2.1 Principal ideas of GMDH: fuzzy model construction

As it well known, the drawbacks of GMDH are the following [3, 4]:


Therefore, in the last 10 years, the new variant of GMDH—fuzzy GMDH—was developed and improved which may work with fuzzy and qualitative input data and is free of classical GMDH drawbacks [3].

As it is well known, GMDH method is based on the following principles [1–3]:

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


Fuzzy GMDH is also based on these principles but construct fuzzy models. Let's consider its main ideas.

In works [3–5], the linear interval regression model was considered:

$$Y = A\_0 Z\_0 + A\_1 Z\_1 + \dots + A\_n Z\_n \tag{1}$$

where Ai is a fuzzy number of triangular form described by a pair of parameters Ai ¼ α<sup>i</sup> ð Þ ;ci , where α<sup>i</sup> is interval center, ci is its width, and ci ≥0, Zi is the input variables.

Then Y is a fuzzy number, parameters of which are determined as follows: The interval center

$$a\_{\mathcal{Y}} = \sum a\_i \mathbf{z}\_i = a^T \cdot \mathbf{z} \tag{2}$$

The interval width

$$\mathbf{c}\_{\mathbf{y}} = \sum \mathbf{c}\_{i} \cdot |\mathbf{z}\_{i}| = \mathbf{c}^{T} |\mathbf{z}| \tag{3}$$

For example, for the partial description (PD) of the kind

$$f(\mathbf{x}\_i, \mathbf{x}\_j) = A\_0 + A\_1 \mathbf{x}\_i + A\_2 \mathbf{x}\_j + A\_3 \mathbf{x}\_i \mathbf{x}\_j + A\_4 \mathbf{x}\_i^2 + A\_5 \mathbf{x}\_j^2 \tag{4}$$

it's necessary to substitute in the general model (1) <sup>z</sup><sup>0</sup> <sup>¼</sup> <sup>1</sup> <sup>z</sup><sup>1</sup> <sup>¼</sup> xi <sup>z</sup><sup>2</sup> <sup>¼</sup> xj <sup>z</sup><sup>3</sup> <sup>¼</sup> xixj <sup>z</sup><sup>4</sup> <sup>¼</sup> <sup>x</sup><sup>2</sup> <sup>i</sup> <sup>z</sup><sup>5</sup> <sup>¼</sup> <sup>x</sup><sup>2</sup> j .

Let the training sample be f g z1; z2; …; zM , y1; y2; …; yM � �. Then for the model (1) to be adequate, it's necessary to find such parameters α<sup>i</sup> ð Þ ;ci i ¼ 1, n, which satisfy the following inequalities:

$$\begin{cases} a^T \mathbf{z}\_k - \mathbf{c}^T \cdot |\mathbf{z}\_k| \le \mathbf{y}\_k \\ a^T \mathbf{z}\_k + \mathbf{c}^T \cdot |\mathbf{z}\_k| \ge \mathbf{y}\_k \end{cases}, \ k = \overline{\mathbf{1}, M} \tag{5}$$

Let's formulate the basic requirements for the linear interval model of a kind (4). It's necessary to find such values of the parameters α<sup>i</sup> ð Þ ;ci of fuzzy coefficients for which:


These requirements lead to the following linear programming (LP) problem [3, 4]:

$$\begin{aligned} &\min(\mathbf{C}\_0 \cdot M + \mathbf{C}\_1 \sum\_{k=1}^M |\mathbf{x}\_{ki}| + \mathbf{C}\_2 \sum\_{k=1}^M |\mathbf{x}\_{kj}| + \mathbf{C}\_3 \sum\_{k=1}^M |\mathbf{x}\_{ki}\mathbf{x}\_{kj}| + \\ &+ \mathbf{C}\_4 \sum\_{k=1}^M |\mathbf{x}\_{ki}^2| + \mathbf{C}\_5 \sum\_{k=1}^M \left| \mathbf{x}\_{kj}^2 \right| \end{aligned} \tag{6}$$

under constraints

$$\begin{aligned} &a\_0 + a\_1 \mathbf{x}\_{ki} + a\_2 \mathbf{x}\_{kj} + a\_3 \mathbf{x}\_{ki} \mathbf{x}\_{kj} + a\_4 \mathbf{x}\_{ki}^2 + a\_5 \mathbf{x}\_{kj}^2 - (\mathbf{C}\_0 + \mathbf{C}\_1 |\mathbf{x}\_{ki}| + \mathbf{C}\_2 |\mathbf{x}\_{kj}| + \mathbf{C}\_3 \mathbf{x}\_{kj} \\ &+ \mathbf{C}\_3 \left| \mathbf{x}\_{ki} \mathbf{x}\_{kj} \right| + \mathbf{C}\_4 \left| \mathbf{x}\_{ki}^2 \right| + \mathbf{C}\_5 \left| \mathbf{x}\_{kj}^2 \right| \le \mathbf{y}\_k \\ &a\_0 + a\_1 \mathbf{x}\_{ki} + a\_2 \mathbf{x}\_{kj} + a\_3 \mathbf{x}\_{ki} \mathbf{x}\_{kj} + a\_4 \mathbf{x}\_{ki}^2 + a\_5 \mathbf{x}\_{kj}^2 + (\mathbf{C}\_0 + \mathbf{C}\_1 |\mathbf{x}\_{ki}| + \mathbf{C}\_2 |\mathbf{x}\_{kj}| + \mathbf{C}\_1 \mathbf{x}\_{ki}) \\ &+ \mathbf{C}\_3 \left| \mathbf{x}\_{ki} \mathbf{x}\_{kj} \right| + \mathbf{C}\_4 \left| \mathbf{x}\_{ki}^2 \right| + \mathbf{C}\_5 \left| \mathbf{x}\_{kj}^2 \right| \ge \mathbf{y}\_{k^\*} \\ &\mathbf{C}\_p \ge 0, \quad p = 0, \ 5 \ k = \overline{1, M} \end{aligned} \tag{8}$$

where k is a number of a point.

As one can easily see, the task (6)–(8) is a LP problem. However, the inconvenience of the model (6)–(8) for the application of standard LP methods is that there are no constraints of non-negativity for variables αi. Therefore for its solution, it's reasonable to pass to the dual LP problem by introducing dual variables f g δ<sup>k</sup> and f g δ<sup>k</sup>þ<sup>M</sup> , k ¼ 1, M. Using simplex method after finding the optimal solution for the dual problem, the optimal solutions α<sup>i</sup> ð Þ ;ci of the initial direct problem will be also found.
