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Fig. 1. ANFIS with the rule base containing two fuzzy if-then rules function must really satisfy is that it must vary between 0 and 1. However, the membership function is usually a parameterized function. By changing its parameters we change its shape. Parameters in the first layer are referred as premise parameters. Fig. 2 shows generalized bell membership function, for three different sets of parameters {a,b,c}. This function is defined by equation (1).

Figure 1 ANFIS with the rule base containing two fuzzy if-then rules function must really satisfy is that it must vary between 0 and 1. However, the membership function is usually a parameterized function. By changing its parameters we change its shape. Parameters in the first layer are referred as premise parameters. Fig. 2 shows generalized bell membership function, for three different sets of parameters {a,b,c}. This function is defined by equation (1).

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Layer 2: Every node in this layer multiplies the incoming signals and sends the product which is referred as firing strength of a rule.
Layer 2: Every node in this layer multiplies the incoming signals and sends the product which is referred as firing strength of a rule.
The goal of a prediction task is to use several past values of x(t) up to the time ¢ to predict the value at time +P in t future. This assumes creating a mapping from n points the signal x(¢) spaced A apart: {x(¢-(1-1)A), ..., x(t-A), x()} to a predicted future value x(4+P). According to t conventional ANFIS architecture with n = 4 inputs depicted in Fig. 4 the standard values A = P = 6 are adopted. T membership functions are of the bell type. Notice that t he of ne ne ne ANFIS structure in Fig. 4 is a simplified version of the five layered structure depicted in Fig. 1, since layers 2, 3 and 4 are integrated into the one rule layer.
The goal of a prediction task is to use several past values of x(t) up to the time ¢ to predict the value at time +P in t future. This assumes creating a mapping from n points the signal x(¢) spaced A apart: {x(¢-(1-1)A), ..., x(t-A), x()} to a predicted future value x(4+P). According to t conventional ANFIS architecture with n = 4 inputs depicted in Fig. 4 the standard values A = P = 6 are adopted. T membership functions are of the bell type. Notice that t he of ne ne ne ANFIS structure in Fig. 4 is a simplified version of the five layered structure depicted in Fig. 1, since layers 2, 3 and 4 are integrated into the one rule layer.
Fig. 4. Conventional four-input structure of ANFIS, used for Mackey: Glass time series prediction; the term E(#+6) is the predicted value of x(t+6) A conventional fuzzy reasoning system, with n inputs each partitioned into p fuzzy sets, requires p” rules. Of course, utilizing human expertise in determining an ANFIS structure is preferable, but as mentioned earlier not always available solution. So using a conventional structure is quite harmonious with ANFIS ability to learn. The number of fitting parameters per membership function is m. The number of premise fitting parameters per input is p-m, so the total number of premise fitting parameters is n-p-m. The number of consequent fitting parameters per rule is n+1 (see Section II — Layer 4), the number of rules is p”, so the total number of consequent fitting parameters is p":(n+1). The total number of fitting parameters (premiset+consequent) is F(n,p,m) = n-p-m + p"(n+1). If we use four inputs (n=4), two fuzzy sets per input (p=2), and the generalized bel function (m3), one can obtain F(4,2,3)=4-2:3+2*.(4+1)=104. To achieve good generalization capability the number of training data points have to be several times larger than the number of parameters to be estimated. In our case this condition is fulfilled if we use 500 training data points: the ratio of training data points and the total number of fitting parameters is 500/104.
Fig. 4. Conventional four-input structure of ANFIS, used for Mackey: Glass time series prediction; the term E(#+6) is the predicted value of x(t+6) A conventional fuzzy reasoning system, with n inputs each partitioned into p fuzzy sets, requires p” rules. Of course, utilizing human expertise in determining an ANFIS structure is preferable, but as mentioned earlier not always available solution. So using a conventional structure is quite harmonious with ANFIS ability to learn. The number of fitting parameters per membership function is m. The number of premise fitting parameters per input is p-m, so the total number of premise fitting parameters is n-p-m. The number of consequent fitting parameters per rule is n+1 (see Section II — Layer 4), the number of rules is p”, so the total number of consequent fitting parameters is p":(n+1). The total number of fitting parameters (premiset+consequent) is F(n,p,m) = n-p-m + p"(n+1). If we use four inputs (n=4), two fuzzy sets per input (p=2), and the generalized bel function (m3), one can obtain F(4,2,3)=4-2:3+2*.(4+1)=104. To achieve good generalization capability the number of training data points have to be several times larger than the number of parameters to be estimated. In our case this condition is fulfilled if we use 500 training data points: the ratio of training data points and the total number of fitting parameters is 500/104.
