2.2 FNN TSK model and hybrid training algorithm

For forecasting of banks bankruptcy risk, the application of fuzzy neural networks (FNN) ANFIS and TSK was suggested [3]. The application of FNN is determined by following reasons:

the capability to work with incomplete and unreliable information under uncertainty; and

the capability to use expert information in the form of fuzzy inference rules.

Let us consider the mathematical model and training algorithm of a fuzzy neural network TSK (Takagi, Sugeno, Kang'a), which is generalization of the neural network ANFIS. The rule base of FNN TSK with M rules and N variables can be written as follows [3]:

Banks Financial State Analysis and Bankruptcy Risk Forecasting with Application of Fuzzy… DOI: http://dx.doi.org/10.5772/intechopen.82534

$$\begin{aligned} R\_1: \text{if } \boldsymbol{\pi\_1} \in A\_1^{(1)}, \boldsymbol{\pi\_2} \in A\_2^{(1)}, \dots, \boldsymbol{\pi\_n} \in A\_n^{(1)} \text{ then } \boldsymbol{\nu\_1} = \boldsymbol{p\_{10}} + \sum\_{j=1}^N \boldsymbol{p\_{j\uparrow}} \boldsymbol{\pi\_j}; \\\\ R\_M: \text{if } \boldsymbol{\pi\_1} \in A\_1^{(M)}, \boldsymbol{\pi\_2} \in A\_2^{(M)}, \dots, \boldsymbol{\pi\_n} \in A\_n^{(M)} \text{ then } \boldsymbol{\nu\_M} = \boldsymbol{p\_{M0}} + \sum\_{j=1}^N \boldsymbol{p\_{Mj}} \boldsymbol{\pi\_j}; \end{aligned}$$

where Að Þ<sup>k</sup> <sup>i</sup> is the value of linguistic variable xi for the rule Rk with membership function (MF) of the form

$$\mu\_A^{(k)}(\mathbf{x}\_i) = \frac{\mathbf{1}}{\mathbf{1} + \left(\frac{\boldsymbol{\kappa}\_i - \boldsymbol{\varepsilon}\_i^{(k)}}{\sigma\_i^{(k)}}\right)^{2b\_i^{(k)}}} \tag{1}$$
 
$$i = \overline{\mathbf{1}, N}; k = \overline{\mathbf{1}, M}.$$

At the intersection of the TSK network rule conditions, Rk MF is defined as a product

$$\mu\_A^{(k)}(\mathbf{x}) = \prod\_{j=1}^N \left[ \frac{\mathbf{1}}{\mathbf{1} + \left( \frac{\mathbf{x}\_j - \mathbf{c}\_j^{(k)}}{\sigma\_j^{(k)}} \right)^{2b\_j^{(k)}}} \right]. \tag{2}$$

With M inference rules, the general output of FNN TSK is determined by the following formula:

$$\boldsymbol{\chi}(\boldsymbol{x}) = \frac{\sum\_{k=1}^{M} w\_{k} \boldsymbol{y}\_{k}(\boldsymbol{x})}{\sum\_{k=1}^{M} w\_{k}},\tag{3}$$

where ykð Þ¼ <sup>x</sup> pk<sup>0</sup> <sup>þ</sup> <sup>∑</sup><sup>N</sup> <sup>j</sup>¼<sup>1</sup> pkjxj. The weights in this expression are interpreted as the degrees of fulfillment of rule antecedents (conditions): wk <sup>¼</sup> <sup>μ</sup>ð Þ<sup>k</sup> <sup>A</sup> ð Þ x , which are given by (2).

The fuzzy neural network TSK, which implements the output in accordance

with (3), represents a multilayer network whose structure is shown in Figure 1. This network has five layers with the following functions:

	- xi, i <sup>¼</sup> <sup>1</sup>, <sup>2</sup>, …, N, defining for each rule the k value MF <sup>μ</sup>ð Þ<sup>k</sup> <sup>A</sup> ð Þ xi in accordance with the fuzzification function, which is described, for example, by Gaussian or bell-wise function. This is a parametric layer with parameters c ð Þk <sup>j</sup> , <sup>σ</sup>ð Þ<sup>k</sup> <sup>j</sup> , bð Þ<sup>k</sup> j , which are subject to adjustment in the learning process.

and estimate novel methods of bank financial state analysis and bankruptcy risk forecasting under uncertainty and compare with classical methods. The implementation and assessment of the efficiency of the suggested methods are performed at the problems of bankruptcy risk forecasting for Ukrainian and European banks.

