Berlin Wu

Shadows of fuzzy sets-a natural approach towards describing 2-D and multi-D fuzzy uncertainty in linguistic terms

Fuzzy information processing systems start with expert knowledge which is usually formulated in terms of words from natural language. This knowledge is then usually reformulated in computer friendly terms of membership functions, and the system transforms these input membership functions into the membership functions which describe the result of fuzzy data processing. It is then desirable to translate this fuzzy information back from computer-friendly membership functions language to human-friendly natural language. In general, this is difficult even in a 1-D case, when we are interested in a single quantity y; however, the fuzzy research community has accumulated some expertise of describing the resulting 1-D membership functions by words from natural language. The problem becomes even more complicated in 2-D and multi-D cases, when we are interested in several quantities y/sub 1/,...,y/sub m/, because there are fewer words which describe the relation between several quantities than words describing a single quantity. To reduce this more complicated multi-D problem to a simpler (although still difficult) 1-D case, Zadeh proposed (1966) to use words to describe fuzzy information about different combinations y=f(y/sub 1/,...,y/sub m/) of the desired variables. This idea is similar to the use of marginal distributions in probability theory. The corresponding terms are called shadows of the original fuzzy set. The main question is: do we lose any information in this translation? Zadeh has shown that under certain conditions, the original fuzzy set can be uniquely reconstructed from its shadows. We prove that for appropriately chosen shadows, the reconstruction is always unique. Thus, if we manage to describe the original membership function by linguistic terms which describe different combinations y, this description is lossless.

How to Select Appropriate Statistical Characteristics

Which is the best way to describe the corresponding probabilistic uncertainty? One of the main objectives of data processing is to make decisions. As we have seen in the previous chapter, a standard way of making a decision is to select the action a for which the expected utility (gain) is the largest possible. This is where probabilities are used: in computing, for every possible action a, the corresponding expected utility. To be more precise, we usually know, for each action a and for each actual value of the (unknown) quantity x, the corresponding value of the utility u a (x). We must use the probability distribution for x to compute the expected value e[u a (x)] of this utility.

Computing Covariance under Interval Uncertainty

When we have two sets of data x1,..., x n and y1,..., y n , we normally compute finite population covariance

Computing Variance under Interval Uncertainty: Efficient Algorithms

We have shown that the problem of computing the upper endpoint (overline{V}) is, in general, NP-hard (later on, in this chapter, we will see that the lower endpoint (underline{V}) can be always computed in feasible (polynomial) time). Since we cannot always efficiently compute the upper endpoint (overline{V}) , we therefore need to consider cases when such an efficient computation may be possible.

Computing Mean under Interval Uncertainty

We have already mentioned that for the interval data x1 = [(underline{x}_{1}), (overline{x}_{1})],..., x n = [(underline{x}_{n}), (overline{x}_{n})], a reasonable estimate for the corresponding statistical characteristic C(x1,...,x n ) is the range

Need to Go Beyond Interval and Fuzzy Uncertainty

Types of uncertainty that we analyzed so far. In the previous chapters, we described the uncertainty of inputs – and the resulting uncertainty in the values of the statistical characteristics and, more generally, the result of data processing. To characterize this uncertainty, we used the following three types of information.

Computing under Interval Uncertainty: Traditional Approach Based on Uniform Distributions

Traditional statistical approach: main idea. In the case of interval uncertainty, we only know the intervals, we do not know the probability distributions on these intervals. The traditional statistical approach to situations in which we have several alternatives with unknown probabilities is to use Laplace Principle of Indifference, according to which,

Beyond Interval Uncertainty: Case of Discontinuous Processes (Phase Transitions)

One of the main tasks of science and engineering is to use the current values of the physical quantities for predicting the future values of the desired quantities. Due to the (inevitable) measurement inaccuracy, we usually know the current values of the physical quantities with interval uncertainty. Traditionally, it is assumed that all the processes are continuous; as a result, the range of possible values of the future quantities is also known with interval uncertainty. However, in many practical situations (such as phase transitions), the dependence of the future values on the current ones becomes discontinuous. We show that in such cases, initial interval uncertainties can lead to arbitrary bounded closed ranges of possible values of the future quantities. We also show that the possibility of such a discontinuity may drastically increase the computational complexity of the corresponding range prediction problem.

Applications to Mechanical Engineering: Failure Analysis under Interval and Fuzzy Uncertainty

One of the main objective of mechanics of materials is to predict when the material experiences fracture (fails), and to prevent this failure. With this objective in mind, it is desirable to use it ductile materials, i.e., materials which can sustain large deformations without failure. Von Mises criterion enables us to predict the failure of such ductile materials. To apply this criterion, we need to know the exact stresses applied at different directions. In practice, we only know these stresses with interval or fuzzy uncertainty. In this chapter, we describe how we can apply this criterion under such uncertainty, and how to make this application computationally efficient.

Computing Statistics under Fuzzy Uncertainty: Formulation of the Problem

Need to process fuzzy uncertainty. In many practical situations, we only have expert estimates for the inputs x i . Sometimes, experts provide guaranteed bounds on the x i , and even the probabilities of different values within these bounds. However, such cases are rare. Usually, the experts’ opinions about the uncertainty of their estimates are described by (imprecise, “fuzzy”) words from natural language. For example, an expert can say that the value x i of the i-th quantity is approximately equal to 1.0, with an accuracy most probably of about 0.1.