Elliptic entropy of uncertain set and its applications

Entropy is commonly used as a way to measure the uncertainty of random variables. In uncertain set theory, a concept of entropy for uncertain sets has been defined by using logarithm. However, such an entropy fails to measure the uncertain degree of some uncertain sets. This paper aims at proposing a concept of elliptic entropy for uncertain sets and investigating its properties such as translation invariance and positive linearity. It also provides some formulas for calculating the elliptic entropy via inverse membership functions. Additionally, elliptic relative entropy for uncertain sets is presented as a measure of the difference between two membership functions, and some applications are considered in portfolio selection and clustering.     

Joint forecasts of Dow Jones stocks under general multivariate loss function

When forecasts are assessed by a general loss (cost-of-error) function, the optimal point forecast is, in general, not the conditional mean, and depends on the conditional volatility—which, for stock returns, is time-varying. In order to provide forecasts of daily returns of 30 DJIA stocks under a general multivariate loss function, the following issues are addressed. We discuss what conditions define a multivariate loss function, and a simple class of such functions is proposed. Based on suitable combinations of univariate losses, the suggested multivariate functions are convenient for practical applications with many variables. To keep the computational aspect tractable, a flexible multivariate GARCH model is employed in estimating the conditional forecast distributions. The model easily copes with large number of series while allowing for skewness, fat tails, non-ellipticity, and tail dependence. Based on Engle’s DCC GARCH, it uses multivariate affine generalized hyperbolic distributions as conditional probability law, and the number of parameters to be estimated simultaneously does not depend on the number of series. The model is fitted using daily data from 2002 to 2007 (keeping data from 2008 for out-of-sample forecasts), and a bootstrap procedure is used to derive point forecasts under several multivariate loss functions of the proposed type.     

Reliability-based design optimization for problems with interval distribution parameters

Traditional reliability-based design optimization (RBDO) generally describes uncertain variables using random distributions, while some crucial distribution parameters in practical engineering problems can only be given intervals rather than precise values due to the limited information. Then, an important probability-interval hybrid reliability problem emerged. For uncertain problems in which interval variables are included in probability distribution functions of the random parameters, this paper establishes a hybrid reliability optimization design model and the corresponding efficient decoupling algorithm, which aims to provide an effective computational tool for reliability design of many complex structures. The reliability of an inner constraint is an interval since the interval distribution parameters are involved; this paper thus establishes the probability constraint using the lower bound of the reliability degree which ensures a safety design of the structure. An approximate reliability analysis method is given to avoid the time-consuming multivariable optimization of the inner hybrid reliability analysis. By using an incremental shifting vector (ISV) technique, the nested optimization problem involved in RBDO is converted into an efficient sequential iterative process of the deterministic design optimization and the hybrid reliability analysis. Three numerical examples are presented to verify the proposed method, which include one simple problem with explicit expression and two complex practical applications.     

Uncertain expected utility function and its risk premium

Other than traditional decision theory, this paper employs uncertainty theory to handle indeterminacy. Uncertain variables are used to represent uncertain choices. Uncertain expected utility function is defined as an increasing function of uncertain choices. Several mathematical properties of the uncertain expected utility functions are derived using inverse uncertainty distributions. In order to compare two different choices, the first order dominance and second order dominance via uncertain expected utility functions are introduced. We also investigate risk aversion attitude and risk premium. Finally, the relationship between risk premium and risk averse attitude is investigated.     

Confidence-based reliability assessment considering limited numbers of both input and output test data

Simulation-based methods can be used for accurate uncertainty quantification and prediction of the reliability of a physical system under the following assumptions: (1) accurate input distribution models and (2) accurate simulation models (including accurate surrogate models if utilized). However, in practical engineering applications, often only limited numbers of input test data are available for modeling input distribution models. Thus, estimated input distribution models are uncertain. In addition, the simulation model could be biased due to assumptions and idealizations used in the modeling process. Furthermore, only a limited number of physical output test data is available in the practical engineering applications. As a result, target output distributions, against which the simulation model can be validated, are uncertain and the corresponding reliabilities become uncertain as well. To assess the conservative reliability of the product properly under the uncertainties due to limited numbers of both input and output test data and a biased simulation model, a confidence-based reliability assessment method is developed in this paper. In the developed method, a hierarchical Bayesian model is formulated to obtain the uncertainty distribution of reliability. Then, we can specify a target confidence level. The reliability value at the target confidence level using the uncertainty distribution of reliability is the confidence-based reliability, which is the confidence-based estimation of the true reliability. It has been numerically demonstrated that the proposed method can predict the reliability of a physical system that satisfies the user-specified target confidence level, using limited numbers of input and output test data.     

