Envisioning uncertainty in geospatial information

Geospatial reasoning has been an essential aspect of military planning since the invention of cartography. Although maps have always been a focal point for developing situational awareness, the dawning era of network-centric operations brings the promise of unprecedented battlefield advantage due to improved geospatial situational awareness. Geographic information systems (GIS) and GIS-based decision support systems are ubiquitous within current military forces, as well as civil and humanitarian organizations. Understanding the quality of geospatial data is essential to using it intelligently. A systematic approach to data quality requires: estimating and describing the quality of data as they are collected; recording the data quality as metadata; propagating uncertainty through models for data processing; exploiting uncertainty appropriately in decision support tools; and communicating to the user the uncertainty in the final product. There are shortcomings in the state-of-the-practice in GIS applications in dealing with uncertainty. No single point solution can fully address the problem. Rather, a system-wide approach is necessary. Bayesian reasoning provides a principled and coherent framework for representing knowledge about data quality, drawing inferences from data of varying quality, and assessing the impact of data quality on modeled effects. Use of a Bayesian approach also drives a requirement for appropriate probabilistic information in geospatial data quality metadata. This paper describes our research on data quality for military applications of geospatial reasoning, and describes model views appropriate for model builders, analysts, and end users.     

Some remarks on functions with values in probabilistic normed spaces

In this paper we consider an enlargement of the notion of the probabilistic normed space. For this new class of probabilistic normed spaces we give some topological properties. By using properties of the probabilistic norm we prove some differential and integral properties of functions with values into probabilistic normed spaces. As special cases, results for deterministic and random functions can be obtained.      

Students’ emergent articulations of uncertainty while making informal statistical inferences

Research on informal statistical inference has so far paid little attention to the development of students?? expressions of uncertainty in reasoning from samples. This paper studies students?? articulations of uncertainty when engaged in informal inferential reasoning. Using data from a design experiment in Israeli Grade 5 (aged 10?C11) inquiry-based classrooms, we focus on two groups of students working with TinkerPlots on investigations with growing sample size. From our analysis, it appears that this design, especially prediction tasks, helped in promoting the students?? probabilistic language. Initially, the students oscillated between certainty-only (deterministic) and uncertainty-only (relativistic) statements. As they engaged further in their inquiries, they came to talk in more sophisticated ways with increasing awareness of what is at stake, using what can be seen as buds of probabilistic language. Attending to students?? emerging articulations of uncertainty in making judgments about patterns and trends in data may provide an opportunity to develop more sophisticated understandings of statistical inference.     

Integral curves from noisy diffusion MRI data with closed-form uncertainty estimates

We present a method for estimating the trajectories of axon fibers through diffusion tensor MRI (DTI) data that provides theoretically rigorous estimates of trajectory uncertainty. We develop a three-step estimation procedure based on a kernel estimator for a tensor field based on the raw DTI measurements, followed by a plug-in estimator for the leading eigenvectors of the tensors, and a plug-in estimator for integral curves through the resulting vector field. The integral curve estimator is asymptotically normal; the covariance of the limiting Gaussian process allows us to construct confidence ellipsoids for fixed points along the curve. Complete trajectories of fibers are assembled by stopping integral curve tracing at locations with multiple viable leading eigenvector directions and tracing a new curve along each direction. Unlike probabilistic tractography approaches to this problem, we provide a rigorous, theoretically sound model of measurement uncertainty as it propagates from the raw MRI data, to the tensor field, to the vector field, to the integral curves. In addition, trajectory uncertainty is estimated in closed form while probabilistic tractography relies on sampling the space of tensors, vectors, or curves. We show that our estimator provides more realistic trajectory uncertainty estimates than a more simplified prior approach for closed-form trajectory uncertainty estimation due to Koltchinskii et al. (Ann Stat 35:1576–1607, 2007) and a popular probabilistic tractography method due to Behrens et al. (Magn Reson Med 50:1077–1088, 2003) using theory, simulation, and real DTI scans.     

A universal algebraic approach for conditional independence

In this paper we show that elementary properties of joint probability density functions naturally induce a universal algebraic structure suitable for studying probabilistic conditional independence (PCI) relations. We call this algebraic system the cain. In the cain algebra, PCI relations are represented in equational forms. In particular, we show that the cain satisfies the axioms of the graphoid of Pearl and Paz (Advances in artificial intelligence. North-Holland, Amsterdam, 1987) and the separoid of Dawid (Ann. Math. Artif. Intell. 32:335–372, 2001), these axiomatic systems being useful for general probabilistic reasoning.     

