Parametric estimation for sub-fractional Ornstein–Uhlenbeck process

Mendy, Ibrahima

doi:10.1016/j.jspi.2012.10.013

Cited by 39 publications

(21 citation statements)

References 35 publications

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“…We note that both the Wiener process and the FBM‐based models have stationary increments, despite being non‐stationary. To extend the range of applications in degradation modelling, we further substitute FBM with sub‐FBM, whose increments are still non‐stationary, and the memory effects are captured by the Hurst exponent though weakened partly . That is to say, sub‐FBM is more moderate than the FBM in the sense of temporal correlation.…”

Section: Degradation Modelling and Identificationmentioning

confidence: 99%

Remaining useful life prediction for fractional degradation processes under varying modes

Zhou

Chen

2019

Can J Chem Eng

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Degradation processes of practical chemical engineering systems are difficult to model accurately because of complicated nonlinearities, non-stationarities, and non-Markovian properties. Traditional prognostics techniques tend to neglect the multi-mode switching issue, and therefore cannot be used for predicting the remaining useful life (RUL) of a piece of long-life equipment, especially when it is continuously operated under varying modes. Specifically, the stationarity of differential data also plays an important role in depicting the evolutionary trend, and further affects the extrapolation procedure for RUL prediction. In this paper, we construct a multi-mode degradation model with regard to fractional diffusion processes by considering both stationary and non-stationary increments. The drift term is represented as a time-related nonlinear function, while the diffusion term is driven by fractional Brownian motion (FBM) or sub-fractional Brownian motion (sub-FBM), according to the results of the stationary test. Based on the monitored data, the model is identified by combining change-point detection and parameter estimation. The closed-form distribution of RUL is then derived through a weak convergence transformation. A numerical example and a case study of a large blast furnace are provided to evaluate the performance of the proposed prognostics framework. K E Y W O R D S degradation, multi-mode, non-Markovian, remaining useful life, stationary test

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Section: Degradation Modelling and Identificationmentioning

confidence: 99%

Remaining useful life prediction for fractional degradation processes under varying modes

Zhou

Chen

2019

Can J Chem Eng

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“…When G is not a fractional Brownian motion, the research for this question is very limited. For µ = 0 and G a sub-fractional Brownian motion, Mendy [13] considered the least squares estimation of θ and studied the consistency and asymptotic behavior. For µ = 0 and G a Gaussian process, El Machkouri et al [14] showed the strong consistency and the asymptotic distribution of the least squares estimatorθ of θ based on the properties of G, and as some examples, the authors also studied the three Vasicek-type models driven by fractional Brownian motion, sub-fractional Brownian motion, and bi-fractional Brownian motion, respectively.…”

Section: Introductionmentioning

confidence: 99%

Parametric Estimation in the Vasicek-Type Model Driven by Sub-Fractional Brownian Motion

Dong

2018

Algorithms

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In the paper, we tackle the least squares estimators of the Vasicek-type model driven by sub-fractional Brownian motion: d X t = ( μ + θ X t ) d t + d S t H , t ≥ 0 with X 0 = 0 , where S H is a sub-fractional Brownian motion whose Hurst index H is greater than 1 2 , and μ ∈ R , θ ∈ R + are two unknown parameters. Based on the so-called continuous observations, we suggest the least square estimators of μ and θ and discuss the consistency and asymptotic distributions of the two estimators.

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“…, nh n , where h n = T n . In [9,17] the so called sub-fractional Ornstein-Uhlenbeck process was studied, where the process B H t in (1) was replaced with a sub-fractional Brownian motion. In [9] the maximum likelihood estimator for such process was constructed, in [17] the estimator (3) was investigated in the case θ > 0.…”

Section: Introductionmentioning

confidence: 99%

“…In [9,17] the so called sub-fractional Ornstein-Uhlenbeck process was studied, where the process B H t in (1) was replaced with a sub-fractional Brownian motion. In [9] the maximum likelihood estimator for such process was constructed, in [17] the estimator (3) was investigated in the case θ > 0. The maximum likelihood drift parameter estimators for fractional Ornstein-Uhlenbeck process and even more general processes involving fBm with Hurst index from the whole interval (0, 1) were constructed and studied in [22].…”

Section: Introductionmentioning

confidence: 99%

Consistency of the drift parameter estimator for the discretized fractional Ornstein–Uhlenbeck process with Hurst index $H\in(0,\frac{1}{2})$

Kubilius¹,

Mishura²,

Ralchenko³

et al. 2015

Electron. J. Statist.

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We consider Langevin equation involving fractional Brownian motion with Hurst index H ∈ (0, 1 2 ). Its solution is the fractional Ornstein-Uhlenbeck process and with unknown drift parameter θ. We construct the estimator that is similar in form to maximum likelihood estimator for Langevin equation with standard Brownian motion. Observations are discrete in time. It is assumed that the interval between observations is n −1 , i.e. tends to zero (high frequency data) and the number of observations increases to infinity as n m with m > 1. It is proved that for positive θ the estimator is strongly consistent for any m > 1 and for negative θ it is consistent when m > 1 2H .60G22, 60F15, 60F25, 62F10, 62F12.

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Parametric estimation for sub-fractional Ornstein–Uhlenbeck process

Cited by 39 publications

References 35 publications

Remaining useful life prediction for fractional degradation processes under varying modes

Remaining useful life prediction for fractional degradation processes under varying modes

Parametric Estimation in the Vasicek-Type Model Driven by Sub-Fractional Brownian Motion

Consistency of the drift parameter estimator for the discretized fractional Ornstein–Uhlenbeck process with Hurst index $H\in(0,\frac{1}{2})$

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