Degradation Data Analysis based on Wiener Process with a Nonlinear Drift and a Stochastic Volatility

The degree of variation of trading prices with respect to time is volatility-measured by the standard deviation of returns. We present the estimation of stochastic volatility from the stochastic differential equation for evenly spaced data. We indicate that, the price process is driven by a semi-martingale and the data are evenly spaced. The results of Malliavin and Mancino [1] are extended by adding a compensated poisson jump that uses a quadratic variation to calculate volatility. The volatility is computed from a daily data without assuming its functional form. Our result is well suited for financial market applications and in particular the analysis of high frequency data for the computation of volatility.

ABSTRACTMany scientific disciplines are faced with the challenge of extracting meaningful information from large, complex and highly structured datasets. A significant portion of contemporary statistical research is dedicated to developing tools for handling such data. This paper introduces a functional linear regression model specifically designed for 3D facial shapes, which are viewed as manifolds. We propose a comprehensive framework that includes converting 3D facial data into functional objects, employing a functional principal component analysis method and utilising a function‐on‐scalar regression model. This framework facilitates computation for high‐dimensional data and is employed to investigate how individual traits, such as age and genetic ancestry, impact the diversity of human facial features.

Wavelet functional principal component analysis for batch process monitoring

We study a generalization of the Heston model, which consists of two coupled stochastic differential equations, one for the stock price and the other one for the volatility. We consider a cubic nonlinearity in the first equation and a correlation between the two Wiener processes, which model the two white noise sources. This model can be useful to describe the market dynamics characterized by different regimes corresponding to normal and extreme days. We analyze the effect of the noise on the statistical properties of the escape time with reference to the noise enhanced stability (NES) phenomenon, that is the noise induced enhancement of the lifetime of a metastable state. We observe NES effect in our model with stochastic volatility. We investigate the role of the correlation between the two noise sources on the NES effect.

Degradation tests are used when there is a quality characteristic related to the life of a product. In this paper, we investigate the model mis-specification effect on the estimation precision of product's mean time to failure (MTTF) and consider a degradation test design problem. The Wiener and inverse Gaussian (IG) processes are two possible models considered. We derive expressions for the mean and variance of the estimated product's MTTF when the true model is an IG process, but is wrongly fitted by a Wiener process. We further discuss the experimental design problem and derive the explicit functional form of the estimation variances. Using the derived functions, optimal degradation test plans assuming a Wiener process model is correctly or wrongly specified are both proposed. The derived plans are applied to a laser data example. We evaluate the test efficiency of the plans derived from a Wiener process assumption when the model is mis-specified. For many optimization criteria, we observe that the obtained plans are robust even when the fitted model is mis-specified. For some criteria that may result in very different test plans under different models, we use a weighted ratio criterion to find practically useful degradation plans under model uncertainty.

A generalized system reliability model based on survival signature and multiple competing failure processes

This paper discusses a time series model which has generalized long memory in the mean process with stochastic volatility errors and develops a new Bayesian posterior simulator that couples advanced posterior maximisation techniques, as well as traditional latent stochastic volatility estimation procedures. Details are provided on the estimation process, data simulation, and out of sample performance measures. We conduct several rigorous simulation studies and verify our results for in and out of sample behaviour. We further compare the goodness of fit of the generalized process to the standard long memory model by considering two empirical studies on the US Consumer Price Index (CPI) and the US equity risk premium (ERP).

Bessel processes with constant negative drift have recently appeared as heavy-traffic limits in queueing theory. We derive a closed-form expression for the spectral representation of the transition density of the Bessel process of orderν> −1 with constant driftμ≠ 0. Whenν> -½ andμ< 0, the first term of the spectral expansion is the steady-state gamma density corresponding to the zero principal eigenvalueλ0= 0, followed by an infinite series of terms corresponding to the higher eigenvaluesλn,n= 1,2,…, as well as an integral over the continuous spectrum aboveμ2/2. When −1 <ν< -½ andμ< 0, there is only one eigenvalueλ0= 0 in addition to the continuous spectrum. As well as applications in queueing, Bessel processes with constant negative drift naturally lead to two new nonaffine analytically tractable specifications for short-term interest rates, credit spreads, and stochastic volatility in finance. The two processes serve as alternatives to the CIR process for modelling mean-reverting positive economic variables and have nonlinear infinitesimal drift and variance. On a historical note, the Sturm–Liouville equation associated with Bessel processes with constant negative drift is closely related to the celebrated Schrödinger equation with Coulomb potential used to describe the hydrogen atom in quantum mechanics. Another connection is with D. G. Kendall's pole-seeking Brownian motion.

Reliability assessment has been widely used and studied as one of the important roles in equipment operation and maintenance. Multivariate competing degradation models have been extensively studied, but some non-competing multivariate degradation processes still exist in reality. In this paper, we present a method of reliability assessment based on Wiener process which considering non-competing relationship. The multivariate degradation process studied in this paper consists of two degradation indicators. The Wiener process was applied to model two degradation indicators separately. Bayesian method is used for parameter estimation of unknown parameters of degradation models and quantification of uncertainty. A case study of spool valve is used for the demonstration and validation of the model presented in this paper.