Accounting and Finance - New Perspectives on Banking, Financial Statements and Reporting

As it is well known, the year 2008 was the crucial year for the bank system of Ukraine. If the first three quarters were periods of fast growth and expansion, the last quarter became the period of collapse in the financial sphere. A lot of Ukrainian

For this research, the quarterly accountancy bank reports used were obtained from National bank of Ukraine site. For analysis, the financial indices of 170 Ukrainian banks were taken up to the date January 01, 2008 and July 01, 2009, that is,

The important problem that occurred before the start of the investigations is which financial indices are to be used for better forecasting of possible bankruptcy. Thus, another goal of this exploration was to detect the most relevant financial

For analysis, the following indicators of banks accountancy were considered:

physical person's entities, juridical person's entities, liabilities, and net incomes

The collected indicators were used for analysis by fuzzy neural networks as well as classic statistical methods. As output data of models for Ukrainian banks were

1, if the significant worsening of bank financial state is not expected in the

For forecasting of banks bankruptcy risk, the application of fuzzy neural networks (FNN) ANFIS and TSK was suggested [3]. The application of FNN is deter-

the capability to work with incomplete and unreliable information under

the capability to use expert information in the form of fuzzy inference rules.

Let us consider the mathematical model and training algorithm of a fuzzy neural network TSK (Takagi, Sugeno, Kang'a), which is generalization of the neural network ANFIS. The rule base of FNN TSK with M rules and N variables can be written

2. Bankruptcy risk forecasting of Ukrainian banks

about two years before crises and just before the start of crises [2].

indicators for obtaining maximal accuracy of forecasting.

assets, capital, financial means, and their equivalents; and

1, if the bank bankruptcy is expected in the nearest future.

2.2 FNN TSK model and hybrid training algorithm

2.1 Problem statement

(losses).

two values:

nearest future

mined by following reasons:

uncertainty; and

as follows [3]:

82

banks faced the danger of coming default.


$$\mathcal{Y}(\mathbf{x}) = \frac{f\_1}{f\_2} = \frac{\sum\_{k=1}^{M} w\_k \mathcal{Y}\_k(\mathbf{x})}{\sum\_{k=1}^{M} w\_k} \tag{4}$$

In the presence of N input variables, each rule Rk formulates ð Þ N þ 1 variable

� � are fixed, e.g., σð Þ<sup>k</sup>

Considering a hybrid learning algorithm which is used for FNN TSK, all parameters can be divided into two groups. The first group includes linear parameters pkj of the third layer, and the second group includes nonlinear parameters (MF) of the

<sup>k</sup> pk<sup>0</sup> þ ∑

With the dimension L of training sample <sup>x</sup>ð Þ<sup>l</sup> ; <sup>d</sup>ð Þ<sup>l</sup> � �, lð Þ <sup>¼</sup> <sup>1</sup>; <sup>2</sup>; …; <sup>L</sup> and replace-

N j¼1

!

pkjxj:

� � , k <sup>¼</sup> <sup>1</sup>, M: (7)

, we get a system of L linear

p<sup>10</sup>

¼

dð Þ<sup>1</sup> dð Þ<sup>2</sup> … dð Þ <sup>L</sup>

(8)

p11 …

p1<sup>N</sup> … pM<sup>0</sup> pM<sup>1</sup> … pMN (6)

In the first stage after fixing the individual parameters of the membership function by solving a system of linear equations, linear parameters of polynomial pkj are calculated. With the known values of MF dependence, input-output can be

<sup>j</sup> and <sup>b</sup>ð Þ<sup>k</sup> j .