Rational approximation of vertical segments

In many applications, observations are prone to imprecise measurements. When constructing a model based on such data, an approximation rather than an interpolation approach is needed. Very often a least squares approximation is used. Here we follow a different approach. A natural way for dealing with uncertainty in the data is by means of an uncertainty interval. We assume that the uncertainty in the independent variables is negligible and that for each observation an uncertainty interval can be given which contains the (unknown) exact value. To approximate such data we look for functions which intersect all uncertainty intervals. In the past this problem has been studied for polynomials, or more generally for functions which are linear in the unknown coefficients. Here we study the problem for a particular class of functions which are nonlinear in the unknown coefficients, namely rational functions. We show how to reduce the problem to a quadratic programming problem with a strictly convex objective function, yielding a unique rational function which intersects all uncertainty intervals and satisfies some additional properties. Compared to rational least squares approximation which reduces to a nonlinear optimization problem where the objective function may have many local minima, this makes the new approach attractive.     

Solution of nonlinear Volterra–Fredholm–Hammerstein integral equations via rationalized Haar functions

Rationalized Haar functions are developed to approximate of the nonlinear Volterra–Fredholm–Hammerstein integral equations. The properties of rationalized Haar functions are first presented, and the operational matrix of integration together with the product operational matrix are utilized to reduce the computation of integral equations into some algebraic equations. The method is computationally attractive, and applications are demonstrated through illustrative examples.     

Optimization on class of operator equations in the probabilistic case setting

In this paper, we introduce a problem of the optimization of approximate solutions of operator equations in the probabilistic case setting, and prove a general result which connects the relation between the optimal approximation order of operator equations with the asymptotic order of the probabilistic width. Moreover, using this result, we determine the exact orders on the optimal approximate solutions of multivariate Preldholm integral equations of the second kind with the kernels belonging to the multivariate Sobolev class with the mixed derivative in the probabilistic case setting.     

Approximate approximations from scattered data

The aim of this paper is to extend the approximate quasi-interpolation on a uniform grid by dilated shifts of a smooth and rapidly decaying function to scattered data quasi-interpolation. It is shown that high order approximation of smooth functions up to some prescribed accuracy is possible, if the basis functions, which are centered at the scattered nodes, are multiplied by suitable polynomials such that their sum is an approximate partition of unity. For Gaussian functions we propose a method to construct the approximate partition of unity and describe an application of the new quasi-interpolation approach to the cubature of multi-dimensional integral operators.     

A methodology for developing Bayesian networks: An application to information technology (IT) implementation

Bayesian Networks (BNs) are probabilistic inference engines that support reasoning under uncertainty. This article presents a methodology for building an information technology (IT) implementation BN from client–server survey data. The article also demonstrates how to use the BN to predict the attainment of IT benefits, given specific implementation characteristics (e.g., application complexity) and activities (e.g., reengineering). The BN is an outcome of a machine learning process that finds the network’s structure and its associated parameters, which best fit the data. The article will be of interest to academicians who want to learn more about building BNs from real data and practitioners who are interested in IT implementation models that make probabilistic statements about certain implementation decisions.     

Constructing and evaluating alternative frames of discernment

We construct alternative frames of discernment from input belief functions. We assume that the core of each belief function is a subset of a so far unconstructed frame of discernment. The alternative frames are constructed as different cross products of unions of different cores. With the frames constructed the belief functions are combined for each alternative frame. The appropriateness of each frame is evaluated in two ways: (i) we measure the aggregated uncertainty (an entropy measure) of the combined belief functions for that frame to find if the belief functions are interacting in interesting ways, (ii) we measure the conflict in Dempster’s rule when combining the belief functions to make sure they do not exhibit too much internal conflict. A small frame typically yields a small aggregated uncertainty but a large conflict, and vice versa. The most appropriate frame of discernment is that which minimizes a probabilistic sum of the conflict and a normalized aggregated uncertainty of all combined belief functions for that frame of discernment.     

An evidential approach to SLAM,path planning,and active exploration

Probability theory has become the standard framework in the field of mobile robotics because of the inherent uncertainty associated with sensing and acting. In this paper, we show that the theory of belief functions with its ability to distinguish between different types of uncertainty is able to provide significant advantages over probabilistic approaches in the context of robotics. We do so by presenting solutions to the essential problems of simultaneous localization and mapping (SLAM) and planning based on belief functions. For SLAM, we show how the joint belief function over the map and the robot's poses can be factored and efficiently approximated using a Rao-Blackwellized particle filter, resulting in a generalization of the popular probabilistic FastSLAM algorithm. Our SLAM algorithm produces occupancy grid maps where belief functions explicitly represent additional information about missing and conflicting measurements compared to probabilistic grid maps. The basis for this SLAM algorithm are forward and inverse sensor models, and we present general evidential models for range sensors like sonar and laser scanners. Using the generated evidential grid maps, we show how optimal decisions can be made for path planning and active exploration. To demonstrate the effectiveness of our evidential approach, we apply it to two real-world datasets where a mobile robot has to explore unknown environments and solve different planning problems. Finally, we provide a quantitative evaluation and show that the evidential approach outperforms a probabilistic one both in terms of map quality and navigation performance.     

