Mixed Uncertainties Research Articles

Omega-CVaR portfolio optimization and its worst case analysis

This paper presents a novel framework for optimizing portfolios using distribution dependent thresholds in Omega ratio to control the downside risk. Portfolios resulting from the maximization of the classical Omega ratio simultaneously maximize the probability of superior performance compared to a threshold point set by an investor and minimize the probability of a worse performance compared to the same threshold. However, there is no mandatory rule or mechanism to choose this threshold point in the Omega ratio optimization model yet. In this paper, we redefine the Omega ratio for a loss averse investor by taking the distribution dependent threshold point as the conditional value-at-risk at an $$\alpha $$ confidence level ( $$ {\mathrm{CVaR}_{\alpha }}$$ ) of the benchmark market. The $$\alpha $$ -value reflects the attitude of an investor towards losses. We then embed this new Omega- $$ {\mathrm{CVaR}_{\alpha }}$$ model in a robust portfolio optimization framework and present its worst case analysis under three uncertainty sets. The robustness is introduced both in the Omega measure and the $$ {\mathrm{CVaR}_{\alpha }}$$ measure. We show that the worst case Omega- $$ {\mathrm{CVaR}_{\alpha }}$$ robust optimization models are linear programs for mixed and box uncertainty sets and a second order cone program under ellipsoidal sets, and hence tractable in all three cases. We conduct a comprehensive empirical investigation of the classical $$ {\mathrm{CVaR}_{\alpha }}$$ model, the STARR $$_{\alpha }$$ model, the Omega- $$ {\mathrm{CVaR}_{\alpha }}$$ model, and robust Omega- $$ {\mathrm{CVaR}_{\alpha }}$$ model under a mixed uncertainty set for listed stocks of the S&P 500. The optimal portfolios resulting from the Omega- $$ {\mathrm{CVaR}_{\alpha }}$$ model exhibit a superior performance compared to the classical $$ {\mathrm{CVaR}_{\alpha }}$$ model in the sense of higher expected returns, Sharpe ratios, modified Sharpe ratios, and lesser losses in terms of $${\mathrm{VaR}_{\alpha }}$$ and $$ {\mathrm{CVaR}_{\alpha }}$$ values. The robust Omega- $$ {\mathrm{CVaR}_{\alpha }}$$ model under mixed uncertainty set is shown to dominate the Omega- $$ {\mathrm{CVaR}_{\alpha }}$$ model in terms of all performance measures. Furthermore, both the Omega- $$ {\mathrm{CVaR}_{\alpha }}$$ and robust Omega- $$ {\mathrm{CVaR}_{\alpha }}$$ model under a mixed uncertainty set yield significantly lower risk compared to STARR $$_{\alpha }$$ model in terms of $$\mathrm{CVaR}_{\alpha }$$ and variance values.

Deciding Degree of Conservativeness in Initial Design Considering Risk of Future Redesign

Early in the design process, there is often mixed epistemic model uncertainty and aleatory parameter uncertainty. Later in the design process, the results of high-fidelity simulations or experiments will reduce epistemic model uncertainty and may trigger a redesign process. Redesign is undesirable because it is associated with costs and delays; however, it is also an opportunity to correct a dangerous design or possibly improve design performance. In this study, we propose a margin-based design/redesign method where the design is optimized deterministically, but the margins are selected probabilistically. The final design is an epistemic random variable (i.e., it is unknown at the initial design stage) and the margins are optimized to control the epistemic uncertainty in the final design, design performance, and probability of failure. The method allows for the tradeoff between expected final design performance and probability of redesign while ensuring reliability with respect to mixed uncertainties. The method is demonstrated on a simple bar problem and then on an engine design problem. The examples are used to investigate the dilemma of whether to start with a higher margin and redesign if the test later in the design process reveals the design to be too conservative, or to start with a lower margin and redesign if the test reveals the design to be unsafe. In the examples in this study, it is found that this decision is related to the variance of the uncertainty in the high-fidelity model relative to the variance of the uncertainty in the low-fidelity model.

