Integration and Conditioning in Numerical Possibility Theory

The paper discusses integration and some aspects of conditioning in numerical possibility theory, where possibility measures have the behavioural interpretation of upper probabilities, that is, systems of upper betting rates. In such a context, integration can be used to extend upper probabilities to upper previsions. It is argued that the role of the fuzzy integral in this context is limited, as it can only be used to define a coherent upper prevision if the associated upper probability is 0–1-valued, in which case it moreover coincides with the Choquet integral. These results are valid for arbitrary coherent upper probabilities, and therefore also relevant for possibility theory. It follows from the discussion that in a numerical context, the Choquet integral is better suited than the fuzzy integral for producing coherent upper previsions starting from possibility measures. At the same time, alternative expressions for the Choquet integral associated with a possibility measure are derived. Finally, it is shown that a possibility measure is fully conglomerable and satisfies Walley's regularity axiom for conditioning, ensuring that it can be coherently extended to a conditional possibility measure using both the methods of natural and regular extension.     

METRIC STRUCTURES ON POSSIBILITY DISTRIBUTIONS

Abstract

Higashi and Klir have defined a metric on the set of possibility distributions, based on the U-uncertainty. We show here that similar metrics can be defined based on Yager's measure of (non) specificity and the imprecision measure of Lamata and Moral. These metrics satisfy almost all properties of the earlier metric, indicating some invariant characteristics of these three measures of (non) specificity. We also disprove some results present in the literature. Finally, we argue in favor of defining these metrics on ordered possibility distributions.     

STATISTICAL INFERENCES BASED ON A SECOND-ORDER POSSIBILITY DISTRIBUTION

Abstract

A new, general method of statistical inference is proposed. It encompasses all the coherent forms of statistical inference that can be derived from a Bayesian prior distribution, Bayesian sensitivity analysis or upper and lower prior probabilities. The method is to model prior uncertainty about statistical parameters in terms of a second-order possibility distribution (a special type of upper probability) which measures the plausibility of each conceivable prior probability distribution. This defines an imprecise hierarchical model. Two,applications are studied: the problem of robustifying Bayesian analyses by forming a neighbourhood of a Bayesian prior distribution, and the problem of combining prior opinions from different sources.     

Existence of solutions for a class of nonlinear Volterra singular integral equations

In this paper we prove some results concerning the existence of solutions for a large class of nonlinear Volterra singular integral equations in the space C[0,1] consisting of real functions defined and continuous on the interval [0,1]. The main tool used in the proof is the concept of a measure of noncompactness. We also present some examples of nonlinear singular integral equations of Volterra type to show the efficiency of our results. Moreover, we compare our theory with the approach depending on the use of the theory of Volterra-Stieltjes integral equations. We also show that the results of the paper are applicable in the study of the so-called fractional integral equations which are recently intensively investigated and find numerous applications in describing some real world problems.     

连续等距区间上积分值的二次样条插值

目的 在现实中,某些插值问题结点处的函数值往往是未知的,而仅仅已知一些区间上的积分值。为此提出一种给定已知函数在连续等距区间上的积分值构造二次样条插值函数的方法。方法 首先,利用二次B样条基函数的线性组合去满足给定的积分值和两个端点插值条件,该插值问题等价于求解n+2个方程带宽为3的线性方程组。然后,运用算子理论给出二次样条插值函数的误差估计,继而得到二次样条函数逼近结点处的函数值时具有超收敛性。最后,通过等距区间上积分值的线性组合逼近两个端点的函数值方法实现了不带任何边界条件的积分型二次样条插值问题。结果 选取低频率函数,对积分型二次样条插值方法和改进方法分别进行数值测试,发现这两种方法逼近效果都是良好的。同样,选取高频率函数对积分型二次样条插值方法进行数值实验,得到数值收敛阶与理论值相一致。结论 实验结果表明,本文算法相比已有的方法更简单有效,对改进前后的二次样条插值函数在逼近结点处的函数值时的超收敛性得到了验证。该方法对连续等距区间上积分值的函数重构具有普适性。     

On the quadratic integral equations and their applications

The class of quadratic integral equations contains, as a special case, numerous integral equations encountered in the theory of radiative transfer, the queuing theory, the kinetic theory of gases and the theory of neutron transport. As a pursuit of this, in the following pages, sufficient conditions are given for the existence of positive continuous solutions to some possibly singular quadratic integral equations. Meanwhile, we prove the existence of maximal and minimal solutions of our problems. The method used here depends on both Schauder and Schauder–Tychonoff fixed point principles. Unlike all previous contributions of the same type, no assumptions in terms of the measure of noncompactness were imposed on the nonlinearity of the given functions. As far as we know, the approach presented in this paper, in particular, the discussion of the existence of maximal and minimal solutions to the quadratic integral equations was never applied in the field of the quadratic integral equations and so is new.     

