Uncertainty Distribution Research Articles

A New Track Association Model Based on Uncertainty Distribution

A New Track Association Model Based on Uncertainty Distribution

Distributionally Robust Optimization With Noisy Data for Discrete Uncertainties Using Total Variation Distance

Stochastic programs, where uncertainty distribution must be inferred from noisy data samples, are considered. They are approximated with distributionally/robust optimizations that minimize the worst-case expected cost over ambiguity sets, i.e., sets of distributions that are sufficiently compatible with observed data. The ambiguity sets capture probability distributions whose convolution with the noise distribution is within a ball centered at the empirical noisy distribution of data samples parameterized by total variation distance. Using the prescribed ambiguity set, the solutions of the distributionally/robust optimizations converge to the solutions of the original stochastic programs when the number of the data samples grow to infinity. Therefore, the proposed distributionally/robust optimization problems are asymptotically consistent. The distributionally/robust optimization problems can be cast as tractable optimization problems.

A Physical-Based Method for Pixel-by-Pixel Quantifying Uncertainty of Land Surface Temperature Retrieval From Satellite Thermal Infrared Data Using the Generalized Split-Window Algorithm

Land surface temperature (LST) is an important physical parameter at the interface between the Earth’s surface and the atmosphere. Accurately quantifying LST uncertainty is essential for the generation of a long-term and consistent LST Climate Data Record (CDR) or Earth System Data Record (ESDR) from either multiple sensors or algorithms. In this study, a physical-based method was proposed to quantify the uncertainty of LST retrieval from satellite thermal infrared data using the generalized split-window (GSW) algorithm. LST uncertainties were parameterized as a function of brightness temperature at the top of the atmosphere (TOA) and surface emissivity in two split-window channels, which are two key input parameters in the GSW algorithm, as well as their uncertainties. The performance of the parameterized uncertainty model was evaluated according to simulation dataset at six prescribed viewing zenith angles (VZAs) of 0°, 33.56°, 44.42°, 51.32°, 56.25°, and 60°, with a root mean squared error (RMSE) of 0.001 K. The coefficients of the parameterized uncertainty model at arbitrary VZA within a sensor’ field of view (FOV) can be obtained by linear interpolation of the coefficients at the six prescribed VZAs. Once the coefficients of the parameterized uncertainty model for each pixel are available, total LST uncertainties can be quantified on a pixel-by-pixel basis. As an example, the parameterized uncertainty model was applied to actual MODIS data for displaying the spatial distribution of LST uncertainties. The results indicate that the parameterized uncertainty model can well characterize the spatial variation in LST uncertainties over various land cover types.

Stochastic Uncertain Degradation Modeling and Remaining Useful Life Prediction Considering Aleatory and Epistemic Uncertainty

Remaining useful life (RUL) prediction based on the degradation model is vital for the maintenance and management of complex equipment. It is necessary to describe the aleatory uncertainty in internal changes during equipment degradation and quantify the epistemic uncertainty caused by a lack of sufficient knowledge simultaneously. This article proposes a novel framework for the degradation model and RUL prediction with aleatory and epistemic uncertainties. First, the probability theory is combined with the uncertainty theory to establish a stochastic uncertain degradation model. Following that, a new stochastic uncertain maximum likelihood estimation (SUMLE) method is proposed based on probability function and uncertainty distribution to identify model parameters. The Bayesian inference is adopted to update the model parameters in the current time. Afterward, the above dual-source uncertainty is incorporated into RUL prediction. The distribution function of the RUL is derived, which can be updated in real-time according to the arrival of new degradation observations. Finally, the experimental studies on gallium arsenide (GaAs) laser and gyroscope degradation data are conducted to interpret the superiority of the proposed method over based on the stochastic process or uncertain process separately for predicting the RUL.

