non-Gaussian Distribution Research Articles

Propagation of material uncertainty in modal parameters and its influence in damage quantification of shear buildings

Structural health monitoring (SHM) plays a crucial role in post-disaster mitigation in investigating the serviceability of an existing structure. The physical properties, namely mass density, cross-sectional area, Poisson’s ratio, modulous of elasticity, etc., vary spatially throughout a real system. These uncertainties propagate in the modal parameters involved in SHM techniques. In this study, the elastic modulous has been considered the primary source of uncertainty. A detailed literature survey has been performed to identify the suitable probability distribution recommended for defining material uncertainty. Although the Gaussian distribution cannot replicate realistic material behaviour, its explicit implementation makes it prevalent. However, based on the principle of maximum entropy, the gamma distribution is the most appropriate recommendation among all non-Gaussian continuous distributions ranging [0, ∞). The Karhunen–Loève (KL) expansion has been adopted to discretize the stochastic field to determine the spectral mass and stiffness of a multi-damaged axially vibrating prismatic cantilever bar-type structure. Closed-form orthogonal pairs of eigensolutions of the KL expansion considering (Adhikari and Friswell, 2010) exponential covariance function have been derived. The expressions for spectral mass and stiffness for a multi-damaged cantilever bar considering material uncertainty have been developed. The uncertainty propagation in the modal properties has been estimated. The complete procedure has been followed to identify uncertainty propagation in damage severity of a common class of civil structures like shear buildings. The influence of material uncertainty in determining the damage severity of a structure with numerable degrees of freedom using a non-destructive approach has been explored. Extensive simulations have been performed to evaluate the effectiveness of the present study in a multi-damaged bar as well as shear building. Statistical parameters, namely mean and standard deviation of natural frequencies, modal displacement and damage severity have been determined. The uncertainty propagation has been reflected in the mean value for varying input stochasticity. However, the standard deviation remained consistent for undamaged as well as damaged structure irrespective of material uncertainty type. Considering gamma distribution in material stochasticity, the spread of uncertainty propagation has been more than the Gaussian distribution. However, experimental studies need to be performed further to investigate the stochastic behaviour in modal parameters of a real structure.

Temperature-mediated diffusion of nanoparticles in semidilute polymer solutions.

The temperature is often a critical factor affecting the diffusion of nanoparticles in complex physiological media, but its specific effects are still to be fully understood. Here, we constructed a temperature-regulated model of semidilute polymer solution and experimentally investigated the temperature-mediated diffusion of nanoparticles using the particle tracking method. By examining the ensemble-averaged mean square displacements (MSDs), we found that the MSD grows gradually as the temperature increases while the transition time from sublinear to linear stage in MSD decreases. Meanwhile, the temperature-dependent measured diffusivity of the nanoparticles shows an exponential growth. We revealed that these temperature-mediated changes are determined by the composite effect of the macroscale property of polymer solution and the microscale dynamics of polymer chain as well as nanoparticles. Furthermore, the measured non-Gaussian displacement probability distributions were found to exhibit non-Gaussian fat tails, and the tailed distribution is enhanced as the temperature increases. The non-Gaussianity was calculated and found to vary in the same trend with the tailed distribution, suggesting the occurrence of hopping events. This temperature-mediated non-Gaussian feature validates the recent theory of thermally induced activated hopping. Our results highlight the temperature-mediated changes in diffusive transport of nanoparticles in polymer solutions and may provide the possible strategy to improve drug delivery in physiological media.

Novel reduced kernel independent component analysis for process monitoring

Kernel independent component analysis (KICA), as a nonlinear extension monitoring method of independent component analysis (ICA), has attracted significant attention. To accomplish different monitoring tasks for nonlinear systems with non-Gaussian data distribution, many modified algorithms based on KICA have also been designed. However, most of the existing methods suffer from defects; for example, the computation time increases with the number of training samples and the models are insensitive to minor faults. Nevertheless, there is currently limited research on addressing these defects, which greatly limits their application in industrial processes. To fill these gaps, a novel reduced kernel independent component analysis (NRKICA) method is proposed to reduce the computation complexity and improve the ability of minor fault detection at the same time. In this approach, an important factor is defined to measure the ability of the samples to represent the properties of the system. In addition, then the top-n important observations are selected to build a data dictionary. To improve the sensitivity to minor faults, the [Formula: see text] and [Formula: see text] statistics are redesigned by introducing information from past observations. Besides, the kernel parameter is optimized by the tabu search algorithm. The proposed method is applied to fault detection with a numerical example and the Tennessee Eastman process (TEP), and the experimental results verify the effectiveness and sensitivity of the proposed method.

