Creating Function Procedures in Java
Part IV Hacking with Plugins
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= X' A' + A' X'. Every year, more than 1 exabyte (1018 bytes) of data is generated worldwide, most of it in digital form [1]. Data-driven methodologies for automatic generation of computational models are probably one of the most important tools needed to be developed to properly process and use that immense quantity of information [2]. In most applications (industry, medicine, nance, business, etc.), these methodologies should be iterative, to process the data as it is being reported in real time, and transparent, building a linguistic model clearly interpretable by humans. eTS (evolving Takagi-Sugeno systems) is one of those methodologies [3]. In the present work, the eTS developed by [4] is improved, increasing the rule-span in state space, and used to build fuzzy systems from data. Traditionally, the most important property of a fuzzy system has been its accuracy in representing the real system (for simulation, prediction, decision making, etc.). However, the obtained fuzzy systems frequently are without practical utility because it is impossible to give some semantic meaning to the rules due to fuzzy sets superposition, and rules that are sometimes redundant and sometimes contradictory, frequently with high complexity. In [5] a comprehensive collection of papers present several approaches for evolving the fuzzy systems to more interpretability (mainly of ine trained or expert-based ones). In recent years, interpretability has been considered to be the key feature of fuzzy models [6, 7], continuing previous important works [8, 9], and can be pursued by rule base simpli cation and reduction methods. There is actually considerable activity concerning this challenging problem. Several perspectives are being developed, for example, by fuzzy set merging using entropy measures [10], by genetic optimization [11 13], by multiobjective evolutionary algorithms [14 16], by manipulating the cost index [17, 18], or by Radial Basis Function Networks [19]. For a more detailed review and treatment, see [22] or [23]. Most of the known methods have been developed for batch processing for an already-existing rule base. Kernel-based learning methods, such as Support Vector Machines (SVMs), Relevance Vector Machines (RVMs), or Kernel PCA, have been studied extensively in the machine-learning community, being among the best learning techniques for many benchmark data sets and real-world applications [24]. The basic idea of these methods is to preprocess the data by some nonlinear mapping (using the kernel trick) and then apply a linear algorithm in a usually higher-dimensional feature space. Such large feature spaces are handled with the simple computational cost of the kernel function. This approach indicates an alternative point of view to the parameters learning of a fuzzy system, with the membership functions being those examples that are critical to solving the given task. In the traditional forms of automatic learning, the functions are simply regarded as clusters. The objective of the present chapter is online learning, for developing models in real time for classi cation problems and dynamic systems. Computational time is a critical issue and simple, ef cient techniques are searched for. A pruning technique is developed and tested to reduce the degree of redundancy and unnecessary complexity arising in the automated building of fuzzy rules. This improves the human semantic interpretability and as a consequence the usefulness of the results, allowing the merging of compatible fuzzy sets and possibly reduction of the number of rules and features. The fuzzy system is based on an improved version of the eTS algorithm of [4], strengthening its capability to
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