6. Adaptation of fuzzy GMDH models

While forecasting by self-organizing methods (fuzzy GMDH, in particular), the problem of adaptation arises in the case of the training sample size increase when it's needed to correct the obtained model in accordance with new available data. Taking into account new information obtained while forecasting adaptation may be done by two approaches. The first one is to correct parameters of a forecasting model with new data assuming that model structure didn't change. The second approach consists in adaptation of not only model parameters but its optimal structure as well. This way demands the repetitive use of full GMDH algorithm and is connected with huge volume of calculations.

The second approach is used if adaptation of parameters doesn't provide good forecast and the new real output values don't drop in the calculated interval for its estimate.

In our work the first approach is used based on adaptation of FGMDH model parameters with new available data. Here the recursive identification methods are preferably used, especially the recursive LSM. In this method the parameter estimates at the next step are determined on the base of estimates at the previous step, model error, and some information matrix which is modified during all estimation process and therefore contains data which may be used at the next steps of adaptation process [5].

Hence, model coefficient adaptation will be simplified substantially. If we store information matrix obtained while identification of optimal model using fuzzy GMDH, then for model parameters adaptation, it will be enough to fulfill only one iteration by recursive LSM method.

#### 6.1 The application of recurrent LSM for model coefficients adaptation

Consider the following model:

$$\boldsymbol{y}(k) = \boldsymbol{\theta}^T \boldsymbol{\Psi}(k) + \boldsymbol{\nu}(k) \tag{32}$$

where y kð Þ is a dependent (output) variable, Ψð Þk is a measurement vector, v kð Þ are random disturbances, and θ is a parameter vector to be estimated.

The parameters estimate θ at the step N is performed due to such formula [5, 6]:

$$
\widehat{\boldsymbol{\theta}}^{\cdot}(\mathbf{N}) = \widehat{\boldsymbol{\theta}}^{\cdot}(\mathbf{N} - \mathbf{1}) + \boldsymbol{\eta}(\mathbf{N}) \left[ \boldsymbol{\eta}(\mathbf{N}) - \widehat{\boldsymbol{\theta}}^{\cdot} \boldsymbol{T}(\mathbf{N} - \mathbf{1}) \boldsymbol{\Psi}(\mathbf{N}) \right] \tag{33}
$$

where γð Þ N is a coefficient vector which is determined by formula

$$\gamma(N) = \frac{P(N-1)\Psi(N)}{1 + \Psi^T(N)P(N-1)\Psi(N)}\tag{34}$$

where P Nð Þ � 1 is so-called an information matrix, determined by formula

$$P(N-1) = P(N-2) - \frac{P(N-2)\Psi(N-1)\Psi^T(N-1)P(N-2)}{1 + \Psi^T(N-1)P(N-2)\Psi(N-1)}\tag{35}$$

As one can easily see in (35), the information matrix may be obtained independent on parameter estimation process and parallel to it. The adaptation of two parameter vectors θ<sup>T</sup> <sup>1</sup> <sup>¼</sup> ½ � <sup>α</sup>1; …; <sup>α</sup><sup>m</sup> ; <sup>θ</sup><sup>T</sup> <sup>2</sup> ¼ ½ � C1; …;Cm ; is performed in such a way using the formulas (35)

$$\begin{aligned} \widehat{\boldsymbol{\theta}}\_{1}(\boldsymbol{N}) &= \widehat{\boldsymbol{\theta}}\_{1}(\boldsymbol{N}-\mathbf{1}) + \boldsymbol{\gamma}\_{1}(\boldsymbol{N}) \left[ \boldsymbol{\eta}(\boldsymbol{N}) - \widehat{\boldsymbol{\theta}}\_{1}^{T}(\boldsymbol{N}-\mathbf{1}) \boldsymbol{\Psi}\_{1}(\boldsymbol{N}) \right] \\ \widehat{\boldsymbol{\theta}}\_{2}(\boldsymbol{N}) &= \widehat{\boldsymbol{\theta}}\_{2}(\boldsymbol{N}-\mathbf{1}) + \boldsymbol{\gamma}\_{2}(\boldsymbol{N}) \left[ \boldsymbol{\eta}\_{c}(\boldsymbol{N}) - \widehat{\boldsymbol{\theta}}\_{2}^{T}(\boldsymbol{N}-\mathbf{1}) \boldsymbol{\Psi}\_{2}(\boldsymbol{N}) \right] \\ \boldsymbol{\eta}\_{c}(\boldsymbol{N}) &= \boldsymbol{\psi}(\boldsymbol{N}) - \boldsymbol{\theta}\_{1}^{T}(\boldsymbol{N}-\mathbf{1}) \boldsymbol{\Psi}\_{1}(\boldsymbol{N}) \end{aligned} \tag{36}$$

where Ψ<sup>T</sup> <sup>1</sup> <sup>¼</sup> ½ � <sup>z</sup>1; …; zm ; <sup>Ψ</sup><sup>T</sup> <sup>2</sup> ¼ j½ � z1j; …; jzmj .

### 7. Experimental investigations of FGMDH in forecasting

The goal of experiments was the forecasting of macroeconomic indicators of Ukraine and estimating of efficiency of suggested FGMDH. In experiments, the database was utilized which contains monthly values of 24 macroeconomic indicators of Ukrainian economy, since July 1995 till 2013. As forecasting variables consumer price index (CPI) and gross national product (GNP) were chosen.

While constructing forecasting models, the technique of sliding window was utilized, whose size was determined automatically by regression analysis. For determination of input variables significant for forecasting, the methods of regression analysis were also used.

The following experiments were performed:

