Multi-criteria based learning of the Choquet integral using Goal programming

In this paper, we explore a new way to learn an aggregation operator for fusion based on a combination of one or more labeled training data sets and information from one or more experts. One problem with learning an aggregation from training data alone is that it often results in solutions that are overly complex and expensive to implement. It also runs the risk of over-fitting and the quality of that solution is based in large on the size and diversity of the data employed. On the other hand, learning an aggregation based on only expert opinion can be overly subjective and may not result in desired performance for some given task. In order to overcome these shortcomings, we explore a new way to combine both of these important sources. However, conflict between data sets, experts or a combination of the two, can (and often do) occur and must be addressed. Herein, weighted Goal programming, an approach from multi-criteria decision making (MCDM), is employed to learn the fuzzy measure (FM) relative to the Choquet integral (CI) for data/information fusion. This framework provides an interesting way in which we can set the priority order of any number of combination of these two sources. Furthermore, it provides a mechanism to preserve the monotonicity constraints of the FM. We demonstrate results from synthetic experiments across a range of different conflicting and combination of sources scenarios.

I. Introduction

The fuzzy integral (FI) has been used time and time again in numerous applications such as signal and image processing, pattern recognition and multi-criteria decision making (MCDM) for data/information aggregation. In 1974, Sugeno introduced the fuzzy measure (FM), a monotone and normal capacity [1]. The FM is significant with respect to the FI because it is in effect what drives (determines what specific function is computed) nonlinear aggregation via Fls like the Choquet integral (CI) and Sugeno integral (SI). Existing approaches either manually specify the FM or attempt to learn it from data according to a criteria such as the sum of squared error (SSE). However, the FM is difficult to specify as it has , for inputs, numbers of “free parameters”. It is extremely rare than an expert can (or would want to) provide such information even for relatively small cases, e.g., has 14 values, has 30 and already has 1, 022 values. Common practice is to specify just the densities, the measure values for only the singletons (individuals). From there, we can use methods such as the S-Decomposable FM, the Sugeno FM [1], and Grabischs k-additive FM and integral [2]. In [3], Grabisch and Roubens proposed a method known as constraint satisfaction that takes the relative importance of the criteria and the type of interaction between them, if any. Alternately, identification of FMs based on data has been explored by numerous authors in different applications; quadratic programming (QP) [4] [5] [6] [7], gradient descent [8], penalty and reward [9], Gibbs sampler [10], and genetic algorithms [11]. However, to the best of our knowledge no method has been put forth to date to learn the FM by taking into account a weighted combination of both experts’ knowledge and data.