Perception-based granular probabilities in risk modeling and decision making

Our focus is on the issue of decision making in risky situations. We discuss the need for using decision functions to aid in capturing the decision maker's preference among these types of uncertain alternatives. The use of fuzzy rule based formulations to model these functions is investigated. We discuss the role of perception based granular probability distributions as a means of modeling the uncertainty profiles of the alternatives. Various properties of this method of describing uncertainty are provided. Tools for evaluating rule based decision functions in the face of perception based uncertainty profiles are presented. Consideration is given to the situation in which our uncertainty profiles are expressed in terms of a cumulative distribution function. We introduce the idea of a perception based granular cumulative distribution and describe its representation in terms of a fuzzy rule-based model.     

广义区间上的泛逻辑自相关系数求解

广义自相关系数的求解是泛逻辑在不确定性推理中需要解决的关键问题之一。称任意[a,b]区间为广义区间,在广义区间上给出了广义N范数、广义N性生成元、广义自相关系数的定义。提出了由复杂系统参数的分布函数求解广义自相关系数的一般方法,给出并证明了重要的直通NLK公式。最后举例说明了求解k值的具体,为从数学模型和逻辑推理两个角度来分析复杂系统参数间的相关性提供了一种新的思路。     

Minimum spanning tree problem of uncertain random network

Minimum spanning tree problem is a typical and fundamental problem in combinatorial optimization. Most of the existing literature is devoted to the case with deterministic or random weights. However, due to lack of data, a proportion of edge weights have to be estimated according to experts’ evaluations, which may be considered as uncertain variables. This paper focuses on the case where some weights are random variables and the others are uncertain variables. The concept of an ideal chance distribution is introduced and its expression is given based on the uncertainty distributions and probability distributions. A model is formulated to find a minimum spanning tree whose chance distribution is the closest to the ideal one. Finally, a numerical example is given to illustrate the modelling idea of the study.     

Distributionally robust fuzzy project portfolio optimization problem with interactive returns

Effective project selection and staff assignment strategies directly impact organizational profitability. Based on critical value optimization criterion, this paper discusses how uncertainty and interaction impact the project portfolio return and staff allocation. Since the exact possibility distributions of uncertain parameters in practical project portfolio problems are often unavailable, we adopt variable parametric possibility distributions to characterize uncertain model parameters. Furthermore, this paper develops a novel parametric credibilistic optimization method for project portfolio selection problem. According to the structural characteristics of variable parametric possibility distributions, we derive the equivalent analytical expressions of credibility constraints, and turn the original credibilistic project portfolio model into its equivalent nonlinear mixed-integer programming models. To show the advantages of the proposed parametric credibilistic optimization method, some numerical experiments are conducted by setting various values of distribution parameters. The computational results support our arguments by comparing with the optimization method under fixed possibility distributions.     

Estimation of body and tail distribution under extreme events for reliability analysis

In the past decades, many reliability analyses have been developed and applied to engineering fields considering uncertainties of input and output random variables as normal distributions. However, when input uncertainty is taken into the system as extreme events such as weather, temperature, environmental conditions etc., output distribution cannot be described by normal distribution. On the other hand, one of distributions to analyze reliability of a system under extreme events is generalized Pareto distribution. Generalized Pareto distribution has been developed and applied for modelling extreme events. However, conventional methods estimate only the shape and scale parameters by assuming that the location parameter is chosen by experiences focused only on the tail distribution. However, since the tail distribution affected by the body distribution and vice versa, both the body and tail distributions should be considered when the parameters of distribution are estimated. In this study, therefore, a new parameter estimation method is proposed to determine shape, scale and location parameters simultaneously by combining likelihood functions of body and tail distributions using Akaike information criterion and generalized Pareto distribution, respectively. Finally, the parameters of body and tail distributions are estimated by maximum likelihood estimation. The proposed method is verified by using mathematical examples with and without inclusion of extreme events. Results show that the proposed method can estimate parameters and distributions for body and tail distributions as well as the more accurate reliability of system under extreme events.     

