Interactions Handling Between the Input Factors in Risk Level Calculation

Interactions Handling Between the Input Factors in Risk Level Calculation

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 risk evaluation of physiological processes there are a lot of input factors and rules, furthermore there are many interactions between them. Handling these interactions is very problematic, because it is difficult to see through the combined effect the large number of factors even the experts. In this paper a possible theoretical method of this problem’s treatment will be presented. The study is based on the authors’ neuro-fuzzy risk calculation model which can be easily expanded with a pre-processing part to handle the interactions.

I. INTRODUCTION

Risk management systems are complex systems with a lot of input factors and rules. The uncertainty, imprecision and subjectivity in data and in the evaluation process justify the use of fuzzy logic-based decision making, but it is not able to adapt to the patient characteristics [10]. This adaptation is absolutely necessary to obtain realistic results, thus there is a need a patient-specific evaluation.

This requirement can be met by applying a neural network. The deficiency of the neural networks is that it regards the evaluation process as a black box and rules cannot be incorporated into the system. Neuro-fuzzy systems have the advantages of both approaches and in this way the evaluation is more efficient. The membership functions can be tuned by a neural network and rules can be incorporated into the system with a fuzzy rule layer [7].

In these kinds of applications there are many interactions between the input factors and their management is very problematic. If the heart rate is taken into account, there are about thirty other factors which can influence it. The most common ones are shown in Table 1.

It is very difficult to see through the combined effect the large number of factors even for the experts therefore there is a need for a method which allows their combined treatment in the simplest and most transparent way. A possible solution of this problem is that the complex interactions lead back to simpler interactions, when one of the factors is selected and it is examined with all the other factors in pairs. This selected factor can be the heart rate with all the factors in Table 1. in pairs. This simplified problem can be easily solved by tuning the membership function parameters according to the patient characteristics for all the factors that affect the selected one. As a result, several input membership functions are obtained for the factor-pairs and these should be aggregated to represent the combined effect of the other factors on the selected factor. In fuzzy logic there are many different aggregation operators that can be used, and they are application-dependent, so it cannot be said in general which one the best operator is. Consequently the

appropriate operator selection is very important. In the following the possible operators will be also introduced.

II. TUNING THE MEMBERSHIP FUNCTIONS IN A USER- SPECIFIC WAY

At this essential stage of the evaluation process, the membership functions are tuned according to the patient characteristics. In all cases only one factor’s effect on the heart rate is considered, but it should be performed for each factor that affects the selected one. The input membership functions are trapezoid shaped and they can be described by (1).

⎪ ⎪

⎪ ⎪

⎩

⎪⎪ ⎪

⎪

⎨

⎧

≤

≤

− ≤

− ≤ ≤

≤

− ≤

− ≤

= 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 (1)

where ai, bi, ci, di are the parameter sets of the membership function. By changing these parameters the membership functions can be tuned [7], [8]. Below are some examples of tuning the membership functions. The studied

TABLE I.

PARAMETERS THAT AFFECT THE HEART RATE [11]

Parameter Effect Age The age increases, the heart rate

decreases

Sex For women it is higher

Weight It is higher for the overweight patients

Time of day It is lower in the morning and increases gradually during the day Medication

It can be reduced or increased by anti-inflammatories, stimulants, sedatives

Smoking Long-term increases it

Special conditions Pregnancy increases, severe loss of consciousness decreases it Training effects After exercise it is higher Physical fitness Decreases it

Air temperature and humidity

Heat, cold, high humidity increases it

parameters are heart rate and the intensity of sport activity.

The tuned membership function belonging to a 20-year- old patient is shown in Fig. 1, and the membership functions belonging to a 50-year-old patient can be seen in Fig. 2. The membership functions based on the predictive maximal heart rate and Polar target heart rate zones in beats per minute (bpm) [12].

