1. Introduction

The problems of banks financial state analysis and bankruptcy risk forecasting are of great importance. The opportune discovery of coming bankruptcy allows top bank managers to make urgent decisions for preventing the bankruptcy. Nowadays, there are a lot of methods and techniques of banks state analysis and determination of bank rating—WEB Money, CAMEL [1], Moody's S&P, etc. But their common drawback is that all of them work with complete and reliable data and cannot give correct results in case of incomplete and unreliable input data. This is especially actual for the Ukrainian banking system where bank managers often provide the incorrect reports about bank financial state to obtain new credits and loans.

Therefore, it is very important to create new methods for banks bankruptcy risk forecasting under uncertainty. The main goal of present investigation is to consider 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.

R<sup>1</sup> : if x<sup>1</sup> ∈ Að Þ<sup>1</sup>

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

RM : if x<sup>1</sup> ∈ Að Þ <sup>M</sup>

where Að Þ<sup>k</sup>

product

following formula:

given by (2).

83

where ykð Þ¼ <sup>x</sup> pk<sup>0</sup> <sup>þ</sup> <sup>∑</sup><sup>N</sup>

This is not a parametric layer.

calculated ykð Þ¼ <sup>x</sup> pk<sup>0</sup> <sup>þ</sup> <sup>∑</sup><sup>N</sup>

function (MF) of the form

<sup>1</sup> , x<sup>2</sup> <sup>∈</sup> <sup>A</sup>ð Þ<sup>1</sup>

<sup>1</sup> , x<sup>2</sup> <sup>∈</sup> <sup>A</sup>ð Þ <sup>M</sup>

μð Þ<sup>k</sup> <sup>A</sup> ð Þ¼ xi

μ ð Þk <sup>A</sup> ð Þ¼ <sup>x</sup> <sup>Y</sup>

<sup>2</sup> , …, xn ∈ Að Þ<sup>1</sup>

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

<sup>2</sup> , …, xn ∈ Að Þ <sup>M</sup>

<sup>i</sup> is the value of linguistic variable xi for the rule Rk with membership

1

ð Þk i σð Þ<sup>k</sup> i � �2bi

1

ð Þk j σð Þ<sup>k</sup> j � �<sup>2</sup>bj

ð Þk

<sup>j</sup>¼<sup>1</sup> pkjxj. The weights in this expression are interpreted as

<sup>1</sup> <sup>þ</sup> xj�<sup>c</sup>

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

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

The fuzzy neural network TSK, which implements the output in accordance with (3), represents a multilayer network whose structure is shown in Figure 1.

with the fuzzification function, which is described, for example, by Gaussian

<sup>1</sup> <sup>þ</sup> xi�<sup>c</sup>

i ¼ 1, N; k ¼ 1, M: At the intersection of the TSK network rule conditions, Rk MF is defined as a

N

j¼1

y xð Þ¼ <sup>∑</sup><sup>M</sup>

the degrees of fulfillment of rule antecedents (conditions): wk <sup>¼</sup> <sup>μ</sup>ð Þ<sup>k</sup>

This network has five layers with the following functions:

which are subject to adjustment in the learning process.

determining the resulting degree of membership wk ¼ μ

carried out determining the rules output functions.

1. The first layer performs fuzzification separately for each variable 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>

or bell-wise function. This is a parametric layer with parameters c

2. The second layer performs the aggregation of individual variables xi,

3. The third layer is a function generator TSK, wherein the output values are

previous layer ykð Þ x on wk are multiplied. This is a parametric layer, wherein the adaptation of linear parameters (weight) pk0, pkj for j ¼ 1,N,k ¼ 1,M, is

<sup>n</sup> then y<sup>1</sup> ¼ p<sup>10</sup> þ ∑

<sup>n</sup> then yM ¼ pM<sup>0</sup> þ ∑

N j¼1 p1j xj;

> N j¼1

ð Þ<sup>k</sup> (1)

: (2)

, (3)

<sup>A</sup> ð Þ x , which are

<sup>A</sup> ð Þ xi in accordance

ð Þk <sup>j</sup> , <sup>σ</sup>ð Þ<sup>k</sup> <sup>j</sup> , bð Þ<sup>k</sup> j ,

<sup>A</sup> ð Þ x for the vector x.

ð Þk

<sup>j</sup>¼<sup>1</sup> pkjxj: At this layer, also functions formed in the

pMjxj,
