Conjunction and Disjunction Operators in Neuro-Fuzzy Risk Calculation Model Simplification

Conjunction and Disjunction Operators in Neuro- Fuzzy Risk Calculation Model Simplification

E. Tóth-Laufer^*, M. Takács^** and I.J. Rudas^**

* Óbuda University/Doctoral School of Applied Informatics, Budapest, Hungary

** Óbuda University/John von Neumann Faculty of Informatics, Budapest, Hungary laufer.edit@bgk.uni-obuda.hu, takacs.marta@nik.uni-obuda.hu, rudas@uni-obuda.hu

Abstract—In this paper a possible simplification of a risk level calculation model’s neuro-fuzzy subsystem will be studied. The basic model has a hierarchical multilevel structure and uses fuzzy logic based decision making for some groups of risk factors and neuro-fuzzy subsystem is based on ANFIS model structure for the other. The simplification of neuro-fuzzy subsystem is based on disjunction fuzzy operator where the operator selection is application-dependent. This is because the three basic operations of crisp sets (negation, conjunction, and disjunction) can be generalized for fuzzy sets in an infinite number of ways. In this work the effect of different conjunction and disjunction operators on the result of simplified structure was compared and analyzed.

I. INTRODUCTION

The neuro-fuzzy model based on the author’s former validated fuzzy logic-based model. Due to the easily expandable hierarchical multilevel decision structure of the fuzzy model, one of the risk groups was effortlessly substituted with a neuro-fuzzy subsystem. Neuro-fuzzy systems combine the advantages of both approaches.

Fuzzy logic-based decision making is very beneficial in medical applications, because it can use linguistic terms to characterize the health state of the patient. Fuzzy approach in risk management is also very advantageous, because it can handle the uncertainty, imprecision and subjectivity in data and in evaluation process [1]. The studied model is used for risk calculation for physical exercise and in this kind of applications there are many interactions between the input factors and there is a need to tune the membership functions and rules according to the patient characteristics. This is not feasible with a fuzzy approach, but it can be handled by a neural network. The neural network regards the evaluation process as a black box, but with the fuzzy rule layer it can handle it. It can be seen that the two approaches are complementary and in this way the evaluation will be more efficient than the fuzzy or the neural network approach itself [2]. Risk management systems are very complex with a high number of input factors and rules, furthermore the interactions between the input factors also should be handled. The size of the rule base is particularly important, because the increase in the number of rules increases the number of the nodes exponentially. This study aims to simplify the structure of the neuro-fuzzy subsystem without rule-base reduction, but with a decrease in the number of the nodes. To achieve this goal an OR layer was incorporated into the system to connect the rules which has the same consequent part. In this layer disjunction operator was

used. Since the three basic operations of crisp sets (negation, conjunction, and disjunction) can be generalized for fuzzy sets in an infinite number of ways, different functions can be the best for different problem implementations [18]. In this paper the Matlab built-in operators, and operators belong to the Hamacher operator family was examined whether all of them are usable in the novel layer as a t-conorm or not.

II. THE MODEL STRUCTURE

The neuro-fuzzy model has a hierarchical multilevel clustered structure, which makes it easy to expand the model structure and also simplifies the evaluation process.

The model based on an AHP-FCE model and its structure can be seen in Fig. 1 [3]. On the left side there is the lowest level with the sub-factors, in the middle of the figure are the groups of the elements and on the right side is the highest level with the problem which one the risk level will be calculated. The evaluation is executed from the left side to the right [4]. The risk level of the “Medical condition” group is calculated with a neuro-fuzzy subsystem, but for the rest of the groups Fuzzy Logic Controllers (FLC) are used. The overall risk level is calculated with an FLC and sent to the Matlab workspace.

The rules were set up with contribution of a trainer and the evaluation is based on Mamdani-type inference system.

Figure 1. The neuro-fuzzy model structure

The first group of risk factors “Medical condition”

provides information about the health state of the patient.