Stochastic stability and stabilization of a class of piecewise-homogeneous Markov jump linear systems with mixed uncertainties

Summary This paper investigates the stochastic stability and stabilization problem for a general class of uncertain, continuous-time Markov jump linear systems (MJLSs). The system under consideration is a piecewise-homogenous Markovian structure subject to piecewise-constant time-varying transition rates (TRs). The time variation of the TRs is characterized by a high-level Markovian signal, which is independent from the low-level Markovian mechanism that governs the switching between the system dynamics. It is assumed that the structure is subject to mixed uncertainties in the form of additive norm-bounded terms. The uncertainties help to consider the effect of imperfections induced by modeling errors for the system dynamics and the TRs of Markovian signals of both levels. This new uncertain, two-level Markovian jump linear system is a more general model than the existing ones and is applicable to more practical situations. Besides, it is capable of being specialized to uncertain piecewise-homogeneous MJLS with arbitrarily varying TRs, as well as the uncertain time-homogeneous MJLS. The stability/stabilizability of this system is first examined by constructing a Lyapunov function which depends on both switching signals. Then, based on the analysis results, the corresponding robust controller gains are synthesized through solving a set of linear matrix inequalities (LMIs). Finally, simulation results for an industrial stirred tank reactor (CSTR) are used to demonstrate the applicability and potentials of the proposed theoretical method. Comparative simulations are also provided to show the superiority of the presented approach to the existing ones. Copyright © 2016 John Wiley & Sons, Ltd.

Discrepancies between modeled and observed nocturnal isoprene in an urban environment and the possible causes: A case study in Houston

Air quality simulations were conducted using the Community Multiscale Air Quality (CMAQ) model for nocturnal isoprene in September 2013 using the United States Environmental Protection Agency's (EPA's) National Emissions Inventory of 2011 (NEI, 2011). The results were evaluated against measurements collected at eight Texas Commission on Environmental Quality (TCEQ) Automated Gas Chromatographs (AutoGCs) monitoring stations. The comparisons demonstrated two distinctive behaviors: overestimation before midnight (20:00–23:00p.m. local time) versus underestimation after midnight (00:00–06:00a.m.). Analyses identify the uncertainties in nitrate radical (NO3) concentration and vertical mixing as the possible minor factors contributing to the underestimation, and the underestimated wind speed as the major factor contributing to the overestimation. Further analysis links isoprene underestimation to the uncertainties in the nocturnal isoprene anthropogenic emissions in the NEI (2011) over industrial areas in Houston. This can be substantiated by the fact that the observed nighttime isoprene concentrations increased when the wind direction veered back from southeast to northeast, placing the stations downwind of industrial facilities. A sensitivity run with adjusted anthropogenic isoprene emissions in the later part of the night (i.e., the emissions were multiplied by the hourly underestimation factors ranging from 3.81 to 14.82) yielded closer isoprene predictions after midnight with slightly improved model mean (0.15 to 0.20ppb), mean error (−0.10 to −0.04ppb), mean absolute error (0.18 to 0.15ppb), root mean squared error (RMSE, 0.27 to 0.25ppb), and index of agreement (IOA, 0.66 to 0.68). The insignificant improvement was likely due to the uncertainties in the location of the high-peaked anthropogenic emissions. The impacts of the nighttime-adjusted isoprene emissions on the isoprene oxidation products, organic nitrate and ozone, were found to be minimal. This study, however, shows that more in-situ surface nighttime measurement data is critical to constrain the underestimated nocturnal isoprene emissions in Houston.

