3. Description of fuzzy GMDH algorithm

Let's present the brief description of the algorithm FGMDH [3, 4].


Investigation of Fuzzy Inductive Modeling Method in Forecasting Problems DOI: http://dx.doi.org/10.5772/intechopen.86348


### 4. Analysis of different membership functions

In the first papers devoted to fuzzy GMDH [3], the triangular membership functions (MFs) were considered. But as fuzzy numbers may also have the other kinds of MF, it's important to consider the other classes of MF in the problems of modeling using FGMDH. In paper [4] fuzzy models with Gaussian and bell-shaped MFs were investigated.

Consider a fuzzy set with Gaussian MF:

$$
\mu\_B(\mathfrak{x}) = e^{-\frac{1^{(\mathfrak{x}-1)^2}{2}}{\mathfrak{x}^2}} \tag{9}
$$

Let the linear interval model for partial description of FGMDH take the form (4). Then the problem is formulated as follows:

Find such fuzzy numbers Bi, with parameters ai ð Þ ;ci , that:


In [4, 6] it was shown that the problem of finding optimal fuzzy model will be finally transformed to the following LP problem:

$$\begin{aligned} &\min(\mathbf{C}\_0 \cdot \mathbf{M} + \mathbf{C}\_1 \sum\_{k=1}^M |\mathbf{x}\_{ki}| + \mathbf{C}\_2 \sum\_{k=1}^M |\mathbf{x}\_{kj}| + \mathbf{C}\_3 \sum\_{k=1}^M |\mathbf{x}\_{ki}\mathbf{x}\_{kj}| + \\ &+ \mathbf{C}\_4 \sum\_{k=1}^M |\mathbf{x}\_{ki}^2| + \mathbf{C}\_5 \sum\_{k=1}^M \left| \mathbf{x}\_{kj}^2 \right| \end{aligned} \tag{10}$$

under constraints

$$\begin{aligned} \left\{ \left. \boldsymbol{a}\_{0} + \boldsymbol{a}\_{1} \boldsymbol{\chi}\_{\bar{n}} + \ldots + \boldsymbol{a}\_{5} \boldsymbol{\chi}\_{\bar{k}\_{\bar{j}}}^{2} + \left( \boldsymbol{C}\_{0} + \boldsymbol{C}\_{1} |\boldsymbol{\chi}\_{\bar{k}i}| + \ldots + \boldsymbol{C}\_{5} \middle| \boldsymbol{\chi}\_{\bar{k}\_{\bar{l}}^{2}}^{2} \right) \right\} \cdot \sqrt{-2 \ln \boldsymbol{a}} \succeq \mathbf{y}\_{\mathbf{k}} \\ \left\{ \left. \boldsymbol{a}\_{0} + \boldsymbol{a}\_{1} \boldsymbol{\chi}\_{\bar{k}i} + \ldots + \boldsymbol{a}\_{5} \boldsymbol{\chi}\_{\bar{k}\_{\bar{j}}^{2}}^{2} - \left( \boldsymbol{C}\_{0} + \boldsymbol{C}\_{1} |\boldsymbol{\chi}\_{\bar{k}i}| + \ldots + \boldsymbol{C}\_{5} \middle| \boldsymbol{\chi}\_{\bar{k}\_{\bar{j}}^{2}}^{2} \right) \right\} \cdot \sqrt{-2 \ln \boldsymbol{a}} \succeq \mathbf{y}\_{\mathbf{k}} \end{aligned} \right\} \left\{ \boldsymbol{k} = \overline{1,M} \quad \text{(11)} \right\} $$

To solve this problem like the case with triangular MF, it's reasonable to pass to the dual LP problem of the form

$$\max \left( \sum\_{k=1}^{M} \mathcal{y}\_k \cdot \delta\_{k+M} - \sum\_{k=1}^{M} \mathcal{y}\_k \cdot \delta\_k \right) \tag{12}$$

with constraints of equalities and inequalities

$$\begin{aligned} \sum\_{k=1}^{M} \delta\_{k+M} - \sum\_{k=1}^{M} \delta\_{k} &= 0, \\ \sum\_{k=1}^{M} X\_{ki} \cdot \delta\_{k+M} - \sum\_{k=1}^{M} X\_{ki} \cdot \delta\_{k} &= 0 \\ \dots & \dots \\ \sum\_{k=1}^{M} X\_{kj}^{-2} \cdot \delta\_{k+M} - \sum\_{k=1}^{M} X\_{kj}^{2} \cdot \delta\_{k} &= 0 \end{aligned} \tag{13}$$
 
$$\sum\_{k=1}^{M} \delta\_{k} + \sum\_{k=1}^{M} \delta\_{k+M} \le \frac{M}{\sqrt{-2\ln\alpha}} $$
 
$$\sum\_{k=1}^{M} |X\_{ki}| \cdot \delta\_{k+M} + \sum\_{k=1}^{M} |X\_{ki}| \cdot \delta\_{k} \le \frac{\sum\_{k=1}^{M} |X\_{ki}|}{\sqrt{-2\ln\alpha}} \Bigg| \tag{14}$$
 
$$\sum\_{k=1}^{M} |X\_{kj}^{2}| \cdot \delta\_{k+M} + \sum\_{k=1}^{M} |X\_{kj}^{2}| \cdot \delta\_{k} \le \frac{\sum\_{k=1}^{M} |X\_{kj}^{2}|}{\sqrt{-2\ln\alpha}} \Bigg| \tag{15}$$
 
$$\delta\_{k} \ge 0, \ k = \overline{1,2M} \tag{15}$$

Analyzing dual LP program (12)–(15), it's easy to notice that this problem is always solvable as there trivial solution δ<sup>k</sup> ¼ 1 k ¼ 1, 2M always exists. Therefore the initial problem (10) and (11) also always has solutions with any data.

Thus, fuzzy GMDH allows to construct fuzzy models and has the following advantages:

