Interval analysis

Statistics: is the science of collection, organization, presentation, analysis, and reasonable interpretation of data. It also deals with methods and techniques that can be used to draw conclusions about the characteristics of a large number of data points--commonly called a population by using a smaller subset of the entire data.
Statistics is sometimes described as the science of decision making under uncertainty and can be divided into two broad areas as follows:
Descriptive Statistics Quantities and techniques used to describe a sample characteristic e.g. mean, standard deviation, box-plot.
Inferential Statistics which covers those statistical procedures used to help draw conclusions or inferences about a population on the basis of a sample of data collected from the population. Important areas inferential statistics include confidence intervals, hypothesis tests, regression analysis and experimental design. Underlying inferential statistics is the idea of probability and probability distributions.

This paper presents a basic tutorial on epistemic uncertainty q uantification methods. Epistemic uncertainty, characterizing lack-of-knowledge, is often prevalent in engineering applications. However, the methods we have for analyzing and propagating epistemic uncertainty are not as nearly widely used or well-understood as methods to propagate aleatory uncertainty (e.g. inherent variability characterized by probability distributions). We examine three methods used in propagating epistemic uncertainties: interval analysis, Dempster-Shafer evidence theory, and second-order probability. We demonstrate examples of their use on a problem in structural dynamics.

A Turbo Pascal program for data acquisition and analysis is presented. The program detects incoming data as TTL pulses through the serial port of an IBM-PC and compatible computers and displays the instantaneous frequency plot and the interval histogram on-line. Novel features of this Pascal program are: (1) no special hardware requirements and (2) on-line analysis of more than one source at a time. No hardware additions to the supplied computer and easy operation make this program a valuable tool, directly useable in an IBM-PC and compatible computers.

This chapter is devoted to introducing the theories of interval algebra to people who are interested in applying the interval methods to uncertainty analysis in science and engineering. In view of this purpose, we shall introduce the key concepts of the algebraic theories of intervals that form the foundations of the interval techniques as they are now practised, provide a historical and epistemological background of interval mathematics and uncertainty in science and technology, and finally describe some typical applications that clarify the need for interval computations to cope with uncertainty in a wide variety of scientific disciplines.

Keywords. Interval mathematics, Uncertainty, Quantitative Knowledge, Reliability, Complex interval arithmetic, Machine interval arithmetic, Interval automatic differentiation, Computer graphics, Ray tracing, Interval root isolation.

Recommended Citation:

Hend Dawood. Interval Mathematics as a Potential Weapon against Uncertainty. In S. Chakraverty, editor, Mathematics of Uncertainty Modeling in the Analysis of Engineering and Science Problems. chapter 1, pages 1-38. IGI Global, Hershey, PA, 2014. ISBN 978-1-4666-4991-0.

The final publication is available at IGI Global via http://dx.doi.org/10.4018/978-1-4666-4991-0.ch001

Scientists are, all the time, in a struggle with uncertainty which is always a threat to a trustworthy scientific knowledge. A very simple and natural idea, to defeat uncertainty, is that of enclosing uncertain measured values in real closed intervals. On the basis of this idea, interval arithmetic is constructed. The idea of calculating with intervals is not completely new in mathematics: the concept has been known since Archimedes, who used guaranteed lower and upper bounds to compute his constant Pi. Interval arithmetic is now a broad field in which rigorous mathematics is associated with scientific computing. This connection makes it possible to solve uncertainty problems that cannot be efficiently solved by floating-point arithmetic. Today, application areas of interval methods include electrical engineering, control theory, remote sensing, experimental and computational physics, chaotic systems, celestial mechanics, signal processing, computer graphics, robotics, and computer-assisted proofs. The purpose of this book is to be a concise but informative introduction to the theories of interval arithmetic as well as to some of their computational and scientific applications.

This paper presents a basic tutorial on epistemic uncertainty quantification methods. Epistemic uncertainty, characterizing lack-of-knowledge, is often prevalent in engineering applications. However, the methods we have for analyzing and propagating epistemic uncertainty are not as nearly widely used or well-understood as methods to propagate aleatory uncertainty (e.g. inherent variability characterized by probability distributions). We examine three methods used in propagating

We begin by constructing the algebra of classical intervals and prove that it is a nondistributive abelian semiring. Next, we formalize the notion of interval dependency, along with discussing the algebras of two alternate theories of intervals: modal intervals, and constraint intervals. With a view to treating some problems of the present interval theories, we present an alternate theory of intervals, namely the "theory of optimizational intervals", and prove that it constitutes a rich S-field algebra, which extends the ordinary field of the reals, then we construct an optimizational complex interval algebra. Furthermore, we define an order on the set of interval numbers, then we present the proofs that it is a total order, compatible with the interval operations, dense, and weakly Archimedean. Finally, we prove that this order extends the usual order on the reals, Moore's partial order, and Kulisch's partial order on interval numbers.