A New Combined Assessment of Mixed Uncertainty in Spatial Models: Conceptualization and Implementation

AbstractUncertainty quantification is not often performed in spatial modeling applications, especially when there is a mixture of probabilistic and non‐probabilistic uncertainties. Furthermore, the effect of positional uncertainty is often not assessed, despite its relevance to geographical applications. Although there has been much work in investigating the aforementioned types of uncertainty in isolation, combined approaches have not been much researched. This has resulted in a lack of tools for conducting mixed uncertainty analyses that include positional uncertainty. This research addresses the issue by first presenting a new, flexible, simulation‐oriented conceptualization of positional uncertainty in geographic objects called F‐Objects. F‐Objects accommodates various representations of uncertainty, while remaining conceptually simple. Second, a new Python‐based framework is introduced, termed Wiggly and capable of conducting mixed uncertainty propagation using fuzzy Monte Carlo simulation (FMCS). FMCS combines both traditional Monte Carlo with fuzzy analysis in a so‐called hybrid approach. F‐Objects is implemented within the Wiggly framework, resulting in a tool capable of considering any combination of: (1) probabilistic variables; (2) fuzzy variables; and (3) positional uncertainty of objects (probabilistic/fuzzy). Finally, a realistic GIS‐based groundwater contamination problem demonstrates how F‐Objects and Wiggly can be used to assess the effect of positional uncertainty.

An extended probabilistic method for reliability analysis under mixed aleatory and epistemic uncertainties with flexible intervals

The reliability analysis approach based on combined probability and evidence theory is studied in this paper to address the reliability analysis problem involving both aleatory uncertainties and epistemic uncertainties with flexible intervals (the interval bounds are either fixed or variable as functions of other independent variables). In the standard mathematical formulation of reliability analysis under mixed uncertainties with combined probability and evidence theory, the key is to calculate the failure probability of the upper and lower limits of the system response function as the epistemic uncertainties vary in each focal element. Based on measure theory, in this paper it is proved that the aforementioned upper and lower limits of the system response function are measurable under certain circumstances (the system response function is continuous and the flexible interval bounds satisfy certain conditions), which accordingly can be treated as random variables. Thus the reliability analysis of the system response under mixed uncertainties can be directly treated as probability calculation problems and solved by existing well-developed and efficient probabilistic methods. In this paper the popular probabilistic reliability analysis method FORM (First Order Reliability Method) is taken as an example to illustrate how to extend it to solve the reliability analysis problem in the mixed uncertainty situation. The efficacy of the proposed method is demonstrated with two numerical examples and one practical satellite conceptual design problem.

Robust adaptive control for dynamic systems with mixed uncertainties

The control problem for single-input single-output (SISO) systems in the presence of mixed uncertainties, both stochastic and deterministic uncertainties, is considered. The stochastic uncertainties are modeled as exogenous noises, while the deterministic uncertainties are time invariant and appear as the unknown parameters which lie in a bounded interval. Based on a subdivision for the continuous interval, a robust adaptive controller is designed. The controller can not only realize the system output to track the desired output, but also learn a more accurate interval which contains the true value of the unknown parameter with a learning error given in advance. An example is given finally to demonstrate the effectiveness of the proposed method.

Time value of delays in unreliable production systems with mixed uncertainties of fuzziness and randomness

This paper proposes a parametric programming approach to address the notion of the time value of delays in the presence of mixed (random and fuzzy) uncertainties that result from unreliable systems. To consider different types of delay time values, the system states are appropriately and carefully identified and defined, and a cost-based fuzzy decision model that incorporates several unreliability factors is constructed. Then, the proposed model is transformed into a pair of nonlinear programs parameterized by the possibility level α to identify the lower and upper bounds on the minimal total cost per unit time at α and thus construct the membership function. To provide analytical expressions, a special case with analytical results is also presented. In contrast to existing studies, the results derived from the proposed solution procedure conserve the fuzziness of the input information, representing a significant difference from the crisp results obtained using approaches based on probability theory. The results indicate that the proposed approach can provide more precise information to managers and improve decision-making in practical system design.

Sustainable design of a closed-loop location-routing-inventory supply chain network under mixed uncertainty

Considering economic, environmental and social impacts, this paper presents a new sustainable closed-loop location-routing-inventory model under mixed uncertainty. The environmental impacts of CO2 emissions, fuel consumption, wasted energy and the social impacts of created job opportunities and economic development are considered in this paper. The uncertain nature of the network is handled using a stochastic-possibilistic programming approach. Furthermore, for large-sized problems, a hybrid meta-heuristic algorithm and lower bounds are developed and discussed. Finally, a real case study is provided to demonstrate the applicability of the model in real-world applications, and several in-depth analyses are conducted to develop managerial implications.

