VIETNAM NATIONAL UNIVERSITY VNU UNIVERSITY OF SCIENCE ROAN THI NGAN SOME NEW MEASURES AND REPRESENTATIONS OF INTUITIONISTIC FUZZY SYSTEMS AND APPLICATIONS PHD DISSERTATION IN MATHEMATICS Hanoi - 2021 VIETNAM NATIONAL UNIVERSITY VNU UNIVERSITY OF SCIENCE ROAN THI NGAN SOME NEW MEASURES AND REPRESENTATIONS OF INTUITIONISTIC FUZZY SYSTEMS AND APPLICATIONS Major: Applied Mathematics Code: 9460112.01 PHD DISSERTATION IN MATHEMATICS RESEARCH SUPERVISOR: 1. Bui Cong Cuong 2. Le Hoang Son Hanoi - 2021 DECLARATION This is to certify that to the best of my knowledge, the content of this thesis is my own work. This thesis has not been submitted for any degree or other purposes.
The experimental datasets in this thesis are clearly derived and published in accordance with regulations. The research results presented in this thesis are honest and objective. This thesis contains no material that is previously published, except that citations that have been clearly specified. The co-authors totally agree for me to use the content of our publications for the purpose of writing and reporting the dissertation at all levels.
Here, if anything goes wrong, I assume full responsibility. Hanoi, 2021 PhD Candidate Roan Thi Ngan ACKNOWLEDGEMENT First of all, I would like to express my sincere and profound gratitude to my research supervisors, Assoc. Bui Cong Cuong and Assoc. Le Hoang Son.
I am extremely grateful for the dedicated and valuable help that the supervisors have given me during the process of implementing the thesis. They have given me a lot of attention, guidance and valuable help not only in scientific research but also in life. I would like to express my sincere thanks to the teachers, research group, and brothers in the Center for High-Performance Computing, University of Science, the laboratory of the Department of Multimedia and Virtual Reality, VNU Information Technology Institute, and the Neuro-Fuzzy Systems with Applications seminar group for their. I would like to express my sincere thanks to Prof.
Pham Ky Anh and the other members in the Department of Computational and Applied Mathe- matics, Faculty of Mathematics, Mechanics and Informatics in particular and the University of Science, VNU in general. The comments after seminar re- ports, as well as the management of training, research environment, and fa- cilitation of the department and the university, help me a lot in completing this thesis. I would like to thank the project 911 of the Ministry of Education and Train- ing and the European Union’s Erasmus Program for giving me the opportu- nity to develop my research. I would like to express my gratitude to Assoc.
Vu Van Manh and Assoc. Juan-Miguel Martinez Rubio, for their support during the scholarship application. I would also like to sincerely thank colleagues in the Faculty of General Sci- ence in particular, Hanoi University of Natural Resources and Environment in general for creating all favorable conditions. Finally, this thesis will not be complete without the encouragement and support in all aspects of the family.
This thesis is to send to my family mem- bers, with all my deepest gratitude. ii CONTENTS Page DECLARATION ACKNOWLEDGEMENT 1i CONTENTS ABBREVIATIONS LIST OF FIGURES LIST OF TABLES 1 INTRODUCTION & PRELIMINARY 11 PROBLEMS.21 Intuitionistic Fuzzy Measure .2 Intuitionistic Fuzzy Representation .1 Intuitionistic Fuzzy Measure .2 Intuitionistic Fuzzy Representation .1 Fuzzy Set and Intuitionistic Fuzzy Set .2 Intuitionistic Fuzzy Order and Operations 1 1. Intuitionistic Fuzzy Relation and Similarity Measures.4 Complex Fuzzy Set and Complex Intuitionistic Fuzzy Set 35 1.5 Intuitionistic Fuzzy Systems. 38 DESIGN NEW METRICS FOR INTUITIONISTIC FUZZY SYSTEMS 39 2.1 d5—Equalities of Intuitionistic Fuzzy Sets.2 d6—Equalities for Intuitionistic Fuzzy Operations and Re- EU.1 H-max Distance Measure of Intuitionistic Fuzzy Sets .2 Distance Measure of IFSs with the Intuitionistic Fuzzy T-normand T-conorm.4 EXPERIMENT ON UCI MEDICAL DATA.
