. They have found many interesting applications in the modeling of several physical phenomena and financial time-series. In this paper, we define the Linnik Lévy process (LLP) through the subordination of symmetric stable Lévy motion with gamma process. We discuss main properties of LLP like probability density function, Lévy measure and asymptotic forms of marginal densities. We also consider the governing fractional-type Fokker–Planck equation. To show practical applications, we simulate the sample paths of the introduced process. Moreover, we give a step-by-step procedure of the parameters estimation and calibrate the parameters of the LLP with the Arconic Inc equity data taken from Yahoo finance. Further, some extensions of the introduced process are also discussed.

Gamma process based optimal mission abort policy

Various safety-critical systems such as aircrafts, submarines and space stations are required to perform missions continually in harsh environment. To enhance the survivability of such systems, a mission can be aborted when the mission completion becomes problematic and a rescue procedure is immediately initiated. This paper investigates the optimal mission abort policy based on a two-stage Gamma process which can be separated into normal and defective stages. Degradation-based abort policy where a mission is aborted if the system degradation level is higher than a threshold and duration-based abort policy where a mission is aborted if the duration in defective stage is larger than a threshold are considered. The mission success probability and system survivability under these two policies are evaluated and the optimal mission abort thresholds balancing the tradeoff between the mission success probability and system survivability are investigated. A numerical example on an unmanned aerial vehicle (UAV) is presented to illustrate the applicability of the proposed abort policies.

Planning gamma accelerated degradation tests with two accelerating variables

AbstractGamma accelerated degradation tests (ADT) are widely used to assess timely lifetime information of highly reliable products with degradation paths that follow a gamma process. In the existing literature, there is interest in addressing the problem of deciding how to conduct an efficient, ADT that includes determinations of higher stress‐testing levels and their corresponding sample‐size allocations. The existing results mainly focused on the case of a single accelerating variable. However, this may not be practical when the quality characteristics of the product have slow degradation rates. To overcome this difficulty, we propose an analytical approach to address this decision‐making problem using the case of two accelerating variables. Specifically, based on the criterion of minimizing the asymptotic variance of the estimated q quantile of lifetime distribution of the product, we analytically show that the optimal stress levels and sample‐size allocations can be simultaneously obtained via a general equivalence theorem. In addition, we use a practical example to illustrate the proposed procedure.

Copula-based reliability analysis of gamma degradation process and Weibull failure time

Purpose The purpose of this paper is the reliability analysis for systems with dependent gamma degradation process and Weibull failure time. Design/methodology/approach Consider a life testing experiment in which a sample of n devices starts to operate at t=0 and the data are available on failure time and failure-evolving process on each individual, called in some contents wear or degradation. Ignoring the between performance characteristics dependency structure may lead us to different reliability estimations, while the dependency justly exists. In previous research, dependency between the degradation process and hard failure time has been studied in limited detail (special closed form expression). Thereafter, the dependency between two degradation processes with the same structure (gamma process) in a system is considered using the copula function. Findings The results indicate that ignoring the dependency structure may lead us to different reliability estimations while the dependency justly exists. Originality/value This study gives some contributions that evaluate reliability metrics with more than one failure mechanism that may not be independent and possibly follow a different distribution function. The authors have used the copula function as a basis to develop a proposal model and analysis methods. In addition, the authors discussed the identifiability of the copula. Finally, simulation data were used to review the suggested approach.

Joint optimal production planning and proactive maintenance policy for a system subject to degradation

PurposeThis paper integrates condition-based maintenance (CBM) with production planning in a single-stage production system that deteriorates with usage during a specified finite planning horizon. The purpose of this paper is to develop an integrated production and maintenance model to minimize the expected total cost over the horizon.Design/methodology/approachA joint production planning and CBM model is proposed. In the model, a set of products must be produced in lots. The system degradation is a stationary gamma process and the degradation level is detected by inspection between production lots. Maintenance actions including imperfect preventive maintenance (PM) should be taken when the failure risk exceeds the maintenance threshold. A fix-iterative heuristic algorithm is proposed to address the joint model.FindingsThe proactive policy expressed as a prognosis maintenance threshold is introduced to integrate CBM with batch production perfectly. Experiments are carried out to conduct sensitivity analysis, which provides some insights to facilitate industrial manufacturing. The superiority of the proposed joint model compared with a separate decision method is demonstrated. The results show an advantage in cost saving.Originality/valueFew studies have been made to integrate production planning and CBM decisions, especially for a multi-product system. Their maintenance decisions are usually based on a periodic review policy, which is not appropriate for batch production system. A prognosis maintenance threshold based on system condition and production quantity is suitable for the integrated decisions. Moreover, the imperfect PM is taken into consideration in this paper. A fix-iterative algorithm is developed to solve the joint model. This work forms a proactive maintenance for batch production.

Truncated random measures

Completely random measures (CRMs) and their normalizations are a rich source of Bayesian nonparametric priors. Examples include the beta, gamma, and Dirichlet processes. In this paper, we detail two major classes of sequential CRM representations—series representations and superposition representations—within which we organize both novel and existing sequential representations that can be used for simulation and posterior inference. These two classes and their constituent representations subsume existing ones that have previously been developed in an ad hoc manner for specific processes. Since a complete infinite-dimensional CRM cannot be used explicitly for computation, sequential representations are often truncated for tractability. We provide truncation error analyses for each type of sequential representation, as well as their normalized versions, thereby generalizing and improving upon existing truncation error bounds in the literature. We analyze the computational complexity of the sequential representations, which in conjunction with our error bounds allows us to directly compare representations and discuss their relative efficiency. We include numerous applications of our theoretical results to commonly-used (normalized) CRMs, demonstrating that our results enable a straightforward representation and analysis of CRMs that has not previously been available in a Bayesian nonparametric context.

Analysis of return on investment in some Niger-Delta oil field projects: using the variance-gamma process

Oil field development projects face market risks, largely because the parameter of key importance, the oil price, fluctuates rapidly over time. The decision to invest or not in an oil field project is therefore very challenging, because information concerning the field is often scarce. Neither the future production, nor sales prices are known with certainly. Constraints on production level also exist, in addition to OPEC quota, which is used in this work as proxy for other production limitations. The price process that best describes the fluctuations in oil price is expected to yield better analysis with respect to expected returns (viability of the project). The Variance Gamma process which is a subordinated Lévy process effectively captures jumps in a process. Thus, Variance Gamma process and a mean reverting process are considered in the analysis. The expected revenue was found to be higher, when the variance Gamma process is used as the model for oil price. This indicates that the variance Gamma process performs better than the Ornstein-Uhlenbeck process as a model for oil price. It therefore provides a good basis for price forecasting, and optimality of investment decision.Keywords: Variance Gamma Process, Mean Reverting Process, Parameters, Niger-Delta Oil Field Projects, Expected Return, OPEC constraint