Distributionally Robust Chance-Constrained Optimal Transmission Switching for Renewable Integration

Increasing integration of renewable generation poses significant challenges to ensure robustness guarantees in real-time energy system decision-making. This work aims to develop a robust optimal transmission switching (OTS) framework that can effectively relieve grid congestion and mitigate renewable curtailment. We formulate a two-stage distributionally robust chance-constrained (DRCC) problem that assures limited constraint violations for any uncertainty distribution within an ambiguity set. Here, the second-stage recourse variables are represented as linear functions of uncertainty, yielding an equivalent reformulation involving linear constraints only. We utilize moment-based (mean-mean absolute deviation) and distance-based ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\infty$</tex-math></inline-formula> -Wasserstein distance) ambiguity sets that lead to scalable mixed-integer linear program (MILP) formulations. Numerical experiments on the IEEE 14-bus and 118-bus systems have demonstrated the performance improvements of the proposed DRCC-OTS approaches in terms of guaranteed constraint violations and reduced renewable curtailment. In particular, the computational efficiency of the moment-based MILP approach, which is scenario-free with fixed problem dimensions, has been confirmed, making it suitable for real-time grid operations.

Uncertain design optimization of automobile structures: A survey

&lt;abstract&gt; &lt;p&gt;In real life, there are a lot of uncertainties in engineering structure design, and the potential uncertainties will have an important impact on the structural performance responses. Therefore, it is of great significance to consider the uncertainty in the initial stage of structural design to improve product performance. The consensus can be reached that the mechanical structure obtained by the reliability and robustness design optimization method considering uncertainty not only has low failure risk but also has highly stable performance. As a large mechanical system, the uncertainty design optimization of key vehicle structural performances is particularly important. This survey mainly discusses the current situation of the uncertain design optimization framework of automobile structures, and successively summarizes the uncertain design optimization of key automobile structures, uncertainty analysis methods, and multi-objective iterative optimization models. The uncertainty analysis method in the design optimization framework needs to consider the existing limited knowledge and limited test data. The importance of the interval model as a non-probabilistic model in the uncertainty analysis and optimization process is discussed. However, it should be noted that the interval model ignores the actual uncertainty distribution rule, which makes the design scheme still have some limitations. With the further improvement of design requirements, the efficiency, accuracy, and calculation cost of the entire design optimization framework of automobile structures need to be further improved iteratively. This survey will provide useful theoretical guidance for engineers and researchers in the automotive engineering field at the early stage of product development.&lt;/p&gt; &lt;/abstract&gt;

Robust Pricing for a Dual-Channel Green Supply Chain Under Fuzzy Demand Ambiguity

The pricing decision is a fundamental problem in a two-echelon dual-channel supply chain, and it is a challenging issue to get the optimal pricing strategy due to high demand ambiguity. Although this problem has been studied by many researchers, the market demand is usually assumed to be deterministic or described as a fuzzy variable with exact distribution information in most of the existing literature. In this article, the uncertain demand is assumed to possess fuzzy uncertainty instead of stochastic nature. In the case that the demand distribution information is partially available, this article proposes a new uncertainty distribution set to depict the demand distribution uncertainty based on the type-2 fuzzy theory. Then, a novel robust pricing game modeling framework is developed for the dual-channel green supply chain with the demand distribution varying in the proposed uncertainty distribution set. Finally, a new method is proposed for obtaining the robust equilibrium decisions of supply chain members. Numerical analysis and comparisons are conducted to demonstrate the impacts of the demand ambiguity on the manufacturer’s equilibrium channel choice strategy. We also demonstrate that our solution results are robust in the setting that the prices of the green products are not equal in the retail and the direct channels.

Sensitivity Analysis and Robust Assignment of Natural Frequencies for Vibrating Systems Using the Receptance Method

This paper presents a new receptance-based method for the robust assignment of natural frequencies. This method aims to address the deviation of the actually achieved natural frequencies for target natural frequencies when assigning natural frequencies, which usually arises due to uncertainties in passive structural modifications (i.e. mass and stiffness modifications). The mathematical expression is derived for the sensitivity of the target natural frequency to the uncertainties in modification parameters, which are subsequently employed to characterize the potential deviations of the target natural frequencies in the total differential form. The obtained deviation expression is integrated into the optimization calculation of the natural frequency assignment as an extra penalty term to boost the acquisition of highly robust modification parameters. The proposed robust assignment method, as a non-probabilistic technique, can be employed without any knowledge of the probabilistic distribution of uncertainty in modification parameters. Moreover, it uses only the measured frequency response functions (FRFs), preserving the advantage of the receptance method not needing a numerical model. Two numerical examples and a comparison with a typical eigenstructure assignment method (reported in the literature) demonstrate that the proposed method can provide highly robust modification parameter solutions. The application of the proposed method to the natural frequency assignment of a laboratory water-filled pipeline system gives experimental evidence of its effectiveness.

