Evaluation of Fuzzy Partitions

The aim of this study is the development of tools dedicated to fuzzy partition evaluation in the field of satellite image classification. While a traditional crisp partition only provides qualitative information, a fuzzy partition represents a large amount of quantitative information. However, such a partition is often evaluated after “defuzzification” (i.e., it is reduced to a crisp partition). The analysis of a traditional confusion matrix, describing the similarities between the computed crisp partition and a control partition, can then be performed. This approach is rather drastic and is far from satisfactory because the quantitative information is lost after the defuzzification. Some methods do not require preliminary defuzzification, but they are not adequate to evaluate nonprobabilistic fuzzy partitions (i.e., fuzzy partitions such that the sum of the membership degrees is not necessarily equal to 1). To solve these issues, we consider the evaluation of any fuzzy partition μ as the evaluation of a still fuzzy new partition: the plausibilistic closure of μ. This approach comes from the theory of evidence. It allows definition of a set of original tools (plausibility matrices, credibility matrices, and overlap degrees) dedicated to fuzzy partition evaluation. A concrete application illustrates our theoretical work and a tutorial is provided in appendix.