CONDITIONAL POSSIBILITY MEASURES

Possibility theory is the formalization of the methods of reasoning about uncertainty and information, derived from the principles of fuzzy sets and systems. Derived possibilistic assignments can be constructed within this theory. This paper defines the notion of conditional possibility—an assignment subject to conditioning by other possibilistic random variables. The proposed definition can be based on one of the two principles: proper interaction with marginal distributions, or minimization of possibilistic information distance between the original and derived distributions. Both approaches independently lead to the same definition, thus strongly suggesting its wider applicability.     

Adams Conditionals and Non-Monotonic Probabilities

Adams' famous thesis that the probabilities of conditionals are conditional probabilities is incompatible with standard probability theory. Indeed it is incompatible with any system of monotonic conditional probability satisfying the usual multiplication rule for conditional probabilities. This paper explores the possibility of accommodating Adams' thesis in systems of non-monotonic probability of varying strength. It shows that such systems impose many familiar lattice theoretic properties on their models as well as yielding interesting logics of conditionals, but that a standard complementation operation cannot be defined within them, on pain of collapsing probability into bivalence.     

Fuzzy random variables—II. Algorithms and examples for the discrete case

The results obtained in part I of the paper are specialized to the case of discrete fuzzy random variables. A more intuitive interpretation is given of the notion of fuzzy random variables. Algorithms are derived for determining expectations, fuzzy probabilities, fuzzy conditional expectations and fuzzy conditional probabilities related to discrete fuzzy random variables. These algorithms are applied to illustrative examples. A sample application to a medical diagnosis problem is briefly discussed.     

Stochastic grid bundling method for backward stochastic differential equations

ABSTRACT

In this work, we apply the Stochastic Grid Bundling Method (SGBM) to numerically solve backward stochastic differential equations (BSDEs). The SGBM algorithm is based on conditional expectations approximation by means of bundling of Monte Carlo sample paths and a local regress-later regression within each bundle. The basic algorithm for solving the backward stochastic differential equations will be introduced and an upper error bound is established for the local regression. A full error analysis is also conducted for the explicit version of our algorithm and numerical experiments are performed to demonstrate various properties of our algorithm.     

多层积分值三次样条拟插值

目的 在实际问题中,某些插值问题结点处的函数值往往是未知的,而仅仅知道一些连续等距区间上的积分值。为此提出了一种基于未知函数在连续等距区间上的积分值和多层样条拟插值技术来解决函数重构。该方法称之为多层积分值三次样条拟插值方法。方法 首先,利用积分值的线性组合来逼近结点处的函数值;然后,利用传统的三次B-样条拟插值和相应的误差函数来实现多层三次样条拟插值;最后,给出两层积分值三次样条拟插值算子的多项式再生性和误差估计。结果 选取无穷次可微函数对多层积分值三次样条拟插值方法和已有的积分值三次样条拟插值方法进行对比分析。数值实验印证了本文方法在逼近误差和数值收敛阶均稍占优。结论本文多层三次样条拟插值函数能够在整体上很好的逼近原始函数,一阶和二阶导函数。本文方法较之于已有的积分值三次样条拟插值方法具有更好的逼近误差和数值收敛阶。该方法对连续等距区间上积分值的函数重构具有普适性。     

CONVERGENCE THEOREMS FOR SEQUENCES OF CHOQUET INTEGRALS

Abstract

The Choquet integral with respect to nonadditive monotone set functions, including imprecise probabilities and fuzzy measures, is a generalization of the classical Lebesgue integral. It is one kind of nonlinear functionals defined on a subspace of all real valued measurable functions. In this paper, several different types of convergence, including the mean convergence that is based on the Choquet integral, for sequences of measurable functions are considered, and the corresponding convergence theorems for sequence of Choquel integrals are demonstrated. Particularly, the theorem of convergence in measure is presented in a form of “necessary and sufficient condition” by using the structural characteristics of nonnegative monotone set functions. As an application of convergence theorems, the stability of a class of nonlinear integral systems is discussed.     

