Towards Fast Algorithms for Processing Type-2 Fuzzy Data: Extending Mendel’s Algorithms From Interval-Valued to a More General Case

—It is known that processing of data under general type-1 fuzzy uncertainty can be reduced to the simplest case – of interval uncertainty: namely, Zadeh's extension principle is equivalent to level-by-level interval computations applied to α-cuts of the corresponding fuzzy numbers. However, type-1 fuzzy numbers may not be the most adequate way of describing uncertainty, because they require that an expert can describe his or her degree of con dence in a statement by an exact value. In practice, it is more reasonable to expect that the expert estimates his or her degree by using imprecise words from natural language – which can be naturally formalized as fuzzy sets. The resulting type-2 fuzzy numbers more adequately represent the expert's opinions, but their practical use is limited by the seeming computational complexity of their use. In his recent research, J. Mendel has shown that for the practically important case of interval-valued fuzzy sets, processing such sets can also be reduced to interval computations. In this paper, we show that Mendel's idea can be naturally extended to arbitrary type-2 fuzzy numbers. I. WHY DATA PROCESSING AND KNOWLEDGE PROCESSING ARE NEEDED IN THE FIRST PLACE Some quantities y we can simply directly measure. For example, when we want to know the current state of a patient in a hospital, we can measure the patient's body temperature, blood pressure, weight, and many other important characteristics. In some situations, we do not even need to measure: we can simply ask an expert, and the expert will provide us with an (approximate) value ỹ of the quantity y. However, many other quantities of interest are dif cult or even important to measure or estimate directly. Examples of such quantities include the amount of oil in a given well or a distance to a star. Since we cannot directly measure the values of these quantities, the only way to learn some information about them is: to measure (or ask an expert to estimate) some other easier-to-measured quantities x1, . . . , xn, and then to estimate y based on the measured values x̃i of these auxiliary quantities xi. For example, to estimate the amount of oil in a given well, we perform seismic experiments: we set up small explosions at some locations and measure the resulting seismic waves at different distances from the location of the explosion. To nd Vladik Kreinovich is with the Department of Computer Science, University of Texas at El Paso, El Paso, TX 79968 (email [email protected]). Gang Xiang is with the Philips Healthcare Informatics (email [email protected]). This work was supported in part by NSF grants HRD-0734825, EAR0225670, and EIA-0080940, by Texas Department of Transportation contract No. 0-5453, by the Japan Advanced Institute of Science and Technology (JAIST) International Joint Research Grant 2006-08, and by the Max Planck Institut für Mathematik. the distance to a faraway star, we measure the direction to the star from different location on Earth (and/or at different seasons) and the coordinates of (and the distances between) the locations of the corresponding telescopes. To estimate the value of the desired quantity y, we must know the relation between y and the easier-to-measure (or easier-to-estimate) quantities x1, . . . , xn. Speci cally, we want to use the estimates of xi to come up with an estimate for y. Thus, the relation between y and xi must be given in the form of an algorithm f(x1, . . . , xn) which transforms the values of xi into an estimate for y. Once we know this algorithm f and the measured values x̃i of the auxiliary quantities, we can estimate y as ỹ = f(x̃1, . . . , x̃n).