Appendix D: Medical Diagnosis by Fuzzy Logic

D.l Idea of Crisp Set and Fuzzy Set

A set that provides an accurate starting point for developing a conception about something is known as a fuzzy set.

And a crisp set is a set defined in such a way that it contains two classes: one contains the elements belonging to a particular set, and the other contains the elements not belonging to the given set.

Let us now understand this with the help of a real-life example. Suppose I take a box of biscuits. Now I want to eat all the biscuits from the box, without knowing how many biscuits are there. Now, this is an example of fuzzy set, as the number of biscuits in the box is undetermined.

Now again if I say that I want to have two biscuits from the box. Then, this becomes a crisp set as the biscuits which I am not eating are the members of the second class which contains the elements not belonging to the set (Table D.l).

D.1.1 Definition of Fuzzy Sets

Fuzzy set in mathematics is a kind of set that contains elements having a certain degree of membership.

A fuzzy set gives us the accuracy of a certain value when it is tending towards the most accurate one (or height of the fuzzy set). This could be properly understood with the implementation of membership function. For example,

TABLE D.1

Differences between Fuzzy Sets and Crisp Sets

where 2 and 6 have memberships 0.5 and 0.3, respectively, whereas 4 has membership 1.

D.1.2 Membership Function

A function that deals with the accuracy of a number when it is tending towards the height of a fuzzy set is known as a membership function.

D.1.3 Different Types of Membership Function D.1.3.1 Intervalued Fuzzy Sets

A fuzzy set where the membership grade of an element is not just a real number that belongs to [0, 1] but an interval that belongs to P[0,l] where P[0,l] denotes all possible partitions/intervals within [0,1].

Where ordinary fuzzy sets are represented by A: X —> [0,1]; these kind of fuzzy sets are represented as A: X —> e[0,l], where e[0,1] С P[0,1].

This can only be done when we don't know the actual values of the membership grades but do know the upper and lower bounds where they can possibly belong.

D.1.3.2 Fuzzy Set of Type 2

These are interval-valued fuzzy sets, where the intervals assigned to the elements are also ordinary fuzzy sets.

D.1.3.3 Normal Fuzzy Sets

These are fuzzy sets whose height is equal to 1: i.e. h(A) = 1 D.1.3.4 Subnormal Fuzzy Sets

These are fuzzy sets where h(A) < 1 are known as subnormal fuzzy sets.

D.2 Intuitionistic Fuzzy Sets

The qualitative measurements in the field of mathematics can be done through fuzzy methods. The idea of these kinds of set was established by Lotfi A. Zadeh and Dieter Klaua in 1965 and extended to the classical notion of set.

Intuitionistic fuzzy sets (IFS) have been introduced by Krassimir Atanassov (1983) as an extension of Lotfi Zadeh's notion of fuzzy set. Many work-life decision-making and applications in different portions of sciences using IFS are observed in the literature. One such application is due to the interpretation of medical sciences.

D.2.1 Basic Concepts of Intuitionistic Fuzzy Sets

IFS are sets whose elements have a degree of membership or nonmembership and even a degree of hesitation or uncertainty. Usually, it is an extension of normal fuzzy sets.

Let a set E be fixed. An IFS A in E is an object having the following form:

where the functions ц_А:Е —>[0,1] and v_A : E —> [0,1 j define the degree of membership and the degree of non-membership, respectively, of the element x e E to the set A, which is a subset of E and for every x e E, 0 < |ia (x) + v_A (x) < 1. The amount я_А (x) = 1 - (p_A (x) + v_A (*)) is called the hesitation part, which may cater to either membership value or nonmembership value or both.

D.2.2 Some Other Properties of IFS

If A and В are two IFSs of the set E, then

Obviously every fuzzy set in an IFS has the form.

D.3 Research Methodology

In this present paper, researchers have showed a novel application of IFS in the number of cases and deaths due to various diseases for a particular year. Flere, researchers have initially fuzzified the crisp data of the table showing the number of cases and deaths due to the diseases in the following years.

D.4 Euclidean Distance

D.4.1 The Normalized Euclidean Distance

d_n - H(A,B) between two IFS.A and В is defined as follows:

where X = {xi,x₂,...,x_n} for i = 1,2,...,n

Now if in an IFS table, each of the value sets represents a point in a 3D space which is of the form (х^у^гЛ

Then, the membership, non-membership, and the hesitation grades represent x₍, y_s and z_u respectively. Now to calculate the distance between two categories, we are calculating the distances between the respective points associated with the categories.

This is done using the formula above, which is very similar to the general distance formula which we have used in analytical 3D geometry. Only what is different is that after calculating the distance, we are dividing it by 2n, i.e. the total number of points. The distance between any two points (xi,yi,Zi) and (x₂, y₂, z₂) in a 3D space is given by

So, here we can use this concept to evaluate the distances between two value sets from two different IFS tables. And hence, get the distance between the two categories.

