Fuzzy Sets As Strongly Consistent Random Sets

It is known that from the purely mathematical viewpoint, fuzzy sets can be interpreted as equivalent classes of random sets. This interpretations helps to teach fuzzy techniques to statisticians and also enables us to apply results about random sets to fuzzy techniques. The problem with this interpretation is that it is too complicated: a random set is not an easy notion, and classes of random sets are even more complex. This complexity goes against the spirit of fuzzy sets, whose purpose was to be simple and intuitively clear. From this viewpoint, it is desirable to simplify this interpretation. In this paper, we show that the random-set interpretation of fuzzy techniques can indeed be simplified: namely, we can show that fuzzy sets can be interpreted not as classes, but as strongly consistent random sets (in some reasonable sense). This is not yet at the desired level of simplicity, but this new interpretation is much simpler than the original one and thus, constitutes an important step towards the desired simplicity. I. FUZZY SETS AND RANDOM SETS: A BRIEF REMINDER AND FORMULATION OF THE PROBLEM Fuzzy sets and random sets: a brief reminder for the knowledgeable readers. It is known that from the purely mathematical viewpoint, fuzzy sets can be interpreted as equivalent classes of random sets; see, e.g., [3]. (For readers who are not fully familiar with this interpretation, its main ideas will be presented in the following text.) This interpretations is useful: • it helps to teach fuzzy techniques to statisticians and • it also enables us to apply results about random sets to fuzzy techniques. The problem with this interpretation is that it is too complicated: a random set is not an easy notion, and classes of random sets are even more complex. This complexity goes against the spirit of fuzzy sets, whose purpose was to be simple and intuitively clear. From this viewpoint, it is desirable to simplify this interpretation. In this paper, we show that the random-set interpretation of fuzzy techniques can indeed be simplified: namely, we can show that fuzzy sets can be interpreted: • not as classes, but • as strongly consistent random sets (in some reasonable sense). This is not yet at the desired level of simplicity, but this new interpretation is much simpler than the original one and thus, constitutes an important step towards the desired simplicity. Fuzzy sets and random sets: an explanation for the general fuzzy-related readers. To explain the problem to a general fuzzy-related reader, a reader who is: • familiar with the concepts of fuzzy sets, but • may not be very familiar with the technical details of random sets, let us describe, in some detail, what are random sets and how they are related to fuzzy sets. Random sets naturally appear in describing our knowledge and about ability to predict future events. So, to adequately describe random sets, we need to start with a brief reminder of the general prediction problem. Predictions are important. One of the main applications of science and engineering is to predict future events – and, in the case of engineering, to come up with designs and controls for which the resulting future situation is the most beneficial. For example, science predicts the position of the Moon in a few months, while engineering not only predicts the position of the spaceship in a month, but also describes the best trajectory correction that would bring the future location as close to the target as possible. Some scientists say – correctly – that the main objective of science is to explain the world. But what does this mean in practical terms? The usual way to prove that a new physical theory explains the world better is to show that it enables us to give more accurate predictions of future events. This is how Einstein’s General Relativity became accepted – when experiments confirmed its prediction of how much the light ray passing near the Sun will be distorted by the Sun’s gravitational field. Perfect knowledge is rarely available: need for set uncertainty. From the prediction viewpoint, perfect knowledge means that we know exactly what will happen in the future. Such a knowledge is rarely available, because that require a full knowledge of all the factors that can affect the future state. Usually, we have only partial knowledge. Thus, instead of a single future state, we have a set of future states. This can be explained, e.g., as follows. One way to predict the future state is to look for similar situations in the past and to see what happened later in these situations. In the case of partial knowledge, we may have several different similar situations in the past. These situations are not identical, they are different, but they differ in the values of the quantities that we do not know, they all fit nicely with whatever information is available. Since the current situation is similar to one of the past one, in this case, all we can do is predict that the future situation will be similar to one of the corresponding outcomes. From set uncertainty to probabilistic uncertainty. When we have many similar situations, we can determine not only which future states are possible, but also how frequent are different future states. For each of the possible future states s1, . . . , sn, the observed frequency of this state serves as a natural estimate for the probability pi of this state. Thus, in this case, we know the set of possible states s1, . . . , sn, and we know the probabilities p1, . . . , pn of different possible states, probabilities adding up to 1: n ∑