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The Effect of Aggregation and Defuzzification Method Selection on the Risk Level Calculation

Edit Tóth-Laufer*,**, Márta Takács***

* Doctoral School, Óbuda University, Budapest, Hungary

** Donát Bánki Faculty of Mechanical and Safety Engineering, Óbuda University, Budapest, Hungary

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

Abstract— In this paper a fuzzy logic-based hierarchical multilevel risk calculation model will be introduced with different model parameters. On each occasion when a fuzzy- based simulation model is constructed, the appropriate aggregation and defuzzification method must be chosen. It is very difficult because it cannot be said generally, which is the best method, its depends on the current application. The model presented in the paper is a model for risk calculation of physical exercise, and it was constructed in Simulink - Fuzzy Logic Toolbox environment with Mamdani-type fuzzy evaluation and different aggregation and defuzzification operators. The test was performed for several typical groups of the patients. The results are compared with a previously implemented Analytic Hierarchy Process with Fuzzy Comprehensive Evaluation based model with similar purposes. The result of the comparison has been analyzed and the best methods have been selected.

I. INTRODUCTION

Sport and physical exercises play a very important role in human life, it can make one healthier and the lack of sport can lead to our health deteriorating. It is however important to be aware that improper movement can be harmful, if it is not the right movements for our capability or not the proper duration, frequency, intensity was chosen. The basic physical information, actual physical status, age and so many other sub-factors should be considered, in order for the sport to be safe and healthy rather than making the situation worse [1].

The authors’ model below constructed in Matlab Fuzzy Logic Toolbox environment try to find the best way to calculate the risk of physical exercise. The model uses fuzzy logic-based decision making, because in risk management it is very beneficial, that the uncertainty, imprecision and subjectivity in data and in evaluation process can be handled. In this way the result is more realistic. Fuzzy based techniques work best in situations where not enough reliable data has been given for statistical model, the cause and effect connection is imprecise or the observations and conditions in linguistic form can be given [2]. In the fuzzy inference process the input parameters are crisp values, and they must be fuzzified. After the fuzzification fuzzy operators (AND or OR) are applied in the rule system and then the implication model was applied from the antecedent to the consequent. The results of the implication process for each rule are the output membership functions and they

are the inputs of the aggregation method. Aggregation models combine the output fuzzy sets into a single fuzzy set - the rule system output. The selection of the aggregation method and operator influences the result of the evaluation, so the proper choice of it is important.

After the aggregation the final step is defuzzification, because crisp value is required as the output of the system. A specific value must be chosen which characterizes the system in the best way. It is an important consideration to select the appropriate defuzzification method, in order to achieve the corresponding result.

The authors tested the model for risk calculation of physical exercise on several combinations of aggregation and defuzzification methods. The test was performed for several typical groups of the patients and the result has been compared with a previously implemented model.

This model was constructed with similar system parameters and based on Analytic Hierarchy Process with Fuzzy Comprehensive Evaluation (AHP-FCE) [3].

II. THE MODEL STRUCTURE

The model was generated in Matlab environment with Simulink and Fuzzy Logic Toolbox based on an AHP- FCE model hierarchical structure [3] with its own membership functions and rules, but without AHP type weighted factors.

The frame of the Simulink model is seen in Fig. 1. The structure follows the logic of the evaluation process. 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. The risk level for each group with a Fuzzy Logic Controller (FLC) is calculated and the next highest level the total risk is determined by the group results with an FLC again. The total risk level in will be sent to the Matlab workspace. The rules were set up with contribution of a trainer and if it is necessary the parameters or the result in each level can be weighted [4],[5].

The membership functions of input data are simple trapeze shaped in each group, but the output functions are given by (4)-(8). The groups of risk factors are Medical condition, Activity load and Environmental condition as the figure of the model shows.