Synthetic aperture radar tomography (TomoSAR) is an important way of obtaining underlying topography and forest height for long-wavelength datasets such as L-band and P-band radar. It is usual to apply nonparametric spectral estimation methods with a large number of snapshots over forest areas. The nonparametric iterative adaptive approach for amplitude and phase estimation (IAA-APES) can obtain a high resolution; however, it only tends to work well with a small number of snapshots. To overcome this problem, this paper proposes the nonparametric iterative adaptive approach based on maximum likelihood estimation (IAA-ML) for the application over forest areas. IAA-ML can be directly used in forest areas, without any prior information or preprocessing. Moreover, it can work well in the case of a large number of snapshots. In addition, it mainly focuses on the backscattered power around the phase centers, helping to detect their locations. The proposed IAA-ML estimator was tested in simulated experiments and the results confirmed that IAA-ML obtains a higher resolution than IAA-APES. Moreover, six P-band fully polarimetric airborne SAR images were applied to acquire the structural parameters of a forest area. It was found that the results of the HH polarization are suitable for analyzing the ground contribution and the results of the HV polarization are beneficial when studying the canopy contribution. Based on this, the underlying topography and forest height of a test site in Paracou, French Guiana, were estimated. With respect to the Light Detection and Ranging (LiDAR) measurements, the standard deviation of the estimations of the IAA-ML TomoSAR method was 2.11 m for the underlying topography and 2.80 m for the forest height. Furthermore, compared to IAA-APES, IAA-ML obtained a higher resolution and a higher estimation accuracy. In addition, the estimation accuracy of IAA-ML was also slightly higher than that of the SKP-beamforming technique in this case study.

Based on empirical analysis of the Capesize and Panamax indices, we propose different continuous-time stochastic processes to model their dynamics. The models go beyond the standard geometric Brownian motion, and incorporate observed effects like heavy-tailed returns, stochastic volatility and memory. In particular, we suggest stochastic dynamics based on exponential Levy processes with normal inverse Gaussian distributed logarithmic returns. The Barndorff-Nielsen and Shephard stochastic volatility model is shown to capture time-varying volatility in the data. Finally, continuous-time autoregressive processes provide a class of models sufficiently rich to incorporate short-term persistence of freight rates. Our proposed dynamical models are fitted to market data. Finally, a numerical Value-at-Risk exercise is provided that illustrate the significance of using more sophisticated models than geometric Brownian motion.

A new age‐ and state‐dependent degradation process is proposed, which can be used in the case the degradation phenomenon under study is not necessarily monotonic increasing. In particular, the degradation increments of the proposed model are assumed to be possibly dependent on each other and negative. The model is obtained by generalizing the well‐known and largely applied Wiener process and partially preserves the mathematically tractability of the Wiener process. In the paper, the main features and properties of the proposed model are first discussed and the maximum likelihood estimators of its parameters are derived. An applicative example is finally developed, which shows the feasibility and the effectiveness of the proposed approach.

Reliability assessment method based on performance degradation data has been proven to be useful when dealing with products characterized as long-life and highly reliable. However, this method can hardly be applied when measurement of performance degradation is technically difficult or costly. To tackle this problem, this paper proposes to utilize the measurement data which are related to the performance degradation, namely, marker data. It is assumed that the degradation process is modeled with Wiener process, and that the marker is the integral of the performance degradation. We first estimate the parameters of the Wiener process using marker data and then conduct the reliability assessment. A simulated example is presented to demonstrate the proposed method.

Remaining useful life (RUL) prediction is a core technology for equipment operation and health management, and the degradation process of most equipment generally has a long-term correlation. To address this issue, we propose a method to quantify the uncertainty of RUL prediction based on a temporal convolutional network (TCN)-transformer- Wiener Process (WP) degradation model. We give a new generalized RUL prediction framework that employs TCN-Transformer to fit device degradation trends, which can adaptively adjust the fitting of degradation trends. WP is introduced, and the concept of first hitting time (FHT) gives an approximate expression for the probability density function (PDF) of the remaining lifetime of the model, further quantifying the uncertainty of the RUL prediction results. Finally, verification and analysis on the bearing dataset and the tool holder power head degradation dataset, and by comparing with other existing popular methods demonstrated the accuracy and stability of the proposed RUL prediction method. This innovative method provides a robust solution for RUL prediction, offering significant contributions to improving maintenance strategies and operational efficiency.

In this paper, a novel approach is proposed to determine target reliability index based on modeling performance degradation of concrete structures as a stochastic process. First, the Wiener process with drift is used to simulate the degradation process of relative dynamic elastic modulus of concrete, and the drift coefficient and diffusion coefficient of Wiener process are estimated according to the experimental data of degradation of relative dynamic elastic modulus. Then, based on the relations between concrete compressive strength, dynamic elastic modulus, and relative dynamic elastic modulus, the relationship between compressive strength and the amount of degradation of relative dynamic elastic modulus are deduced, and then, the statistical parameters of concrete compressive strength under different service time can be calculated. Finally, based on the established limit state equation of flexural capacity of the weak section and the statistical parameters of compressive strength at different times, the reliability index of a structure under different service life is calculated. On the other hand, according to the drift Wiener process model of relative dynamic elastic modulus degradation of concrete, the time‐varying failure probability of structural durability degradation is calculated based on the first‐passage failure probability formula of drift Wiener process. Taking a concrete plant structure as an example, the time‐varying reliability index calculated based on the limit state equation and the time‐varying failure probability obtained based on the first‐passage failure probability analysis of drift Wiener process are compared, and then the recommended values of target reliability index of structural bearing capacity are comprehensively determined.