Fuzzy and Rough Techniques in Medical Diagnosis and Medication

200 Pages·2007·1.3 MB·English

by  Elisabeth Rakus-Andersson

ElisabethRakus-Andersson
FuzzyandRoughTechniquesinMedicalDiagnosisandMedication
StudiesinFuzzinessandSoftComputing, Volume212
Editor-in-chief
Prof.JanuszKacprzyk
SystemsResearchInstitute
PolishAcademyofSciences
ul.Newelska6
01-447Warsaw
Poland
E-mail:[email protected]
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Elisabeth Rakus-Andersson
Fuzzy
and Rough Techniques
in Medical Diagnosis
and Medication
ABC
ElisabethRakus-Andersson
DocentinMathematics
BlekingeInstituteofTechnology
DepartmentofMathematicsandScience
37179Karlskrona
Sweden
E-mail:[email protected]
LibraryofCongressControlNumber:2006937349
ISSNprintedition:1434-9922
ISSNelectronicedition:1860-0808
ISBN-10 3-540-49707-2SpringerBerlinHeidelbergNewYork
ISBN-13 978-3-540-49707-3SpringerBerlinHeidelbergNewYork
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Contents
List of Figures......................................................................................................IX
List of Tables........................................................................................................XI
1 Introduction....................................................................................................1
2 Fundamental Items.......................................................................................3
2.1 Introduction............................................................................................3
2.2 Fuzzy Sets..............................................................................................4
2.3 Basic Operations on Fuzzy Sets...........................................................12
2.4 Linguistic Variables.............................................................................19
2.5 Fuzzy Relations....................................................................................23
3 Medical Diagnosis........................................................................................31
3.1 Introduction..........................................................................................31
3.2 The Modus Ponens Law in Medical Diagnosis...................................31
3.3 The Patient – Symptom Relation ........................................................33
3.3.1 Simple Qualitative Biological Parameters.....................................33
3.3.2 Compound Qualitative Biological Features...................................34
3.3.3 Compound Quantitative Biological Features.................................40
3.4 The Symptom – Diagnosis Relation ....................................................45
3.4.1 Numerical Representations of Linguistic Variables......................46
3.4.2 Relations of “Presence” and “Decisive Character”.......................52
3.5 The Patient – Diagnosis Relation........................................................55
4 Complementary Solutions in Diagnostic Models.....................................63
4.1 Introduction..........................................................................................63
4.2 OWA Operators in Decision Relations................................................63
4.3 Fuzzy Set Distances in Diagnostic Decisions......................................71
4.4 Diagnostic Processes Extended in Time Intervals...............................80
4.5 Rough Set Theory in the Classification of Diagnoses.........................87
5 Evaluation of Medicine Action Levels......................................................93
5.1 Introduction..........................................................................................93
5.2 Theoretical Assumptions of Eigen Fuzzy Problem.............................93
5.3 Eigen Sets in Medicine Effectiveness Levels....................................100
VIII Contents
5.4 Order Operations on Fuzzy Numbers...............................................104
5.5 Eigen Fuzzy Sets with Fuzzy Numbers............................................114
5.6 The L-R Fuzzy Numbers as Drug Efficiency Intervals....................119
6 The Choice of Optimal Medicines...........................................................127
6.1 Introduction........................................................................................127
6.2 Fuzzy Utilities in Decision-Making Models.....................................127
6.2.1 Jain’s Utility Matrix as the Drug – Symptom Table.....................128
6.2.2 The Solution of Jain’s Decision Case..........................................134
6.3 Group Decision-Making in the Selection of Drugs...........................140
6.4 Unequal Objectives in the Choice of Medicines................................144
6.4.1 The Design of Objectives-Constraints.........................................145
