A.Edward Samuel International Journal of Engineering Research and Application www.ijera.com ISSN : 2248-9622, Vol. 8, Issue 6 (Part -I) June 2018, pp 26-29 RESEARCH ARTICLE
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Cosine Exponential Distance of Single valued neutrosophic multi sets in medical diagnosis A.Edward Samuel *, R.Narmadhagnanam ** *Ramanujan Research Centre, P.G. & Research Department of Mathematics, GAC(A),Kumbakonam,TN,India.
Email:aedward74_thrc@yahoo.co.in ** (Ramanujan Research Centre, P.G. & Research Department of Mathematics, GAC(A), Kumbakonam, TN, India.
Email:narmadhagnanam03@gmail.com Corresponding Author : A.Edward Samuel
ABSTRACT In this paper, cosine exponential distance among single valued neutrosophic multi sets is proposed and some of its properties are discussed herein. The concept of the above method is an essential tool for dealing with uncertainties and shortcomings that affect the existing methods. Implementation of medical diagnosis is presented to find out the disease impacting the patient. Keywords - Single valued neutrosophic set, Single valued neutrosophic multi sets, cosine exponential distance, medical diagnosis. --------------------------------------------------------------------------------------------------------------------------------------Date of Submission: 25-05-2018 Date of acceptance:10-06-2018 --------------------------------------------------------------------------------------------------------------------------------------biological systems as well as in their characterization. To model an expert doctor it is I. INTRODUCTION imperative that it should not disallow uncertainty as A number of real life problems in it would be then inapt to capture fuzzy or incomplete engineering, medical sciences, social sciences, knowledge that might lead to the danger of fallacies economics etc., involve imprecise data and their due to misplaced precision. solution involves the use of mathematical principles As medical diagnosis contains lots of based on uncertainty and imprecision. Such uncertainties and increased volume of information, uncertainties are being dealt with the help of topics available to physicians from new medical like probability theory, fuzzy set theory [15], rough technologies, the process of classifying different set set theory [8] etc., Healthcare industry has been of symptoms under a single name of disease trying to complement the services offered by becomes difficult. In some practical situations, there conventional clinical decision making systems with is the possibility of each element having different the integration of fuzzy logic techniques in them. As truth membership, indeterminate and false it is not an easy task for a clinician to derive a fool membership functions. The unique feature of single proof diagnosis it is advantageous to automate few valued neutrosophic multi set is that it contains multi initial steps of diagnosis which would not require truth membership, indeterminate and false intervention from an expert doctor. Neutrosophic set membership. By taking one time inspection, there which is a generalized set possesses all attributes may be error in diagnosis. Hence, multi time necessary to encode medical knowledge base and inspection, by taking the samples of the same patient capture medical inputs. at different times gives the best diagnosis. So, single As medical diagnosis demands large valued neutrosophic multi sets and their applications amount of information processing, large portion of play a vital role in medical diagnosis. which is quantifiable, also intuitive thought process In 1965, Fuzzy set theory was initially involve rapid unconscious data processing and given by Zadeh[1] which is applied in many real combines available information by law of average, applications to handle uncertainty. Sometimes the whole process offers low intra and inter person membership function itself is uncertain and hard to consistency. So contradictions, inconsistency, be defined by a crisp value. So the concept of indeterminacy and fuzziness should be accepted as interval valued fuzzy sets was proposed to capture unavoidable as it is integrated in the behavior of
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DOI: 10.9790/9622-0806012629
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A.Edward Samuel International Journal of Engineering Research and Application www.ijera.com ISSN : 2248-9622, Vol. 8, Issue 6 (Part -I) June 2018, pp 26-29 the uncertainty of membership grade. In 1982, Pawlak [2] introduced the concept of rough set, as a formal tool for modeling and processing incomplete information in information systems. In 1986, Atanassov [3] introduced the intuitionistic fuzzy sets which consider both truth-membership and falsitymembership. Later on, intuitionistic fuzzy sets were extended to the interval valued intuitionistic fuzzy sets. Intuitionistic fuzzy sets and interval valued intuitionistic fuzzy sets can only handle incomplete information not the indeterminate information and inconsistent information which exists commonly in belief systems. Neutrosophic set (generalization of fuzzy sets, intuitionistic fuzzy sets and so on) defined by Florentin Smarandache [4] has capability to deal with uncertainty, imprecise, incomplete and inconsistent information which exists in real world from philosophical point of view. Wang et al[5] proposed the single valued neutrosophic set. Pinaki Majumdar and S.K. Samanta [6] proposed the similarity and entropy of neutrosophic sets. Jun Ye and Jing Fu [7] proposed the tangent similarity measure of single valued neutrosophic sets. Jun Ye [8] proposed the cotangent similarity measure of single valued neutrosophic sets. Shan Ye and Jun Ye [9] introduced the concept of single valued neutrosophic multi sets. In this paper, by using the notion of single valued neutrosophic multi set, it was provided an exemplary for medical diagnosis. In order to make this a new method was executed. Rest of the article was 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 contain methodology, algorithm and case study related to medical diagnosis respectively. Conclusion was given in Section 7.