<sup>j</sup> of linear dependence ykð Þ x . If M inference rules are present, then M Nð Þ þ 1 linear network parameters are obtained. In turn, each MF uses three parameters ð Þ c; σ; b , which are subject to adjustment. With M inference rules, three MN nonlinear parameters are obtained. In total, this gives Mð Þ 4N þ 1 linear and nonlinear parameters that must be determined in the learning process. This is a very large value. In order to reduce the number of parameters for adaptation, we operate

Banks Financial State Analysis and Bankruptcy Risk Forecasting with Application of Fuzzy…

with fewer number of MF. In particular, it can be assumed that some of the

<sup>A</sup> xj

2.2.1 Hybrid learning algorithm for fuzzy neural networks

represented as a linear form with respect to the parameters pkj:

ykð Þ¼ x ∑

w0 <sup>k</sup> ¼

ment of the network output by expected value dð Þ<sup>l</sup>

11xð Þ<sup>1</sup>

21xð Þ<sup>2</sup>

Lxð Þ <sup>L</sup> <sup>N</sup> w<sup>0</sup>

vector x<sup>ℓ</sup>. This expression can be written in matrix form:

<sup>N</sup> :… w<sup>0</sup>

<sup>N</sup> … w<sup>0</sup>

…… … … … …

<sup>1</sup><sup>M</sup> w<sup>0</sup>

<sup>2</sup><sup>M</sup> w<sup>0</sup>

LM w<sup>0</sup>

1Mxð Þ<sup>1</sup>

2Mxð Þ<sup>2</sup>

LMxð Þ <sup>L</sup> <sup>1</sup> w<sup>0</sup>

<sup>1</sup> … w<sup>0</sup>

<sup>1</sup> … w<sup>0</sup>

<sup>ℓ</sup><sup>i</sup> means normalized weight of the i-th rule at presentation of ℓ-th input

1Mxð Þ<sup>1</sup> N

2Mxð Þ<sup>2</sup> N

LMxð Þ <sup>L</sup> N

M k¼1 w0

Q<sup>N</sup> <sup>j</sup>¼<sup>1</sup> <sup>μ</sup>ð Þ<sup>k</sup> <sup>A</sup> xj � �

∑<sup>M</sup> r¼1 Q<sup>N</sup> <sup>j</sup>¼<sup>1</sup> <sup>μ</sup>ð Þ<sup>r</sup> <sup>A</sup> xj

pð Þ<sup>k</sup>

where

equations of the form

11xð Þ<sup>1</sup> <sup>1</sup> … w<sup>0</sup>

21xð Þ<sup>2</sup>

L1xð Þ <sup>L</sup>

<sup>1</sup> … w<sup>0</sup>

<sup>1</sup> w<sup>0</sup>

w0 <sup>11</sup> w<sup>0</sup>

w0 <sup>21</sup> w<sup>0</sup>

w0 <sup>L</sup><sup>1</sup> w<sup>0</sup>

where w<sup>0</sup>

85

parameters of one function MF μð Þ<sup>k</sup>

DOI: http://dx.doi.org/10.5772/intechopen.82534

first layer. Adaptation occurs in two stages.

This is also nonparametric layer.

From this description follows that TSK fuzzy network contains only two parametric layers: first and third, the parameters of which are determined in the training process. Parameters of the first layer c ð Þk <sup>j</sup> ; <sup>σ</sup>ð Þ<sup>k</sup> <sup>j</sup> ; <sup>b</sup>ð Þ<sup>k</sup> j � �, we call nonlinear, and the parameters of the third layer pkj n o—linear weights. The general expression for the functional dependence (4) for the network TSK is defined as follows:

$$\mu\_{\mathcal{N}}(\boldsymbol{\kappa}) = \frac{1}{\sum\_{i=1}^{M} \prod\_{s=1}^{N} \mu\_{\mathcal{A}\_s}^{(s)}(\boldsymbol{x}\_{s})} \sum\_{i=1}^{M} \left(\mu\_{\boldsymbol{\kappa}\boldsymbol{\alpha}} + \sum\_{j=1}^{N} \mu\_{\boldsymbol{\kappa}\boldsymbol{\beta}} \boldsymbol{x}\_{j}\right) \prod\_{j=1}^{N} \mu\_{\mathcal{A}\_s}^{(s)}(\boldsymbol{x}\_{s})$$

If we assume that at any given time moment, the nonlinear parameters are fixed, then the function y xð Þ would be linear with respect to the variable xj:

Figure 1. The structure of TSK fuzzy neural network.