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Figure 1. The model structure in Simulink

The Medical condition is a basic risk group, because it fundamentally determines the load-ability. The sub- factors of this group are Disease condition (dis_con), Current physical status (phy_sta) and Basic physical information (bas_inf). Disease condition is maybe the most important among the sub-factors. This group includes such persistent diseases as hypertension, diabetes, cardiac diseases among others. In all cases the personalized medical recommendations of these sub- factors should be considered. Possible values are very bad, bad, medium, healthy and very healthy. Current physical status is used to assess the actual physical condition. Parameters such as pulse or blood pressure are measured. Monitoring these values in real time in the future allows not only for the initial level of risk to be calculated, but the person ought to be controlled continuously. Possible values of this sub-factor are very bad, bad, medium, good and very good. Basic physical information is about age, sex and the living conditions such as occupation (stress and activity). Possible values of this sub-factor are inactive, moderately inactive, medium, active and very active. Inputs are for each sub- factor in this group between 0 and 1.

The Activity load refers to the current activity with sub- factors Intensity (int), Duration (dur) and Frequency (freq). Intensity is given in percentage between 0 and 100 and the membership function called very light, light, medium, intensive and very intensive. Duration is between 0 and 120 minutes per occasion and the possible values are very short, short, medium, long, very long.

Finally Frequency means, how many times per week does the person do this physical activity, it can be between 0

The third main group is Environmental condition first of all at outdoor sports is important, but humidity and temperature together can influence the risk level indoor too. The first sub-factor is air temperature and humidity together (TH), because of their combination influence of thermal sensation. Possible values are low, moderately low, medium, high and very high. The other sub-factor is air temperature and wind (TW) important together also because of their effect on thermal sensation. Values are weak, moderately weak, medium, strong and very strong.

Both sub-factors have values between 0 and 1.

The output membership functions in all levels of the hierarchy are scaled in the same form. Risk level calculation for the actual group of risk factors is given by (4)-(8) with results between 0 and 1. The possible values of risk level are very safe, moderately safe, medium, moderately dangerous and very dangerous [1],[3].

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III. AGGREGATION METHODS

The results of the implication process for each rule are the output membership functions and they are the inputs of the aggregation process. Aggregation combines these fuzzy sets into a single fuzzy set. In this section the basic aggregation methods will be introduced and represented.

These methods are included in the Matlab Fuzzy Logic Toolbox.

A. Max

This method forms a single fuzzy set from the output fuzzy sets of the implication rules with their union which is calculated by (9).

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s =

(9)

B. Sum

The Sum method creates the aggregation output set simply by the calculation the bounded sum of each implication rule's output fuzzy set. The calculation method is given by (10).

( , ) min( ,1)

s a b = a b+ (10)

C. Probor

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 (11) from the truncated output functions of implication.

( ) a , b a b ab

s = + −

(11)

IV. DEFUZZIFICATION METHODS

In the case when crisp value is required as the output of the system instead of a fuzzy set, the defuzzification method must be used. This means that a specific value must be chosen which best characterizes the system. In order to achieve the corresponding result, the defuzzification method has to be selected depending on the application. In this section the basic defuzzification methods will be presented. These methods are included in the Matlab Fuzzy Logic Toolbox. The defuzzification methods are shown at the end of this section in Fig. 2.

A. Centroid of area (COA)

The result of this method is the centre of the area under the curve of the aggregated membership functions. It is one of the most commonly used defuzzification techniques. The crisp result of the centroid defuzzification method (yCOA) can be calculated by (12), where B* is the conclusion. The disadvantage of this method is that the calculation of crisp value for the complex-shaped partial conclusions is very difficult.

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B. Bisector of area (BOA)

The result of the bisector method is the vertical line that will divide the region into two sub-regions of equal area.

This method can be calculated by (13), where α=min{y;

y∈B*}, β=max{y; y∈B*}. The vertical line y=BOA partitions the region between y=α, y=β, z=0 and z=B*(y) into two regions with the same area. It sometimes, but not always coincides with the centroid line [6].