Diagnosis of single and multiple-source faults of chiller sensors using EWEEMD-ICKNN by time sequence denoising and non-Gaussian distribution feature extraction

Currently, fault detection of chiller sensors primarily focuses on single-source fault (SSF) sensors. These methods are typically ineffective in multiple source faults (MSF) cases. These methods assume a Gaussian distribution of the chiller data and ignore the coupling between the sensors. Therefore, we propose temporal feature denoising and non-Gaussian distribution-based methods in this study. First, the chiller sensor signals are denoised based on exponential weighted moving average ensemble empirical mode decomposition (EWEEMD). Subsequently, an independent component analysis k-nearest neighbour (ICKNN) classifier is proposed based on the non-Gaussian distribution features of the chiller sensor data. A residual vector combined with independent component analysis was used for feature extraction to propose the ICKNN classifier. Finally, according to their characteristics, different SSF and MSF were applied to the chiller sensors. Experimental comparisons with other methods were performed for four single-source faults and six multisource faults. The EWEEMD-ICKNN was experimentally confirmed to have excellent fault detection performance in the SSF and MSF chiller sensors. The experimental results demonstrate that the fault diagnosis rate of EWEEMD-ICKNN ranges from 99.90% to 86.80%. Additionally, the fault diagnosis efficiency of EWEEMD-ICKNN remains consistently stable across various fault states, exhibiting an average F1 score of 0.9589 with F1 score statistics of 0.038.

Assessment of train running safety on railway bridges based on velocity-related indices under random near-fault ground motions

Random analysis has emerged as the dominant approach for analyzing train-bridge coupled (TBC) systems, replacing deterministic analysis. This shift is driven by considerations of system economy and randomness. However, the seismic train running safety assessment with multiple non-Gaussian distribution parameters faces the double dilemma of low computational efficiency and inaccuracy for TBC systems. In this study, an approach is applied by the combination of the new point estimate method and moment expansion approximation (NPEM-MEA) to address this problem. Furthermore, the application of this approach is illustrated by the random dynamic responses of a three-dimensional TBC system subjected to random near-fault ground motions. Validation of the NPEM-MEA’s feasibility is conducted using the Monte Carlo method, and comparative analysis confirms the accuracy and efficiency of the method. Additionally, the influences of random near-fault ground motions are discussed based on the random dynamic responses and train running safety indices in the TBC system. The simulation results underscore the necessity of simultaneously considering the randomness of multiple parameters in seismic train running safety assessment. This methodology holds potential for safety assessments involving non-Gaussian random processes in complex systems of a similar nature.

Bivariate sea-ice assimilation for global-ocean analysis–reanalysis

Abstract. In the last decade, various satellite missions have been monitoring the status of the cryosphere and its evolution. Besides sea-ice concentration data, available since the 1980s, sea-ice thickness retrievals are now ready to be used in global operational prediction and global reanalysis systems. Nevertheless, while univariate algorithms are commonly used to constrain sea-ice area or volume, multivariate approaches have not yet been employed due to the highly non-Gaussian distribution of sea-ice variables together with the low accuracy of thickness observations. This study extends a 3DVar system, called OceanVar, which is routinely employed in the production of global/regional operational/reanalysis products, to process sea-ice variables. The tangent/adjoint versions of an anamorphosis operator are used to locally transform the sea-ice anomalies into Gaussian control variables and back, minimizing in the latter space. The benefit achieved by such a transformation is described. Several sensitivity experiments are carried out using a suite of diverse datasets. The sole assimilation of the CryoSat-2 provides a good spatial representation of thickness distribution but still overestimates the total volume that requires the inclusion of Soil Moisture and Ocean Salinity (SMOS) mission data to converge towards the observation estimates. The intermittent availability of thickness data can lead to potential jumps in the evolution of the volume and requires a dedicated tuning. The use of the merged L4 product CS2SMOS shows the best skill score when validated against independent measurements during the melting season when satellite data are not available. This new sea-ice module is meant to simplify the future coupling with ocean variables.