Interval spectral stochastic finite element analysis of structures with aggregation of random field and bounded parameters

SummaryThis paper presents the study on non‐deterministic problems of structures with a mixture of random field and interval material properties under uncertain‐but‐bounded forces. Probabilistic framework is extended to handle the mixed uncertainties from structural parameters and loads by incorporating interval algorithms into spectral stochastic finite element method. Random interval formulations are developed based on K–L expansion and polynomial chaos accommodating the random field Young's modulus, interval Poisson's ratios and bounded applied forces. Numerical characteristics including mean value and standard deviation of the interval random structural responses are consequently obtained as intervals rather than deterministic values. The randomised low‐discrepancy sequences initialized particles and high‐order nonlinear inertia weight with multi‐dimensional parameters are employed to determine the change ranges of statistical moments of the random interval structural responses. The bounded probability density and cumulative distribution of the interval random response are then visualised. The feasibility, efficiency and usefulness of the proposed interval spectral stochastic finite element method are illustrated by three numerical examples. Copyright © 2016 John Wiley & Sons, Ltd.

Reliability-based design optimization under mixture of random, interval and convex uncertainties

In view of the significant influences of multi-source uncertainties on structural safety, which generally exist in practical engineering (such as the dispersion of material, the uncertainty of external load and the error of processing technology), more academic research and engineering applications had paid attention to uncertainty in recent years. However, due to the complexity of the structural problems, there may be multiple uncertain parameters. Traditional methods of optimal design by single-source uncertainty model, particularly the one derived from probability theory, may no longer be feasible. This paper investigates a new formulation and numerical solution of reliability-based design optimization (RBDO) of structures exhibiting random and uncertain-but-bounded (interval and convex) mixed uncertainties. Combined with the non-probabilistic set-theory convex model and the classical probabilistic approach, the mathematical definition of hybrid reliability is firstly presented for a quantified measure of the safety margin. The reliability-based optimization incorporating such mixed reliability constraints is then formulated. The PSO algorithm is employed to improve the convergence and the stability in seeking the optimal global solution. Additionally, by introducing the general concept of the safety factor, the compatibility between the proposed hybrid RBDO technique and the safety factor-based model is further discussed. By virtue of the above two methods, two numerical examples of typical components (the cantilever structure and the truss structure) as well as one complex engineering example (the hypersonic wing structure) are performed, subjected to the strength or stiffness criteria. The accuracy and effectiveness of the present method are then demonstrated.

Omega-CVaR Portfolio Optimization and Its Worst Case Analysis

This paper presents a novel framework for optimizing portfolios using stochastic thresholds in Omega ratio to control the downside risk. Portfolios resulting from the maximization of the classical Omega ratio simultaneously maximize the probability of superior performance compared to a threshold point set by an investor and minimize the probability of a worse performance compared to the same threshold. However, there is no mandatory rule or mechanism to choose this threshold point in the Omega ratio optimization model; until now it has been left to an investor to decide an appropriate threshold point. In this paper, we redefine the Omega ratio for a loss averse investor by taking the stochastic threshold point as the conditional value-at-risk at an alpha confidence level (CVaR) of the benchmark market. We call this model the Omega-CVaR model. The alpha-value reflects the attitude of an investor towards losses; higher values of indicate a higher loss averse attitude of an investor. We then embed the Omega-CVaR model in a robust portfolio optimization framework and present its worst case analysis under three uncertainty sets, namely mixed, box, and ellipsoidal, for discrete distribution under consideration. The robustness is introduced both in the Omega measure as well as the CVaR measure by respectively maximizing and minimizing their worst scenarios over the uncertainty sets. We show that the worst case Omega-CVaR robust optimization models are linear programs for mixed and box uncertainty sets and a second order cone program under the ellipsoidal set, and hence tractable in all three cases. We conduct a comprehensive empirical investigation to analyze the performance of the Omega-CVaR portfolios and robust Omega-CVaR portfolios under a mixed uncertainty set for listed stocks of the S&P 500 index as of June 2015. We perceive that the optimal portfolios resulting from the Omega-CVaR model exhibit a superior performance compared to the classical CVaR model in the sense of higher expected returns, Sharpe ratios, modified Sharpe ratios, and lesser losses in terms of VaR and CVaR values. Also, robust Omega-CVaR model under mixed uncertainty set is shown to dominate the Omega-CVaR model in terms of all performance measures. Furthermore, the classical CVaR model, the Omega-CVaR model and its robust Omega-CVaR model under mixed uncertainty sets have significantly less losses in terms of CVaR values compared to a naive 1/m portfolio strategy of constituents of the S&P 500 index.