Figure 1. The tuned membership functions (20-year-old patient)

Figure 2. The tuned membership functions (50-year-old patient) III. AGGREGATION OF THE INPUT MEMBERSHIP

FUNCTIONS

The user-specific tuned membership functions should be aggregated to obtain their combined effect on the selected input factor. Therefore there is a need to select the appropriate aggregation operator to combine fuzzy sets into a single fuzzy set. In this section several possible aggregation operators will be introduced. Aggregated results in each case are shown in the figure with a thick line. The theoretical membership functions used in the examples are shown in Fig. 3.

Figure 3. The input membership functions

A. Matlab built-in aggregation opoerators

In Matlab there are some built-in aggregation operators these are the most common ones in practical applications and these are described in the following section.

The Max method forms a single fuzzy set from the input fuzzy sets with their union which is calculated by (2) and the result is shown in Fig. 4.

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

s =

⁽²⁾

Figure 4. Maximum aggregation

The Sum method creates the aggregation output set simply by the calculation the bounded sum of each input fuzzy set. The calculation method is given by (3) and the result is illustrated in Fig. 5.

( ^{a b , 1} )

min ) b , a (

s = +

⁽³⁾

Figure 5. Sum aggregation

Probor is the abbreviation of the probabilistic OR which is also known as the algebraic sum. This method forms the single fuzzy set as the output of the aggregation by (4) from the input functions of implication. The result of the aggregation can be seen in Fig. 6 [2].

( ) ^{a , b a b ab}

s = + −

⁽⁴⁾

Figure 6. Probor aggregation B. Averaging aggregation operators

In Matlab custom aggregation operators can also be defined. In this study averaging operators were examined thus their definition and some of their classes will be described.

Definition: h: [0,1]ⁿ→[0,1] (n≥2) is an aggregation operator, where n is the number of the fuzzy sets. If the arguments A1(x), …, An(x) are fuzzy sets, where X is the basic set, then h generates a fuzzy set from the membership degree of the arguments for all x∈X which means that

A(x)=h(A1(x), …, An(x)) (5)

A well-defined aggregation operator must satisfy the following axioms:

Axiom 1: (boundary condition)

h(0,…,0)=0 and h(1,…,1)=1 (6) Axiom 2: h monotonically increases in its each argument, namely if there are two arbitrary n-tuple 〈a₁,…,an〉 and

〈b₁,…,bn〉 where a_i,bi ∈ [0,1] and ai≤bi ∀i∈[1,n], then

h(a1,…,an) ≤h(b1,…,bn) (7)

Axiom 3: h is a continuous function.

In addition further restrictions can be given:

Axiom 4: h is symmetric in its each argument, namely h(a1,…,an) = h(ap(1),…,ap(n)) (8) where p is an arbitrary permutation of 1,…,n.

Axiom 5: h is idempotent, namely h(a,…,a)=a, ∀a∈ [0,1].

Theorem: For the aggregation operators which satisfy the above five axioms

min(a1,…,an) ≤ h(a1,…,an) ≤ max(a1,…,an) (9)

∀〈a1,…,an〉∈[0,1]ⁿ [1].

One of the averaging operators is the general power mean, which covers the full interval between the minimum and maximum. It can be defined by (10) and the result is shown in Fig. 7. if α=10.

( )

^{α α α}

α

⎟⎟

⎠

⎜⎜ ⎞

⎝

⎛ + +

=

1 n 1

n

1

n

a ...

a a ,..., a

h

(10)

where α≠0 and

∏

ⁿⁱ⁼¹

^a

ⁱ

^{≠ 0}

^{if α<0.}

Figure 7. General power mean aggregation

There are some special values, for which famous means are obtained. If α→0 then h_α is converges to the geometric mean, if α=1 then this is the arithmetic mean and in the case when α=-1 this is the harmonic mean.