Scenario reduction and scenario tree generation for stochastic programming using Sinkhorn distance

Scenario-based stochastic programming is a widely used method for optimization under uncertainty. The solution quality of this approach is dependent on the approximation of the underlying uncertainty distribution. Therefore, the optimal generation of scenarios (or scenario trees) is a pertinent research objective in stochastic programming. In this work, we approach the scenario reduction and scenario tree generation problem through the perspective of optimal transport, specifically entropy-regularized optimal transport. The availability of an iterative procedure to compute the optimal entropy-regularized transport plan between support sets, using the Sinkhorn–Knopp algorithm in lieu of conventional linear programming-based optimal transport, is found to decrease solution time appreciably, with a decrease in memory burden as well. We present algorithms for optimal scenario reduction and multistage scenario tree generation, and illustrate their use through two case studies. We show that the proposed approach generates high-quality scenarios whose use in stochastic programming offers solutions with good accuracy.

Probabilistic seismic hazard analysis of the North-East India towards identification of contributing seismic sources

The North East (NE) India is regarded as one of the most seismically active regions of the globe. It is surrounded by two active plate boundaries (the Indian-Eurasian Plate boundary and the Indian-Burmese plate boundary), followed by rotation in the central part. As per the Seismic zonation map of India, most parts of NE India is classified under single seismic zone (zone V). However, numerous studies have shown significant variations in seismic activity as well as seismic hazard values across the NE. Present work attempts deaggregation of seismic hazard to identify seismic sources which control probabilistic seismic hazard values across the NE. With dominating fault mechanisms, seismic activity, and non-uniform past earthquake (EQ) records across NE, identifying controlling sources is crucial in order to understand the present and future seismic scenarios. Spatial distribution of seismic hazard levels and uncertainties are estimated using the Monte Carlo approach to sample a logic tree with alternate methods and models. As per the present findings, Manipur and Nagaland have highest seismic hazard in the NE, controlled by Indo-Burmese ranges. Further, seismic hazard of Guwahati, Shillong and Gangtok are significantly influenced by great EQs. In other parts, regional major to strong EQ sources control the seismic hazard values. Additionally, EQs with magnitude lesser than 4.5 have lower contributions to regional seismic hazards.

Near-Linear Orbit Uncertainty Propagation Using the Generalized Equinoctial Orbital Elements

This paper addresses the problem of minimizing the impact of nonlinearities when dealing with uncertainty propagation in the perturbed two-body problem. The recently introduced generalized equinoctial orbital elements set is employed as a means to reduce nonlinear effects stemming from and higher-order gravity field harmonics. The uncertainty propagation performance of the proposed set of elements in different Earth orbit scenarios, including low-thrust orbit raising, is evaluated using a Cramér–von Mises test on the Mahalanobis distance of the uncertainty distribution. A considerable improvement compared to all sets of elements proposed so far is obtained.

Uncertain mean-chance model for portfolio selection with multiplicative background risk

Theoretical and empirical studies all show that there are many situations where we cannot treat the indeterminate quantities as random variables. Then we need to explore using a new tool other than probability theory to make investment decisions. Using uncertainty theory, this paper discusses a portfolio selection problem considering inflation, a most popular multiplicative background risk, in such situations, and analyses the effect of uncertain inflation on investment in risky and risk-free assets. Treating risky asset return and inflation rate as uncertain variables rather than random variables, we first propose an uncertain mean-chance model. Then we give its deterministic form and compare the model with the one that ignores inflation. Next, we provide the optimal solution, which can advise the investors on how to allocate capital between risky and risk-free assets under the influence of uncertain inflation. Then by discussing the properties of the model when the return of risky asset and inflation rate obey linear uncertainty distributions, we show how the optimal investment proportion changes when the inflation rate and risky asset return change. The experiment results show that the proposed method brings investors greater wealth than the existing method using probability theory.