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C. Mean of Maximum (MOM)

This method is used mostly in the case when the universe has a finite number of elements. This method determines the mean of maximum value by the aggregated membership functions by (13). The advantage of this method is that it is easy to calculate. The disadvantage is that the slight change of the observation can cause large differences in the outcome [7].

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D. Largest of Maximum (LOM)

The LOM method determines the largest of the

maximum value of the area under the curve of the

aggregated membership functions.

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E. Smallest of Maximum (SOM)

The SOM method determines the smallest of the maximum value of the area under the curve of the aggregated membership functions.

If the aggregate membership function has a unique maximum, then the MOM, SOM, and LOM all take on the same value [6].

Figure 2. Defuzzification methods [6]

V. COMPARISION OF THE METHODS COMBINATIONS A. Test environment and input data selection

The models were tested in Matlab environment by the appropriate functions programming and the evaluation was simulated with Simulink model Fig. 1 use Fuzzy Logic Toolbox.

Test groups were selected based on several typical parameters of people such as average healthy adult, middle-aged person with light cardiac disease, healthy senior over 65 and person between 50 and 65 years old in bad condition. The theoretical parameters for Medical condition that have been specified for the test groups are shown in Table I. Optimal values were established by the American Heart Association (AHA) Guidelines [8],[9],[10],[11],[12]. Every group was tested based on several input parameters in both models to examine the extreme and normal values too [1].

The authors have tested the model with these test groups for all combinations of aggregation and defuzzification methods that are included in the Matlab environment.

B. The method of the comparision and its result

After the pairing of the possible aggregation and defuzzification methods fifteen different combinations are received. The test has been executed for four hundred eighty different cases for each test group for the above mentioned fifteen different combinations. Several values have been calculated to characterize the deviation of these results from the result of the reference model (AHP- FCE). These characteristic values are the minimum, maximum, mean and dispersion of the deviation and the correlation coefficient of the two models. Since the range for each risk level is 0.2, it was also counted the number

of cases when the deviation is greater than or equal to 0.1.

The goal was to find the combination of the aggregation and defuzzification methods which can be used reliably in all groups of the patients. As a result of the calculations it is obtained that there is no combination of aggregation and defuzzification methods that is clearly the best in all groups. Nevertheless there are two combinations that work in almost all groups better than the others. In both combinations the aggregation method is Sum and the defuzzification methods are Bisector of area and Centroid of area. Based on the correlation coefficient (Corr) for three groups they are the best methods. These groups are Average healthy adult (Table II.), Healthy senior over 65 and Person between 50 and 65 (Table III.). The only group where their correlation coefficient is somewhat worse is Middle-aged person with light cardiac disease. If the mean and dispersion (Disp) of deviation are considered, these combinations are every time among the best again. In addition to the testing of the minimum (Min) and maximum (Max) deviations, furthermore number of cases when the deviation is greater than or equal to 0.1 (≥0.1) were also examined. As it was explained above the maximum value with the number of cases when the deviation is greater than or equal to 0.1 are more important than the minimum value of the examination. This is because these values should not be too large on account of the range of risk levels. In this part of the examination it has been found again that the sum aggregation method with centroid or bisector defuzzification, are among the best.

The examination showed that there is a significant difference between the results depending on which methods' and operators' combination is used. It has been observed that the difference between the mean deviation of chosen combination and the result of methods with the largest mean deviation is between 0.1222 and 0.2025.

The results for the other calculated characteristic values are similar in magnitude. The chosen methods combinations’ difference from the worst combinations for maximum deviation is between 0.2100 and 0.3012.When examining the number of cases, and when the deviation is greater than or equal to 0.1, the difference is also very big. Between the chosen combination and the worst combinations it is between 337 and 404 cases, while the number of total cases is 480. It can be seen from the results that the proper choice of the methods combination suited for the application is essential.