Medical Diagnosis based on the 6—Equality Measure .2 Medical Diagnosis based on the H-max Measure.5 EXPERIMENT ON DENTAL IMAGE DATA .1 Diagnosis Method based on the Modified H-max Measure 80 2.2 Experimental Environment and Datasets. 86 NEW REPRESENTATIONS OF INTUITIONISTIC FUZZY SYSTEMS UNDER COMPLEX SET 87 31 INTRODUCHON. IFS-C: INTUITIONISTIC FUZZY SYSTEMS BASED ON COM- PLEXNUMBERS.1 Polar Representation and a New Order Relation of IFSs .2 A New Distance Measure of IFS-C. REPRESENTING COMPLEX INTUITIONISTIC FUZZY SET BY QUATERNIONNDMBERS.1 A New Representation of Complex Intuitionistic Fuzzy Sets based on Quaternion Numbers .2 Logic and Algebraic Operations.
Quaternion Distance Measure.4 EXPERIMENT ON BENCHMARK MEDICAL DATASETS .1 PDM Decision-Making Model .2 Decision-Making Model based on Quaternion Distance Measures.5 CONCLUDING REMARKS 125 CONCLUSION 127 LIST OF PUBLICATIONS 129 REFERENCES 130 APPENDIX 145 ABBREVIATIONS APC Affinity Propagation Clustering BCW Breast Cancer Wisconsin CFC Complex Fuzzy Class CFSs Complex Fuzzy Sets CIFSs Complex Intuitionistic Fuzzy Sets CIFSs-Q Complex Intuitionistic Fuzzy Sets-Quaternion CM-SPA Cloud Model-Set Pair Analysis C-ODM Cartesian-ODM CTG Cardiotocography DDS Dental Diagnosis System Dermal Dermatology DIHM Diagnosis from Image based on H-Max Measure DIMHM Diagnosis from Image based on Modified H-Max Measure DRD Diabetic Retinopathy Debrecen EEI Edge-value and Intensity FIS Fuzzy Inference System FKNN Fuzzy K-Nearest Neighbors FSs Fuzzy Sets GCK Kruskal spanning tree GCP Prim spanning tree GRA Gradient Feature H-max Hamming-max HS Haberman’s Survival IFR(X x Y) The set of all IFRs on X x Y IFRs Intuitionistic Fuzzy Relations IFS(U) The set of all IFSs on U IFSs Intuitionistic Fuzzy Sets IFSs-C Intuitionistic Fuzzy Systems-Complex ILPD Indian Liver Patient I-TSFIS Intuitionistic Time Series Fuzzy Inference System LBP Local Patterns Binary Feature LD Liver-Disorders MAE Mean Absolute Error MSE Mean Squared Error NaN Not-a-Number PDM P-Distance Measure PIDD Pima Indians Diabetes Data Set P-QDM Polar representation of QDM QDM Quaternion Distance Measure RGB Red-Green-Blue Sec Seconds SPA Set Pair Analysis SVM Support Vector Machine t-conorm triangle-conorm t-norm triangle-norm TOPSIS Technique for Order of Preference by Similarity to Ideal Solution t-representable triangle-representable UCI University of California, Irvine Machine Learning Repository WX-H-max Wang and Xin-Hamming-max LIST OF FIGURES 1.1 The change between IFSs.2 The order relation between two IFSs AandB. The histogram of the 1° attribute of the ILPD Data.4 The histogram of the 8" attribute of the ILPD Data.5 The dental X-ray images with the corresponding diseases.6 The white-black colorstrip[77].7 Basic structure of a fuzzy system[53]. 37 21 Optimizing the thresholdvalue.2 The proposed model for medical diagnosis .3 Optimizing the disease threshold in proposed diagnosis model 75 2.4 The proposed model for medical diagnosis. eee ee eee 83 2.
