An Enterprise Financial Evaluation Model Based

An Enterprise Financial Evaluation Model Based on Rough Set theory with Information Entropy Ming-Chang Lee

An Enterprise Financial Evaluation Model Based on Rough Set theory with Information Entropy Ming-Chang Lee Department of Information Management, Fooyin University, Taiwan [email protected] doi: 10.4156/jdcta.vol3.issue1.lee

Abstract Enterprise financial evaluation is an important issue. Rough set is a new technique for data mining domain application. Information Entropy, as a measurement of the average amount of information contained in an information system, is used in the classification of objectives and the analysis of information systems. In this study, establish a financial evaluation model using rough Information entropy approach. Therefore, we firstly define the attribute reduction based on rough set theory. Secondly, we define information entropy. Using this definition, we can easily find the important of attributes. Thirdly, we discuss the reducible indiscernible matrix and decision rule generation. The empirical research which is based on the latest data of Taiwan’s listed company, the result shows that this method is high accurate. Using the unmatched and unbalanced training data and test data, rough set theory with information entropy, shows the best overall prediction accuracy level at 83.4%.

Keywords Rough set theory, Information entropy, Enterprise Financial Evaluation

1. Introduction Enterprise financial evaluation has been a focal point of issue in financial analysis. Use of financial data or financial ratio to evaluate enterprise financial distress/failure has been the major methodology for this research topic (Odom and Sharda, 1990; Tam and Kiang, 1992; Coats and Fant, 1993; Altman et al., 1994; Koh and Tan, 1999; Shin et al., 2005). Recently, the theory of rough sets has emerged as another method for dealing with uncertainty using from inexact or incomplete information (Pawlak, 1991; Pawlak, 1982). Rough sets theory belongs to the family of concepts concerning the modeling and representing of

incomplete knowledge (Pawlak, 1984). Hu, et al. (2003) presented a new rough set model based on database systems. Wong and Ziako (1987) presented the probabilistic rough set model, and Wei and Zhang (2003) studied fuzziness in probabilistic rough sets using fuzzy sets. The entropy used in thermodynamics is more or less closely related to the concept of information as used in communication theory (Shannon, 1948). Information Entropy, as a measurement of the average amount of information contained in an information system, is used in the classification of objectives and the analysis of information systems. When the information entropy of information system is not equal zero, the set of attributes in the information system is not expressible enough to distinguish objects from each other (Yao, 2003). This paper proposed a Rough set method for enterprise financial model based on information entropy. This approach is divided into three tasks to be fulfilled. (1) Create knowledge system (2) Find the minimal reduction based on rough set theory with information entropy, (3) Decision rule generation, (4) Result and discussion (Shiu et al., 2001). This approach utilizes information gain and important attribute for distinguishing importance of attributes. Then, by applying rough set approach, a decision table can reduced by removing redundant attributes without any information loss. Decision rules can be extracted from the equivalence classes. Finally, by using two error types (TypeＩerror and Type Π error), an accuracy of this approach can be calculated. TypeＩerror refer to the situation when matched data is classified as unmatched one, and Type Π error refer to unmatched data is classified into matched data.

2. The proposed method 2.1 Concept of rough set Let U be a nonempty set, and let R be an

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International Journal of Digital Content Technology and its Applications Volume 3, Number 1, March 2009

indiscernible relation or equivalence relation on U.

called as a decision system or decision table.

(U, R) is called a Pawlak approximation space.

every minimum set B (B ⊂ A) and IND (A) = IND (B),

Let

the concept X be a subset of U, the lower approximation of X in (U, R), denoted as

X , is

B is called a reduce set in knowledge system. For example, from table1, U= {1, 2, 3, 4 },

X = { u | [u] R ⊆ x }

= φ and A = C∪D. x ij is called as the ith conditional

(1)

attribute with jth sample value, y j is called jth sample

And the upper approximation of X in (U, R), denoted as X , is defined as

decision value.