Importance analysis for model with mixed uncertainties

In engineering, sparse or imprecise data often leads to epistemic uncertainty about the distribution parameters of the aleatory input variables. Separating and estimating the individual contributions of the aleatory variability and epistemic parameter uncertainties to the uncertainty in the model output can assist in resource allocation for data collection, as natural variability is irreducible whereas parameter uncertainty is reducible. For structural systems involving aleatory inputs with epistemic distribution parameters, a new kind of importance measure is proposed to distinguish and quantify the individual contributions of these two kinds of uncertainties, in which probability distribution is used to describe the aleatory uncertainty, and fuzzy membership function is employed to represent the epistemic distribution parameter. The mathematical properties of the proposed importance measures are discussed and proved. The defined importance measures are easy to apprehend, and can evaluate the contributions of the aleatory variability and epistemic parameter uncertainties even when the information of the epistemic parameters is very sparse. Thus, they can measure the importance of the two kinds of uncertainties more reasonably. For efficiently estimating the proposed importance measures, a monotonicity analysis is conducted for model function and the probability transformation process. The results indicate that the extreme values of model function at each membership level can generally be obtained by combining the bounds of membership intervals of the fuzzy distribution parameters. Based on the monotonicity analysis, an efficient algorithm is then formulated to compute the proposed importance measures. Several examples demonstrate the rationality and effectiveness of the proposed importance measures and the efficiency of the presented algorithm.

A Novel Multiobjective Programming Model for Coping with Supplier Selection Disruption Risks under Mixed Uncertainties

Supply chain has become more and more vulnerable to disruption since it is suffering widespread risk issues from inside or outside. Higher uncertainties in the supplier selection problem have gone beyond the traditional cost minimization concern. These uncertainties are related to an ever increasing product variety, more demanding customers, and a highly interconnected distribution network. This paper focuses on the supplier selection problem with disruption risks and mixed uncertainties. A novel multiobjective optimization model with mixed uncertain coefficients is developed, which maximizes the total profits and minimizes the percentage of items delivered late, percentage of items rejected, and total loss cost due to supplier dysfunction. Meanwhile, we also consider the customer demand to be a random fuzzy variable and the unit purchase cost to be a fuzzy variable. By examining a numerical example, we found that the confidence level and demand of customers have impact on the quantities purchased by customers from suppliers although the distribution of suppliers will not change. The cost, quality, and service also influence the selection of suppliers. The superevents have little influence on the distribution of supplier selection; however, when unique event occurs, the distribution of supplier selection will change.

Quantification of Margins and Uncertainties Approach for Structure Analysis Based on Evidence Theory

Quantification of Margins and Uncertainties (QMU) is a decision-support methodology for complex technical decisions centering on performance thresholds and associated margins for engineering systems. Uncertainty propagation is a key element in QMU process for structure reliability analysis at the presence of both aleatory uncertainty and epistemic uncertainty. In order to reduce the computational cost of Monte Carlo method, a mixed uncertainty propagation approach is proposed by integrated Kriging surrogate model under the framework of evidence theory for QMU analysis in this paper. The approach is demonstrated by a numerical example to show the effectiveness of the mixed uncertainty propagation method.

