Visualization and learning of the Choquet integral with limited training data

The fuzzy integral (FI) is a nonlinear aggregation operator whose behavior is defined by the fuzzy measure (FM). As an aggregation operator, the FI is commonly used for evidence fusion where it combines sources of information based on the worth of each subset of sources. One drawback to FI-based methods, however, is the specification of the FM. Defining the FM manually quickly becomes too tedious since the number of FM terms scales as 2ⁿ, where n is the number of sources; thus, an automatic method of defining the FM is necessary. In this paper, we review a data-driven method of learning the FM via minimizing the sum-of-squared error (SSE) in the context of decision-level fusion and propose an extension allowing knowledge of the underlying FM to be encoded in the algorithm. The algorithm is applied to real-world and toy datasets and results show that the extension can improve classification accuracy. Furthermore, we introduce a visualization strategy to simultaneously show the quantitative information in the FM as well as the FI.

I. Introduction

In many fields, we are often faced with the task of making decisions based on a set of feature-vector data . This data is typically accompanied by a set of training labels for each feature-vector, giving the pair , where is a vector of labels such that is the label of feature-vector . This problem can be considered a classification task, and is typically tackled by training a classifier such that it can accurately predict the class label of a new sample of data where the label is not known. More concretely, the data are used to learn some prediction function such that we can accurately predict the label of feature vectors as .