Table 1 Knowledge system

(2)

Conditional attribute

Where [u] R is an equivalence class of R containing Pawlak (1985) regards the group of subsets of U

with the same upper and lower approximations in (U, Let ( X , X ) denote the

R) as a rough set in (U, R). rough set of X.

For example, let (U, R) be a Pawlak

approximation space.

Table 1 is showed the knowledge

system.

X = { u |[u] R ∩ x ≠ φ }

2}, {3, 4}}.

A=

{x 1 , x 2 ,…, x 4 , y}, C = { x 1 , x 2 ,…, x 4 }, D= {y}, C∩D

defined as

u.

For

U = {1, 2, 3, 4} and R = {{1,

Assume that X = {1, 2, 3}.

Let R(X) be

the rough set of X. then R(X) = {{1, 2},{1, 2, 3, 4}} =

Decision attribute

U

x1

x2

x3

X4

y

1 2 3 4

x 11 x 21 x 31 x 41

x 12 x 22 x 32 x 42

x 13 x 23 x 33 x 43

x 14 x 24 x 34 x 44

y1 y2 y3 y4

Step 2: Search minimal reduction Let S = {U, A} be the information system, and U is nonempty set, C is the conditional attribution’s set. {U, A} is in the state of equilibrium, if for x, y ∈ U, if x

{{1, 2, 3},{1, 2, 4}}, where{1, 2}is the lower

≠y then there is at least one a ∈ A such that x(a) ≠

approximation of X, {1, 2, 3, 4} is the upper

y(a).

approximation of X, {{1, 2, 3},{1, 2, 4}}is the family of all sets having {1, 2}and {1, 2, 3, 4}as their lower and upper approximations.

({1, 2}, {1, 2, 3, 4}) is the

rough set of {1, 2, 3}.

2.2

Decision

Definition 1: Information entropy If X = {X 1 , X 2 , …, X n } is a n equivalence relation on U.

H(X) is called as information resource

X’s Information entropy.

attribute

reduce

based

on

information entropy

n

H (X)=-∑ p(X i )(log p(X i ))= i=1

1 n ∑ Xi log Xi n i=1 (3)

Step 1: Create Knowledge system Let S = (U, A, V, f) be a knowledge system with rough set theory, C is called a set of conditional

Where log base on 2, n = U , p (X i ) is the probability of x i , and

n

∑ p( X ) = 1 . i =1

i

attribute, D is called a set of decision attribute and

Definition 2: The information entropy of conditional

C∩D = φ and A = C∪D.

attribution’s set C is denoted H(C).

information function. attribute x i .

f: U×A

V i is an

V i is a range domain of

. C = {X 1 , X 2 , …, X n }

T = (U, A, C, D) be a decision table. T is

17


n

H(C)= -∑ p(X i )log p(X i )

(4)

values from patterns used in equivalence classes.

i=1

Where log based on 2,

p ( xi ) is the

n

An accuracy of this approach can be calculated by

∑ p( x ) = 1

probability of xi , n is U and

i =1

Step 4: Calculate the number of error by using test data

i

using two error types (TypeＩerror and Type Π error).

Definition 3: The conditional information entropy of D with C is denoted as H (D|C).

Where D is decision

attribution’s set and C is conditional attribution’s set. D is {Y 1 , Y 2 , …, Y m }, C is {X 1 ,X 2 , …, X n } n

m

i=1

j=1

H(D|C)=-∑ p(X i )∑ p(Yj |X i )log(Yj |X i )

3.1 Enterprise financial distress index (5)

Where p(Yj |X i ) = Y j ∩ X i / X i

i = 1, 2,…,n

3. Illustration- enterprise financial evaluation

j =1, 2, …,m

In financial reporting analysis, it has five factors for enterprise financial failure (Gibson, 2006). In this study, using rating ability, debt paying ability, earning ability and cash flow are measured attribute. Table 2 is showed as enterprise financial evaluation index.

Definition 4: Let Let S = (U, A, V, f) be a knowledge system with rough theory, C is called a set of

Table 2 Enterprise financial evaluation index

attribute and C∩D = φ and A = C∪D. f: U×U = V i is an information function.