actual values (solid line), x(¢+6), and the predicted ones (dashed), &(¢+6), are indistinguishable, since the error is less than 1%, as depicted in Fig. 5 (b). Fig. 5. Short-time prediction of the MG time series using conventional 4- input ANFIS: (a) Actual and predicted time series (b) Prediction error
actual values (solid line), x(¢+6), and the predicted ones (dashed), &(¢+6), are indistinguishable, since the error is less than 1%, as depicted in Fig. 5 (b). Fig. 5. Short-time prediction of the MG time series using conventional 4- input ANFIS: (a) Actual and predicted time series (b) Prediction error
Fig. 6. Long-time prediction of the MG time series using conventional 4- HIE LOUIE LO Udit ait SAINT LO GLCUUY GCOPeHs LOL UNS number so we must be careful if we increase it. Especially sensible is the forward pass in an epoch due to the presence of exponential term in the number of consequent parameters. It seems logical to increase the number of inputs (which corresponds to using more samples from past to predict the future value) in order to decrease prediction error. But, in this case even a little increment of 7 leads to large increase of F(n,p,m) producing an increasing in computing time, too. For instance, F(4,2,3)=104 but F°(5,2,3)=222. Increasing of fitting parameters requires more training data points, as well. Certainly, we can always increase the number of training data but this would lead inevitably to additional increase of computing time. The possible way for prediction error reducing is to increase the number of membership functions per input (parameter p) but this approach will lead us to practically the same problem — for instance, F(4,3,3) = 441. The lowest influence on the total number of fitting parameters is that of the number of membership functions' parameters, m, which depends on the membership function type. However, this approach usually does not bring significant improvement. It could even lead to degradation of efficiency. For example, if instead of generalized bell function (m=3) we use two-sided Gaussian function (m=4), the number of fitting parameters is F(4,2,4) = 112, but after training and testing we shall obtain RMSE = 0.0017 and NDEI = 0.0076 for the same number of 100 epochs.
Fig. 6. Long-time prediction of the MG time series using conventional 4- HIE LOUIE LO Udit ait SAINT LO GLCUUY GCOPeHs LOL UNS number so we must be careful if we increase it. Especially sensible is the forward pass in an epoch due to the presence of exponential term in the number of consequent parameters. It seems logical to increase the number of inputs (which corresponds to using more samples from past to predict the future value) in order to decrease prediction error. But, in this case even a little increment of 7 leads to large increase of F(n,p,m) producing an increasing in computing time, too. For instance, F(4,2,3)=104 but F°(5,2,3)=222. Increasing of fitting parameters requires more training data points, as well. Certainly, we can always increase the number of training data but this would lead inevitably to additional increase of computing time. The possible way for prediction error reducing is to increase the number of membership functions per input (parameter p) but this approach will lead us to practically the same problem — for instance, F(4,3,3) = 441. The lowest influence on the total number of fitting parameters is that of the number of membership functions' parameters, m, which depends on the membership function type. However, this approach usually does not bring significant improvement. It could even lead to degradation of efficiency. For example, if instead of generalized bell function (m=3) we use two-sided Gaussian function (m=4), the number of fitting parameters is F(4,2,4) = 112, but after training and testing we shall obtain RMSE = 0.0017 and NDEI = 0.0076 for the same number of 100 epochs.
The relationship between computing complexity and ANFIS dimensionality is one of the drawbacks of a conventional fuzzy approach that is commonly referred to as the “curse of dimensionality” [17]. To dismount the curse obviously we must think of changing the ANFIS structure, so now we will reconsider a modified ANFIS (MANFIS) architecture shown in Fig. 7. Fig. 7. A modified ANFIS (MANFIS) architecture based on the approximation that the number of fuzzy rules can be represented by the number of input fuzzy sets
The relationship between computing complexity and ANFIS dimensionality is one of the drawbacks of a conventional fuzzy approach that is commonly referred to as the “curse of dimensionality” [17]. To dismount the curse obviously we must think of changing the ANFIS structure, so now we will reconsider a modified ANFIS (MANFIS) architecture shown in Fig. 7. Fig. 7. A modified ANFIS (MANFIS) architecture based on the approximation that the number of fuzzy rules can be represented by the number of input fuzzy sets
MAMNFIS EFFICIENCY DEPENDING ON THE NUMBER OF INPUTS AND THE NUMBER OF EPOCHS REFERENCES
MAMNFIS EFFICIENCY DEPENDING ON THE NUMBER OF INPUTS AND THE NUMBER OF EPOCHS REFERENCES
Fig. 9. Long-time prediction error of the MG time series using proposed MANFIS architecture shown in Fig. 8. Fig. 8. A Short-time prediction error of the MG time series using proposed MANFIS architecture shown in Fig. 8.
Fig. 9. Long-time prediction error of the MG time series using proposed MANFIS architecture shown in Fig. 8. Fig. 8. A Short-time prediction error of the MG time series using proposed MANFIS architecture shown in Fig. 8.
Related topics:
Signal ProcessingFuzzy set theoryadaptive neuro fuzzy inference system (ANFIS)Fuzzy Inference System
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