Keywords. Classical interval arithmetic; Machine interval arithmetic; Interval dependency; Constraint intervals; Modal intervals; Classical complex intervals; Optimizational intervals; Optimizational complex intervals; S-field algebra; Ordering interval numbers.

In this paper, a new interval version of Runge-Kutta methods is proposed for time discretization and solving of optimal control problems (OCPs) in the presence of uncertain parameters. A new technique based on interval arithmetic is introduced to achieve the confidence bounds of the system. The proposed method is based on the new forward representation of Hukuhara interval difference and combining it with Runge-Kutta method for solving the OCPs with interval uncertainties. To perform the proposed method on OCPs, the Lagrange multiplier method is first applied to achieve the necessary conditions and then, using some algebraic manipulations, they are converted to an ordinary differential equation to achieve the interval optimal solution for the considered OCP with uncertain parameters. Shooting method is also employed to cover the Runge-Kutta methods restrictions in solving the OCPs with boundary values. The simulation results are applied to some practical case studies for demonstrating the effectiveness of the proposed method.

This paper presents the kinematic analysis and the development of a 4-degree-of-freedom serial-parallel mechanism for large commercial vehicle driving simulators. The degrees of freedom are selected according to the target maneuvers and the structure of human motion perception organs. Several kinematic properties of parallel part of the mechanism under study are investigated, including the inverse and the forward kinematics problems, workspace determination, singularity, and kinematic sensitivity analysis. The workspace of the parallel part of the mechanism is obtained by interval analysis. Moreover, using elimination theory, a univariate expression representing the forward kinematics solution of the parallel part is obtained.

Crystals were detected in glass bottles containing autoclaved (standard cycle: 1218 C, 17 lb/in2, for 20 min) tap water. Bottles were fitted with a sipper tube and stopper. Crystals were observed twice at approximately a 6-month interval. Analysis of crystals and crystal-laden water by transmission electron microscopy and inductively coupled plasma and energy X-ray analysis revealed that the crystals were silicon. Subsequently, procedures and processes involved with preparation of water bottles were analyzed to determine the source or factors involved with silicon contamination. Analyses of silicon concentrations were performed on samples of tap water (0.88 to 1.20 ppm), tap water autoclaved in polycarbonate (0.89 to 1.20 ppm) and glass water bottles (0.84 to 11.0 ppm), glass and polycarbonate bottles after purposeful contamination of tap water with various amounts of alkaline detergent (1.10 to 1.70 ppm), and on steam condensate from the autoclave (0.23 and 0.47 ppm) to ascertain t...

1 Le Mésotonique Tempéré de Bach (texte extensif) Relaté à « Das wohltemperirte Clavier » Résumé Les battements d'harmoniques des quintes et tierces majeures de tempéraments historiques bien connus sont analysés. Il apparait que les battements d'harmoniques pourraient être le facteur réel et déterminant pour les tempéraments baroques, surtout parce que les battements d'harmoniques tiennent une grande importance pour le musicien interprète pour ce qui concerne l'harmonie et les affects musicaux possibles, et pour l'accordeur pour le confort et la qualité de l'accord. D'autre part, il n'est pas toujours clair si les rapports, cents ou commas publiés sont déduits à partir de calculs théoriques ou à partir de résultats concrets obtenus de mesures ou réglages faits au monocorde. L'importance révélée des caractéristiques des battements apporte des arguments supplémentaires pour l'acceptation de la proposition de Jobin relative au tempérament probable pratiqué par Bach, ou d'alternatives quasi identiques basées sur le calcul de battements. Une nouvelle hypothèse est proposée concernant les spirales dessinées sur la page de titre du « Clavier bien tempéré » de Jean Sebastien Bach. Mots clé Baroque ; bien tempéré ; mésotonique ; intervalle ; comma ; battement ; harmonique ; rapport ; cent ; Bach Mes plus sincères remerciements à Mr. A Calvet, qui a minutieusement vérifié, amélioré et corrigé la version française de ce texte, non seulement du point de vue linguistique, mais aussi de son point de vue en tant qu'accordeur professionnel. Symboles utilisés : Les symboles utilisés dans ce texte sont ceux généralement utilisés dans les publications anglophones : • Notes : les notes sont nommées selon pour l'octave qui contient la note A4, pour laquelle on a en général : ~ 415 < A4 < ~ 466 Hz. • Nombres décimaux : le point est utilisé comme séparateur entre les nombres entiers et les décimales.