The important measure of

attribute “a” is defined: SGF(a, C, D)= H(D|C) - H(D|C- {a}) Where a ∈ C

(6)

If SGF (a, C, D) > SGF (b, C, D) then attribute “a” is more importance than b in condition C. Definition 5::Important of attribute S(σ) is called the important of attribute is

Enterprise financial Evaluation index

conditional attribute, D is called a set of decision

Earning ability

Rating ability Debt Paying ability Cash flow

X 1 :ROA (Return on total assets) X 2 :ROE (Return on Stockholders equity) X 3 :Profit margin X 4 :`Account receivable turnover ration X 5 :Average collection period X 6 :Inventory turnover ratio X 7 :Average days to sell the Inventory X 8 :Current ratio X 9 :Quick ratio X 10 :Time interest earned X 11 :Cash flow ratio X 12 :Cash flow adequacy ratio

3.2 Knowledge system In order to test and verify rough-set model, it use a

defined as: S(σ) = |H (C) – H (C-{σ})|, ∀ σ ∈ C.

(7)

When S (σ) > 0, it is denoted that σ is need.

securities

firm’s

data

base

in

Taiwan.

45

experimental samples are random select from database

When S (σ)= 0, σ is redundant attribution, that is, σ

and 12 test data.

can leave out from the attribution’s set.

S = (U, A, V, f) be a knowledge system, where U = {1,

Step 3: Decision rule generation After constructed the decision table, then the decision rules will be generate. We may extract decision rules in IF-Then form.

The rules are

generation through union of attributes conditional

2, …, 45}, D= {bad, middle, good, }={1, 2, 3 }, A = {C, D}, C ={ X 1 , X 2 , X 3 , X 4 , X 5 , X 6 , X 7 , X 8 , X 9 , X 10 , X 11 , X 12 }.

Let X i be denoted as order 1, 2, 3,

i.e. X i ∈ {bad, middle, good} = {1, 2, 3}.

3.3 Obtain reducible indiscernible matrix

18


Therefore, {X 1 , X 4 , X 6 , X 9 , X 11 } is said a financial

From (5) (6), the mutual information measure of conditional attribute C and decision attribute D is

evaluation index.

H(D|C)

constructed as Table 3.

=0.6245.

We

have

H(D|C-{X 2 })

=

The equivalence classes are

H(D|C-{X 3 })=H(D|C-{X 5 })=H(D|C-{X 7 }) =H(D|C-{X 8 })=

H(D|C-{X 10 })=

0.645.

SGF (X 2 , C, D) = SGF (X 3 , C, D)=

That is

H(D|C-{X 12 })

Table 3 Indiscernible matrix

=

class 1 2 3 4 5 6 7 8 9

SGF( X 5 , C, D)= SGF( X 7 , C, D)= SGF (X 8 , C, D) = SGF (X 10 , C, D)= SGF X 12 , C, D) = 0, then the attribute X 2 , X 3 , X 5 , X 7 ,X 8 , X 10 , X 12 can be remove in conditional attribute set C. The core of C relative to

Number of sample 5 5 5 6 4 4 5 4 7

X1

X4

X6

X9

X 11

D

1 2 2 1 3 3 3 2 2

2 2 2 2 3 2 2 3 2

2 2 3 2 3 3 3 3 2

1 1 2 2 3 2 3 3 2

1 1 2 1 3 2 3 3 2

bad bad middle bad good middle good good middle

D is CORE D (C) is {X 1 , X 4 , X 6 , X 9 , X 11 }.

3.4 Decision rule generation

From (4) (7), the information entropy of conditional attribution’s set C is H(C) = 0.4559.

We have S (X 1 )

According to Table 4, we build the discerning matrix M = ( mij )9×9 .

