7.3 Comparison of crisp and fuzzy GMDH

For more comprehensive efficiency comparison of crisp and fuzzy GMDH, existing implementation of GMDH was extended by inclusion of new types of PD orthogonal polynomials: Chebyshev's and trigonometric and ARIMA models as PD.

Figure 1. Forecasting accuracy of PCI with different MF.

#### Figure 2.

Forecasting accuracy of GNP with different MF.

#### Figure 3.

As adaptation algorithm stochastic approximation and recurrent LSM were implemented. In Figures 4 and 5, the mean RMSE values for crisp and fuzzy GMDH in the whole range of data variation are presented for different types of PD without adaptation and with adaptation algorithms.

As one can easily see from presented results, the fuzzy algorithm GMDH shows better forecasting accuracy than classic GMDH for all adaptation algorithms.

So the results of experiments have confirmed indisputable advantages of fuzzy GMDH over classic GMDH for problem of forecasting macroeconomic indicators. In the next experiments, the comparison of fuzzy GMDH with results of neural network (NN) backpropagation was performed. The final results—MSE values on five forecasting points while forecasting CPI and GNP—are presented in Table 1.

Summing the experimental results, the following conclusions were made:


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

#### Figure 4.

Forecasting accuracy of classical and fuzzy GMDH for PCI.

#### Figure 5.

Forecasting accuracy of classical and fuzzy GMDH for GDP.

doesn't lead to significant changes of forecasting quality, but the best results were obtained with bell-wise and Gaussian MF.


In [7, 8] the generalization of fuzzy GMDH for case when input data are also fuzzy was considered. Then a linear interval regression model takes the following form:

$$Y = A\_0 Z\_0 + A\_1 Z\_1 + \dots + A\_n Z\_n$$


1 Neural network constructed with Neural Networks Toolbox 4.0.6 (MathWorks). 2 Neural network constructed with Alyuda Forecaster 1.6 (Alyuda Research).

#### Table 1.

Forecasting accuracy (MSE) for different forecasting methods.

Consider the case of symmetrical membership function for parameters Ai, so they can be described by the pair of parameters (ai, ci), where

Ai ¼ ai � ci, Ai ¼ ai þ ci, ci is the interval width, ci ≥ 0,

and Zi is input variable which is also a fuzzy number of triangular shape, defined by three parameters Zi; <sup>Z</sup><sup>i</sup> ; Zi , where Zi is a lower border, <sup>Z</sup><sup>i</sup> is a center, and Zi is an upper border of fuzzy number.

It was shown that corresponding model is also LP problem, and corresponding algorithm FGMDH was developed for such case [7, 8].