Gain-Scheduled Model-Matching Flight Controller Using Inexact Scheduling Parameters

This paper addresses the design problem of Gain-Scheduled (GS) flight controller for the lateral-directional motions of a research airplane. The GS flight controller, which is scheduled by an inexactly provided scheduling parameter, viz. equivalent air speed measured by Pitot tube, is required to realize model-matching as well as gust suppression under the uncertainties related to onboard actuators. The uncertainty in the provided scheduling parameter is supposed to be represented as neither pure proportional nor pure absolute uncertainties, but as mixed uncertainties of the two types of uncertainties. Two GS flight controllers are designed for our problem; one is designed with the consideration of the uncertainty in the provided scheduling parameter, and the other is designed with the supposition that exact scheduling parameter is given. Flight tests in real environment well illustrate the practicality and the effectiveness of the former GS controller compared to the latter GS controller.

Polynomial chaos expansion with random and fuzzy variables

A dynamical uncertain system is studied in this paper. Two kinds of uncertainties are addressed, where the uncertain parameters are described through random variables and/or fuzzy variables. A general framework is proposed to deal with both kinds of uncertainty using a polynomial chaos expansion (PCE). It is shown that fuzzy variables may be expanded in terms of polynomial chaos when Legendre polynomials are used. The components of the PCE are a solution of an equation that does not depend on the nature of uncertainty. Once this equation is solved, the post-processing of the data gives the moments of the random response when the uncertainties are random or gives the response interval when the variables are fuzzy. With the PCE approach, it is also possible to deal with mixed uncertainty, when some parameters are random and others are fuzzy. The results provide a fuzzy description of the response statistical moments.

Probabilistic interval stability assessment for structures with mixed uncertainty

A new hybrid probabilistic and interval computational scheme is proposed to robustly assess the stability of engineering structures involving mixture of random and interval variables. Such hybrid approach possesses noticeable flexibility by directly implementing primitive information on uncertain system parameters, thus the validity of the structural safety assessment against uncertainties can be improved. By implementing the different types of uncertainties, the presented approach is able to separately investigate the effects of random and interval variables acting upon the overall structural stability. A unified interval stochastic sampling (UISS) approach is proposed to calculate the statistical characteristics (i.e., mean and standard deviation) of the lower and upper bounds of the linear bifurcation buckling load of engineering structure involving hybrid uncertain system parameters. Consequently, the stability profile of engineering structure against various uncertainties can be parametrically established, such that the bounds on the maximum structural buckling load at any particular percentile of probability can be effectively determined. Both academic and practically motivated engineering structures have been thoroughly investigated by rigorously establishing the corresponding structural stability profiles, so the applicability and accuracy of the proposed method can be critically justified.

Design and Evaluation of an Automatic Extraventricular Drainage Control System

Until this day, the drainage of cerebrospinal fluid, as necessary in case of an acute increased intracranial pressure, is conducted manually by adjusting the hydrostatic height of an external drainage bag. The associated problems with the manual open-loop control strategy are an increasing pressure error over time periods involving no correction and the inherent risk of an overdrainage, which may occur, for example, after changes of the patient’s upper body inclination angle. In this paper, an automatic control strategy is suggested to alleviate these problems thereby increasing the patient’s safety and the overall quality of the therapy. The automatic controller presented in this paper is designed for our newly developed intelligent external ventricular drainage system. The proposed controller has to guarantee robustness and performance in face of uncertain patient intracranial dynamics and nonlinearities associated with the actuator. The controller is thus designed to guarantee robust performance using a mixed uncertainty modeling approach and extended by a self-scheduling approach to compensate for input nonlinearities. Controller performance is validated in nonlinear simulations, an experimental test setup, and animal experiments, involving pigs with an artificially induced hydrocephalus.

Mixed Uncertainty Analysis of Pole Placement and H∞ Controllers for Directional Drilling Attitude Tracking

This paper describes the design of attitude-hold controllers and their subsequent stability and performance analysis for directional drilling tools as typically used in the oil industry. Based on an input transformation developed in earlier work that partially linearizes and decouples the plant dynamics of the drilling tool, the resulting plant model is used as the basis for both pole placement and optimal H∞ controller designs. A mixed uncertainty stability and performance analysis is then performed on each of the controller designs. Results for a transient simulation of the proposed controller are also presented.