High-Fidelity Computational Assessment of Aero-Thermal Performance and the Reynolds’ Analogy for Additively Manufactured Anisotropic Surface Roughness

Abstract Direct numerical simulations of incompressible turbulent forced convection over irregular, anisotropic surface roughness in a pressure-driven plane channel flow have been performed. Heat transfer was simulated by solving the passive scalar transport equation with Prandtl number Pr = 0.7. The roughness topographies under investigation here are based on an X-ray computed tomography scan of an additively manufactured internal cooling passage, which had an irregular, multiscale and mildly non-Gaussian height distribution. Three different roughness topographies and three different friction Reynolds numbers (Reτ = 395, 590, 720) were considered, along with reference smooth-wall simulations at matched Reτ. By systematically varying the roughness topography and flow conditions, a direct computational assessment of aero-thermal performance (pressure losses and heat transfer) and the Reynolds analogy factor, i.e., 2Ch/Cf, where Ch is the heat-transfer coefficient (Stanton number) and Cf is the skin-friction coefficient, was conducted. The results highlight the profound impact that the roughness orientation (relative to the flow direction) has upon the aero-thermal performance of additively manufactured internal passages, with transverse-aligned roughness augmenting heat transfer by as much as 33%, relative to its streamwise-aligned counterpart. An interrogation of velocity and temperature statistics in the near-wall region was also performed, which underlined the growing dissimilarity between heat transfer and drag as fully rough conditions are approached.

Enhanced square root CKF with mixture correntropy loss for robust state of charge estimation of lithium-ion battery

The measurement data of lithium-ion battery collected in complex operating conditions may be contaminated by non-Gaussian noises (or outliers), a novel robust state of charge (SOC) estimation approach that enhances the robustness of square-root cubature Kalman filter (SRCKF) by incorporating a mixture correntropy loss (MCL) is proposed to overcome the issue of non-Gaussian measurement noise interference. Although the SRCKF can perform non-destructive state estimation for nonlinear system, it is sensitive to the non-Gaussian noises (or outliers) due to the use of mean square error (MSE) loss. To overcome this limitation, the MCL is used as a cost to replace the MSE in CKF framework, which is constructed by mixture of two Gaussian kernel functions with different kernel widths and has outstanding robustness. Moreover, we establish an effective nonlinear regression model that incorporates noise and state error information into the MCL function, and then a fixed-point iteration method is utilized to update the posterior state estimation. Additionally, to further suppress the influence of abnormal noise on the posterior state estimation, we introduce the exponential function of the innovation vector to adjust the measurement covariance matrix in real-time. Accordingly, an enhanced MCL-SRCKF is developed for SOC estimation considering the measurement noises with non-Gaussian distribution. Results from various simulation environments demonstrate that the proposed method achieves high estimation accuracy in different cases.

Stochastic analysis of the nonlinear dynamics of oscillating water columns: A frequency domain approach

This paper investigates the first and second-order stochastic responses of oscillating water columns (OWCs) under random waves. The OWCs’ nonlinear dynamics are computed in the frequency domain, where sources of nonlinearities are replaced by equivalent polynomial terms up to second order by minimising their difference in a mean-square sense. This procedure is known as the statistical quadratisation (SQ) technique. In such an approach, the linear and quadratic coefficients are obtained using an iterative procedure and non-Gaussian distributions based on Gram–Charlier expansions, and the dynamics are solved using the Volterra theory. The results are compared against a statistical linearisation model (SL), and nonlinear time-domain simulations (TD) to illustrate the capabilities of the method. The result demonstrated an excellent agreement for the first and second-order motions of the water column obtained using statistical quadratisation compared to nonlinear time-domain simulations in terms of spectral response and probability distribution. Along with the good accuracy, the statistical quadratisation has the advantage of being approximately two orders of magnitude faster than nonlinear time-domain simulations. For the proposed systems, the nonlinearity from the variable mass system (inertial type) is shown to be the most important source of second-order effects driving the oscillating water column dynamics based on the environmental conditions and drafts investigated in this work.

