Cross-entropy for the evaluation of the accuracy of a fuzzy land cover classification with fuzzy ground data

Fuzzy classifications have been used to represent land cover when pixels may have multiple and partial class membership. A fuzzy classification can be derived by softening the output of a conventional “hard” classification. Thus, for example, the probabilities of class membership may be derived from a conventional probability-based classification and mapped to represent the land cover of a site. The accuracy of the representation provided by a fuzzy classification is, however, difficult to evaluate. Conventional measures of classification accuracy cannot be used since they are appropriate only for “hard” classifications. The accuracy of a classification may, however, be indicated by the way in which the probability of class membership is partitioned between the classes and this may be expressed by entropy measures. Here cross-entropy is proposed as a means of evaluating the accuracy of a fuzzy classification, by illustrating how closely a fuzzy classification represents land cover when multiple and partial class membership is a feature of both the remotely sensed and ground data sets. Cross-entropy is calculated from the probability distributions of class membership derived from the remotely sensed and ground data sets. The use of cross-entropy as an indicator of classification accuracy was investigated with reference to land cover classifications of two contrasting test sites. The results show that cross-entropy may be used to indicate the accuracy of the representation of land cover when the classification of the remotely sensed data and ground data are both fuzzy.