=0.254; S (X 2 ) = 0.019; S (X 3 ) = 0.025; S (X 4 ) = 0.321; S (X 5 ) = 0.061;S (X 6 ) = 0.214;

S (X 7 ) =

Definition 6: Discernible matrix M, the element of M is mij definite as:

0.055; S (X 8 ) = 0.043 ; S (X 9 ) = 0.185; S (X 10 ) =0.325; S (X 11 ) = 0.228; S (X 12 ) =0.073. We used 0.150 as threshold value.

{a ∈ C ; a ( xi ) ≠ a ( x j )} wh ne d ( xi ) ≠ d ( x j ) mij =  otherwise ∅

Since S (X 2 ),

S (X 3 ), S (X 5 ), S (X 7 ), S (X 8 ) and S (X 12 ) small than

(8)

threshold value, X 2 , X 3 , X 5 , X 7 , X 8 , X 12 may be excluded due to their less importance.

From definition 6, we calculate discerning matrix, and the result is showed in Table 4.

We have the same result by using the important measure of attribute and Important of attribute.

Table 4. Discerning matrix of the equivalence class E-1 E-1

∅

E-2

E-3

E-4

E-5

E-6

E-7

E-8

E-9

Rule

(1)

(1,6,9,11)

(9)

(1,4,6,9,11)

(1,4,9,11)

(1,4,9,11)

(1,4,6,9,11)

(1,9,11)

R-1

(1,11)

(1,4,6,9,11)

(1,6,9,11)

(1,6,9,11)

(4,6,9,11)

(1,9,11)

R-2

(1,6,9,11)

(1,4,6,9,11)

(1)

(1,9,11)

(4,9,11)

(6)

R-3

(1,4,6,9,11)

(1,6,9,11)

(1,6,9,11)

(1,4,6,9,11)

(1,11)

R-4

(4,9,11)

(4,6,9,11)

(1)

(1,4,6,9,11)

R-5

(9,11)

(1,4)

(1,6)

R-6

(1,4)

(1,6,9,11)

R-7

E-2

(1)

∅

E-3

(1,6,9,11)

(2,9,11)

(2,9,11)

∅

E-4

(9)

(1,11)

(1,6,9,11)

∅

E-5

(1,4,6,9,11)

(1,4,6,9,11)

(1,4,6,9,11)

(1,4,6,9,11)

∅

E-6

(1,4,9,11)

(1,6,9,11)

(1)

(1,6,9,11)

(1,6,9,11)

∅

E-7

(1,4,9,11)

(1,6,9,11)

(1,9,11)

(1,6,9,11)

(1,6,9,11)

(9,11)

∅

E-8

(1,4,6,9,11)

(4,6,9,11)

(4,9,11)

(1,4,6,9,11)

(1,4,6,9,11)

(1,4)

(1,4)

∅

E-9

(1,9,11)

(1,9,11)

(6)

(1,11)

(1,11)

(1,4)

(1,6,9,11)

(4,6,9,11)

(4,6,9,11)

∅

R-8 R-9

Note: (1) denote X 1 , (1,6,9,11) denote X 1 ∪X 6 ∪X 9 ∪ X 11. We generate rules as the following calculation (Boolean operation):

R-1: X 1 ∩(X 1 ∪X 6 ∪X 9 ∪X 11 ) ∩X 9 ∩(X 1 ∪X 4

19


“middle” AND inventory turnover ratio = “good” AND quick ratio = “middle” AND cash flow ratio = “middle” THEN decision =”middle”. R-7: IF return on total assets = “good” AND account receivable turnover ration = “middle” AND quick ratio = “good” AND cash flow ratio = “good” THEN decision =”good”. R-8: IF return on total assets = “middle” AND account receivable turnover ration = “good” AND quick ratio = “good” AND cash flow ratio = “good” THEN decision =”good”. R-9: IF return on total assets = “middle” AND account receivable turnover ration = “middle” AND quick ratio = “middle” AND cash flow ratio = “middle” THEN decision =”middle”.

