2.6 Adaptive neuro-fuzzy inference system or adaptive network-based fuzzy inference system (ANFIS)

ANFIS is a hybrid learning procedure which employs the linguistic concept of fuzzy systems (human knowledge) and the training power of the ANN to solve a regression problem [41]. All ANFIS works reported here are based on the Takagi-Sugeno fuzzy inference system [42], where the fuzzy rule applied has the form: if x is A and y is B then z ¼ f xð Þ ; y . Other fuzzy methods are Mamdani-type or Tsukamoto-type [42].

Figure 6 depicts a typical ANFIS architecture. Square nodes (adaptive nodes) have parameters, while circle nodes (fixed nodes) do not. The first and the fourth layers contain the parameters that can be modified over time. A particular learning method was required to update these parameters.

In layer 1, every node is adaptive and associated with an appropriate continuous and piecewise differentiable function such as Gaussian, generalized bell-shaped, trapezoidal-shaped, and triangular-shaped functions.

In layer 2, every node is fixed and represents the firing strength of each rule. This is calculated by the fuzzy and connective method of the "product" of the incoming signals, that is, O<sup>2</sup> <sup>i</sup> ¼ wi ¼ μAið Þ x ∗ μBið Þ x , i\_1, 2:

In layer 3, every node is also fixed, showing the normalized firing strength of each rule. The ith node calculates the ratio of the ith rule's firing strength to the summation of two rules' firing strengths.

In every adaptive node of layer 4 (consequent nodes) is a function indicating the contribution of the ith rule to the overall output: O4,i ¼ wif ¼ wi pi þ qi þ ri , where wi is the output of layer 3 and pi , qi , ri is the parameter set. Finally, layer 5 (output node) is a single node that computes the overall output of the ANFIS as: <sup>O</sup>5, <sup>1</sup> <sup>¼</sup> <sup>∑</sup>wifi <sup>¼</sup> <sup>∑</sup><sup>i</sup> wifi <sup>∑</sup><sup>i</sup> wi .

One of the most important steps in developing a satisfactory forecasting model is the selection of the input variables. These variables determine the structure of the forecasting model and affect the weighted coefficients and the results of the model

Figure 6. Architecture of an adaptive network-based fuzzy inference system (ANFIS). function in layer 2. As the number of parameters increases with the fuzzy rule increment, the model structure becomes more complicated. A very good description of ANFIS is presented in [43, 44].