Banks Financial State Analysis and Bankruptcy Risk Forecasting with Application of Fuzzy… DOI: http://dx.doi.org/10.5772/intechopen.82534

In the presence of N input variables, each rule Rk formulates ð Þ N þ 1 variable pð Þ<sup>k</sup> <sup>j</sup> of linear dependence ykð Þ x . If M inference rules are present, then M Nð Þ þ 1 linear network parameters are obtained. In turn, each MF uses three parameters ð Þ c; σ; b , which are subject to adjustment. With M inference rules, three MN nonlinear parameters are obtained. In total, this gives Mð Þ 4N þ 1 linear and nonlinear parameters that must be determined in the learning process. This is a very large value. In order to reduce the number of parameters for adaptation, we operate with fewer number of MF. In particular, it can be assumed that some of the parameters of one function MF μð Þ<sup>k</sup> <sup>A</sup> xj � � are fixed, e.g., σð Þ<sup>k</sup> <sup>j</sup> and <sup>b</sup>ð Þ<sup>k</sup> j .

## 2.2.1 Hybrid learning algorithm for fuzzy neural networks

Considering a hybrid learning algorithm which is used for FNN TSK, all parameters can be divided into two groups. The first group includes linear parameters pkj of the third layer, and the second group includes nonlinear parameters (MF) of the first layer. Adaptation occurs in two stages.

In the first stage after fixing the individual parameters of the membership function by solving a system of linear equations, linear parameters of polynomial pkj are calculated. With the known values of MF dependence, input-output can be represented as a linear form with respect to the parameters pkj:

$$\mathcal{Y}\_k(\mathbf{x}) = \sum\_{k=1}^{M} w\_k' \left( p\_{k0} + \sum\_{j=1}^{N} p\_{kj} \mathbf{x}\_{j\cdot} \right) \tag{6}$$

where

4.The fourth layer consists of two summing neurons, one of which calculates the weighted sum of the signals ykð Þ x , and the second one calculates the sum

Accounting and Finance - New Perspectives on Banking, Financial Statements and Reporting

5. The fifth layer is composed of a single output neuron. In it, weight normalizing is performed and the output signal determined in accordance with the

<sup>¼</sup> <sup>∑</sup><sup>M</sup>

From this description follows that TSK fuzzy network contains only two parametric layers: first and third, the parameters of which are determined in the training

If we assume that at any given time moment, the nonlinear parameters are fixed,

ð Þk <sup>j</sup> ; <sup>σ</sup>ð Þ<sup>k</sup> <sup>j</sup> ; <sup>b</sup>ð Þ<sup>k</sup> j � �

<sup>k</sup>¼<sup>1</sup>wkykð Þ <sup>x</sup> ∑<sup>M</sup> <sup>k</sup>¼<sup>1</sup>wk

(4)

, we call nonlinear, and the

—linear weights. The general expression for the

y xð Þ¼ <sup>f</sup> <sup>1</sup> f 2

n o

functional dependence (4) for the network TSK is defined as follows:

then the function y xð Þ would be linear with respect to the variable xj:

of the weights ∑<sup>M</sup>

expression:

Figure 1.

84

The structure of TSK fuzzy neural network.

<sup>k</sup>¼<sup>1</sup>wk.

This is also nonparametric layer.

process. Parameters of the first layer c

parameters of the third layer pkj

$$w\_k' = \frac{\prod\_{j=1}^N \mu\_A^{(k)}(\mathbf{x}\_j)}{\sum\_{r=1}^M \prod\_{j=1}^N \mu\_A^{(r)}(\mathbf{x}\_j)}, k = \overline{\mathbf{1}, M}. \tag{7}$$

With the dimension L of training sample <sup>x</sup>ð Þ<sup>l</sup> ; <sup>d</sup>ð Þ<sup>l</sup> � �, lð Þ <sup>¼</sup> <sup>1</sup>; <sup>2</sup>; …; <sup>L</sup> and replacement of the network output by expected value dð Þ<sup>l</sup> , we get a system of L linear equations of the form