∪X 6 ∪X 9 )∩(X 1 ∪X 4 ∪X 9 ∪X 11 )∩(X 1 ∪X 4 ∪X 9 ∪X 11 )∩(X 1 ∪X 4 ∪X 6 ∪X 9 ∪ X 11 ) ∩(X 1 ∪X 9 ∪X 11 ) = X 1 ∩X 9 R-2:

X 11

R-3:

X 1 ∩X 6 ∩X 9 ∩X 11

R-4:

X 1 ∩X 11 ∩X 9

R-5:

X 4 ∩X 9 ∩X 11

R-6

X 1 ∩X 4 ∩X 6 ∩X 9 ∩X 11

R-7:

X 1 ∩X 4 ∩X 9 ∩X 11

R-8:

X 1 ∩X 4 ∩X 9 ∩X 11

R-9:

X 1 ∩X 4 ∩X 9 ∩X 11

Calculation result is show in Table 5. Table 5 Decision table for rule extraction class 1 2 3 4 5 6 7 8 9

# of sample 5 5 5 6 4 4 5 4 7

X1

X4

X6

X9

X 11

D

1 × 2 1 × 3 3 2 2

× × × × 3 2 2 3 2

× × 3 × × 3 × × ×

× × 2 2 3 2 3 3 2

1 1 2 1 3 2 3 3 2

bad bad middle bad good middle good good middle

Finally, the decision table can be built and rule extraction from decision table. According to Table 6, we may extract decision rules in IF-THEN form. R-1: IF return on total assets = “bad” AND Cash flow ratio =”bad” THEN decision = ”bad”> R-2: IF cash flow ratio = “bad” THEN decision = ”bad”. R-3: IF return on total assets = “middle” AND inventory turnover ratio = “good” AND quick ratio = “middle” AND cash flow ratio =”middle” THEN decision =”middle”. R-4: IF return on total assets = “bad” AND quick ratio = “middle” AND cash flow ratio = ”bad” THEN decision = ”bad”. R-5: IF account receivable turnover ration = “good” AND quick ratio = “good” AND cash flow ratio = ”good ” THEN decision = ”good”. R-6: IF return on total assets = “good” AND account receivable turnover ration =

3.5 Calculate the number of error by using test data We compare the accuracy of different approaches by introducing two error types. TypeＩerror refer to the situation when matched data is classified as unmatched one, and Type Π error refer to unmatched data is classified into matched data. The result is listed in Table 6. Rough approach based on information entropy has the better performance in minimizing Type Ｉ error, while having satisfactory Type Π. Using the unmatched and unbalanced training data and test data, rough set theory with information entropy, shows the best overall prediction accuracy level at 83.4% (see Table 7), when using the rough set theory based on information entropy. Table 6

The prediction result with test data

Decision by this algorithm

U

X1

X4

X6

X9

X 11

Test data

1

3

2

3

2

2

Middle (error)

2

2

2

2

1

1

bad

bad

3

2

2

3

2

2

middle

Good (error)

4

1

2

2

1

1

bad

bad

5

3

3

3

3

3

good

6

1

2

2

2

1

7

2

2

2

1

1

good middle (error) bad

8

2

2

2

1

1

middle

middle

good

bad bad

9

2

3

3

3

3

good

good

10

3

2

3

3

3

good

good

11

2

2

2

1

1

bad

bad

20


Table 7 Result comparison with rough approach with TypeＩand Type Π error Method

Error

Type Π error

Rough approach with information entropy

TypeＩ error

2/12 (17%)

0/12

2/12(16..6%)

4. Conclusion This study constructed an enterprise financial distress evaluation model based on rough set theory. We find the performance factors are return on total assets, account receivable turnover ration, cash flow ratio, Inventory turnover ratio, and Quick ratio. The paper proposes to utilize information entropy for distinguishing importance among attributes. After doing the empirical research which is based on the latest data of Taiwan’s list firm, the result proves the validity of our model. Therefore, we make scientific evaluation and get the right forecasting. Using the unmatched and unbalanced training data and test data, shows the best overall prediction accuracy level at 83.4%, when using the rough set theory based on information entropy. We can use this methodology to perform alliance perform index on other property.

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