6.4.2 The Power-Importance of Objectives..........................................148
6.4.3 Minimization of Regret................................................................151
7 Approximation of Clock-like Point Sets.................................................155
7.1 Introduction........................................................................................155
7.2 Fitting of (cid:83)-functions to Clock-like Polygons...................................155
7.3 Rough Sets in Classifying of Clock-like Polygons............................165
7.4 s-functionsinFittingtoLetter-shapedPolygons...............................168
7.5 The Classification of Letter-shaped Polygons...................................179
References...........................................................................................................183
Index....................................................................................................................189
List of Figures
2.1 The characteristic function of the crisp set C = [4, 8]……………..……...4
2.2 Fuzzy set A = {(x,(cid:80)(x))},x(cid:143) [0, 10]……………………………………6
A
2.3 The continuous fuzzy set “young”………………………………………..7
2.4 The function s(x, 25, 37.5, 50)……………………………........................8
2.5 The membership function of the set “young” as the s-class function…….9
2.6 The function (cid:83)(x,20,45)…………………………………………………10
2.7 The fuzzy sets “young”, “middle-aged” and “old” in X = [0, 100]……...11
2.8 The membership function of the set “body temperature”……………….12
2.9 The union of fuzzy sets A and B from Ex. 2.10…………………………13
2.10 TheintersectionofAandBfromEx.2.10……………………………....14
2.11 ThecomplementofthesetBfromEx.2.10…………………………….15
2.12 The membership function of the set “young or experienced”..................16
2.13 Themembershipfunctionoftheset“youngandexperienced”…………17
2.14 The constraint R(“often”) over the reference set A = [0, 100]…..............21
2.15 The fuzzy constraints of terms representing “presence of S in D”……...22
2.16 “Astrong and proportional construction of the man’s body”…………..25
3.1 The membership function of the compound qualitative symptom S……37
j
3.2 The membership function of the symptom S …………………………...42
8
3.3 The membership function of the symptom S with modifications………43
8
3.4 The membership functions of fuzzy variables generated by “seldom”….49
3.5 The membership functions of fuzzy variables generated by “often”……50
3.6 The terms from the lists “presence” and “decisive character”………….51
4.1 The comparison of sets PD and AP(PD ) for patient P………………...73
3 3
4.2 The distance between AP(PD ) and AP(PD )………………………...74
3 D 3
1
4.3 The distance between AP(PD ) and AP(PD )………………………..74
3 D 3
3
4.4 The comparison between PD (or PD ) and JP(PD ) (or JP(PD ))…….76
4 5 4 5
5.1 A(cid:142)B for (cid:80)(x) = s(x, 30, 50, 70) and (cid:80)(x) = s(x, 10, 50, 90)………….95
A B
5.2 The fuzzy number N = (40, 2, 3)……………………………………….105
5.3 Minimum for N = (25, 2, 3) and N = (40, 1, 5) according to (5.13)….106
1 2
5.4 Minimum for N = (40, 2, 3) and N = (42, 1, 5) made by (5.14)…...…107
1 2
5.5 Maximum for N = (25, 2, 3) and N = (40, 1, 5) due to (5.15)………..108
1 2
5.6 Maximum for N = (40, 2, 3) and N = (42, 1, 5) computed by (5.16)...109
1 2
5.7 MinimumforN = (40, 7, 8) and N = (42, 3, 2) as the result of (5.14)..113
1 2
X List of Figures
5.8 Minimum for N = (40, 7, 8) and N = (42, 3, 2) due to (5.17)………...114
1 2
5.9 The type-1 fuzzy set B = {(10, 0.2), (20, 0.3), (30, 0.15)}……………116
1
5.10 The type-2 set B = {(10, (20, 3, 2)), (20, (30, 5, 7)), (30, (15, 3, 4))}...117
2
5.11 Terms of drug effectiveness expressed as fuzzy numbers……………..120
5.12 Fuzzy numbers A(S ),A(S ),A(S ) of the eigen set A of the relation R..126
1 2 3
6.1 The fuzzy constrains R –R …………………………………………... 131
1 11
6.2 The 0.5-level of A characterized by (cid:83)(x, 20, 50)……………………….141
6.3 The set R as the directed graph………………………………………..143
1
6.4 The set R as the directed graph……………………………………...143
0.83
6.5 The set R as the directed graph……………………………………...144
0.67
7.1 The polygon reflecting A = {(x,y)}……………………………………156
7.2 The polygon representing A …………………………………………...157
1
7.3 The (cid:83)-function for (cid:68) = 30, (cid:74) = (cid:68) = 32.5, (cid:74) = 35 and (cid:72) = 0.25……...159
1 1 2 2
7.4 The approximation of A by the truncated (cid:83)-function…………………162
1
7.5 The approximation of A* by truncated (cid:83) functions……………………163
7.6 The sampled (cid:83) in the approximation of A……………………………...165
7.7 (cid:83)–(cid:83) in approximation of polygons A –A …………………………….167
1 6 1 6
7.8 The example of a letter-shaped polygon reflecting A = {(x,y)}……….169
7.9 The approximation of A by “sampled truncated s”…………………….173
7.10 The approximated polygons A1–A5…………………………………….175
7.11 The curves A1–A5 with their start points at the origin………………….176
7.12 The curves A1–A5 over the common interval [0, 1]…………………….178