This volume is a collection of papers presented at the international conference on Nonlinear Mathematics for Uncertainty and Its Applications (NLMUA2011), held at Beijing University of Technology during the week of September 7--9, 2011. The conference brought together leading researchers and practitioners involved with all aspects of nonlinear mathematics for uncertainty and its applications. Over the last fifty years there have been many attempts in extending the theory of classical probability and statistical models to the generalized one which can cope with problems of inference and decision making when the model-related information is scarce, vague, ambiguous, or incomplete. Such attempts include the study of nonadditive measures and their integrals, imprecise probabilities and random sets, and their applications in information sciences, economics, finance, insurance, engineering, and social sciences. The book presents topics including nonadditive measures and nonlinear integrals, Choquet, Sugeno and other types of integrals, possibility theory, Dempster-Shafer theory, random sets, fuzzy random sets and related statistics, set-valued and fuzzy stochastic processes, imprecise probability theory and related statistical models, fuzzy mathematics, nonlinear functional analysis, information theory, mathematical finance and risk managements, decision making under various types of uncertainty, and others.
In the viewpoint of Knightian uncertainty, this paper deals with option pricing with jump volatility. First, we prove that the jump volatility model is a Knightian uncertainty problem; then we identify the factors which reflect the Knighitan uncertainty based on k-Ignorance. We find that the option price under Knightian uncertainty is not unique but an interval. Through theoretical analysis and simulation, we conclude that the intensity of Poisson, the jump size, and the maturity date determine the price interval.
Measurement of the value of intellectual capital is increasingly receiving more and more attention. However, the evaluation of intellectual capital is complex, it involves many factors, such as peoples’ utility (human subjective recognition) and the economic efficiency etc. that are very difficult to compute by traditional methods. In this paper we propose an integrated fuzzy evaluation procedure to measure intellectual capital. The main methods used in this research are fuzzy statistical analysis, fuzzy-weighting and fuzzy ranking. This integrated procedure is aimed at yielding appropriate and reasonable rankings and value of intellectual capital. Empirical study shows that fuzzy statistics with soft computing are more realistic and reasonable in the intellectual capital evaluation.
Drawing inferences from general lower probabilities on finite possibility spaces usually involves solving linear programming problems. For some applications this may be too computationally demanding. Some special classes of lower probabilities allow for using computationally less demanding techniques. One such class is formed by the completely monotone lower probabilities, for which inferences can be drawn efficiently once their Möbius transform has been calculated. One option is therefore to draw approximate inferences by using a completely monotone approximation to a general lower probability; this must be an outer approximation to avoid drawing inferences that are not implied by the approximated lower probability. In this paper, we discuss existing and new algorithms for performing this approximation, discuss their relative strengths and weaknesses, and illustrate how each one works and performs.
This paper takes the mean monitoring data of Wenyu river basin in Beijing during 2006-2010 as an example. Based on the characteristics of origin data, the water quality assessment (WQA) is analyzed by factor analysis, which focuses on five aspects: data standardization, applicability evaluation, principle factor extraction, principle factor interpretation and factor scores. Results show that, by objectively and reasonably using the factor analysis to evaluate water quality, the development tendency and variation law of regional water quality can be further understood, which can be a reference for contamination control planning of a region environmental system.
Using a sequence of subsets of an index set for a family of quasinonexpansive mappings, we propose an iterative scheme generated by the shrinking projection method for finding their common fixed point. We prove strong convergence of this scheme under appropriate conditions.
An OWA-operator (ordered weighted averaging aggregation operator) can be seen as a discrete Choquet integral with respect to a symmetric monotone measure. Based on this representation and using universal integrals, several modifications of OWA-operators are introduced and discussed.
Rough set theory established by Pawlak in 1982 plays an important role in dealing with uncertain information and to some extent overlaps fuzzy set theory. The key notions of rough set theory are approximation spaces of pairs (U,R) with R being an equivalence relation on U and approximation operators \(\underline{R}\) and \(\overline{R}\). Let \(\mathcal{R}\) be the family {( \(\underline{R}X,\overline{R}X)|X \subseteq U\)} of approximations endowed with the pointwise order of set-inclusion. It is known that \(\mathcal{R}\) is a complete Stone lattice with atoms and is isomorphic to the family of rough sets in the approximation space (U, R). This paper is devoted to investigate algebraicity and completely distributivity of \(\mathcal{R}\) from the view of domain theory. To this end, completely compact elements, compact elements and atoms of \(\mathcal{R}\) are represented. In terms of the representations established in this paper, it is proved that \(\mathcal{R}\) is isomorphic to a complete ring of sets, consequently \(\mathcal{R}\) is a completely distributive algebraic lattice. An example is given to show that \(\mathcal{R}\) is not atomic nor Boolean in general. Further, a sufficient and necessary condition for \(\mathcal{R}\) being atomic is thus given.