Statistical analysis of stochastic magnetic fluctuations in space plasma based on theMMSmission

ABSTRACTBased on the Magnetospheric Multiscale (MMS) mission we look at magnetic field fluctuations in the Earth’s magnetosheath. We apply the statistical analysis using a Fokker–Planck equation to investigate processes responsible for stochastic fluctuations in space plasmas. As already known, turbulence in the inertial range of hydromagnetic scales exhibits Markovian features. We have extended the statistical approach to much smaller scales in space, where kinetic theory should be applied. Here we study in detail and compare the characteristics of magnetic fluctuations behind the bow shock, inside the magnetosheath, and near the magnetopause. It appears that the first Kramers–Moyal coefficient is linear and the second term is a quadratic function of magnetic increments, which describe drift and diffusion, correspondingly, in the entire magnetosheath. This should correspond to a generalization of Ornstein–Uhlenbeck process. We demonstrate that the second-order approximation of the Fokker–Planck equation leads to non-Gaussian kappa distributions of the probability density functions. In all cases in the magnetosheath, the approximate power-law distributions are recovered. For some moderate scales, we have the kappa distributions described by various peaked shapes with heavy tails. In particular, for large values of the kappa parameter this shape is reduced to the normal Gaussian distribution. It is worth noting that for smaller kinetic scales the rescaled distributions exhibit a universal global scale invariance, consistently with the stationary solution of the Fokker–Planck equation. These results, especially on kinetic scales, could be important for a better understanding of the physical mechanism governing turbulent systems in space and astrophysical plasmas.

A novel uncertainty evaluation method based on the particle filter and beta distribution for data with unknown distribution.

Uncertainty evaluation for unknown distribution data is a key problem to be solved in uncertainty evaluation theory. To evaluate the measurement uncertainty of data with unknown distributions, a novel uncertainty evaluation method based on the particle filter (PF) and beta distribution is proposed in this paper. A beta distribution with wide adaptability was adopted as the distribution type of measurement results, the parameters of the beta distribution were taken as the parameters to be estimated, and a state-space model was established. The PF method, suitable for non-Gaussian data, was utilized to obtain the estimates of the parameters of the beta distribution according to the measurement results. Finally, the best estimates of the measurement results and their uncertainty were calculated using the beta distribution parameters. Simulation results show that the proposed method is adaptive to accurately evaluate the measurement uncertainties of data, especially for non-Gaussian distribution data or asymmetrically distributed data. Multiple evaluation results show that the method has good robustness. The experimental results for the drift errors of a laser interferometer show that the uncertainty result of the proposed method is consistent with the Monte Carlo method. This method is suitable for a variety of distribution types that can be characterized through beta distribution and can solve the optimal estimation and uncertainty evaluation of most measurement results with unknown distribution types.

Geostatistical analysis of groundwater levels in a mining area with three active mines

AbstractMining activities can significantly impact groundwater reservoirs in their vicinity. Different approaches have been employed, with varying success, to investigate the spatial variability of groundwater levels in mining areas. Typical problems include the small sample size, the non-Gaussian distribution of the data, and the clustering of sample locations near the mines. These conditions complicate the estimation of spatial dependence. Under sparse and irregular sampling conditions, stochastic methods, which can provide estimates of prediction uncertainty, are preferable to deterministic ones. This research focuses on the comparison of two stochastic methods, stochastic local interactions (SLI) and universal Kriging (UK), using water level data from 72 locations around three mines in Northern Greece. UK is a well-known, variogram-based geostatistical method, while SLI is a computationally efficient kernel-based method that can cope with large spatial datasets. The non-Gaussian distribution of the data is handled by means of a flexible, data-driven Gaussian anamorphosis method that uses kernel functions. The spatial prediction performance of both methods is assessed based on cross-validation. UK performs better than SLI, due to the fact that the former incorporates a linear trend function. On the other hand, a comparison of the two methods using data from a single mine that contains only 28 measurement locations shows that SLI performs slightly better than UK. The prediction uncertainties for both methods are also estimated and compared. The results suggest that SLI can provide better estimates than classical geostatistical methods for small sample sizes that do not allow reliable estimation of the variogram model.

Effect of the nature of uncertainty on the optimization of collision avoidance maneuvers

