Evaluation of Fuzzy Partitions

The aim of this study is the development of tools dedistage, although commonly overlooked (Hammond and Verbyla, 1996). Evaluating a partition generally consists cated to fuzzy partition evaluation in the field of satellite image classification. While a traditional crisp partition of comparing it with a control partition and measuring only provides qualitative information, a fuzzy partition repthe degree of association of two variables that may be resents a large amount of quantitative information. Howbinary, qualitative, ordinal, or quantitative. Traditional ever, such a partition is often evaluated after “defuzzificaproducts of image classification are crisp partitions (i.e., tion” (i.e., it is reduced to a crisp partition). The analysis of hard, ordinary, classical partitions), thus only the case of a traditional confusion matrix, describing the similarities qualitative variables has really held interest. The literabetween the computed crisp partition and a control parture on the analysis of confusion matrices is quite abuntition, can then be performed. This approach is rather dant. It generally describes methods that summarize the drastic and is far from satisfactory because the quantitamatrices by a single index, hence allowing different clastive information is lost after the defuzzification. Some sifications to be comparable (Congalton, 1991; Gong and methods do not require preliminary defuzzification, but Howarth, 1990; Ma and Redmond, 1995; Zhuang et al., they are not adequate to evaluate nonprobabilistic fuzzy 1995). This abundance of literature is justified by the nupartitions (i.e., fuzzy partitions such that the sum of the merous issues related to accuracy assessment, issues for membership degrees is not necessarily equal to 1). To solve which clarifications (Stehman, 1997; Stehman and these issues, we consider the evaluation of any fuzzy parCzaplewski, 1998; Congalton and Green, 1999; Stehman, tition l as the evaluation of a still fuzzy new partition: 2000) are still welcome and necessary after three dethe plausibilistic closure of l. This approach comes from cades of satellite image classification. the theory of evidence. It allows definition of a set of To complicate even more this domain, the last deoriginal tools (plausibility matrices, credibility matrices, cade has seen the emergence of new approaches and and overlap degrees) dedicated to fuzzy partition evaluaproducts (i.e., fuzzy partitions) intended to overcome the tion. A concrete application illustrates our theoretical intrinsic limitations of crisp partitions (e.g., Bezdek and work and a tutorial is provided in appendix. Elsevier Pal, 1992; Key et al., 1989; Wang, 1990; Simpson and Science Inc., 2000 Keller, 1995; Bastin, 1997; Raffy, 1997; Smith et al., 2000). Indeed, traditional crisp partitions may be inappropriate for representing heterogeneous, complex areas, INTRODUCTION and fuzzy partitions have proven useful to postclassification processing, the detection of high-risk confusion The evaluation of the accuracy of a partition arises natuzones, and the correction of flagrant misclassification rally in the field of image classification. It is a mandatory (Harris, 1985; Maselli et al., 1994; Andréfouët et al., 2000). They also open up new prospects for describing * Computer Engineering and Computer Science Department, the spatial structure of ecological systems without reducUniversity of Missouri-Columbia, 201 Engineering Building West, Coing them to an assemblage of pixels assigned to arbitrary lumbia, MO. pure classes (Palubinskas et al., 1995; Foody, 1996; Bur† Laboratoire de Géosciences Marines et Télédétection, Université de la Polynésie Française, BP 6570 Faaa-Aéroport, Tahiti, rough et al., 1997; Andréfouët and Roux, 1998). Such French Polynesia achievements are possible because fuzzy partitions proThe present address of S. Andréfouët is Department of Marine vide ordinal quantitative membership degrees, while Science, Remote Sensing and Biological Oceanography, University of South Florida. crisp partitions only provide qualitative assignments. DeAddress correspondence to S. Andréfouët, University of Florida, spite these characteristics, fuzzy partitions have often Dept. of Marine Science, 140 7th Avenue S., St. Petersburg, FL 33701, been condemned to undergo immediate “defuzzification” USA. E-mail: [email protected] Received 10 June 1998; revised 13 April 2000. before their assessment (Foody and Trodd, 1993). Re-