$$
\begin{bmatrix}
w'\_{11} & w'\_{11}\mathbf{x}\_1^{(1)} \dots & w'\_{11}\mathbf{x}\_N^{(1)} \dots & w'\_{1M}\mathbf{w}'\_{M\mathbf{x}}\mathbf{x}\_1^{(1)} \dots & w'\_{1M}\mathbf{x}\_N^{(1)} \\
w'\_{21} & w'\_{21}\mathbf{x}\_1^{(2)} \dots & w'\_{21}\mathbf{x}\_N^{(2)} \dots & w'\_{2M}\mathbf{w}'\_{M\mathbf{x}}\mathbf{x}\_1^{(2)} \dots & w'\_{2M}\mathbf{x}\_N^{(2)} \\
\dots & \dots & \dots & \dots & \dots \\
w'\_{L1} & w'\_{L1}\mathbf{x}\_1^{(L)} & w'\_{L}\mathbf{x}\_N^{(L)} & w'\_{L M}\mathbf{w}'\_{L1}\mathbf{x}\_1^{(L)} & w'\_{L M}\mathbf{x}\_N^{(L)} \end{bmatrix} \times \begin{bmatrix}
p\_{10} \\
p\_{11} \\
\dots \\
p\_{1N} \\
\dots \\
p\_{M0} \\
p\_{M1} \\
p\_{M1} \\
p\_{M1} \\
p\_{M\mathbf{M}} \end{bmatrix} = \begin{bmatrix}
d^{(1)} \\
d^{(2)} \\
\dots \\
d^{(L)} \\
\dots \\
d^{(L)} \\
\end{bmatrix} \tag{8}
$$

where w<sup>0</sup> <sup>ℓ</sup><sup>i</sup> means normalized weight of the i-th rule at presentation of ℓ-th input vector x<sup>ℓ</sup>. This expression can be written in matrix form:

$$Ap = d.$$

Matrix A dimension is equal to L Nð Þ þ 1 M: By thus, a number of rows L usually is much greater than a number of columns ð Þ N þ 1 M: The solution of this equations system may be obtained by conventional methods as well as using pseudoinverse matrix A at one step:

$$p = A^{+}d,$$

where A<sup>þ</sup> is a pseudoinverse matrix for matrix A.

In the second stage, after fixing the values of linear parameters pkj, the actual output signals <sup>y</sup>ð Þ <sup>ℓ</sup> , <sup>ℓ</sup> <sup>¼</sup> <sup>1</sup>, <sup>2</sup>, …, L are determined using a linear equations system:

$$\mathbf{y}^{(L)} = \mathbf{A}\mathbf{p}.\tag{9}$$

∂E ∂c ð Þk j

DOI: http://dx.doi.org/10.5772/intechopen.82534

∂E ∂σð Þ<sup>k</sup> j

∂E ∂bð Þ<sup>k</sup> j

should be repeated many times in each cycle.

influence on forecasting accuracy were performed.

ruptcy was forecasted at the beginning of 2010.

risk forecasting are presented below.

forecasting

Experiment No. 1:

87

Number of rules = 5.

¼ ð Þ y xð Þ� d ∑

¼ ð Þ y xð Þ� d ∑

¼ ð Þ y xð Þ� d ∑

efficient in comparison with conventional gradient-based methods.

3. The application of fuzzy neural networks for financial state

Training sample—120 Ukrainian banks, test sample—50 banks.

Input data—financial indices (taken from bank accountant reports):

The results of application of FNN TSK are presented in Table 1.

assets, capital, cash (liquid assets), households deposits, liabilities.

A special software kit was developed for FNN ANFIS and TSK application in bankruptcy risk forecasting problems. As input data, the financial indicators of Ukrainian banks in financial accountant reports were used in the period of

2008–2009 [2] . As the output values were used +1, for bank nonbankrupt and �1, for bank bankrupt. In the investigations, various financial indicators were analyzed, and different number of rules for FNN and the analysis of data collection period

The results of experimental investigations of FNN application for bankruptcy

In the first series of experiments, input data at the period of January 2008 were used (that is for two years before possible bankruptcy) and possible banks bank-

M r¼1

Banks Financial State Analysis and Bankruptcy Risk Forecasting with Application of Fuzzy…

M r¼1

M r¼1

In the practice of the hybrid learning method implementation, the dominant factor in adaptation is considered to be the first stage in which weights pkj are determined using pseudoinverse in one step. To balance its impact, the second stage

It is worth to note that the described hybrid algorithm is one of the most effective ways of training fuzzy neural networks. Its principal feature is the division of the process into two stages separated in time. Since the computational complexity of each nonlinear optimization algorithm depends nonlinearly on the number of parameters subject to optimization, the reduction in the dimensions of optimization significantly reduces the total amount of calculations and increases the speed of convergence of the algorithm. Due to this, hybrid algorithm is one of the most

pr<sup>0</sup> þ ∑ N j¼1 prjxj

pr<sup>0</sup> þ ∑ N j¼1 prjxj

pr<sup>0</sup> þ ∑ N j¼1 prjxj

!

!

!