With the soaring number of space debris, collision avoidance maneuvers have become crucial to ensure the safety of any operating satellite. In real-world scenario, the problem gets a little more complicated due to the presence of uncertainty in the values of the states of the space objects. Also, the uncertainty distribution assigned to a given state variable evolves over time. This paper studies the effects of linear and nonlinear propagation of the uncertainty distributions on the collision avoidance maneuver optimization. These different methods for error propagation lead to differing shapes and sizes of the error distribution, resulting in differing regions to be avoided during the encounter, thus resulting in different optimal Δv requirements. In the method using nonlinear propagation of uncertainty distribution, the uncertainty of the passive object is propagated using Monte Carlo samples while that of the operating spacecraft is evolved using Unscented Transform. The sigma points from the Unscented Transform of the operating spacecraft are required to lie outside the stretch of most of the debris uncertainty while implementing cumulative density function. This also results in defining a new probabilistic measure which is like a modified Mahalanobis distance but for a non-Gaussian distribution. The other approach uses linearly propagating the error covariances and a combined error ellipsoid to ensure the maintenance of a desired Mahalanobis distance of the hard body from the uncertainty distribution to ascertain safety. Both the approaches consider nonlinear state dynamics while incorporating the perturbing forces to find the optimal thrust profile and the corresponding steer angles. The approach implementing nonlinear propagation of uncertainty and using cumulative density to ensure that the sigma points of the satellite avoid the debris’ uncertainty distribution is found to require lower Δv compared to the other method while also representing the error propagation better. However, the method implementing linear propagation of uncertainty and using Mahalanobis distance of the hard body from the combined error distribution to frame the avoidance constraint proves to be computationally much faster.

A novel variational robust filter with Gaussian mixture model for unknown non-Gaussian noises

In numerous practical applications, the presence of unpredictable abnormal behaviors often leads to random measurement outliers, causing the random measurement noises are no longer Gaussian but unknown non-Gaussian distributions, and it is intractable to obtain accurate a priori information about the noises. To address this challenge and enable nonlinear state estimation under such measurement noises, we propose a novel robust filter based on Gaussian mixture model (GMM). Firstly, a GMM is introduced to model the unknown non-Gaussian measurement likelihood probability density function (ML-PDF). Without relying on any prior noise statistics, the GMM can accurately model the ML-PDF utilizing a set of model parameters adaptively learned by the variational Bayesian (VB) method. Then, considering the high computational burden associated with the existing variational iterative process, we develop an improved VB method, which independently calculates the posterior distributions of state and unknown model parameters, thereby improving the computational efficiency. Finally, a new robust filter is derived by integrating the GMM with the improved VB approach, The effectiveness and superiority of the filter are demonstrated through both numerical simulations and real-world data analysis.

ADS-B overlapping signals separation based on minimum dispersion criterion

As a new generation of air traffic management (ATM) monitoring system, the airborne automatic dependent surveillance-broadcast (ADS-B) transmitters will broadcast their position, speed, and other information to enhance the efficiency and safety of air traffic. However, due to the asynchronous nature of ADS-B transmitters broadcasting their information on the same frequency, it is inevitable for multiple ADS-B signals from different aircraft to overlap. This paper proposes an ADS-B signal separation algorithm based on the minimum dispersion criterion (MD), which takes advantage of the non-Gaussian distribution properties exhibited by ADS-B signals. Firstly, the non-Gaussian distribution properties of overlapping ADS-B signals are illustrated by analyzing their probability density distribution and kurtosis. Secondly, the separation problem is formulated as a non-convex optimization problem based on the minimum dispersion criterion. Finally, the alternating direction method of multipliers (ADMM) is employed to tackle the formulated optimization problem. Simulations will demonstrate the effectiveness of the proposed algorithm.

Bayesian inference with finitely wide neural networks.

The analytic inference, e.g., predictive distribution being in closed form, may be an appealing benefit for machine learning practitioners when they treat wide neural networks as Gaussian process in a Bayesian setting. The realistic widths, however, are finite and cause weak deviation from the Gaussianity under which partial marginalization of random variables in a model is straightforward. On the basis of multivariate Edgeworth expansion, we propose a non-Gaussian distribution in differential form to model a finite set of outputs from a random neural network, and derive the corresponding marginal and conditional properties. Thus, we are able to derive the non-Gaussian posterior distribution in Bayesian regression task. In addition, in the bottlenecked deep neural networks, a weight space representation of a deep Gaussian process, the non-Gaussianity is investigated through the marginal kernel and the accompanying small parameters.

