Adaptive Neuro-Fuzzy Inference System (ANFIS) with first order Sugeno consequent is used widely in modeling applications. Though it has the advantage of giving good modeling results in many cases, it is not capable of modeling highly non-linear systems with high accuracy. In this paper, an efficient way for using ANFIS with Sugeno second order consequents is presented. Better approximation capability of Sugeno second order consequents compared to lower order Sugeno consequents is shown. Subtractive clustering is used to determine the number and type of membership functions. A hybrid-learning algorithm that combines the gradient descent method and the least squares estimate is then used to update the parameters of the proposed Second Order Sugeno-ANFIS (SOS-ANFIS). Simulation of the proposed SOS-ANFIS for two examples shows better results than that of lower order Sugeno consequents. The proposed SOS-ANFIS shows better initial error, better convergence, quicker convergence and much better final error value.
In this study Active Learning Method (ALM) as a novel fuzzy modeling approach is compared with optimized Support Vector Machine (SVM) using simple Genetic Algorithm (GA), as a well known datadriven model for long term simulation of daily streamflow in Karoon River. The daily discharge data from 1991 to 1996 and from 1996 to 1999 were utilized for training and testing of the models, respectively. Values of the Nash-Sutcliffe, Bias, R2 , MPAE and PTVE of ALM model with 16 fuzzy rules were 0.81, 5.5 m3 s -1, 0.81, 12.9%, and 1.9%, respectively. Following the same order of parameters, these criteria for optimized SVM model were 0.8, -10.7 m3 s-1, 0.81, 7.3%, and -3.6%, respectively. The results show appropriate and acceptable simulation by ALM and optimized SVM. Optimized SVM is a well-known method for runoff simulation and its capabilities have been demonstrated. Therefore, the similarity between ALM and optimized SVM results imply the ability of ALM for runoff modeling. In addition, ALM training is easier and more straightforward than the training of many other data driven models such as optimized SVM and it is able to identify and rank the effective input variables for the runoff modeling. According to the results of ALM simulation and its abilities and properties, it has merit to be introduced as a new modeling method for the runoff modeling. and Cieľom štúdie bolo porovnať možnosti dlhodobej simulácie denných prietokov v rieke Karoon pomocou novovyvinutej fuzzy metódy aktívneho učenia (Active Learning Method - ALM) a známej metódy vektormi podporených strojov (Support Vector Machine - SVM), optimalizovanej genetickým algoritmom (GA). Na tréning a testovanie modelov boli použité časové rady denných prietokov za obdobie rokov 1991 až 1996 a 1996 až 1999. Hodnoty parametrov Nash-Sutcliffe, Bias, R2 , MPAE a PTVE pre model ALM boli 0,81; 5,5 m3 s-1; 0,81; 12,9% a 1,9%. Parametre v tom istom poradí pre model SVM boli 0,8 -10,7 m3 s-1, 0,81; 7,3%; a -3,6%. Z výsledkov simulácií vyplýva, že aplikáciou metód ALM a SVM možno získať porovnateľné a akceptovateľné výsledky. Podobnosť výsledkov medzi ALM a SVM implikuje vhodnosť novovyvinutej metódy ALM pre simuláciu odtoku. Tréning ALM je ľahší a jednoduchší ako je tréning ďalších dátami riadených modelov podobného typu. Navyše algoritmus ALM je schopný identifikovať a zoradiť efektívne vstupné premenné pre modelovanie odtoku. Na základe dosiahnutých výsledkov možno metódu ALM zaradiť medzi nové, alternatívne metódy modelovania odtoku.