International Journal of Mathematics Trends and Technology (IJMTT) – Volume 58 Issue 1 – June 2018
Execution of N-Valued interval neutrosophic sets in medical diagnosis A.Edward Samuel#1, R.Narmadhagnanam*2 #
Ramanujan Research Centre, P.G. & Research Department of Mathematics, GAC(A),Kumbakonam,TN,India.
Abstract In this paper, cosecant similarity measure among n-valued interval neutrosophic sets are proposed and some of its properties are discussed herein. Finally, an application of medical diagnosis is presented to find out the disease impacting the patient. Keywords N-valued interval neutrosophic set, cosecant similarity measure, medical diagnosis. I. INTRODUCTION A number of real life problems in engineering, medical sciences, social sciences, economics etc., involve imprecise data and their solution involves the use of mathematical principles based on uncertainty and imprecision. Such uncertainties are being dealt with the help of topics like probability theory, fuzzy set theory [1], rough set theory [2] etc., Healthcare industry has been trying to complement the services offered by conventional clinical decision making systems with the integration of fuzzy logic techniques in them. As it is not an easy task for a clinician to derive a fool proof diagnosis, it is advantageous to automate few initial steps of diagnosis which would not require intervention from an expert doctor. Neutrosophic set which is a generalized set possesses all attributes necessary to encode medical knowledge base and capture medical inputs. As medical diagnosis demands large amount of information processing, large portion of which is quantifiable, also intuitive thought process involve rapid unconscious data processing and combines available information by law of average, the whole process offers low intra and inter personal consistency. So contradictions, inconsistency, indeterminacy and fuzziness should be accepted as unavoidable as they are integrated in the behaviour of biological systems as well as in their characterization. To model an expert doctor it is imperative that it should not disallow uncertainty as it would be then inapt to capture fuzzy or incomplete knowledge that might lead to the danger of fallacies due to misplaced precision. As medical diagnosis contains lots of uncertainties and increased volume of information available to physicians from new medical technologies, the process of classifying different sets of symptoms under a single name of disease becomes difficult. In some practical situations, there is the possibility of each element having different truth membership, indeterminate and false membership functions. The unique feature of n-valued interval neutrosophic set is that it contains multi truth membership, indeterminate and false membership. By taking one time inspection, there may be error in diagnosis. Hence, multi time inspection, by taking the samples of the same patient at different times gives the best diagnosis. So, n-valued interval neutrosophic sets and their applications play a vital role in medical diagnosis. In 1965, Fuzzy set theory was firstly given by Zadeh [1] which is applied in many real applications to handle uncertainty. Sometimes membership function itself is uncertain and hard to be defined by a crisp value. So the concept of interval valued fuzzy sets was proposed to capture the uncertainty of grade of membership. In 1986, Atanassov [3] introduced the intuitionistic fuzzy sets which consider both truth-membership and falsitymembership. De et al [4] presented an application of intuitionistic fuzzy set in medical diagnosis. Jun Ye [5] introduced the concept of cosine similarity measures for intuitionistic fuzzy sets. Tian Maoying [6] presented the cotangent similarity function for intuitionistic fuzzy sets. Later on, intuitionistic fuzzy sets