A tropical cyclone intensity prediction model using conditional generative adversarial network

This study proposes a model for predicting tropical cyclone (TC) intensity evolution by utilizing the conditional Wasserstein generative adversarial network with gradient penalty (CWGAN-GP) for TC-related risk assessment. The TC intensity evolution is treated as a random variable, which is the output of a nonlinear system conditioned on a series of input random variables that characterize both the TC state (e.g., translation speed and potential intensity) and environment (e.g., vertical wind shear and ocean mixed layer depth). To represent the high-dimensional non-Gaussian probabilistic characteristics, the system is modeled by the CWGAN-GP and calibrated concerning the dataset of 1010 historical TCs from the western North Pacific basin. The over-ocean and overland branches of the model are calibrated separately and then merged. The suitability of the established model is validated by comparing the predicted TC intensity to the observations and the results of existing models such as those based on linear regression. Numerical results indicate that the proposed model successfully replicates the probabilistic properties such as the non-Gaussian marginal distribution of the intensity change and the input-output joint distribution and moments between the TC intensity and predictors, which the linear regression model cannot provide. Further application examples are also carried out by using the model to simulate the evolution of historical TC events and the extreme TC intensities in Southern China, demonstrating its potential role in the TC hazard assessment.

Fast robust enhanced trend filter: A promising tool for automatically extracting high precision friction coefficient under unknown noise

Friction coefficient is the core parameter of material friction performance test. However, it often suffers from complex unknown noise, which frequently follows non-Gaussian distribution, such as outliers during the test phase. Moreover, the amount of test data is often very large that lead to the heavy computational burden. Thus, how to efficiently and accurately extract the friction coefficient is becoming a challenging task. To address this challenge, this paper proposes a fast trend filtering method based on Robust Enhanced Trend Filtering (RobustETF) which combines mix of Gaussian (MoG) and non-convex sparsity-inducing functions to extend l1 trend filtering, called Fast RobustETF. First, the sliding window strategy is used to segment the original data, and the wear stage is distinguished via judging the data margin factor inside the window. Then the window data with the largest and smallest margins is selected as the characteristic data segment of the stable and severe wear stage, and we consider the trend kurtosis and variance into the selection of optimal regularization parameters for different wear stages. Then, RobustETF is used in each window for extracting high precision friction coefficient under unknown noise. Finally, the trend signals in all windows are integrated and reconstructed to complete the precise extraction of the friction coefficient. Finally, simulation signals and real friction coefficient signals are used to verify the performance of Fast RobustETF by the comparison with other filtering methods.

LIFT of cell spheroids: Proof of concept

The application of spheroids in tissue engineering has a number of advantages over conventional cell suspensions and 2D cultures. One of the methods for tissue and organ fabrication from spheroids is bioprinting. As one of bioprinting methods, laser-induced forward transfer (LIFT) has received much attention in terms of cell printing, while its potential has not been realized for spheroid patterning yet. In this paper, the authors have shown for the first time the practical applicability of LIFT for spheroid transfer with high survival rates and printing precision. For this, a special optical device, a piShaper, was used to change the laser energy distribution to non-Gaussian profile which allowed for mitigating the negative effects of laser radiation on the spheroids during LIFT. The authors showed that non-Gaussian energy distribution in the laser spot in the form of double ring led to higher post-printing viability of spheroids than in case of conventional Gaussian energy distribution in laser beam. Subsequently, using the double ring laser spot geometry, the spheroids were bioprinted in the form of simple geometric figures: line, triangle, and square. Overall, LIFT bioprinting of spheroids has demonstrated a strong potential as the precise, safe, and reproducible method for biofabrication that can be potentially used for making tissue-engineered bioequivalents or building specific organ-on-a-chip platforms.

Performance degradation assessment for mechanical system based on semi-analytical solution of self-similar stable distribution process

To more accurately predict remaining useful life (RUL) and quantitatively evaluate the uncertainty of the predicted results, a performance degradation assessment framework based on semi-analytical solution of self-similar stable distribution process is proposed. The established performance degradation model based on adaptive fractional Lévy stable motion (AFLSM) is more flexible in revealing the long-range dependence, non-Gaussian, and heavy-tailed distribution properties of the incremental behavior. The corresponding stable distribution parameters are estimated through characteristic function method, and Hurst exponent is calculated based on the generalized Hurst exponent approach with narrower confidence interval. Aiming at the difficulties in solving the exact analytical solution and the excessive computation of the numerical solution in the whole process, based on Mellin-Stieltjes transform and direct integration, a semi-analytical solution of RUL distribution function is proposed, which can be readily implemented in practical equipment operations. The proposed performance degradation assessment framework is validated by the novel truck transmission dataset and the benchmark rolling bearing dataset. Experimental results indicate that the developed framework is more effective and superior than other state-of-the-art approaches in terms of RUL prediction.