Medical diagnosis with the aid of using fuzzy logic and intuitionistic fuzzy logic

Abstract

The objective of the present study is to develop/establish a web-based medical diagnostic support system (MDSS) by which health care support can be provided for people living in rural areas of a country. In this respect, this research provides a novel approach for medical diagnosis driven by integrating fuzzy and intuitionistic fuzzy (IF) frameworks. Subsequently, based on the proposed approach a web-based MDSS is developed. The proposed MDSS comprises of a knowledge base (KB) and intuitionistic fuzzy inference system (IFIS). Based on the observation that medical data cannot be described with both precision and certainty, a medical KB is constructed in the form of a set of if-then decision rules by employing both fuzzy and IF logics. After constructing the medical KB, a new set of patients is considered for diagnosing the diseases. For each patient, linguistic values of the patients’ symptoms are considered as inputs of the proposed IFIS and modeled by using the generalized triangular membership functions. Subsequently, integrated fuzzy and IF rule-based inference system is used to find a valid conclusion for the new set of patients. In a nutshell, in this paper fuzzy rule-based and IFS based inference systems are combined for better and more realistic representation of uncertainty of the medical diagnosis problem and for more accurate diagnostic result. The method is composed of following four steps: (1) the modeling of antecedent part of the rules, which consist of linguistic assessments of the patients’ symptoms provided by the doctors/medical experts with their corresponding confidence levels, by using generalized fuzzy numbers; (2) the modeling of consequent part, which reveals the degree of association and the degree of non-association of diseases into the patient, by using IFSs; (3) the use of IF aggregation operator in inference process; (4) the application of relative closeness function to find the final crisp output for a given diagnosis. Finally, the applicability of the proposed approach is illustrated with a suitable case study. This article has also justified the proposed approach by using similarity measurement.

Appendices

Appendix A:

Here, we review definitions and concepts of fuzzy number, GFN and IFS that are required for developments of the present study. We start by recalling the definition of a fuzzy number.

Definition A.1

[64] A fuzzy set defined on real line $\mathbb {R}$ is called a fuzzy number if it is convex, normal and having a piecewise continuous membership function with bounded support.

To represent expert’s opinion, in a more flexible manner, GFN is better to use as it captures confidence level of expert. The definition of GFN is given as follows:

Definition A.2

[50] A GFN $\widetilde {A}=[(a,b,c,d); w]$, where a,b,c,d are real numbers and 0<w≤1, is a fuzzy subset of the real numbers $\mathbb {R}$ with a membership function μ _A (u) which satisfies the following conditions:

• i) $\mu _{\widetilde {A}}(u)$ is a continuous function from $\mathbb {R}$ to the closed interval [0,w].

• ii) $\mu _{\widetilde {A}}(u)=0,$where −∞<u≤a.

• iii) $\mu _{\widetilde {A}}(u)$ is monotonic increasing function in [a,b].

• iv) $\mu _{\widetilde {A}}(u)=w,$where b≤u≤c.

• v) $\mu _{\widetilde {A}}(u)$ is monotonic decreasing function in [c,d].

• vi) $\mu _{\widetilde {A}}(u)=0,$ where d≤u<∞.

If $\mu _{\widetilde {A}}(u)$ is a linear function of u and also b = c then $\widetilde {A}$ is called a GTFN. Without any loss of generality, throughout the work to represent the linguistic assessments of patients’ symptoms, provided by the medical experts, in a flexible manner, we consider GTFNs. When a GTFN $\widetilde {A}$ is used to represent the expert’s opinion, the value w indicates the level of confidence of the medical expert.

Definition A.3

[15] An IFS A in the universe $\mathcal {U}$ can be expressed as a set of ordered triple, $A=\{(u,\mu _{A}(u),\nu _{A}(u)):u\in \mathcal {U}\}$, where $\mu _{A}(u):\mathcal {U}\rightarrow [0,1]$ is the degree of belongingness and $\nu _{A}(u):\mathcal {U}\rightarrow [0,1]$ is the degree of non-belongingness of u in A. They satisfy the relation 0≤μ _A (u) + ν _A (u)≤1∀u∈U. The quantity π _A (u)=1−μ _A (u)−ν _A (u) is called the degree of hesitation (indeterminacy) of u in A.

A brief description of the concepts of IFS operations and aggregation operators for IFSs based on continuous Archimedean t-norm and t-conorm, are briefly reviewed in Appendix B.