were extended to the interval valued intuitionistic fuzzy sets. Intuitionistic fuzzy sets and interval valued intuitionistic fuzzy set scan only handle incomplete information not the indeterminate information and inconsistent information which exists commonly in belief systems. So, Neutrosophic set (generalization of fuzzy sets, intuitionistic fuzzy sets and so on) defined by Florentin Smarandache [7] has capability to deal with uncertainty, imprecise, incomplete and inconsistent information which exists in real world from philosophical point of view. Wang et al [8] proposed the single valued neutrosophic set. Similarity and entropy between neutrosophic sets were proposed by Mamjumdar and Samanta [9]. Wang et al [10] proposed the set theoretic operations on an instance of neutrosophic set is called interval valued neutrosophic set which is more flexible and practical than neutrosophic
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International Journal of Mathematics Trends and Technology (IJMTT) – Volume 58 Issue 1 – June 2018
set. Similarity measures between interval valued neutrosophic sets were proposed by Jun Ye [11]. Interval valued neutrosophic soft sets were introduced by Irfan Deli [12]. Sebastian and Ramakrishnan [13] studied a new concept called fuzzy multi sets (FMS), which is the generalization of multi sets. Shinoj and John [14] extended the concept of fuzzy multi sets by introducing intuitionistic fuzzy multi sets (IFMS). Rajarajeswari and Uma [15] proposed the normalized hamming similarity measure between them. However, the concepts of FMS & IFMS are not capable of dealing with indeterminacy. Shan Ye and Jun Ye[16]introduced the concept of single valued neutrosophic multi sets. Distance based similarity measures between them were introduced by Ye et al [17]. Florentin Smarandache [18] extended the classical neutrosophic logic to n-valued refined neutrosophic logic, by refining each neutrosophic component into respectively, T 1 , T 2 , , T m , I 1 , I 2 , , I p & F1 , F 2 , , F r . Deli et al [19] studied a new concept called neutrosophic refined sets. Broumi and Deli [20] proposed the correlation measure between them. Broumi et al [20] generalize the concept of n-valued neutrosophic sets to the case of n-valued interval neutrosophic sets. Edward Samuel and Narmadhagnanam [21] proposed few novel methods in n-valued interval neutrosophic sets. In this paper, using the notion of n-valued interval neutrosophic set it was provided an exemplary for medical diagnosis. In order to make this, a novel method was implemented. Rest of the article is structured as follows. In Section 2, the basic definitions were briefly presented. Section 3 deals with proposed definitions and some of its properties. Sections 4,5&6 contains methodology, algorithm and case study related to medical diagnosis respectively. Conclusion was given in Section 7. II. PRELIMINARIES A. Definition [10] Let X be a space of points (objects), with a generic element in X denoted by x. An interval neutrosophic set (INS) A in X is characterized by the truth-membership function TA , indeterminacy-membership function I A and falsity-membership function FA T A x , I A x & F A x are real standard or non-standard subsets of
0,1 ,
is TA : X
IA : X
0,1 ,
FA : X
of T A x , I A x & F A x , so 0 T A x I A x F A x 3
0,1 .
0,1 . That
There is no restriction on the sum
B. Definition [20] Let X be a universe, a n-valued interval neutrosophic set on X can be defined as follows: x, inf T 1A x , sup T 1A x , inf T 2A x , sup T 2A x ,, inf T Ap x , sup T Ap x , 1 1 2 2 p p A inf I A x , sup I A x , inf I A x , sup I A x ,, inf I A x , sup I A x , where inf F 1A x , sup F 1A x , inf F 2A x , sup F 2A x ,, inf F Ap x , sup F Ap x : x X
1 A
2 A
P A
1 A