Finite population estimation under generalized linear model assistance

Finite population estimation is the overall goal of sample surveys. When information regarding auxiliary variables are available, one may take advantage of general regression estimators (GREG) to improve sample estimates precision. GREG estimators may be derived when the relationship between interest and auxiliary variables is represented by a normal linear model. However, in some cases, such as when estimating class frequencies or counting processes means, Bernoulli or Poisson models are more suitable than linear normal ones. This paper focuses on building regression type estimators under a model-assisted approach, for the general case in which the relationship between interest and auxiliary variables may be suitably described by a generalized linear model. The finite population distribution of the variable of interest is viewed as if generated by a member of the exponential family, which includes Bernoulli, Poisson, gamma and inverse Gaussian distributions, among others. The resulting estimator is a generalized linear model regression estimator (GEREG). Its general form and basic statistical properties are presented and studied analytically and empirically, using Monte Carlo simulation experiments. Three applications are presented in which the GEREG estimator shows better performance than the GREG one.     

MIP-based approaches for robust storage loading problems with stacking constraints

In this paper, we consider storage loading problems under uncertainty where the storage area is organized in fixed stacks with a limited height. Such problems appear in several practical applications, e.g., when loading container terminals, container ships or warehouses. Incoming items arriving at a partly filled storage area have to be assigned to stacks under the restriction that not every item may be stacked on top of every other item and taking into account that some items with uncertain data will arrive later. Following the robust optimization paradigm, we propose different MIP formulations for the strictly and adjustable robust counterparts of the uncertain problem. Furthermore, we show that in the case of interval uncertainties the computational effort to find adjustable robust solutions can be reduced. Computational results are presented for randomly generated instances with up to 480 items. The results show that instances of this size can be solved in reasonable time and that including robustness improves solutions where uncertainty is not taken into account.     

[0,∞]区间N范数的定义及生成定理

柔性逻辑学的研究目标是探索逻辑的一般规律,它指出命题真值误差用连续变化的广义自相关系数k∈[0,1]来刻画。在柔性逻辑的不确定推理中,N范数是一级运算的数理模型。由于在现实生活中,很多逻辑推理控制必须在其自身的定义域内完成,因此以三角范数作为柔性逻辑学研究的数学工具,定义了[0,∞]区间上的N范数和N性生成元,并研究了相关主要性质;证明了N范数生成定理;给出了广义自相关系数的计算方法;证明了[0,∞]区间上指数(幂)型N性生成元为N性生成元完整簇;从而为柔性逻辑中[0,∞]区间的一级运算模型提供了重要的理论基础。     

Numerical solution of multi-variable order fractional integro-differential equations using the Bernstein polynomials

In practice, computer simulations cannot be perfectly controlled because of the inherent uncertainty caused by variability in the environment (e.g., demand rate in the inventory management). Ignoring this source of variability may result in sub-optimality or infeasibility of optimal solutions. This paper aims at proposing a new method for simulation–optimization when limited knowledge on the probability distribution of uncertain variables is available and also limited budget for computation is allowed. The proposed method uses the Taguchi robust terminology and the crossed array design when its statistical techniques are replaced by design and analysis of computer experiments and Kriging. This method offers a new approach for weighting uncertainty scenarios for such a case when probability distributions of uncertain variables are unknown without available historical data. We apply a particular bootstrapping technique when the number of simulation runs is much less compared to the common bootstrapping techniques. In this case, bootstrapping is undertaken by employing original (i.e., non-bootstrapped) data, and thus, it does not result in a computationally expensive task. The applicability of the proposed method is illustrated through the Economic Order Quantity (EOQ) inventory problem, according to uncertainty in the demand rate and holding cost.

     

Data visualization via latent variables and mixture models: a brief survey

In the literature, data visualization is extensively studied via diverse parametric probabilistic distributions for the exploration of continuous, binary, and counting data. An overview of the existing methods for non-symmetric data matrices is presented in an unified framework via the Bernoulli law and binary variables. An extension to continuous or counting variables is available by using instead any another univariate distribution such as the Poisson or Gaussian one. Several approaches are possible when the model is with a distribution on the rows, the columns, the row clusters, the column clusters, the cells, the blocks, or a transformed matrix of the distances from the pairs of rows or columns. The objective functions are presented with their full expressions in separated sections, one for each method: Kohonen’s map and related methods of self-organizing maps, generative topographic mapping as a probabilistic self-organizing map, linear principal component analysis and related matricial methods (non-negative factorization, factorization), probabilistic parametric embedding, probabilistic latent semantic visualization, latent cluster position model, t-distributed stochastic neighbor embedding. The conclusion is a discussion of the contribution with perspectives.     