� ∂w<sup>0</sup> r ∂c ð Þk j

� ∂w0 r ∂σð Þ<sup>k</sup> j

� ∂w0 r ∂bð Þ<sup>k</sup> j

(14)

(15)

(16)

Then, the error vector ε ¼ ð Þ y � d and the criterion E are calculated:

$$E = \frac{1}{2} \sum\_{\ell=1}^{L} \left( \mathbf{y} \left( \mathbf{x}^{(\ell)} \right) - d^{(\ell)} \right)^2. \tag{10}$$

The error signals are sent through the network backward according to the method of back propagation until the first layer, at which gradient vector components of the objective function with respect to parameters c ð Þk <sup>j</sup> ; <sup>σ</sup>ð Þ<sup>k</sup> <sup>j</sup> ; <sup>b</sup>ð Þ<sup>k</sup> j are calculated.

After calculating the gradient vector, a step of gradient descent method is made. The corresponding formulas (for the simplest method of the steepest descent) are the following:

$$c\_j^{(k)}(n+1) = c\_j^{(k)}(n) - \eta\_c \frac{\partial E(n)}{\partial c\_j^{(k)}} \tag{11}$$

$$
\sigma\_j^{(k)}(n+1) = \sigma\_j^{(k)}(n) - \eta\_\sigma \frac{\partial E(n)}{\partial \sigma\_j^{(k)}} \tag{12}
$$

$$b\_j^{(k)}(n+1) = b\_j^{(k)}(n) - \eta\_b \frac{\partial E(n)}{\partial b\_j^{(k)}} \tag{13}$$

where n is a number of iterations.

After verifying the nonlinear parameters, the process of adaptation of linear parameters TSK (first phase) restarts and nonlinear parameters are further adapted (second stage). This cycle continues until all the parameters will be stabilized.

Formulas (11)–(13) require the calculation of the gradient of the objective function with respect to the parameters of the MF. The final form of these formulas depends on the type of MF. For example, if using the generalized bell-wise functions:

$$\mu\_A(\mathbf{x}) = \frac{1}{1 + \left(\frac{\mathbf{x} - \mathbf{c}}{\sigma}\right)^{2b}}$$

the corresponding formulas for gradient of the objective function for one pair of data ð Þ x; d take the form [3]:

Banks Financial State Analysis and Bankruptcy Risk Forecasting with Application of Fuzzy… DOI: http://dx.doi.org/10.5772/intechopen.82534

$$\frac{\partial E}{\partial c\_j^{(k)}} = (\mathcal{y}(\mathbf{x}) - d) \sum\_{r=1}^{M} \left( p\_{r0} + \sum\_{j=1}^{N} p\_{rj} \mathbf{x}\_j \right) \cdot \frac{\partial w\_r'}{\partial c\_j^{(k)}} \tag{14}$$

$$\frac{\partial E}{\partial \sigma\_j^{(k)}} = (\boldsymbol{\jmath}(\boldsymbol{\varkappa}) - \boldsymbol{d}) \sum\_{r=1}^{M} \left( p\_{r0} + \sum\_{j=1}^{N} p\_{rj} \boldsymbol{\varkappa}\_j \right) \cdot \frac{\partial \boldsymbol{w}\_r'}{\partial \sigma\_j^{(k)}} \tag{15}$$

$$\frac{\partial E}{\partial b\_j^{(k)}} = (\boldsymbol{\jmath}(\boldsymbol{\varkappa}) - \boldsymbol{d}) \sum\_{r=1}^{M} \left( p\_{r0} + \sum\_{j=1}^{N} p\_{rj} \boldsymbol{\varkappa}\_j \right) \cdot \frac{\partial \boldsymbol{w}\_r'}{\partial b\_j^{(k)}} \tag{16}$$

In the practice of the hybrid learning method implementation, the dominant factor in adaptation is considered to be the first stage in which weights pkj are determined using pseudoinverse in one step. To balance its impact, the second stage should be repeated many times in each cycle.

It is worth to note that the described hybrid algorithm is one of the most effective ways of training fuzzy neural networks. Its principal feature is the division of the process into two stages separated in time. Since the computational complexity of each nonlinear optimization algorithm depends nonlinearly on the number of parameters subject to optimization, the reduction in the dimensions of optimization significantly reduces the total amount of calculations and increases the speed of convergence of the algorithm. Due to this, hybrid algorithm is one of the most efficient in comparison with conventional gradient-based methods.