inf T ( x), inf T ( x),..., inf T ( x), inf I ( x), inf I A2 ( x),..., inf I AP ( x), inf FA2 ( x), inf FA2 ( x),..., inf FAP , sup TA1 ( x), sup TA2 ( x),..., sup TAP ( x), sup I 1A ( x), sup I A2 ( x),...,
sup I AP ( x), sup FA1 ( x), sup FA2 ( x),..., sup FAP 0,1
such that 0 sup T Aj x sup I Aj x sup F Aj x 3 j 1, 2, 3,, p C. Definition [20] A n-valued interval neutrosophic set A is contained in the other n-valued interval neutrosophic set B , denoted by A B , if and only if {inf T 1A x inf T 1B x, inf T 2A x inf T 2B x, , inf T Ap x inf T Bp x, supT 1A x supT 1B x, supT 2A x supT 2B x,, supT Ap x supT Bp x, inf I 1A x inf I 1B x , inf I 2A x inf I 2B x,, inf I Ap x inf I Bp x , sup I 1A x sup I 1B x ,
sup I 2A x sup I 2B x,, sup I Ap x sup I Bp x , inf F1A x inf F1B x, inf F 2A x inf F 2B x ,, inf F Ap x inf F Bp x,
sup F 1A x sup F 1B x , sup F 2A x sup F 2B x,, sup F Ap x sup F Bp x x X }
(1)
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III.PROPOSED DEFINITION A. Definition
x, inf TA1 x , sup TA1 x , inf TA2 x , sup TA2 x , , inf TAp x , sup TAp x , Let A = inf I 1A x , sup I 1A x , inf I A2 x , sup I A2 x , , inf I Ap x , sup I Ap x , & p p 1 1 2 2 inf FA x , sup FA x , inf FA x , sup FA x , , inf FA x , sup FA x : x X x, inf TB1 x , sup TB1 x , inf TB2 x , sup TB2 x ,, inf TBp x , sup TBp x , B = inf I 1B x , sup I 1B x , inf I B2 x , sup I B2 x ,, inf I Bp x , sup I Bp x , be two n-valued interval p p 1 1 2 2 inf FB x , sup FB x , inf FB x , sup FB x ,, inf FB x , sup FB x : x X neutrosophic sets then the cosecant similarity measure is defined as
CSM NVINS ( A, B)
1 n2
1 inf T j x inf T j x sup T j x sup T j x inf I j x inf I j x i i i i B B B i A A A i n 1 (2) cos ec j j j j j j 10 24 j 1i 1 sup I A x i sup I B x i inf F A x i inf F B x i sup F A x i sup F B x i p
Proposition 1
(i) CSM NVINS A, B 0 (ii) CSM NVINS A, B CSM NVINS B, A (iii)If A B C then
CSM NVINS ( A, C ) CSM NVINS ( A, B) & CSM NVINS ( A, C ) CSM NVINS ( B, C )
Proof (i) The proof is straightforward. (ii) The proof is straightforward (iii) By (1), inf T Aj xi inf T Bj xi inf T Cj xi sup T Aj xi sup T Bj xi sup T Cj xi inf I Aj xi inf I Bj xi inf I Cj xi
sup I Aj xi sup I Bj xi sup I Cj xi
inf F Aj xi inf F Bj xi inf F Cj xi
sup F Aj xi sup F Bj xi sup F Cj xi Hence, inf T Aj xi inf T Bj xi inf T Aj xi inf T Cj xi
supT Aj xi supT Bj xi supT Aj xi supT Cj xi inf I Aj xi inf I Bj xi inf I Aj xi inf I Cj xi
sup I Aj xi sup I Bj xi sup I Aj xi sup I Cj xi inf F Aj xi inf F Bj xi inf F Aj xi inf F Cj xi sup F Aj xi sup F Bj xi sup F Aj xi sup F Cj xi
inf T Bj xi inf T Cj xi inf T Aj xi inf T Cj xi
supT Bj xi supT Cj xi supT Aj xi supT Cj xi inf I Bj xi inf I Cj xi inf I Aj xi inf I Cj xi sup I Bj xi sup I Cj xi sup I Aj xi sup I Cj xi
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International Journal of Mathematics Trends and Technology (IJMTT) – Volume 58 Issue 1 – June 2018
inf F Bj xi inf F Cj xi inf F Aj xi inf F Cj xi sup F Bj xi sup F Cj xi sup F Aj xi sup F Cj xi Here, the cosecant similarity measure is a decreasing function. CSM NVINS ( A, C) CSM NVINS ( A, B) & CSM NVINS ( A, C) CSM NVINS ( B, C) IV. METHODOLOGY In this section, it was presented an application of n-valued interval neutrosophic set in medical diagnosis. In a given pathology, suppose S is a set of symptoms, D is a set of diseases and P is a set of patients and let Q be a n-valued interval neutrosophic relation from the set of patients to the symptoms. i.e., Q( P S ) and R be an interval neutrosophic relation from the set of symptoms to the diseases i.e., R(S D) and then the methodology involves three main jobs: 1. Determination of symptoms 2. Formulation of medical knowledge based on n-valued interval neutrosophic sets & interval neutrosophic sets 3. Determination of diagnosis on the basis of new computation technique of n-valued interval neutrosophic sets.
Step 1: Step 2: Step 3: Step 4:
V. ALGORITHM The Symptoms of the patients are given to obtain the patient-symptom relation Q and are noted in Table 1 The medical knowledge relating the symptoms with the set of diseases under consideration are given to obtain the symptom-disease relation R and are noted in Table 2. The computation T of the relation of patients and diseases is found using (2) and are noted in Table 3. Finally, maximum value from Table 3 of each row were selected to find the possibility of the patient affected with the respective disease and then it was concluded that the patient Pk k 1,2 & 3 was suffering from the disease Dr r 1,2,3 & 4 . VI. CASE STUDY [20] Let there be three patients P P1 , P2 , P3 and the set of symptoms S { S1 =Temperature, S 2 =
Cough, S 3 = Throat pain, S 4 = Headache, S 5 = Body pain}.The n-valued interval neutrosophic relation D Q( P S ) is given as in Table 1. Let the set of diseases D ={ 1 =Viral fever, D2 = Tuberculosis, D3 =Typhoid, D4 =Throat disease}.The interval neutrosophic relation R(S D) is given as in Table 2. Table 1:The relation between Patients and Symptoms (Using step1) Q
P1 P2
P3
Temperature [0.3,0.4],[0.4,0.5],[0.3,0.7] [0.0,0.3],[0.1,0.3],[0.0,0.5] [0.0,0.6],[0.4,0.5],[0.3,0.4] [0.2,0.3],[0.4,0.5],[0.1,0.2] [0.4,0.5],[0.2,0.5],[0.0,0.3] [0.6,0.7],[0.4,0.5],[0.4,0.5] [0.1,0.3],[0.0,0.5],[0.4,0.6] [0.1,0.2],[0.3,0.4],[0.2,0.5] [0.2,0.4],[0.4,0.5],[0.3,0.7]
Cough [0.1,0.2],[0.3,0.6],[0.6,0.8] [0.0,0.5],[0.4,0.7],[0.4,0.5] [0.2,0.3],[0.0,0.5],[0.4,0.6] [0.5,0.7],[0.0,0.4],[0.7,0.8] [0.6,0.7],[0.0,0.5],[0.4,0.5] [0.4,0.6],[0.2,0.7],[0.0,0.3] [0.2,0.3],[0.0,0.7],[0.1,0.4] [0.5,0.6],[0.0,0.3],[0.3,0.5] [0.3,0.5],[0.2,0.5],[0.4,0.6]
Throat pain [0.0,0.5],[0.2,0.6],[0.0,0.4] [0.3,0.4],[0.2,0.3],[0.3,0.4] [0.0,0.7],[0.3,0.7],[0.3,0.5] [0.5,0.6],[0.0,0.6],[0.2,0.3] [0.4,0.7],[0.4,0.6],[0.3,0.4] [0.1,0.3],[0.2,0.3],[0.5,0.7] [0.2,0.4],[0.3,0.6],[0.0,0.6] [0.4,0.5],[0.0,0.3],[0.3,0.4] [0.5,0.7],[0.4,0.6],[0.3,0.7]
Headache [0.2,0.3],[0.3,0.5],[0.0,0.7] [0.4,0.5],[0.4,0.7],[0.3,0.6] [0.2,0.6],[0.0,0.6],[0.3,0.4] [0.2,0.5],[0.5,0.6],[0.1,0.5] [0.2,0.3],[0.2,0.5],[0.5,0.6] [0.1,0.3],[0.3,0.4],[0.4,0.5] [0.2,0.3],[0.5,0.6],[0.4,0.5] [0.2,0.4],[0.0,0.4],[0.2,0.7] [0.4,0.5],[0.2,0.3],[0.3,0.5]
R Temperature Cough Throat pain Headache
Table 2:The relation among Symptoms and Diseases (Using step2) Viral fever Tuberculosis Typhoid [0.2,0.4],[0.3,0.5],[0.3,0.7] [0.1,0.4],[0.2,0.6],[0.6,0.7] [0.0,0.3],[0.4,0.6],[0.0,0.2] [0.2,0.4],[0.2,0.3],[0.0,0.5] [0.3,0.4],[0.2,0.5],[0.7,0.8] [0.3,0.4],[0.2,0.3],[0.1,0.2] [0.0,0.4],[0.2,0.4],[0.2,0.4] [0.0,0.2],[0.3,0.6],[0.6,0.7] [0.1,0.2],[0.4,0.5],[0.3,0.4] [0.4,0.7],[0.0,0.3],[0.3,0.5] [0.1,0.2],[0.0,0.5],[0.0,0.6] [0.3,0.4],[0.2,0.3],[0.2,0.5]
Body pain
[0.1,0.4],[0.2,0.5],[0.3,0.4]
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[0.5,0.7],[0.4,0.5],[0.2,0.5]
[0.2,0.3],[0.2,0.4],[0.2,0.3]
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Body pain [0.0,0.4],[0.6,0.7],[0.2,0.5] [0.2,0.4],[0.4,0.5],[0.1,0.2] [0.1,0.3],[0.1,0.3],[0.2,0.3] [0.2,0.4],[0.4,0.6],[0.1,0.4] [0.0,0.5],[0.2,0.4],[0.5,0.6] [0.5,0.7],[0.0,0.7],[0.2,0.4] [0.0,0.6],[0.4,0.7],[0.2,0.3] [0.2,0.3],[0.2,0.3],[0.1,0.2] [0.0,0.6],[0.2,0.4],[0.4,0.6]
Throat disease [0.3,0.4],[0.2,0.5],[0.0,0.6] [0.4,0.5],[0.1,0.3],[0.0,0.5] [0.2,0.4],[0.2,0.5],[0.3,0.7] [0.0,0.3],[0.3,0.6], [0.2,0.5] [0.0,0.4],[0.1,0.2], [0.1,0.3]
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International Journal of Mathematics Trends and Technology (IJMTT) – Volume 58 Issue 1 – June 2018
T
Viral fever 0.9054
Table 3: Cosecant similarity measure(Using step3 &step4) Tuberculosis Typhoid 0.7879 0.8374
Throat disease 0.8514
P1 0.7888 0.7596 0.8216 P2 0.7847 0.8717 0.9243 P3 From Table 3, it is obvious that if the doctor agrees, then P1 & P3 suffers from Viral fever Throat disease.