基于Pythagorean不确定语言的扩展VIKOR多属性群决策方法

针对属性值为Pythagorean不确定语言变量,属性权重和专家权重完全未知的群决策问题,提出一种扩展VIKOR多属性群决策方法.首先,给出Pythagorean不确定语言变量的概念,提出考虑语义变化的Pythagorean不确定语言变量运算规则、大小比较方法和Hamming距离测度;其次,提出基于Pythagorean 不确定语言模糊熵的属性权重确定方法和基于相似度的专家权重确定方法,进而提出一种新的扩展VIKOR方法;最后,通过国内航空公司服务质量评估实例验证所提出方法的有效性和可行性.     

Uncertain programming models for portfolio selection with uncertain returns

In an indeterminacy economic environment, experts’ knowledge about the returns of securities consists of much uncertainty instead of randomness. This paper discusses portfolio selection problem in uncertain environment in which security returns cannot be well reflected by historical data, but can be evaluated by the experts. In the paper, returns of securities are assumed to be given by uncertain variables. According to various decision criteria, the portfolio selection problem in uncertain environment is formulated as expected-variance-chance model and chance-expected-variance model by using the uncertainty programming. Within the framework of uncertainty theory, for the convenience of solving the models, some crisp equivalents are discussed under different conditions. In addition, a hybrid intelligent algorithm is designed in the paper to provide a general method for solving the new models in general cases. At last, two numerical examples are provided to show the performance and applications of the models and algorithm.     

滞后离散广义系统的鲁棒严格耗散控制

研究确定的及不确定的滞后离散广义系统的无记忆状态反馈严格耗散控制器设计问题.利用线性矩阵不等式(LMI)方法,首先给出滞后离散广义系统容许(即正则、稳定、因果)且严格耗散的条件,然后通过矩阵不等式(MIs)得到无记忆状态反馈严格耗散控制器的存在条件和设计方法;进而针对除E外其余系数矩阵均具有范数有界不确定性的滞后离散广义系统,利用矩阵不等式的解设计鲁棒严格耗散控制器,保证闭环系统广义二次稳定且严格耗散.     

Type-2 fuzzy Gaussian mixture models

This paper presents a new extension of Gaussian mixture models (GMMs) based on type-2 fuzzy sets (T2 FSs) referred to as T2 FGMMs. The estimated parameters of the GMM may not accurately reflect the underlying distributions of the observations because of insufficient and noisy data in real-world problems. By three-dimensional membership functions of T2 FSs, T2 FGMMs use footprint of uncertainty (FOU) as well as interval secondary membership functions to handle GMMs uncertain mean vector or uncertain covariance matrix, and thus GMMs parameters vary anywhere in an interval with uniform possibilities. As a result, the likelihood of the T2 FGMM becomes an interval rather than a precise real number to account for GMMs uncertainty. These interval likelihoods are then processed by the generalized linear model (GLM) for classification decision-making. In this paper we focus on the role of the FOU in pattern classification. Multi-category classification on different data sets from UCI repository shows that T2 FGMMs are consistently as good as or better than GMMs in case of insufficient training data, and are also insensitive to different areas of the FOU. Based on T2 FGMMs, we extend hidden Markov models (HMMs) to type-2 fuzzy HMMs (T2 FHMMs). Phoneme classification in the babble noise shows that T2 FHMMs outperform classical HMMs in terms of the robustness and classification rate. We also find that the larger area of the FOU in T2 FHMMs with uncertain mean vectors performs better in classification when the signal-to-noise ratio is lower.     

Asymptotic stabilization of a class of uncertain composite systems

A class of stabilizing, generalized feedback controls is presented for a class of uncertain dynamical systems. The uncertain systems are based on a composite prototype system consisting of two nonlinearly coupled subsystems, with a non-asymptotically-stabilizable linearization. To encompass all possible realizations of uncertainty, a problem formulation based on differential inclusions is adopted. The generalized feedback controls, described in terms of set-valued maps, have practical analogues in the form of discontinuous feedback controls. An intrinsic element of the control strategy is designed to render a prescribed nonlinear manifold, containing the state origin, invariant and globally finite-time attractive. For each uncertain system, the generalized feedback controls guarantee global asymptotic stability of the zero state.