This is the most important and the most complex group of all, because personal conditions basically determine the load capacity and in this group there are the most interactions between the input factors. This is the reason why the risk level of this group is calculated with a neuro- fuzzy subsystem. This group is most in need of tuning the membership functions and rules and it is possible with a combination of fuzzy logic and neural network in the future. The first input factor of this group is “Disease condition” with such chronic diseases as hypertension, diabetes, cardiac diseases among others [4]. The second input factor is “Current physical status,” which provides information about the actual status of the patient by measured parameters as pulse, or blood pressure.

Monitoring these values in real-time in the future allows not only for the initial level of risk to be calculated, but to be given a continuous control [4]. The input factor “Basic physical information” is used to typify basic characteristics and living conditions of the patient.

The second main group “Activity load” is used to characterize the sport activity. The sub-factors describe how intensively (Intensity), how long per occasion (Duration) and how many times per week (Frequency) the patient does this activity [5].

The third main group “Environmental condition” uses combined sub-factors. The temperature is combined with humidity (TH) and wind (TW) because of their influence on thermal sensation. This group is mainly important regarding outdoor sports, but humidity and temperature together can influence the risk level indoor too [5].

III. THE NEURO-FUZZY SUBSYSTEM

The risk level evaluation of the “Medical condition”

group performs with a neuro-fuzzy subsystem, which is based on ANFIS model structure with Mamdani-type inference system. This group has three input factors as the previous section was described and they are denoted by x, y, z and one output, is denoted by V in the following. The model can be represented by following type of rules:

IF x is Ai and y is Bj and z is Ck then V=Dl (1) where i,j,k,l∈[1,5]. Five input membership functions belong to each input factor and the rule premises are obtained from the all possible combinations of these membership functions. In this way in this group it has one hundred and twenty-five rules in the subsystem of the basic model. The processing elements of the network are organized into layers according to their tasks. Neurons located in the same layer have the same or similar local processing [6]. The output of each layer is described in the following, where the output of the j^th node in the i^th layer is denoted as Oi,j. The structure of this system is shown in Fig. 2. [7],[19].

Figure 2. Neuro-fuzzy subsystem

The input layer is the first with input factors of the

“Medical condition” group, from which the rule-premise is created. For the next layer these inputs must be fuzzified. From the input x the layer output can be calculated by (2),(3). The output for the other two input parameters also can be calculated based on this formula with the appropriate substitution. In the model trapezoid shape membership functions are used.

( )

^x

O_1i, =μ_Ai ⁽²⁾

⎪ ⎪

⎪ ⎪

⎩

⎪⎪ ⎪

⎪

⎨

⎧

≤

≤

− ≤

− ≤ ≤

≤

− ≤

− ≤

= μ

x d 0

d x c c

d x d

c x b 1

b x a a

b a x

a x 0

i i i

i i i

i i

i i

i i

i

i

A (3)

where ai, bi, ci, di are the parameter sets of the membership function. These parameters for the time are fixed, but later by changing that, the membership functions can be tuned [5], [12].

The second layer is the rule-premise layer it calculates the firing strength of the rules using a conjunction operator. In Fig. 2. the minimum operator is used, which is calculated by (4).

( ) ( ) ( )

(

^{x , y, z}

)

min w

O2,i = i = μAi μBi μCi ⁽⁴⁾

The third is the normalization layer, which calculates the normalized firing strength of the rules by (5).

= ∑

=

i i i i

i ,

3

w

w w

O

(5)

The consequent layer is the fourth and it obtains the consequent part of the rules and uses the centre of gravity method (6). Output membership functions are very safe, safe, medium, moderately dangerous and very dangerous, they are given respectively by (7)-(11) [20].

