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Fig. 1. Scheme of a Mamdani Fuzzy inference System

Figure 1 Scheme of a Mamdani Fuzzy inference System

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Fig. 1. Scheme of a Sugeno Fuzzy inference System Following the same assumptions as for the Mamdani FIS, a scheme for the Sugeno FIS is depicted in Figure 2 (adapted from [7]): In this sort of fuzzy inference system only the antecedents of the rules are fuzzy, and so, the rules act as an inference
Fig. 1. Scheme of a Sugeno Fuzzy inference System Following the same assumptions as for the Mamdani FIS, a scheme for the Sugeno FIS is depicted in Figure 2 (adapted from [7]): In this sort of fuzzy inference system only the antecedents of the rules are fuzzy, and so, the rules act as an inference
The optimization problem can thus be formulated as follows: YVUICIE Po PENTEL TAY Ce PIVUUIEI IS sdlG WO Ue all Unconstrained Linear Least Square problem and can be solved by many heuristic algorithms, such as genetic
The optimization problem can thus be formulated as follows: YVUICIE Po PENTEL TAY Ce PIVUUIEI IS sdlG WO Ue all Unconstrained Linear Least Square problem and can be solved by many heuristic algorithms, such as genetic
Table I shows results of tuning a Sugeno type FIS using least mean square method. The second column, for example, contains, from top to bottom, the number of input vectors in the data set (50), the MSE which measures the difference between the Mamdani outputs from the training data and the As mentioned before, our objective is to find a first order Sugeno FIS that produces similar results to the given Mamdani FIS. This optimization problem uses equation (3) and formalization (4). Figures 5 shows the Mandani FIS modeled for the case where quality of food and quality of service are both classified with 5 (figure done in matlab). Figure 6 shows the rules and output surface for the same problem and classification. As can be observed in Figure 5, only rule 2 is triggered and the final result (after deffuzzification with centroid) is to give a tip of 15 dollars.
Table I shows results of tuning a Sugeno type FIS using least mean square method. The second column, for example, contains, from top to bottom, the number of input vectors in the data set (50), the MSE which measures the difference between the Mamdani outputs from the training data and the As mentioned before, our objective is to find a first order Sugeno FIS that produces similar results to the given Mamdani FIS. This optimization problem uses equation (3) and formalization (4). Figures 5 shows the Mandani FIS modeled for the case where quality of food and quality of service are both classified with 5 (figure done in matlab). Figure 6 shows the rules and output surface for the same problem and classification. As can be observed in Figure 5, only rule 2 is triggered and the final result (after deffuzzification with centroid) is to give a tip of 15 dollars.
Sugeno FIS parameters for 6 sets of input data Figure 7 shows the plot with comparative results for Mamdani FIS and corresponding Sugeno FIS, obtained as explained above, for the training set with 100 input vectors. TABLE I
Sugeno FIS parameters for 6 sets of input data Figure 7 shows the plot with comparative results for Mamdani FIS and corresponding Sugeno FIS, obtained as explained above, for the training set with 100 input vectors. TABLE I
In both cases, it can be easily seen that the difference between the MSE of the training set and those of randomly generated data sets will go to zero by increasing the size of data set. This implies that the training process (finding system parameters) becomes more stable and accurate for larger training data sets.
In both cases, it can be easily seen that the difference between the MSE of the training set and those of randomly generated data sets will go to zero by increasing the size of data set. This implies that the training process (finding system parameters) becomes more stable and accurate for larger training data sets.
Parameters Of Genetic Algorithm Table IV, shows the resulting values for each parameter
Parameters Of Genetic Algorithm Table IV, shows the resulting values for each parameter
Table II presents two calculations for the MSE. One obtained as previously explained, for expected and obtained outputs computed using the inputs present in the trained set; the second MSE, also compares expected outputs and obtained outputs but for randomly generated inputs i.e. not the ones used in the optimization process. Fig. 7. Mamdani and Sugeno FIS Outputs for 100 data set
Table II presents two calculations for the MSE. One obtained as previously explained, for expected and obtained outputs computed using the inputs present in the trained set; the second MSE, also compares expected outputs and obtained outputs but for randomly generated inputs i.e. not the ones used in the optimization process. Fig. 7. Mamdani and Sugeno FIS Outputs for 100 data set
Sugeno Parameters using 10000 set of data for the three algorithms Table V shows the respective MSE calculated for each algorithm, based on random input data (called test data set) — not the one used for training.
Sugeno Parameters using 10000 set of data for the three algorithms Table V shows the respective MSE calculated for each algorithm, based on random input data (called test data set) — not the one used for training.
In the next step we must import historical data to ANFIS main page (in our case the input/output values of corresponding Mamdani FIS) and set the parameters (error tolerance, num. epochs, and optim. method.) and by training the system the modified FIS will be available. Figure 2 shows the ANFIS main page and parameters for tipper case.
In the next step we must import historical data to ANFIS main page (in our case the input/output values of corresponding Mamdani FIS) and set the parameters (error tolerance, num. epochs, and optim. method.) and by training the system the modified FIS will be available. Figure 2 shows the ANFIS main page and parameters for tipper case.
AMFIS [11] is a tool for constructing an input/output mapping based on both human knowledge and historical data. In an ANFIS architecture, the FIS is parameterized with respect to membership function values and the values of these parameters are modified by an optimization algorithm to increase FIS ability to approximate the data in the historical data. The optimization algorithm minimizes the approximation error criteria and does not change the architecture of the FIS. Least square, gradient descent or both of them together (i.e. hybrid algorithm) can be used for the parameter optimization. For our example (Tipper) Figure 9 shows the structure of the Tipper Sugeno FIS as an AMFIS.
AMFIS [11] is a tool for constructing an input/output mapping based on both human knowledge and historical data. In an ANFIS architecture, the FIS is parameterized with respect to membership function values and the values of these parameters are modified by an optimization algorithm to increase FIS ability to approximate the data in the historical data. The optimization algorithm minimizes the approximation error criteria and does not change the architecture of the FIS. Least square, gradient descent or both of them together (i.e. hybrid algorithm) can be used for the parameter optimization. For our example (Tipper) Figure 9 shows the structure of the Tipper Sugeno FIS as an AMFIS.
Related topics:
adaptive neuro fuzzy inference system (ANFIS)First-Order LogicFuzzy Inference System
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