II. PRELIMINARIES 2.1 Definition [5] Let X be a space of points (objects) with a generic element in X denoted by x . A single valued neutrosophic set A in X is characterized by truth membership function TA , indeterminacy function
I A and falsity membership function FA .For each point x in X , is X TA x , I A x , FA x 0,1 When continuous, a SVNS A can be written as A T x , I x , F x / x, x X When X is discrete, a x
SVNS
A can
n
be
A T xi , I xi , F xi /xi , xi X i 1
2.2 Definition [9]
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written
as
Let X be a nonempty set with generic elements in X denoted by x . A single valued neutrosophic multi sets(SVNM) A drawn from X is characterized by the three functions: count truthmembership of CTA , count indeterminacymembership of CI A and count falsity-membership of CFA such that CTA ( x) : X R , CI A ( x) : X R , CFA ( x) : X R for x X , where R is the set of all real number multi sets in the real unit interval [0,1]. Then a SVNM A is denoted by x, (T A1 ( x), (T A2 ( x),...T Aq ( x)), ( I 1A ( x), ( I A2 ( x),...I Aq ( x)), A FA1 ( x), ( FA2 ( x),...FAq ( x)) / x X
where
the
truth-membership sequence the indeterminacy-
((TA1 ( x), (TA2 ( x),...TAq ( x)) ,
membership sequence (( I 1A ( x), ( I A2 ( x),...I Aq ( x)) and the falsity-membership sequence (( FA1 ( x), ( FA2 ( x),...FAq ( x)) may be in decreasing or increasing order, and the sum of TAi (x), I Ai (x), FAi ( x) [0,1] satisfies the condition 0 sup TAi ( x) sup I Ai ( x) sup FAi ( x) 3 for x X and i 1,2,, q. For convenience, a SVNM can be denoted by the simplified form:
A
x,T (x), I i A
i i A ( x), FA ( x)
x X , i 1,2,, q
III. PROPOSED DEFINITION 3.1 Definition Let A
x , T ( x ), I j
i A
j
i A
( x j ), FAi ( x j ) | x j X , i 1,2, , q
i B
( x j ),FBi ( x j ) | x j X , i 1,2, , q
and B
x , T ( x ), I j
i B
j
be any two SVNMs in X x1 , x2 , xn . Then the cosine exponential distance is defined as 1 CEDSVNMS ( A, B) 3n 1 i ( x ) F i ( x ) F i ( x ) (1) q n T i ( x j ) TBi ( x j ) I i ( x j ) I B j B j A A A j cos e i 1 j 1 Proposition 1 (i) CEDSVNMS ( A, B) 0 (ii) CEDSVNMS ( A, B) CEDSVNMS ( B, A) If A B C then
CED SVNMS ( A, C ) CED SVNMS ( A, B) & CED SVNMS ( A, C ) CED SVNMS ( B, C ) Proof (i) The proof is straightforward
DOI: 10.9790/9622-0806012629
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A.Edward Samuel International Journal of Engineering Research and Application www.ijera.com ISSN : 2248-9622, Vol. 8, Issue 6 (Part -I) June 2018, pp 26-29 (ii) We know that,