The notion of a weak approximable concept is introduced in this paper, and the lattice consisting of weak approximable concepts is investigated. It is shown that this concept lattice is a completely algebraic lattice, and each completely algebraic lattice is isomorphic to such a concept lattice.
This paper aims at investigating the dynamics of fractional-order model of love in the fact that fractional-order derivatives could possess memories by which romantic relationships are naturally impacted. Based on the discussions of properties including the stability of equilibrium points, chaotic behaviors and typical bifurcations, we found rich dynamics exhibited by the fractional-order love system with proper fractional order and model parameters. Besides, the control problems were studied theoretically and the simulation results illustrated the effectiveness of the proposed methods.
In this paper, we using semi-discrete method, transformed convection-diffusion equation into a ODEs: \(\frac{dU(t)}{dt}\) = AU(t), then we get the solution of the ODEs: U(t) = e tA U0. Furthermore, we give a numerical approximation for e tA and get a special difference scheme for solving the convection-diffusion equation which improve the accuracy order and stability condition greatly. The accuracy order is fourth order and second order in space and time direction respectively. Finally, numerical result shows that this method is effective.
We propose a novel quantization method for human likelihood or preference using rank data. We estimate an evaluation function by assuming that the data is sorted on the basis of observed function values and that the observation includes some errors. We use the difference in the rank order to estimate the corresponding function. Then, its performance will vary depending on the settings. Our data are normally distributed random numbers. First, function values are given as normally distributed random numbers, and after that, observed values are given by adding normally distributed random errors. The performance will depend on the difference in the variances of two normal distributions: for the function values and for the error values. We present a test to evaluate the proposed method by using random data.
Let \(Y_{i} = {x}^{\prime}_{i}\beta + e_{i}, 1 \leq i \leq n, n \geqslant 1\), be a linear regression model. Denote by λ n and μ n the smallest and largest eigenvalues of \(\sum\limits^{n}_{i=1} x_{i}x^{\prime}_{i}\). Assume that the random errors e1, e2, ⋯ are iid, Ee 1 = 0 and E ∣ e 1 ∣ < ∞. Under the restriction that μ n = O(λ n ), this paper obtains the necessary and sufficient condition for the LS estimate of β to be strongly consistent.
This paper is one of many attempts to introduce graphicalMarkov models within Dempster-Shafer theory of evidence. Here we take full advantage of the notion of factorization, which in probability theory (almost) coincides with the notion of conditional independence. In Dempster-Shafer theory this notion can be quite easily introduced with the help of the operator of composition. Nevertheless, the main goal of this paper goes even further. We show that if a belief network (a D-S counterpart of a Bayesian network) is to be used to support decision, one can apply all the ideas of Lauritzen and Spiegelhalter's local computations.
This paper investigates the use of partially reliable information elicited from multiple experts to improve the diagnosis of a railway infrastructure device. The general statistical model used to perform the diagnosis task is based on a noiseless Independent Factor Analysis handled in a soft-supervised learning framework.
We propose a notion of nonspreading mapping with respect to a Bregman distance in a Banach space. We investigate the existence of a fixed point of such a mapping. The proposed class of mappings is related to zero point problems for monotone operators in Banach spaces.
Exploiting the properties of set-valued stochastic trajectory integrals we consider a notion of fuzzy stochastic Lebesgue–Stieltjes trajectory integral and a notion of fuzzy stochastic trajectory integral with respect to martingale. Then we use these integrals in a formulation of fuzzy stochastic integral equations. We investigate the existence and uniqueness of solution to such the equations.
An SI 1 I 2 RS epidemic model is studied. We derive the sufficient conditions on the system parameters which guarantee that the equilibrium points of the system are locally asymptotically stable or globally asymptotically stable.
An index is introduced to measure the risk aversion of a g-expected utility maximizer.
