Fuzzy inference systems (FIS) are widely used for process simulation or control. They can be designed either from expert knowledge or from data. For complex systems, FIS based on expert knowledge only may suffer from a loss of accuracy. This is the main incentive for using fuzzy rules inferred from data. Designing a FIS from data can be decomposed into two main phases: automatic rule generation and system optimization. Rule generation leads to a basic system with a given space partitioning and the corresponding set of rules. System optimization can be done at various levels. Variable selection can be an overall selection or it can be managed rule by rule. Rule base optimization aims to select the most useful rules and to optimize rule conclusions. Space partitioning can be improved by adding or removing fuzzy sets and by tuning membership function parameters. Structure optimization is of a major importance: selecting variables, reducing the rule base and optimizing the number of fuzzy sets. Over the years, many methods have become available for designing FIS from data. Their efficiency is usually characterized by a numerical performance index. However, for human-computer cooperation another criterion is needed: the rule interpretability. An implicit assumption states that fuzzy rules are by nature easy to be interpreted. This could be wrong when dealing with complex multivariable systems or when the generated partitioning is meaningless for experts. This paper analyses the main methods for automatic rule generation and structure optimization. They are grouped into several families and compared according to the rule interpretability criterion. For this purpose, three conditions for a set of rules to be interpretable are defined.

Abstract. The goal of this study was to test the usefulness of high-spatial resolution information provided by airborne imagery and soil electrical properties to define plant water restriction zones within-vineyards. The main contribution of this is to propose a study on a large area representing the regions' vineyard diversity (different age, different varieties and different soils) located in southern France (Languedoc-Roussillon region, France). Nine non-irrigated plots were selected for this work in 2006 and 2007. In each plot, different zones were defined using the high-spatial resolution (1m 2 ) information provided by airborne imagery (Normalised Difference Vegetation Index, NDVI). Within each zone, measurements were conducted to assess: (i) vine water status (Pre-dawn Leaf Water Potential, PLWP), (ii) vine vegetative expression (vine trunk circumference and canopy area), (iii) soil electrical resistivity and, (iv) harvest quantity and quality. Large differences were observed for vegetative expression, yield and plant water status between the individual NDVI-defined zones. Significant differences were also observed for soil resistivity and vine trunk circumference, suggesting the temporal stability of the zoning and its relevance to defining vine water status zones. The NDVI zoning could not be related to the observed differences in quality, thus showing the limitations in using this approach to assess grape quality under non-irrigated conditions. The paper concludes with the approach that is currently being considered: using NDVI zones (corresponding to plant water restriction zones) in association with soil electrical resistivity and plant water status measurements to provide an assessment of the spatial variability of grape production at harvest.

In this paper, we propose a new method to construct fuzzy partitions from data. The procedure generates a hierarchy including best partitions of all sizes from n to two fuzzy sets. The maximum size n is determined according to the data distribution and corresponds to the finest resolution level. We use an ascending method for which a merging criterion is needed. This criterion is based on the definition of a special metric distance suitable for fuzzy partitioning, and the merging is done under semantic constraints. The distance we define does not handle the point coordinates, but directly their membership degrees to the fuzzy sets of the partition. This leads to the introduction of the notions of internal and external distances. The hierarchical fuzzy partitioning (HFP) is carried independently over each dimension, and, to demonstrate the partition potential, they are used to build fuzzy inference system using both fuzzy decision trees and a simple selection mechanism. Due to the merging technique, all the fuzzy sets in the various partitions are interpretable as linguistic labels. The trade-off between accuracy and interpretability constitutes the most promising aspect in our approach. Well known data sets are investigated and the results are compared with those obtained by other authors using different techniques. The method is also applied to real world agricultural data, the results are analyzed and weighed against those achieved by other methods, such as fuzzy clustering or discriminant analysis.

No abstract

a b s t r a c tFuzzy inference systems (FIS) are likely to play a significant part in system modeling, provided that they remain interpretable following learning from data. The aim of this paper is to set up some guidelines for interpretable FIS learning, based on practical experience with fuzzy modeling in various fields. An open source software system called FisPro has been specifically designed to provide generic tools for interpretable FIS design and learning. It can then be extended with the addition of new contributions. This work presents a global approach to design data-driven FIS that satisfy certain interpretability and accuracy criteria. It includes fuzzy partition generation, rule learning, input space reduction and rule base simplification. The FisPro implementation is discussed and illustrated through several detailed case studies.