( )

∫ ( )

∫

μ μ

=

=

y D y

D i i ,

4

y dy

ydy y f

O

i i

(6)

( )

( )

⎪⎩

⎪⎨

⎧

>

≤

≤

−

<

= μ

3 . 0 x 0

3 . 0 x 1 . 0 2

. 0 x 3 . 0

1 . 0 x 1

) x

( ^1.2

1 (7)

( )

( )

( )

( )

⎪⎪

⎪

⎩

⎪⎪

⎪

⎨

⎧

≥

≤

≤

−

≤

≤

≤

≤

−

<

= μ

5 . 0 x 0

5 . 0 x 325 . 0 175

. 0 x 5 . 0

325 . 0 x 275 . 0 1

275 . 0 x 1 . 0 175

. 0 1 . 0 x

1 . 0 x 0

) x (

2 . 1 2 . 1

2

(8)

( )

( )

( )

( )

⎪⎪

⎪

⎩

⎪⎪

⎪

⎨

⎧

≥

≤

≤

−

≤

≤

≤

≤

−

<

= μ

7 . 0 x 0

7 . 0 x 525 . 0 175

. 0 x 7 . 0

525 . 0 x 475 . 0 1

475 . 0 x 3 . 0 175

. 0 3 . 0 x

3 . 0 x 0

) x (

2 . 1 2 . 1

3

(9)

( )

( )

( )

( )

⎪⎪

⎪

⎩

⎪⎪

⎪

⎨

⎧

≥

≤

≤

−

≤

≤

≤

≤

−

<

= μ

9 . 0 x 0

9 . 0 x 725 . 0 175

. 0 x 9 . 0

725 . 0 x 675 . 0 1

675 . 0 x 5 . 0 175

. 0 5 . 0 x

5 . 0 x 0

) x (

2 . 1 2 . 1

4

(10)

( )

( )

⎪⎩

⎪⎨

⎧

>

≤

≤

−

<

= μ

9 . 0 x 1

9 . 0 x 7 . 0 2

. 0 7 . 0 x

7 . 0 x 0

) x

( ^1.2

5 (11)

The fifth layer calculates the overall risk level for the group as the weighted average of the outputs of the fourth layer where the firing strength was calculated in the second layer is used as weights (12).

∑

∑

⁼

∑

=

i i i

i i

i i i i

,

5 w

f w f

w

O (12)

IV. THE SIMPLIFIED NEURO-FUZZY SUBSYSTEM

There are some rules in the system whose rule-premise belongs to the same consequent part. This is the basis of the structure simplification, because these rule-premises can be connected together with a novel layer incorporated into the system. This layer is a disjunction operator layer and it builds in the system between the normalization layer and the consequent layer. In this way the number of the nodes in the consequent layer can be significantly decreased. In the basic neuro-fuzzy subsystem there are one hundred and twenty-five nodes, because each rule has a node in the consequent layer. In the simplified structure the output membership functions can be represented sufficiently with five nodes. The simplified structure is shown in Fig. 3. In the figure the maximum operator is used as OR method in the disjunction layer. It can be calculated by (13).

( )

^Di

i i ,

4

w max w

O = =

⁽¹³⁾

Figure 3. Simplified neuro-fuzzy subsystem structure

The simplification with using the maximum operator as a conjunction method can be performed, because of the Mamdani type inference system rule evaluation method.

In this process the overall rule output is obtained as a union of each rule output. Consequently the rules with the same consequent part have the same output membership function and as a result of the union, these trapezoids overlap each other. Therefore, among the rules which have the same consequent part, the output will be applied which has the higher supremum, this includes the others.

The maximum method is equivalent to this, therefore this disjunction operator can be used in the OR layer. In this case only the output with higher supremum will be considered. Fig. 4 shows the evaluation of the Mamdani type inference system with two input parameters, three rules and the applied minimum operator as AND method.

In this figure the first and the third rules have the same consequent part, but the supremum of the rules are different, the first is higher than the third. Therefore the first rule output will be considered and combined with the second rule output during the calculation of the overall output.

Figure 4. Mamdani inference system

V. CONJUNCTION AND DISCUNCTION OPERATORS

The three basic operations for fuzzy sets can be generalized an infinite number of ways, therefore different functions can be the best for different problem implementations. In this work Fuzzy Logic Toolbox built- in methods and the Hamacher operator family were studied. In this section these operators will be introduced.