The order relation <œonL”. Graphical representation of the products of unit quaternions as a 90°-rotation in 4D-space .4 Graphical representation of the complex degrees .5 The graphical representation of complex degrees in Polar form .6 The order relation <,onQ*. Optimizing the assessment threshold.8 Decision Making Model based on P-Distance Measure.10 Optimizing the assessment threshold for the QDM model.11 A diagram oftheQDM model.12 An illustration of the functions Z;, B „t0, and |.13 The MAE results of the considered algorithms on ILPD Data .14 The MAE results of the considered algorithms on Diabetes Data 124 3.15 The MAE results of the considered algorithms on HS Data .16 The MAE results of the considered algorithms on Ecoli Data .17 The MAE results of the considered algorithms on BCW Data. 125 LIST OF TABLES 1.1 The descriptions of experimental datasets.1 The values of dyam,de,dHay,dwxe.2 The values of dpam,de,dHan, dwx dome.3 Q, is intuitionistic fuzzy relation between the set of patients P and the set of symptoms S with the data from the first group of decision makers.4 Q¿ is intuitionistic fuzzy relation between the set of patients P and the set of symptoms S with the data from the second group of decision makers.5 Q3 is intuitionistic fuzzy relation between the set of patients P and the set of symptoms S with the data from the third group of decision makers.7 R is intuitionistic fuzzy relation betweenSandD.8 Ro Qis intuitionistic fuzzy relation between PandD.9 SRoo (p,d) where red values show the most severe diseases of apatent.10 SroQ, (p,d) where red values show the most severe diseases of apatient.11 The MAE values of the algorithms are compared on 5 selected datasetS.12 The computational time of the algorithms are compared on 5 selected datasets.
ee ee ee 2.13 Symptoms characteristic for the patients considered.14 Symptoms characteristic for the diagnoses considered.15 Diagnosed results for the proposed distance measure đr;„ given by (2.16 Comparison of all algorithms.17 The MAE values of the algorithms are compared on 11 selected datasets.18 The computational time of the algorithms are compared on 11 selected datasets.19 The fuzzified dataset.20 The performance of 7methods.1 The MAEs and computational time of the PDM and the related methods on 3 considered datasets.2 m records of a dataset encoded in the Cartesian form of quater- nionnumbers.0 00 eee eee eee 116 3.3 Records on Viral Fever Iisease.4 The MAE values of the methods on the benchmark medical data.5 Total time (sec) of the methods on the benchmark medical data. 123 Chapter 1 INTRODUCTION & PRELIMINARY 1.1 PROBLEMS Logic and sets are among the fundamental concepts that are considered as the building blocks of mathematical foundations. To solve practical prob- lems that contain ambiguity and uncertainty, the concept of fuzzy sets, born in 1965, replaces the crisp set by describing the membership function whose range is in [0,1]. In 1983, in order to access information with indecisive fac- tors, fuzzy sets were extended by the concept of intuitionistic fuzzy sets by defining the function of non-membership.
Up to now, the theory of intuition- istic fuzzy systems has been studied and applied to many fields, but there are still some problems, including two major ones as follows. Problem 1: Intuitionistic fuzzy measure. - Some measures have not yet been extended for intuitionistic fuzzy sets, such as proximity measure of Pappis [80]. - Some distance measures do not satisfy the condition regarding the inclu- sion relation between intuitionistic fuzzy sets on a finite space of points, that is, if A, B and C are three intuitionistic fuzzy sets on a finite space of points X = {x1,x2,.,Xm} and A C BC C, thend (A,B) < d(A,C) and d(B,C) < d(A,C).
For instance, for A, B and C being three intuitionistic fuzzy sets on X = {x}, where their membership degrees are 4 = 0.3, and their non-membership degrees are v4 = 0.6, and d is the Euclidean measure [109], a(A,B) = ( ((fa — Ho)? + (va — vp)? + (xà — 7t8))) 1, 10 where 7r = 1 — #— 1, then đ(B,C) = 0. - Further, some existing measures give an equal rating that is not tight. For instance, for A, B and C being three intuitionistic fuzzy sets on X = {x}, d is the Hausdorff measure [40], d(A,B) = max {|MA — p|,|UA — vel}, and va = 0. However, in this case d(A, B) should be smaller than d(A,C) because the change from A to C is more soundness than that from A to B.
Specifically, from A to C, the mem- bership degree (e. the support degree in the election) decreases and at the same time the non-membership degree (e. the opposition degree in the election) increases. This is illustrated in Figure 1.1, where the green part includes the points that have the degrees of membership and non- membership being both less than those of the point A.1: The change between IFSs Problem 2: Intuitionistic fuzzy representation.
- To build a measure between objects, we firstly need to define the concept of the order relation between them. On the existing representations of intuitionistic fuzzy sets in the literature, building a total order relation is of remarkable complication since various evaluation steps are employed along the way, such as the order of Xu and Yager [131] must be based on 11 the intermediate functions, or the score function, S=p-v, and the accuracy function, H=pt+v. For instance, to compare two intuitionistic fuzzy sets A and B on X = {x}, where pa = 0.2, we need to evaluate the score values, Sa = Sz = 0.4, and the accuracy values, Ha = 0.8, and then A is smaller than B (see Figure 1.