D.4.2 Sample Selection

In the given study, let D = {ACUTE DIARRHOEAL DISEASES, MALARIA, ACUTE RESPIRATORY INFECTION, JAPANESE ENCEPHALITIS, C} represent the set of diseases; Y = {2006, 2007, 2008, 2009, 2010) be the set of given years for which the number of cases and deaths occurred due to a particular disease; and S = {KERALA, KARNATAKA, MADHYA PRADESH, ANDHRA PRADESH, TAMIL NADU, WEST BENGAL) be the states that also show the number of cases and deaths occurred in that particular state due to a particular disease.

The intuitionistic fuzzy set is used as a tool with the membership degree p (the degree of surety that a particular disease caused the given number of deaths), the non-membership degree v (the degree of surety that

TABLE D.2

Table Showing the Number of Deaths in Various Years

a particular disease did not cause the given number of deaths), and the hesitation degree n (the degree of unsurety of the number of deaths in a particular year).

Table D.2 shows the relationship between the numbers of deaths in the given years.

Table D.3 shows the membership, non-membership, and hesitation parts between the states and diseases:

Table D.4 is the final table that shows the membership, non-membership, and hesitation parts between States and Diseases.

TABLE D.3

Table Showing Membership, Non-Membership, and Hesitation Parts between States and Diseases

TABLE D.4

Table Showing Membership, Non-Membership, and Hesitation Parts between States and Diseases

D.5 Conclusion

In this chapter, the normalized Euclidean distance method was used in IFS to determine the distribution of different states in India with the infection of different diseases like acute diarrheal diseases, malaria, acute respiratory infection, Japanese encephalitis, and varicella (c). At the end of the case study, we found that most of the infection occurs in West Bengal, Tamil Nadu, and Madhya Pradesh. A significant number of malaria patients are still present in these states.

There is a reasonable chance for the presence of a nonzero hesitation grade [h(x)] in Tables D.2 and D.3, which is predicting the confusion in the society or ancillary systems of approach.

Bibliography

Asplund, Kiell. "Clinometries in stroke research". Stroke 18, no. 2 (1987): 528-530.

Atanassov, Krassimir T. "Intuitionistic fuzzy sets". Fuzzy Sets and Systems 20 (1986): 87-96.

Atanassov, Krassimir, and Christo Georgeiv. "Intuitionistic fuzzy prolog". Fuzzy Sets and Systems 53 (1993): 121-128.

Djatna, Taufik. et al. "An intuitionistic fuzzy diagnosis analytics for stroke disease". journal of Big Data 5 (2018): 35,1-14.

Doubois, Didier, and Henri Prade. Fuzzy Sets and Systems: Theory and Application. Academic Press, New York (1980).

Kacprzyk, Janusz, and Eulalia. Szmidt. "Medical applications of intuitionistic fuzzy sets". NIFS 7, no. 4 (2001): 58-64.

Klaua, David J. "An analysis of programming techniques". Teaching Machines and Programmed Learning II. Washington, DC: National Education Association 7 (1965): 118-161.

Martino, R., N. Foley, and S. Bhogal. "Dysphasia after stroke, incidence and diagnosis". Stroke 36, no. 12 (2005): 2756-2763.

Meena, K., and К. V. Thomas. "An application of intuitionistic fuzzy sets in choice of discipline of study". Global Journal of Pure and Applied Mathematics 14, no. 6 (2018): 867-871.

Ottenbacher, Kenneth J., and Sylvie Jannell. "The results of clinical trials in stroke rehabilitation research". Archives of Neurology 50, no. 1 (1993): 37-44.

Robins, M., and H. M. Baum. "The national survey of stroke". Stroke 12, no. 2 (Suppl 1) (1981): 145-157.

Sanchez, Elie. "Solutions in composite fuzzy relation equations: application to medical diagnosis in Brouwerian logic". Readings in Fuzzy Sets for Intelligent Systems, Morgan Kaufmann, (1993) pp. 159-165..

Sullivan, Joe., and William H. Woodall. "A comparison of fuzzy forecasting and Markov modelling". Fuzzy Sets and Systems 64, no.3 (1994): 279-293.

Szmidt, Eulalia, and Janusz Kacprzyk. "Remarks on some applications of intuitionistic fuzzy sets in decision making". Note on IFS (1996).

Szmidt, Eulalia, Janusz Kacprzyk, and Pawel Bujnowski. "How to measure the amount of knowledge conveyed by Atanassov's intuitionistic fuzzy sets." Information Sciences (2014): 257,276-285.

Thrift, Amanda G., et al. "Global stroke statistics". International Journal of Stroke 9, no. 1 (2014): 6-18.

Yu, Dejian, and Huchang Liao. "Visualization and quantitative research on intuitionistic fuzzy studies". Journal of Intelligent and Fuzzy Systems 30, no. 6 (2016): 3653-3663.

Zadeh, Lotfi A. "Fuzzy sets". Information and Control 8, no. 3 (1965): 338-353.

Zimmermann, Hans-Jurgen. Fuzzy Set Theory and Applications (2nd edition). Kluwer Academic Publishers, Amsterdam (1985).