Hung T. Nguyen

Computing Statistics under Interval and Fuzzy Uncertainty: Applications to Computer Science and Engineering

In many practical situations, we are interested in statistics characterizing a population of objects: e.g. in the mean height of people from a certain area. Most algorithms for estimating such statistics assume that the sample values are exact. In practice, sample values come from measurements, and measurements are never absolutely accurate. Sometimes, we know the exact probability distribution of the measurement inaccuracy, but often, we only know the upper bound on this inaccuracy. In this case, we have interval uncertainty: e.g. if the measured value is 1.0, and inaccuracy is bounded by 0.1, then the actual (unknown) value of the quantity can be anywhere between 1.0 - 0.1 = 0.9 and 1.0 + 0.1 = 1.1. In other cases, the values are expert estimates, and we only have fuzzy information about the estimation inaccuracy. This book shows how to compute statistics under such interval and fuzzy uncertainty. The resulting methods are applied to computer science (optimal scheduling of different processors), to information technology (maintaining privacy), to computer engineering (design of computer chips), and to data processing in geosciences, radar imaging, and structural mechanics.

How to Bargain: An Interval Approach

In many real-life situations, we need to bargain. What is the best bargaining strategy? If you are already in a negotiating process, your previous offer was α, the sellers last offer was (average)α＞α, what next offer α should you make? A usual commonsense recommendation is to ”split the difference,” i.e., to offer α=(α+(average)α)/2, or, more generally, to offer a linear combination α=k⋅(average)α+(1-k)•α (for some parameter k ∈ (0, 1)). The bargaining problem falls under the scope of the theory of cooperative games. In cooperative games, there are many reasonable solution concepts. Some of these solution concepts-like Nashs bargaining solution that recommends maximizing the product of utility gains-lead to offers that linearly depend on α and (average)α; other concepts lead to non-linear dependence. From the practical viewpoint, it is desirable to come up with a recommendation that would not depend on a specific selection of the solution concept-and on specific difficult-to-verify assumptions about the utility function etc. In this paper, we deliver such a recommendation: specifically, we show that under reasonable assumption, we should always select an offer that linearly depends on α and (average)α.

What is the Right Context for an Engineering Problem: Finding Such a Context is NP-Hard

In the general case, most computational engineering problems are NP-hard. So, to make the problem feasible, it is important to restrict this problem. Ideally, we should use the most general context in which the problem is still feasible. In this paper, we prove that finding such most general context is itself an NP-hard problem. Since it is not possible to find the appropriate context by utilizing some algorithm, it is therefore necessary to be creative - i.e., To use some computational intelligence techniques. On three examples, we show how such techniques can help us come up with the appropriate context. Our analysis explains why it is beneficial to take knowledge about causality into account when processing data, why sometimes long-term predictions are easier than short-term ones, and why often for small deviations, a straightforward application of a seemingly optimal control only makes the situation worse.

50 Years of Fuzzy: from Discrete to Continuous to -- Where?

While many objects and processes in the real world are discrete, from the computational viewpoint, discrete objects and processes are much more difficult to handle than continuous ones. As a result, a continuous approximation is often a useful way to describe discrete objects and processes. We show that the need for such an approximation explains many features of fuzzy techniques, and we speculate on to which promising future directions of fuzzy research this need can lead us.

Why Lattice-valued fuzzy values? A mathematical justification

To take into account that experts degrees of certainty are not always comparable, researchers have used partially ordered set of degrees instead of the more traditional linearly (totally) ordered interval (0, 1). In most cases, it is assumed that this partially ordered set is a lattice, i.e., every two elements have the greatest lower bound and the least upper bound. In this paper, we prove a theorem explaining why it is reasonable to require that the set of degrees is a lattice.

Algorithm Complexity: Two Simple Examples

We start in this first lesson with the concept of algorithm complexity and use two simple examples to illustrate the applicability of continuous mathematics.

Expert Systems and the Basics of Fuzzy Logic

This lesson describes an area from Artificial Intelligence: expert systems. We describe the essentials of fuzzy logic in modeling of expert knowledge. We also touch upon the field of fuzzy control; the general methodology of fuzzy control will be given in the next Lesson 14.

Program Testing: A Problem

Now that we know how to solve simple functional equations, we will start the discussion of another computer science area where these equations are useful: the problem of software reliability. In this lesson, we will briefly describe the problem, and describe its current (semi-heuristic) solutions.

Optimal Program Testing

In this lesson, we re-formulate the problem of program testing as a mathematically precise optimization problem, and solve the simplest case of the corresponding optimization problem.

Intelligent and Fuzzy Control

In this lesson, we continue the description of mathematical methods for dealing with expert knowledge; namely, we describe, step-by-step, the fuzzy control methodology that transforms the control rules (that expert operators formulate by using words of a natural language) into a precise control strategy.