I iA x j I iB x j I iB x j I iA x j F iA x j F iB x j F iB x j F iA x j
T iA x j T iB x j T iB x j T iA x j
CEDSVNMS ( A, B) CEDSVNMS ( B, A)
(iii)We know that, T iA x j T iB x j T iC x j
x j I iB x j I iC x j F iA x j F iB x j F iC x j I iA
A B C Hence,
TAi ( x j ) TBi ( x j ) TAi ( x j ) TCi ( x j )
TBi ( x j ) TCi ( x j ) TAi ( x j ) TCi ( x j ) I Ai ( x j ) I Bi ( x j ) I Ai ( x j ) I Ci ( x j )
I Bi ( x j ) I Ci ( x j ) I Ai ( x j ) I Ci ( x j ) FAi ( x j ) FBi ( x j ) FAi ( x j ) FCi ( x j )
FBi ( x j ) FCi ( x j ) FAi ( x j ) FCi ( x j ) Here, our cosine an increasing function
exponential
distance
is
CED SVNMS ( A, C ) CED SVNMS ( A, B) & CED SVNMS ( A, C ) CED SVNMS ( B, C ) IV. METHODOLOGY
In this section, we present an application of single valued neutrosophic multi 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 single valued neutrosophic multi relation from the set of patients to the symptoms. i.e., Q( P S ) and R be a single valued neutrosophic relation from the set of
Q
P1 P2 P3 P4
Temperature (0.8,0.6,0.5) (0.3,0.2,0.1) (0.4,0.2,0.1) (0.5,0.4,0.3) (0.3,0.3,0.2) (0.5,0.4,0.4) (0.2,0.1,0.1) (0.3,0.2,0.2) (0.8,0.7,0.6) (0.5,0.5,0.4) (0.3,0.2,0.2) (0.4,0.4,0.3)
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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 single valued neutrosophic multi sets & single valued neutrosophic sets 3. Determination of diagnosis on the basis of new computation technique of single valued neutrosophic multi sets V. ALGORITHM Step 1: The Symptoms of the patients are given to obtain the patient-symptom relation Q and are noted in Table 1 Step 2: 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. Step3: The computation T of the relation of patients and diseases is found using (1) and are noted in Table 3. Step 4: Finally, minimum 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 & 4) was suffering from the disease Dr (r 1,2,3 & 4). VI. CASE STUDY [9] Let there be four patients P P1 , P2 , P3 , P4 and the set of symptoms S ={Temperature, Cough, Throat pain, Headache, Body pain }.The Single valued neutrosophic multi relation Q( P S ) is given as in Table 1.Let the set of diseases D = {Viral fever, Tuberculosis, Typhoid, Throat disease}.The Single valued neutrosophic relation R(S D) is given as in Table 2.