In some cases the factors effect are not equally weighted on the selected factor. Therefore the other studied averaging operator is Ordered Weighted Averaging (OWA), which also covers the full interval between the minimum and maximum. Let the weight vector be

w = w

₁

,..., w

_n ^{, ∀wi∈[0,1]}

∑

ⁿ_i=₁

w

i

= 1

(11)

then the OWA operator belonging to the weight vector w is defined by (12)

(

1 n

)

1 1 n n w

a ,..., a w b ... w b

h = + +

⁽¹²⁾

where bi is the i-th largest element of a1,…,an, namely the

〈b₁,…,bn〉 vector is the descending ordered permutation of

〈a₁,…,an〉, b_i≥bj if i<j, i,j∈[1,n] [1]. The result of the aggregation can be seen in Fig. 8. if w=〈0.3, 0.7〉.

Figure 8. OWA aggregation

IV. THE MODEL STRUCTURE

The studied model is a subsystem of the authors’

hierarchical multilevel risk level calculation model. The overall model structure based on an AHP-FCE model structure [13] is shown in Fig. 9. The lowest level of the hierarchy is on the left side with sub-factors, which are grouped according to the type of the factors. “Activity load” and “Environmental condition” group have a Fuzzy Logic Controller (FLC) to evaluate their risk and for

“Medical condition” a neuro-fuzzy subsystem is used. The results are transmitted to the next highest level, where the total risk level is calculated by an FLC. The rules of the FLC-s were set up with a contribution of a trainer. The first group of risk factors is Medical condition with sub- factors Disease condition (dis_con), Current physical status (phy_sta) and Basic physical information (bas_inf).

Disease condition includes such persistent diseases as hypertension, diabetes and cardiac diseases among others.

Current physical status provides information about the actual status of the patient by measured parameters as heart rate, 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 [2]. The studied subsystem belongs to this group.

The Basic physical information is about age, sex and the living conditions, such as occupational stress and activity.

The next group refers to the characteristics of the sport activity. These parameters describe how intensively (Intensity), how long per occasion (Duration) and how many times per week (Frequency) the patient does this activity.

The third main group is Environmental condition, this is mainly important regarding outdoor sports, but humidity and temperature together can influence the risk level indoor too. Here at the sub-factors temperature is combined with two other parameters, these are humidity and wind. The reason for this combination is that they can influence the thermal sensation together [9].

The model uses Mamdani-type inference system and based on the authors’ former study sum method is used as aggregation, and bisector method for defuzzification [2].

Figure 9. The overall model structure

The studied subsystem structure is based on ANFIS model structure, but it is not equivalent to it. If the system is considered with three input parameters (x, y, and z) and one output, is denoted by V the Mamdani-type inference system can be represented by the following type of rules:

IF x is Di and y is Ej and z is Fk then V=Gl (13) where i,j,k,l∈[1,5]. In this example five input membership functions belong to each input factor and the rule premises are obtained from all possible combinations of these membership functions. The processing elements of the network are organized into layers according to their tasks.

Neurons located in the same layer have the same local processing [3]. 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. 10. [4],[5].

Figure 10. Neuro-fuzzy subsystem basic structure [5]

This structure is different from the ANFIS model structure in the consequent layer, because it was simplified by a connection layer and the structure is extended with a pre-processing part [6]. This part is optional depending on whether the interactions between the input factors justify the pre-processing or not. In the first layer of this part the membership functions are tuned according to the patient characteristics. Input membership functions are trapezoid shape and they can be calculated by (14),(1).

A i,

O

1

= μ

⁽¹⁴⁾

These tuned membership functions are the inputs of the next layer, which aggregates them to represent their combined effect. One of the aggregation operators can be used here. In this example this is the OWA operator, which is described by (15).

O

2i,

= h

w

( A

i

, B

i

, C

i

) = w

1

b

1

+ w

2

b

2

+ w

3

b

3 ⁽¹⁵⁾

where bi is the i-th largest element of Ai, Bi, Ci.