Characterization of rotational symmetry and noise in multifocal contact lens power maps

AbstractPurpose: Spherical multifocal contact lenses are typically designed with a rotationally symmetrical power profile. Imperfections in the raw material, manufacturing and measurement process can break this symmetry, altering the intended lens design. The purpose of this work is to develop a method to separate the intended symmetric design profile from the asymmetries and to give the spatial distribution of the measurement noise uncertainty in diopters.Methods: NIMO TR1504 (Lambda‐X) version V.2.13.2 has been used to measure the local power map of various multifocal contact lenses. From the map, a set of radial power profiles is obtained for different angles (0°, 180°), (0°, 360°). A symmetrical lens should show a constant radial profile for different angles, while asymmetries manifest as differences in the set of radial power profiles. They are analysed using a Principal Components Method (PCA) implemented in MATLAB.Results: The method displays a map of the symmetric radial profile of the lens, another map of the deviations with spatial structure indicating the angles where the deviations are stronger (rotational asymmetries), and noise with three separated components (spatially random variations). Asymmetries of the order of 0.17D (with maximums of 0.4D), and random noise values of 0.12D were found in some lenses.Conclusions: The characterization of multifocal contact lens prescription is important in visual optics. The prescription perceived by the patient depends on the distribution of the radial power profile on the pupil. That is the reason why asymmetries and imperfections estimation is important. The method developed in this work allows to calculate a diopter equivalent number for them.

Collision Detection for Cobots: A Back-Input Compensation Approach

The generalized momentum observer (GMO) is a widely used collision detection technique for collaborative robots (cobots). However, a consensus amongst the robotics community is that the GMO has limited bandwidth and is, therefore, insensitive to hard collisions (rapid occurrence). This article proposes a Back-Input compensation ( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">BICom</i> ) approach that can effectively detect both soft (slow) and hard collisions. The basic idea is to utilize the velocity-related residual distribution of model uncertainties and compensate for the deviation between computed and measured torques of robot dynamics, whereas deviations caused by collisions might exceed the upper limits of compensation and then be detected. Besides, the window detector module mitigates false positives of collision detection, and overcomes two common defects of the anti-false alarm scheme, namely the detection delay and the false negatives. Compared with the high-pass filter-based detection technique, <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">BICom</i> can not only respond instantaneously to a hard collision but also monitor the quasi-static contact, which improves detection sensitivity. Experiments on a 6-DoF cobot validate the effectiveness of our method.

Portfolio Optimization of Uncertain Returns Based on Tsallis Entropy

In uncertainty theory, entropy is a measure to capture the level of unpredictability associated with an uncertain variable. In this paper, we introduce an alternative type of entropy named Tsallis entropy as a generalized form of logarithmic entropy in an uncertain environment. Then, we provide a formula to calculate the Tsallis entropy via inverse uncertainty distribution. In addition, we study some of its mathematical properties such as translation invariance and positive linearity. Furthermore, we establish a mean-CVaR-Tsallis entropy portfolio selection model, in which security returns are regarded as uncertain variables and we derive its equivalent form. Finally, a numerical example is given and the effect of Tsallis entropy and uncertainty distribution on the proposed model is discussed, including a comparative study.

Distributionally robust frequency dynamic constrained unit commitment considering uncertain demand-side resources

Demand-side resources (DSR) have been considered as flexible frequency reserve providers in power systems. Given their uncertain nature in the reserve deployment and the load recovery periods, it is necessary to consider these uncertainties in the frequency dynamic constrained unit commitment (FDUC) model. This paper proposes a two-stage distributionally robust FDUC model to minimize the worst-case cost expectation under a set of probability distributions of DSR uncertainties, which are restricted by the supports and moments generated from statistical information. In addition, the cumulative frequency deviation during the entire regulation period is exploited as a frequency dynamic criterion. Particularly, it can quantify the regulation efficacy of the reserve resources with diverse frequency response characteristics, such as the dead-band time and the reserve delivery time. Simulations based on the IEEE 118-bus and 300-bus systems validate the effectiveness of the proposed method in hedging against the DSR uncertainties.