Table 1: Patient-Symptom relation Cough Throat pain Headache (0.5,0.4,0.3) (0.2,0.1,0.0) (0.7,0.6,0.5) (0.4,0.4,0.3) (0.3,0.2,0.2) (0.3,0.2,0.1) (0.6,0.3,0.4) (0.8,0.7,0.7) (0.4,0.3,0.2) (0.9,0.8,0.7) (0.6,0.5,0.4) (0.6,0.4,0.3) (0.2,0.1,0.1) (0.3,0.2,0.2) (0.4,0.1,0.1) (0.2,0.1,0.0) (0.4,0.3,0.3) (0.7,0.7,0.3) (0.3,0.2,0.2) (0.8,0.8,0.7) (0.3,0.2,0.2) (0.4,0.2,0.2) (0.2,0.2,0.2) (0.3,0.3,0.3) (0.7,0.6,0.5) (0.1,0.1,0.0) (0.7,0.6,0.6) (0.4,0.3,0.1) (0.7,0.1,0.0) (0.6,0.5,0.3) (0.8,0.3,0.1) (0.7,0.2,0.3) (0.6,0.2,0.1) (0.7,0.5,0.3) (0.7,0.7,0.6) (0.6,0.4,0.3)
DOI: 10.9790/9622-0806012629
Body pain (0.4,0.3,0.2) (0.6,0.5,0.5) (0.6,0.4,0.4) (0.8,0.7,0.5) (0.4,0.3,0.1) (0.7,0.2,0.1) (0.4,0.4,0.3) (0.4,0.3,0.2) (0.7,0.7,0.5) (0.5,0.1,0.1) (0.3,0.3,0.2) (0.6,0.5,0.4)
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A.Edward Samuel International Journal of Engineering Research and Application www.ijera.com ISSN : 2248-9622, Vol. 8, Issue 6 (Part -I) June 2018, pp 26-29
Table 2: Symptom-Disease relation Viral fever Tuberculosis Typhoid Throat disease (0.8,0.1,0.1) (0.2,0.7,0.1) (0.5,0.3,0.2) (0.1,0.7,0.2) (0.2,0.7,0.1) (0.9,0.0,0.1) (0.3,0.5,0.2) (0.3,0.6,0.1) (0.3,0.5,0.2) (0.7,0.2,0.1) (0.2,0.7,0.1) (0.8,0.1,0.1) (0.5,0.3,0.2) (0.6,0.3,0.1) (0.2,0.6,0.2) (0.1,0.8,0.1) (0.5,0.4,0.1) (0.7,0.2,0.1) (0.4,0.4,0.2) (0.1,0.8,0.1) [3] Krassimir,T.Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 20 (1), Table 3: Cosine exponential distance 1986,87-96. [4] Florentin Smarandache, A unifying field in Viral Tuberculo Typhoi Throat T logics, Neutrosophy: Neutrosophic fever sis d disease probability, set& logic, Rehoboth : American P1 0.9059 0.9376 0.8845 0.9766 Research Press, 1998. P2 0.8975 0.9225 0.9016 0.9750 [5] Haibin Wang, Florentin Smarandache, Yan P3 0.9161 0.9507 0.8850 0.9770 Qing Zhang & Rajshekar Sunderraman,. Single valued neutrosophic sets, Multi space P4 0.9145 0.8873 0.9470 0.8693 and Multistructure,4, 2010,410-413.. [6] Pinaki Majumdar and S.K. Samanta, On From Table 3, it is obvious that, if the similarity and entropy of neutrosophic sets. doctor agrees, then both P1 & P3 is suffering from Journal of intelligent and fuzzy systems, 26, Typhoid, P2 is suffering from Viral fever and P4 is 2014, 1245-1252. [7] Jun Ye and Jing Fu, “Multi-period medical suffering from Tuberculosis. diagnosis method using a single valued neutrosophic similarity measure based on VII. CONCLUSION tangent function, Computer Methods and Our propounded technique is most reliable Programs in Biomedicine, 123, 2016, 142to handle medical diagnosis problems quiet 149. comfortably. The recommended method can invade Jun Ye, Single valued neutrosophic similarity in other areas such as clustering, image processing [8] measures based on cotangent function and etc., In future, we will enhance this method to other their application in the fault diagnosis of types of neutrosophic sets steam turbine, Soft computing,21(3) ,2017,817-825. REFERENCES Shan Ye and Jun Ye, Dice similarity measure [1] Zadeh. L.A, Fuzzy sets, Information and [9] between single valued neutrosophic multi sets Control, 8, 1965, 338-353. and its application in medical diagnosis, [2] Pawlak.Z,Rough sets, International Journal of Neutrosophic sets and systems,6, 2014, 48-53 Information and Computer Sciences, 11,1982, 341-356..
R Temperature Headache Stomach pain Cough Chest pain
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DOI: 10.9790/9622-0806012629
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