Precise Fuzzy Inference Systems
Fuzzy inference is the process of mapping an input to an output using fuzzy logic. It consists of membership functions which are responsible for the fuzzification of inputs, fuzzy operators and IF-THEN rules. Fuzzy inference systems have been successfully applied in many different fields, such as automatic control, data classification, decision analysis, expert systems, and computer vision. And because of this multidisciplinary characteristic, fuzzy inference systems can also be mentioned as fuzzy rule based systems, fuzzy expert systems, fuzzy modeling, fuzzy associative memory, fuzzy logic controllers, and fuzzy systems. Fuzzy inference systems are usually divided into two areas: linguistic fuzzy model which focus on interpretability, mainly Mamdani models, and precise fuzzy models which focus on accuracy, mainly Takagi-Sugeno-Kang (TSK) model. Compared with Mamdani model, the first two step of fuzzy inference process of TSK method, fuzzifying inputs and applying the fuzzy operator, are exactly the same. The most fundamental difference between these two models are the way the crisp output is generated from the fuzzy inputs . TSK network uses weighted average to compute the crisp output instead of the defuzzification step in Mamdani model which is time consuming. Then together with the fact that in most cases TSK model has less fuzzy rules than Mamdani method make TSK model a more computational efficient fuzzy inference system. On the other hand, known as an universal approximator , TSK can do smooth piece-wise linear approximation for a nonlinear function which makes it a precise fuzzy inference system. There are two types of learning algorithms of TSK fuzzy inference system: offline and online. Using offline algorithm, the structure of system (like the number of rules) is prefixed before the learning process. The inputs are in a batch mode and can be repeatedly accessed for training parameters. After iteratively training process, an output for function to be approximated can be generated. And if the output is not good enough, the structure can be changed, more rules can be added and then a new learning process takes place. On the contrast, there are no pre-existing rules before the training process in online methods. Rules will be generated after learning input data. Inputs here are in a sequential fashion, usually one by one. After learning a input, it cannot be hold and will be lost and another new input will be used. Adaption to data generated as time goes by makes online methods more powerful. Moreover, some online methods developed recently can not only add rules to fuzzy inference system but also remove redundant ones so that the computational cost will be reduced while the accuracy of approximation can be kept. Also, the membership functions and consequent parameters can be learnt by online algorithm. In this article, we first discuss how to extract rules for zero-order TSK systems from artificial neural networks. Then we will look at some well-established models for implementation of offline and online algorithms, such as ANFIS, Genetic Algorithm based approaches and Immune-based learning systems for offline methods, and DENFIS, SAFIS and several others using online learning. II. ZERO-ORDER TSK-SYSTEMS
Artificial neural networks adjust their internal parameters to match the inputs to their corresponding outputs from the training datasets. These parameters won’t give any semantic understand to us, humans. Thus extracting rules from ANNs has gained a lot of attention since the last decade. Heuristically speaking, the Zero order TSK systems are functionally equivalent to the ANNs as they map the fuzzy input space to crisp values map. Thus the mapping function used in Zero Order TSK systems should resemble the function realized by ANNs by moulding the connection weight of internal neurons during the training. Thus rules can be extracted from the ANNs...
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