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 bad

condition 0,25 0,3 0,5

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After the calculation of the deviances’ characteristic values, the two selected combinations of aggregation and defuzzification methods are further examined. It is important that small changes in the values do not cause a large gap in the risk level during the evaluation. For this reason the control surface of the risk level calculation has been viewed in Fuzzy Logic Toolbox. Analyzing the two selected combination of the methods it is observed that the surface is consistent as it is shown in Fig. 3 and Fig.

4. There (is) are no great level jumps, so they meet the above explained criteria. An example for the case when this condition is not met is shown in Fig. 5. This is a risk level calculation with the same model where the aggregation is also the sum method, but the defuzzification method is SOM. Based on these figures it can be seen how important the method of choice is.

Figure 3. Surface of risk level with sum and centroid methods

Figure 4. Surface of risk level with sum and bisector methods TABLE III.

PERSON BETWEEN 50 AND 65 YEARS OLD IN BAD CONDITION

Method Min Max 0,1 Mean Disp Corr Centroid

max 0,0002 0,1126 2 0,0336 0,0258 0,7673 Bisector

max 0,0005 0,1302 15 0,0375 0,0294 0,7840 MOM

max 0,0014 0,2044 181 0,0874 0,0550 0,6067 LOM

max 0,0008 0,3656 296 0,1528 0,1101 0,8019 SOM

max 0,0256 0,2544 337 0,1371 0,0556 0,6095 Centroid

sum 0,0001 0,0834 0 0,0304 0,0210 0,8077 Bisector

sum 0,0004 0,0835 0 0,0306 0,0226 0,8066 MOM

sum 0,0003 0,1312 21 0,0419 0,0313 0,7812 LOM

sum 0,0001 0,1384 36 0,0487 0,0311 0,7479 SOM

sum 0,0009 0,2409 197 0,0961 0,0577 0,7588 Centroid

probor 0,0001 0,0834 0 0,0332 0,0219 0,7649 Bisector

probor 0,0002 0,0926 0 0,0320 0,0219 0,7779 MOM

probor 0,0003 0,1194 21 0,0433 0,0312 0,6740 LOM

probor 0,0000 0,1384 34 0,0477 0,0309 0,7678 SOM

max 0,0009 0,2209 192 0,0924 0,0503 0,7067 TABLE II.

AVERAGE HEALTHY ADULT

Method Min Max 0,1 Mean Disp Corr Centroid

max 0,0001 0,2011 182 0,0760 0,0498 0,8267 Bisector

max 0,0001 0,1894 145 0,0799 0,0391 0,8495 MOM

max 0,0001 0,1493 33 0,0537 0,0300 0,6533 LOM

max 0,0004 0,5006 455 0,2658 0,1205 0,7886 SOM

max 0,0001 0,1499 49 0,0411 0,0362 0,6556 Centroid

sum 0,0001 0,2001 70 0,0644 0,0413 0,8717 Bisector

sum 0,0001 0,1994 64 0,0633 0,0414 0,8668 MOM

sum 0,0001 0,2493 189 0,0836 0,0593 0,8320 LOM

sum 0,0006 0,2493 244 0,1017 0,0560 0,8109 SOM

sum 0,0002 0,2493 112 0,0696 0,0594 0,8055 Centroid

probor 0,0000 0,1922 136 0,0701 0,0456 0,8408 Bisector

probor 0,0001 0,1894 140 0,0689 0,0449 0,8475 MOM

probor 0,0001 0,2132 152 0,0687 0,0498 0,7692 LOM

probor 0,0006 0,2832 252 0,1080 0,0654 0,8157 SOM

max 0,0001 0,1636 44 0,0477 0,0365 0,7637

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Figure 5. Surface of risk level with sum and SOM methods C. Improvement possibilities

− Further studies using aggregation operators

− Using weighted aggregation operators in the evaluation process to point out important output functions of the resolution and implication

− To use weights in our model at factors and sub- factors to point out the major elements in the risk calculation and to find the best way to determine these weights

VI. CONCLUSION

The authors have tested a hierarchical fuzzy-based model that is used for body physical exercise risk level calculation. The test has been executed in Matlab environment by the appropriate functions programming and the evaluation was simulated with Simulink and Fuzzy Logic Toolbox. The test groups were selected based on several typical theoretical parameters of people.