Line Sampling Procedure for Extensive Planetary Protection Analysis

Exploration missions to other planets have to satisfy planetary protection requirements to limit the probability of impacts between mission-related objects and celestial bodies, with the goal of reducing the risk of contaminating them with biological material coming from Earth. The verification of these requirements can become a lengthy and computationally expensive task when addressed with common methods such as Monte Carlo simulations, as they involve analysing the interplanetary trajectories and the uncertainties associated with them for time spans up to 100 years, and estimating small probabilities with strict confidence levels. This paper presents novel improvements of the line sampling method, already introduced for the verification of planetary protection requirements as a way to estimate the impact probabilities more efficiently and with greater accuracy than achieved with standard Monte Carlo. These newly developed techniques are presented, with the aim of making the analysis with Line Sampling more effective, and providing more information about the distribution of impacts in the initial uncertainty distribution: an algorithm to identify the time intervals where most close approaches are clustered, and an algorithm to improve the determination of the main sampling direction and increase the accuracy of the probability estimation.

An Approach for Numerical Solutions of Caputo–Hadamard Uncertain Fractional Differential Equations

This paper is devoted to investigating a numerical scheme for solving the Caputo–Hadamard uncertain fractional differential equations (UFDEs) arising from nonlinear uncertain dynamic systems. In our approach, we define an α-path, which is a link between a Caputo–Hadamard UFDE and a Caputo–Hadamard fractional differential equation and is the inverse uncertainty distribution of a Caputo–Hadamard UFDE. Then, a formula for calculating the expected value of the Caputo–Hadamard UFDE is studied. With the help of the modified predictor–corrector method, some numerical algorithms for the inverse uncertainty distribution and the expected value of the solution of Caputo–Hadamard UFDEs are designed. Corresponding numerical examples are given to confirm the validity and accuracy of the proposed algorithms.

Simple Optimal Sampling Algorithm to Strengthen Digital Soil Mapping Using the Spatial Distribution of Machine Learning Predictive Uncertainty: A Case Study for Field Capacity Prediction

Machine learning models are now capable of delivering coveted digital soil mapping (DSM) benefits (e.g., field capacity (FC) prediction); therefore, determining the optimal sample sites and sample size is essential to maximize the training efficacy. We solve this with a novel optimal sampling algorithm that allows the authentic augmentation of insufficient soil features using machine learning predictive uncertainty. Nine hundred and fifty-three forest soil samples and geographically referenced forest information were used to develop predictive models, and FCs in South Korea were estimated with six predictor set hierarchies. Random forest and gradient boosting models were used for estimation since tree-based models had better predictive performance than other machine learning algorithms. There was a significant relationship between model predictive uncertainties and training data distribution, where higher uncertainties were distributed in the data scarcity area. Further, we confirmed that the predictive uncertainties decreased when additional sample sites were added to the training data. Environmental covariate information of each grid cell in South Korea was then used to select the sampling sites. Optimal sites were coordinated at the cell having the highest predictive uncertainty, and the sample size was determined using the predictable rate. This intuitive method can be generalized to improve global DSM.

An enhanced two‐quantile Wilks methodology for engineering uncertainty analysis

AbstractComplex computational engineering uncertainty analyses have become more prevalent. When input parameters of such engineering models are uncertain, the output metric's uncertainty distribution is of an unknown parametric form. Since Wilks' method, named after the seminal paper by SS Wilks in 1941 entitled “Determination of sample sizes for setting tolerance limits”, is a nonparametric statistical procedure, it has received renewed interest, in particular in nuclear and chemical safety engineering. Unfortunately, the prevailing Wilks' method applied relies on arbitrary specification of order statistics' ranks with undue influence on the sample size recommendations that follow. Herein, a novel modification of Wilks' method involving two quantiles is proposed resolving that arbitrary rank selection. Together with a confidence level to be exceeded, these quantiles uniquely determine the parameters of an order statistics' beta distribution which drive the selection of symmetric tolerance limits. The modified procedure is demonstrated in two illustrative engineering uncertainty analysis examples drawn from the nuclear and chemical engineering domains.