In the fuzzy inference system the conjunction operator (t- norm) and disjunction operator (t-conorm) are used if the rule premise has more than one part to obtain one number as a result of the antecedent [11].

A. Fuzzy Logic Toolbox built-in methods

In the Fuzzy Logic Toolbox there are some built-in operators for conjunction as AND and for disjunction as OR, furthermore, custom operators can be defined if it is necessary. The built-in AND methods are minimum (min) and product (prod), OR methods are maximum (max) and probabilistic OR (probor), whose definition is described below in [8].

Min operator is Zadeh’s t-norm:

( ) ^{a , b min} ( ) ^{a , b}

t =

⁽¹⁴⁾

Algebraic product:

( ) ^{a , b ab}

t =

⁽¹⁵⁾

The Max operator is Zadeh’s t-conorm:

( ) ^{a , b max} ( ) ^{a , b}

s =

⁽¹⁶⁾

Probabilistic OR is also known as the algebraic sum:

ab b a ) b , a (

s = + − ⁽¹⁷⁾

B. Hamacher operator family

There is some so-called super-norm which aims to include a wide-range of the operators. These types of norms are parameterized and thus can be varying depending on the problem. For each fixed parameter value there is a special operator belonging to it [9]. This kind of

norm is the Hamacher operator. The general form of the Hamacher operator in a multivariate case can be calculated by (18).

( )

α

=

α α

α ⎟⎟

⎟

⎠

⎞

⎜⎜

⎜

⎝

⎛

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ ⎟⎟−

⎠

⎞

⎜⎜

⎝

⎛

⎟⎟⎠

⎜⎜ ⎞

⎝

⎛ −

γ γ +

+

=

∏

1 n

1

i i

i n

1

x 1 x 1 1

1 1 x 1 ,..., x

t

(18) where the type of the operator depends on the α parameter. If α=1 then this is a t-norm operator and γ1=γc, but if α=-1 then this is a t-conorm operator and γ-1=γd. By varying the γ parameter in general form of the Hamacher operator different renowned operators are obtained (19)- (21).

The product operator is a t-norm: γc=1 and α=1.

(

^x1,...,xn

)

⁼

∏

_i=ⁿ₁ ^xi

t (19)

The probabilistic or operator is a t-conorm: γd=1 and α=-1.

(

^x1,...,xn

)

⁼¹⁻

∏

_i=ⁿ₁

(

^1−xi

)

s (20)

The Einstein operator: γ =2.

( )

α

=

α

⎟⎟

⎟

⎠

⎞

⎜⎜

⎜

⎝

⎛

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ ⎟⎟−

⎠

⎞

⎜⎜

⎝

⎛

⎟⎟⎠

⎜⎜ ⎞

⎝

⎛ − + +

=

∏

1 n

1

i i

i n

1

x 1 x 2 1 2 1

1 1 x 1 ,..., x

s ⁽²¹⁾

if α=1 the Einstein operator is a t-norm, if α=-1 it can be used as a t-conorm operator [10].

VI. COMPARISON OF THE MODELS

The comparison was performed in a Matlab environment with Simulink and Fuzzy Logic Toolbox.

Several typical groups of the patient were examined which are characterized with the “Medical condition” group parameters as it is shown in Table 1. The parameters are theoretical and based on the American Heart Association Guidelines [13],[14],[15], [16],[17]. The test was executed both of the basic neuro-fuzzy structure and the simplified structure also. The conjunction and disjunction operators described in the previous section were implemented into the system and the aim was to study whether all of them are usable in the novel layer as a t-conorm or not.

Taking into account only the “Medical condition”

group, it has been determined that using the maximum and product operator the calculated risk level is the same for both the basic and simplified neuro-fuzzy structure. Using other conjunction-disjunction pairs, the result is higher with the simplified structure than with the full. These

differences are shown in Table II. The results of both the above structures compared with the author’s fuzzy logic- based model seen, if the calculated risk level of full and simplified structures is different, then the result of the full structure is closer to the fuzzy model result than the simplified one. These differences are given in Table III.