Generalized information theory for hints

This paper develops a new uncertainty measure for the theory of hints that complies with the established semantics of statistical information theory and further satisfies all classical requirements for such a measure imposed in the literature. The proposed functional decomposes into conversant uncertainty measures and therefore discloses a new interpretation of the latters as well. By abstracting to equivalence classes of hints we transport the new measure to mass functions in Dempster-Shafer theory and analyse its relationship with the aggregate uncertainty, which currently is the only known functional for the Dempster-Shafer theory of evidence that satisfies the same set of properties. Moreover, the perspective of hints reveals that the standard independence notion in Dempster-Shafer theory called non-interactivity corresponds to an amalgamation of probabilistic independence and qualitative independence between frames of discernment. All results in this paper are developed for arbitrary families of compatible frames generalizing the very specialized multi-variate systems that are usually studied in information theory.     

Numerical solution of nonlinear stochastic Itô‐Volterra integral equations driven by fractional Brownian motion

In this paper, an efficient numerical technique is applied to provide the approximate solution of nonlinear stochastic Itô‐Volterra integral equations driven by fractional Brownian motion with Hurst parameter . The proposed method is based on the operational matrices of modification of hat functions (MHFs) and the collocation method. In this approach, by approximating functions that appear in the integral equation by MHFs and using Newton's‐Cotes points, nonlinear integral equation is transformed to nonlinear system of algebraic equations. This nonlinear system is solved by using Newton's numerical method, and the approximate solution of integral equation is achieved. Some theorems related to error estimate and convergence analysis of the suggested scheme are also established. Finally, 2 illustrative examples are included to confirm applicability, efficiency, and accuracy of the proposed method. It should be noted that this scheme can be used to solve other appropriate problems, but some modifications are required.     

On the Evaluation of Uncertain Courses of Action

We consider the problem of decision making under uncertainty. The fuzzy measure is introduced as a general way of representing available information about the uncertainty. It is noted that generally in uncertain environments the problem of comparing alternative courses of action is difficult because of the multiplicity of possible outcomes for any action. One approach is to convert this multiplicity of possible of outcomes associated with an alternative into a single value using a valuation function. We describe various ways of providing a valuation function when the uncertainty is represented using a fuzzy measure. We then specialize these valuation functions to the cases of probabilistic and possibilistic uncertainty.     

Approximate Operators and Dual Extremum Principles for Linear Problems

The use of approximate operators in the theory of dual extremumprinciples is investigated for certain linear problems. If second-ordertrial functions to certain approximate Lagrangian equationscan be found, bounds which are improvements on the dual boundingfunctions can be constructed. Some simple applications to apartial differential equation and an integral equation are given.It is shown that both upper and lower bounds follow from onetrial function.     

Theory of truth degrees of propositions in two-valued logic

By means of infinite product of evenly distributed probabilistic spaces of cardinal 2 this paper introduces the concepts of truth degrees of formulas and similarity degrees among formulas, and a pseudo-metric on the set of formulas is derived therefrom, this offers a possible framework for developing an approximate reasoning theory of propositions in two-valued logic.     

A CQM-based BEM for transient heat conduction problems in homogeneous materials and FGMs

A method for numerical solution of time-domain boundary integral formulations of transient problems governed by the heat equation is presented. The heat conduction problem is analyzed considering homogeneous and non-homogeneous media. In the case of the non-homogeneous media, the conductor material is assumed to be a functionally graded material, i.e., the material properties vary spatially according to known smooth functions. For some specific spatial variations of the material properties, the fundamental solution and the boundary integral equation of the problem are obtained thanks to a change of variables that transforms the original problem to the standard heat conduction problem for homogeneous materials. For the treatment of time-dependent terms, the convolution quadrature method is adopted to approximate numerically the integral equation of the time-domain boundary element method. In the case that the responses are required at a large number of interior points, the convolution performed to calculate them is very time consuming. It is shown that the discrete convolution of the proposed formulation can be computed by means of the fast Fourier transform technique, which considerably reduces the computational complexity. Results for some transient heat conduction examples are presented to validate the numerical techniques studied.     

Generalizing inference rules in a coherence-based probabilistic default reasoning

In this paper we first recall some notions and results on the coherence-based probabilistic treatment of uncertainty. Then, we deepen some probabilistic aspects in nonmonotonic reasoning, by generalizing OR, CM, and Cut rules. We also illustrate the degradation of these inference rules when the number of premises increases. Finally, we show that the lower bounds obtained when applying OR and Quasi-Conjunction inference rules coincide, respectively, with Hamacher and Lukasiewicz t-norms; the upper bounds in both rules coincide with Hamacher t-conorm.     

Nature plays with dice – terrorists do not: Allocating resources to counter strategic versus probabilistic risks

Probabilistic uncertainty is caused by “chance”, whereas strategic uncertainty is caused by an adverse interested party. Using linear impact functions, the problems of allocating a limited resource to defend sites that face either probabilistic risk or strategic risk are formulated as optimization problems that are solved explicitly. The resulting optimal policies differ – under probabilistic risk, the optimal policy is to focus the investment of resources on priority sites where they yield the highest impact, while under strategic risk, the best policy is to spread the resources so as to decrease the potential damage level of the most vulnerable site(s). Neither solution coincides with the commonly practiced proportionality allocation scheme.