Appendix B

It is well known that, a strict Archimedean t-norm has a representation by means of its additive generator [56] g:[0,1]→[0,∞] such that g(1)=0 as T(x,y) = g ⁻¹(g(x) + g(y))

Similarly, a strict Archimedean t-conorm is expressed through its additive generator [56] h:[0,1]→[0,∞] as S(x,y) = h ⁻¹(h(x) + h(y))

With the above analysis, pointwise operations for IFSs based on a given strict Archimedean t-norm T (generated by an additive generator g) and strict Archimedean t-conorm S (generated by an additive generator h) are introduced as follows:

Definition B.1

[54, 56] Let $a_{1}=(\mu _{a_{1}},\nu _{a_{1}})$ and $a_{2}=(\mu _{\widetilde {a}_{2}},\nu _{\widetilde {a}_{2}})$ be two IFSs and α≥0 be a scalar, and let a strict t-norm T and strict t-conorm S be generated by additive generators g and h, respectively. Then

• (i) $a_{1}\oplus a_{2}=\left (S(\mu _{a_{1}},\mu _{a_{2}}), T(\nu _{a_{1}},\nu _{a_{2}})\right )$

  $=\left (h^{-1}\!\left (h(\mu _{a_{1}})\,+\, h(\mu _{a_{2}})\right )\!, g^{-1}\!\left (g(\nu _{a_{1}})\,+\,g(\nu _{a_{2}})\right )\right )$

• (ii) $a_{1}\otimes a_{2}=\left \langle T(\mu _{a_{1}},\mu _{a_{2}}), S(\nu _{a_{1}},\nu _{a_{2}})\right )$

  $=\!\left (g^{-1}\!\left (g(\mu _{a_{1}})\,+\,g(\mu _{a_{2}})\right )\!, h^{-1}\!\left (h(\nu _{a_{1}})\,+\,h(\nu _{a_{2}})\right )\right )$

• (iii) $\alpha a_{1}=\left (h^{-1}\left (\alpha h(\mu _{a_{1}})\right ), g^{-1}\left (\alpha g(\nu _{a_{1}})\right )\right )$

• (iv) $a_{1}^{\alpha } =\left (g^{-1}\left (\alpha g(\mu _{a_{1}})\right ), h^{-1}\left (\alpha h(\nu _{a_{1}})\right )\right )$

To combine several IFSs into a single IFS, an aggregation function [53] is required which can be defined as: a function f:[0,1]ⁿ→[0,1] is called aggregation function of n arguments if it satisfies the properties (i) f(0,0,...,0)=0 and f(1,1,...,1)=1; (ii) if x≤y then f(x)≤f(y)∀x,y∈[0,1]ⁿ. In this work, we use intuitionistic fuzzy weighted arithmetic mean (IWAM) operator to aggregate IFSs. Based on arithmetic operations of IFSs defined in Definition B.1, IWAM operator [56] is defined as

Definition B.2

[56] Let $a_{i}=(\mu _{a_{i}},\nu _{a_{i}}) (i=1,2,...,n)$ be a collection of IFSs. An IWAM operator is a mapping IWAM $:(\mathbb {R^{+}})^{n}\rightarrow \mathbb {R^{+}}$ which is defined by

$$\begin{array}{@{}rcl@{}} IWAM(a_{1}, a_{2},...,a_{n})=\omega_{1}a_{1}+\omega_{2}a_{2}+...+\omega_{n}a_{n} \\ =\left( h^{-1}\left( \displaystyle{\sum}_{i=1}^{n}\omega_{i}h(\mu_{a_{i}})\right), g^{-1}\left( \displaystyle{\sum}_{i=1}^{n}\omega_{i}g(\nu_{a_{i}})\right)\right) \end{array} $$

(11)

where ω=(ω ₁,ω ₂,..,ω _n )^T be the weight vector of a _i (i=1,..,n),ω _i >0,and $\displaystyle \sum \limits _{i=1}^{n}\omega _{i}=1$.

Definition B.3

[56] If in particular, h ⁻¹ = h=1−I d and g ⁻¹ = g = I d on [0, 1], and that $\displaystyle \sum \limits _{i=1}^{n}\omega _{i}=1$ ensures the argument of h ⁻¹∈[0,1], then IWAM and IAM operator reduce to the following forms:

$$\begin{array}{@{}rcl@{}} &&IWAM(a_{1}, a_{2},...,a_{n})\\&&{\kern12pt}=\left( 1-\left( \displaystyle{\sum}_{i=1}^{n}\omega_{i}(1-\mu_{a_{i}})\right), \displaystyle\sum\limits_{i=1}^{n}\omega_{i}\nu_{a_{i}}\right) \end{array} $$

(12)

If the weight vector $\omega =(\frac {1}{n},\frac {1}{n},...,\frac {1}{n})^{T}$, then IWAM operator reduces to IAM operator and consequently (12) becomes

$$ IAM(a_{1}, a_{2},...,a_{n})\,=\,\left( 1\,-\,\left( \frac{1}{n}\displaystyle\sum\limits_{i=1}^{n}(1-\mu_{a_{i}})\right), \frac{1}{n}\displaystyle\sum\limits_{i=1}^{n}\nu_{a_{i}}\right) $$

(13)