The Problem of Classification when the Data are Non-precise

Abstract

Non-precise data arise in a natural way in several contexts. For example, the water level of a river does not usually consist of a single number as can be seen from the intensity of the wetness as a function of depth of a survey rod. The temperature of a room varies as a function of distance from a reference point. The color intensities associated with a pixel which describe observations from remote sensing are non-precise numbers because they vary as a function of the reflection from the sun. In these examples, it is the imprecision of the observation itself that is of interest rather than the uncertainty due to statistical variation. Even in the absence of stochastic error, there would still be an imprecision in the measurement. Viertl (1997) developed the subject of statistical inference for such non-precise data and associated it very closely to fuzzy set theory. Precise data can be described by an indicator function whereas non-precise data is described by characterizing functions. In this article, we first review the notation and then consider the problems of classification for non-precise data.