We propose an algorithm to compute the limit cycle set of uncertain non-rational nonlinear systems with nonlinear parametric dependencies. The proposed algorithm computes the limit cycles for a wide class of uncertain nonlinear systems, where the transfer function of the linear element and describing function of the nonlinear element need to be only continuous with respect to the parameters and continuously differentiable with respect to the amplitude and frequency of periodic input signal. The proposed algorithm guarantees that the limit cycles are reliably computed to a prescribed accuracy, and that none of the actual limit cycle point is missed out irrespective of the tightness of the prescribed accuracy. Moreover, for a prescribed accuracy, the proposed algorithm computes all the limit cycles in a finite number of iterations, and an upper bound for this number is also computable. The algorithm is demonstrated on a challenging non-rational example with nonlinear parametric dependencies. Copyright © 2005 John Wiley & Sons, Ltd.

For the given arbitrary sequence of real numbers {xi} n i=1 we construct several lower and upper bound converging sequences. Our goal is to localize the absolute value of the sequence maximum. Also we can calculate the value of such numbers. Since the proposed algorithms are iterative, asymptotical convergence theorems are proved. The presented task seems to be pointless from

Interval analysis is a growing branch of computational mathematics where operations are carried out on intervals instead of real numbers. This paper presents the first application of this method to robotic mechanisms for the solution of inverse kinematics. As shown in this paper, it is possible to potentially compute all solutions of the inverse kinematics problem using this method. This paper describes the preliminaries of interval analysis, the numerical algorithm, the computational complexity, and illustrations with examples.

We give a constructive and exhaustive definition of Kochen-Specker (KS) vectors in a Hilbert space of any dimension as well as of all the remaining vectors of the space. KS vectors are elements of any set of orthonormal states, i.e., vectors in an n-dimensional Hilbert space, {\cal H}^n, n\ge 3 , to which it is impossible to assign 1s and 0s in such a way that no two mutually orthogonal vectors from the set are both assigned 1 and that not all mutually orthogonal vectors are assigned 0. Our constructive definition of such KS vectors is based on algorithms that generate MMP diagrams corresponding to blocks of orthogonal vectors in {\bb R}^n , on algorithms that single out those diagrams on which algebraic 0-1 states cannot be defined, and on algorithms that solve nonlinear equations describing the orthogonalities of the vectors by means of statistically polynomially complex interval analysis and self-teaching programs. The algorithms are limited neither by the number of dimensions nor by the number of vectors. To demonstrate the power of the algorithms, all four-dimensional KS vector systems containing up to 24 vectors were generated and described, all three-dimensional vector systems containing up to 30 vectors were scanned, and several general properties of KS vectors were found.

In this paper, the interval nature of fuzzy numbers is revealed by showing that many interesting results from classical interval analysis transfer also into the fuzzy case. The paper deals with a solution of a fuzzy interval system of linear equations, i.e., a system in which fuzzy intervals (numbers) appear instead of crisp numbers.

Map matching algorithms are used to integrate an initial estimated position with digital road network data for computing the vehicle position on a road map. In this paper, a map matching algorithm based on belief function theory is proposed. This method provides an accurate estimation of vehicle position relative to a digital road map using belief function theory and interval

In the article we present an interval difference scheme for solving a general elliptic boundary value problem with Dirichlet’ boundary conditions. The obtained interval enclosure of the solution contains all possible numerical errors. A numerical example we present confirms that the exact solution belongs to the resulting interval enclosure.

The classification problem is concerned with categorizing observations into different groups. The performance of the classification process is dependent on how well the discriminant function for the specific problem performs. A discriminant function is developed to minimize the ...

An Interval Partitioning Method (IPM) is proposed to solve the (nonconvex) Mixed Integer Nonlinear Programming Problem (MINLP). The MINLP is encountered in many application areas and solving this problem bears practical importance. This paper proposes an IPM where two tree search strategies (breadth first and mixed breadth/depth first) and three variable subdivision methods are implemented. Two proposed variable subdivision methods are novel and they proritize variables hierarchically according to several features. The IPM is implemented on a set of nonconvex MINLP instances extracted from the MINLP benchmarks and numerical results show that its performance is quite promising.