Fuzzy measures and generalized Möbius transform

Generalized Möbius transform is recalled and applied in some special cases. The relationship with the standard Möbius transform is shown. By means of the generalized Möbius transform, a general concept of k -order additivity independent of the cardinality of the underlying space is introduced. The relationship of the Choquet integral and the Lebesgue integral by means of the generalized Möbius transform is clarified. Also possibilistic Möbius transform and k -order possibility measures are introduced. Finally, some examples are given, including the characterization of de Finetti's discrete lower probabilities.     

Independence and Possibilistic Conditioning

There is not a unique definition of a conditional possibility distribution since the concept of conditioning is complex and many papers have been conducted to define conditioning in a possibilistic framework. In most cases, independence has been also defined and studied by means of a kind of analogy with the probabilistic case. In [2,4], we introduce conditional possibility as a primitive concept by means of a function whose domain is a set of conditional events. In this paper, we define a concept of independence associated with this form of conditional possibility and we show that classical properties required for independence concepts are satisfied.     

A numerical method for solving systems of linear and nonlinear integral equations of the second kind by hat basis functions

In this paper, we use hat basis functions to solve the system of Fredholm integral equations (SFIEs) and the system of Volterra integral equations (SVIEs) of the second kind. This method converts the system of integral equations into a linear or nonlinear system of algebraic equations. Also, we consider the order of convergence of the method and show that it is O(h²). Application of the method on some examples show its accuracy and efficiency.     

Meadows and the equational specification of division

The rational, real and complex numbers with their standard operations, including division, are partial algebras specified by the axiomatic concept of a field. Since the class of fields cannot be defined by equations, the theory of equational specifications of data types cannot use field theory in applications to number systems based upon rational, real and complex numbers. We study a new axiomatic concept for number systems with division that uses only equations: a meadow is a commutative ring with a total inverse operator satisfying two equations which imply 0^?1=0. All fields and products of fields can be viewed as meadows. After reviewing alternate axioms for inverse, we start the development of a theory of meadows. We give a general representation theorem for meadows and find, as a corollary, that the conditional equational theory of meadows coincides with the conditional equational theory of zero totalized fields. We also prove representation results for meadows of finite characteristic.     

Approximate solution of a system of linear integral equations by the Taylor expansion method

A simple and efficient approximate technique is developed to obtain the solution to a system of linear integral equations. This technique is based on the Taylor expansion. The method has been successfully applied to determine approximate solutions of a system of Fredholm integral equations and Volterra integral equations of not only the second kind but also the first kind. The mth order approximation of the solution is exact up to a polynomial of degree equal to or less than m. Several illustrative examples are presented to show the effectiveness and accuracy of this method.     

specsim: A Fortran-77 program for conditional spectral simulation in 3D

A Fortran 77 program, specsim, is presented for conditional spectral simulation in 3D domains. The traditional Fourier integral method allows generating random fields with a given covariance spectrum. Conditioning to local data is achieved by an iterative identification of the conditional phase information. A flowchart of the program is given to illustrate the implementation procedures of the program. A 3D case study is presented to demonstrate application of the program. A comparison with the traditional sequential Gaussian simulation algorithm emphasizes the advantages and drawbacks of the proposed algorithm.     

A Taylor Collocation Method for the Solution of Linear Integro-Differential Equations

In this study, a matrix method called the Taylor collocation method is presented for numerically solving the linear integro-differential equations by a truncated Taylor series. Using the Taylor collocation points, this method transforms the integro-differential equation to a matrix equation which corresponds to a system of linear algebraic equations with unknown Taylor coefficients. Also the method can be used for linear differential and integral equations. To illustrate the method, it is applied to certain linear differential, integral, and integro-differential equations and the results are compared.     

Decision Making with Second‐Order Imprecise Probabilities

In decision analysis, uncertainty is usually described in the framework of probability. However, a large number of experimental and theoretical studies showed that a single nature of probability does not accurately capture human preferences. To avoid this drawback, they use imprecise probabilities. But, as decision maker is usually uncertain about first‐order imprecise probabilities, imprecise hierarchical probability models are used. For most of such models, the second levels are precise. There also exist studies on two‐level imprecise hierarchical models, which use imprecise probabilities or possibilities at the second level. Most of these works are based on lower prevision theory leading to a large number of optimization problems. In the present paper, we propose an imprecise hierarchical decision‐making model where the first and the second level are described by interval probabilities. The method associates with the construction of a nonadditive measure as a lower prevision and uses this capacity in Choquet integral for constructing a utility function.