The inputs of the third layer are the risk factors in all cases and also the aggregated membership functions if there is a pre-processing part. For the next layer these

inputs must be fuzzified. The shape of the membership function depends on whether there was a pre-processing or not. If there was a pre-processing part, the output can be calculated by (16), otherwise by (14),(1).

( ) ^y

( )

( ) ^y

O

3i,

= μ

Ei

= μ

hw Ai,Bi,Ci ⁽¹⁶⁾

The forth layer is the rule-premise layer, it calculates the firing strength of the rules using the minimum operator, which is calculated by (17).

( ) ( ) ( )

( ^{x , y , z} )

min w

O

_4i,

=

_i

= μ

_Di

μ

_Ei

μ

_Fi ⁽¹⁷⁾

The fifth is the normalization layer, which calculates the normalized firing strength of the rules by (18).

= ∑

=

i i i i

i,

5

w

w w

O

(18)

The sixth layer is the disjunction or the connection layer, where rule-premises are connected together which belong to the same consequent part. In this layer the maximum operator is used which can be calculated by (19).

( )

^Gi

i i,

6

w max w

O = =

₍₁₉₎

The consequent layer is the seventh, it obtains the consequent part of the rules and uses the centre of gravity method (20). Output membership functions are very safe, safe, medium, moderately dangerous and very dangerous.

These are given respectively by (21)-(25) [9].

( )

∫ ( )

∫

μ μ

=

=

y G y

G i

i,

7

y dy

ydy y f

O

i i

(20)

( )

( )

⎪⎩

⎪⎨

⎧

>

≤

≤

−

<

= μ

3 . 0 x 0

3 . 0 x 1 . 0 2

. 0 x 3 . 0

1 . 0 x 1

) x

( ^1.2

1 (21)

( )

( )

( )

( )

⎪⎪

⎪

⎩

⎪⎪

⎪

⎨

⎧

≥

≤

≤

−

≤

≤

≤

≤

−

<

= 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

(22)

( )

( )

( )

( )

⎪⎪

⎪

⎩

⎪⎪

⎪

⎨

⎧

≥

≤

≤

−

≤

≤

≤

≤

−

<

= 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

(23)

( )

( )

( )

( )

⎪⎪

⎪

⎩

⎪⎪

⎪

⎨

⎧

≥

≤

≤

−

≤

≤

≤

≤

−

<

= 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

(24)

( )

( )

⎪⎩

⎪⎨

⎧

>

≤

≤

−

<

= μ

9 . 0 x 1

9 . 0 x 7 . 0 2

. 0 7 . 0 x

7 . 0 x 0

) x

( ^1.2

5 (25)

The last layer calculates the overall risk level for the group as the weighted average of the outputs of the seventh layer where the firing strength was calculated in the forth layer is used as weights (26).

∑

∑ ⁼ ∑

=

i i

i i i

i i i

i,

8

w

f w f

w

O

(26)

V. FUTURE TASKS

The above method used for interaction handling between the input factors was described in theoretically, but it should be tested in practical application with detailed analysis of the interactions. Furthermore, in this way its practical applicability should be proven.

T-norm and t-conorms in multivariate case can be used also as aggregation operators therefore the range of the studied aggregation operators can be expanded [1].

VI. CONCLUSION

In complex systems there is a problem to handle the interactions between the numerous input factors. Even the experts are not able to see through the combined effect of the factors. The study aimed to solve this problem with a pre-processing method incorporated into their neuro-fuzzy risk calculation model for physical activity. The basic model can be easily expanded with a novel part for pre- processing to handle the interactions between the input factors. The most important step in this part is the simplification of the complex interactions until it becomes transparent. The membership functions can be tuned according to the patient characteristics which is a very important consideration in these kinds of applications. To calculate the combined effect of the several factors different aggregation operators can be used for the tuned membership functions. Thereafter this aggregated function is used as input membership function representing the interacting factors. Operator selection is application dependent and it depends on the factors characteristics. In the author’s model, due to the different weights of the factors’ effect, the weighted aggregating operators can be the best.

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).

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