The overall risk level was calculated for four hundred and eighty cases for each group of patients with simplified structure. To characterize the difference between this model and the fuzzy model several metrics were used.

These metrics are mean, minimum and maximum as in the above comparisons, furthermore standard deviation, average absolute deviation of the average and correlation coefficient. The goal of this comparison was to find out which operator-pair’s results are closer to the former validated fuzzy model. The product conjunction operator and probabilistic OR disjunction operator proved to be the best for all tested groups of the patients for all calculated

metrics. The result of the comparison for the “50-65 years old in poor condition” group for min-max and prod-probor operator pairs is shown in Table IV.

Summarizing the results it can be concluded, that simplification cannot be performed for all the studied disjunction operators, only for maximum and probor.

Compared to the author’s former validated model, the prod-probor operator pair has proven to be the best.

VII. CONCLUSIONS

This study aimed to simplify the structure of the neuro- fuzzy subsystem without rule-base reduction, but with a decrease in the number of the nodes. It is important, because the increase in the number of the rules increases the number of the nodes exponentially. A new layer was integrated into the basic subsystem to connect the rule- premises which have the same consequent part. This is an OR layer and contains a conjunction operator. The basic neuro-fuzzy model structure and the simplified structure were compared for several typical groups of the patients.

During the study the model was tested with several possible disjunction operators. It can be concluded that the author’s basic neuro-fuzzy model structure can be simplified without rule-base reduction, but only the maximum and product operators are usable in the connection layer, the rest of the studied t-conorms are not.

ACKNOWLEDGMENT

The authors gratefully acknowledge the grant provided by the project TÁMOP-4.2.2/B-10/1-2010-0020, Support of the scientific training, workshops, and establish talent management system at Óbuda University and the Hungarian Scientific Research Fund (OTKA K 105846).

REFERENCES

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TABLE IV.

COMPARISON WITH THE FUZZY MODEL FOR OVERALL RISK LEVEL

t-norm:min, t-conorm: max

t-norm : prod, t- conorm : probor

Mean 0.0105 0.0010

Minimum 0 0

Maximum 0.0500 0.0100

Deviation 0.0119 0.0030

Av. D. Av. 0.0071 0.0019

Correlation 0.9801 0.9990

TABLE III.

COMPARISON WITH THE FUZZY MODEL FOR MEDICAL CONDITION GROUP

Operator Structure Mean Min Max Einstein Full 0.0048 0.0020 0.0102 Einstein Simplified 0.0475 0.0198 0.0774 Hamacher

(γc=3, α=1) Full 0.0029 0.0025 0.0056 Hamacher

(γc=3, α=1) Simplified 0.0586 0.0084 0.1056 Hamacher

(γc=1000, α=1)

Full 0.0008 0.0001 0.0025 Hamacher

(γc=1000, α=1)

Simplified 0.0743 0.0725 0.0751 TABLE I.

MEDICAL CONDITION PARAMETERS

Group Dis_con Phy_sta Bas_inf Average healthy adult 0.5 0.5 0.5

Middle-aged with light

cardiac disease 0.4 0.4 0.7

Healthy senior 0.64 0.5 0.3

50-65 years old in poor

condition 0.25 0.3 0.5

TABLE II.

DIFFERENCES FOR THE MEDICAL CONDITION GROUP

T-norm T-conorm Mean Min Max

Minimum Maximum 0 0 0

Product Probor 0 0 0

Einstein Einstein 0.0427 0.0168 0.0734 Hamacher

(γc=3, α=1)

Hamacher (γc=3,

α=1)

0.0556 0.0036 0.1000 Hamacher

(γc=1000, α=1)

Hamacher (γc=1000, α=1)

0.0750 0.0750 0.0750

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10.1109/SAMI.2012.6208943

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