0.8572 0.8929
and P2 suffers from
VII. CONCLUSIONS Our proposed method plays a vital role in handling medical diagnosis problems . In future, we will enhance these methods to other types of neutrosophic sets. ACKNOWLEDGEMENT We sincerely acknowledge the suggestions of the anonymous reviewers which improve the quality of this article. The input of the editor is highly appreciated. REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]
Zadeh L.A, “Fuzzy sets”, Information and Control, vol. 8, pp.338-353,1965. Pawlak Z, “Rough sets”, International Journal of Information and Computer Sciences, vol. 11, pp.341-356, 1982. Krassimir T.Atanassov, “Intuitionistic fuzzy sets”, Fuzzy Sets and Systems, vol. 20 (1), pp. 87-96, 1986. Supriya Kumar De, Ranjit Biswas and Akhil Ranjan Roy, “An application of intuitionistic fuzzy sets in medical diagnosis”, Fuzzy Sets & Systems, vol. 117(2), pp.209- 213, 2001. Jun Ye, “ Cosine similarity measures for intuitionistic fuzzy sets and their applications”, Mathematical and Computer Modelling, vol.53, pp.91-97,2011. Tian Maoying, “A new fuzzy similarity measure based on cotangent function for medical diagnosis”, Advanced Modeling and Optimization, vol .15(2), pp.151-156,2013. Florentin Smarandache, A unifying field in logics, Neutrosophy: Neutrosophic Probability, set & logic, Rehoboth : American Research Press,1998. Haibin Wang, Florentin Smarandache, Yan Qing Zhang & Rajshekar Sunderraman, “Single valued neutrosophic sets”, Multi space and Multi structure, vol. 4,pp.410-413,2010. Majumdar,P & S.K.Samanta,“On similarity and entropy of neutrosophic sets”, Journal of Intelligent and Fuzzy Systems, vol. 26(3), pp.1245-1252,2014. Haibin Wang, Florentin Smarandache, Yan Qing Zhang & Rajshekar Sunderraman, Interval Neutrosophic Sets and Logic: Theory and Applications in Computing. Hexis, Arizona, Neutrosophic Book Series; 5, 2005. Jun Ye, “Similarity measures between interval neutrosophic sets and their applications in multi criteria decision-making”, Journal of Intelligent and Fuzzy Systems,26(1), pp. 165-172,2014. Irfan Deli, “Interval valued neutrosophic soft sets and its decision making”, International Journal of Machine learning and Cybernetics, vol. 8(2),pp.665-676, 2017. Sebastian, S. & T.V. Ramakrishnan, “Multi fuzzy sets”, International Mathematical Forum, vol. 5(50), pp.2471-2476, 2010. Shinoj T. K & Sunil Jacob John, “Intuitionistic fuzzy multi sets and its application in medical diagnosis”, World Academy of Science, Engineering and Technology, vol. 6, pp.1418-1421,2012. P.Rajarajeswari & Uma.N, “Normalized hamming similarity measure for intuitionistic fuzzy. Multi sets and its application in medical diagnosis”, International Journal of Mathematics, Trends and Technology, vol. 5(3), pp.219-225, 2014. Shan Ye &Jun Ye, “Dice similarity measure between single valued neutrosophic multi sets and its application in medical diagnosis”, Neutrosophic Sets &Systems, vol. 6,pp.48-52,2014. Shan Ye, Jing Fu & Jun Ye, “Medical diagnosis using distance-based similarity measures of single valued neutrosophic multi sets”, Neutrosophic Sets & Systems, vol. 7,pp.47-52,2015. Florentin Smarandache, “N-valued refined neutrosophic logic& its application in Physics”. Progress in Physics, vol. 4, pp.143146,2013. Irfan Deli, Said Broumi & Florentin Smarandache, “On neutrosophic refined sets and their applications in medical diagnosis”, Journal of New Theory, vol. 6, pp.88-982015. Said Broumi, Irfan Deli & Florentin Smarandache, “N-valued interval neutrosophic sets and their application in medical diagnosis”. Critical Review, vol. 10,pp.45-692015. A.Edward Samuel and R.Narmadhagnanam, “Innovative approaches for n-valued interval neutrosophic sets and their execution in medical diagnosis”, Journal of Applied Sciences, vol. 17(9), pp. 429-440, 2017.
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