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.
Highlights
Non-precise data arise in a natural way in several contexts
Referring to the example on the water level of a river, let w(h) represent the intensity of the wetness of a survey rod as a function of the depth h1 ≤ h ≤ h2, where h1, h2 provide the range of values
As an application of a classification problem involving non-precise data, we may wish to identify the species of fish on the basis of echo sounder measurements
Summary
We shall draw heavily on the analogy with inference for precise data. Definition 4 Let g : Rn → R be a real valued continuous function whose arguments are non-precise vectors x∗ with characterizing function ξ. The first term is constant, and choosing x = ω, the second term is 0, so that ψ(y) = exp Setting x1 = · · · = xn−1 = ω ± y2/n and xn = nω − (n − 1) ω ± y2/n yields this optimum In both examples, the characterizing functions decrease exponentially fast as the sample size increases. The basis for inference involving non-precise data is the construction of the characterizing function ξ(·) of the n-dimensional non-precise vector describing the combined sample x∗. We make use of this procedure in the problem of classification
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