The results have been computed for these groups for different combination of aggregation and defuzzification methods included in the Matlab environment. Several values have been calculated to characterize the deviation of these results from the result of the reference model (AHP-FCE). The goal was to find the combination of the aggregation and defuzzification methods which can be used reliably in all groups of the patients. As a result of the calculations it is obtained that there are two combinations that work in almost all groups better than the others. In both combinations the aggregation method is Sum and the defuzzification methods are Bisector of area or Centroid of area. Examining the difference of

characteristic values between the chosen combinations and the worst combinations, a large difference can be seen. That means that the proper choice of the methods combination suit for the application is essential. When the surface of the risk level calculation is observed in Fuzzy Logic Toolbox, it also supports this conclusion.

ACKNOWLEDGMENT

The research was supported by the Research Grant of Óbuda University (ÓE-RH 1186/2-2011) and Vojvodina Secretary of the Science and Technological development (title of the project: Mathematical Models for Decision Making under Uncertain Conditions and Their Applications).

REFERENCES

[1] E. Tóth-Laufer, M. Takács, Risk Level Calculation for Body Physical Exercise with Different Fuzzy Based Methods, 12th IEEE International Symposium on Computational Intelligence and Informatics (CINTI 2011), Budapest, Hungary, November 21-22, 2011, ISBN: 978-1-4577-0043-9, pp:583-586

[2] Y. Kleiner, B. Rajani, R. Sadiq, Failure risk management of buried infrastructure using fuzzy-based techniques, Journal of Water Supply Research and Technology: Aqua, Vol. 55, no. 2, pp.81-94, March (2006)

[3] Y. Wu, Y. Ding, H. Xu, Comprehensive Fuzzy Evaluation Model for Body Physical Exercise, Risk Life System Modeling and Simulation Lecture Notes in Computer Science, 2007, Volume 4689/2007, pp.227–235, DOI: 10.1007/978-3-540-74771-0_26.

[4] M. Takács, Multilevel Fuzzy Approach to the Risk and Disaster Management, Acta Polytechnika Hungarica, Vol. 7, Issue No.4.

(2010).

[5] M. Takács, Extended Fuzzy Methods in Risk Management, Proc., of 14th WSEAS International Conference on Applied Mathematics, ISBN 978-960-474-138-0, pp-300-304.

[6] Fuzzy Logic Toolbox – Defuzzification Methods, Mathworks, http://www.mathworks.com/products/fuzzy-

logic/demos.html?file=/products/demos/shipping/fuzzy/defuzzdm.

html#2

[7] T. Kóczy, D. Tikk, Fuzzy rendszerek, Kempelen Farkas Tankönyvtár, (2001)

[8] J American Heart Association Exercise Guidelines, doi:

10.1161/01.CIR.96.1.345.

[9] Physical activity and Public health Guideline, http://circ.ahajournals.org/content/116/9/1094.full.pdf

[10] American Heart Association Recommended Exercise, http://www.livestrong.com/article/124077-american-heart-

association-recommended-exercise/.

[11] American Heart Association & Exercise, http://circ.ahajournals.org/content/116/9/1081.full.pdf.

[12] AHA Guidelines on Exercise for Seniors, http://www.livestrong.com/article/529168-aha-guidelines-

onexercise-for-seniors/.

Ábra

Figure 1.   The model structure in Simulink
Figure 2.   Defuzzification methods [6]
Figure 3.   Surface of risk level with sum and centroid methods
Figure 5.   Surface of risk level with sum and SOM methods  C.  Improvement possibilities

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