Efficacy of Fuzzy-Stat Modelling in Classification of Gynaecologists and Patients

1 Introduction

The application of the fuzzy set theory in medical diagnosis was initiated by researchers [1–3, 8, 9, 12, 21]. The closely related application with clinical monitoring is clinical diagnosis using fuzzy automata. Kuncheva and Steimann [11] used the concepts of fuzzy input, fuzzy reasoning, and fuzzy classes to study clinical diagnosis. Some authors [4] used hierarchical rule-based monitoring and fuzzy logic control for neuromuscular block. A monitoring framework was developed [24] that allows the construction of problem-oriented diagnostic monitors. It allows time in the data model. The model detects the trend and tracks the disease history. Some work has been done on the fuzzy logic in medicine [18]. This work is mainly focused on lung diseases and includes rule-based fuzzy systems in medicine. Some authors studied the intuitionistic fuzzy set theory [27] and described the case study of some patients [5]. The max-min method is applied here for it is an intuitive recipe and is easy to use. The composition considers the extreme values. Sanchez [21] represented the physician’s medical knowledge as a fuzzy relation between symptoms and diseases. This approach was further elaborated by Steimann and Adlassnig [26]. Steimann and Adlassnig [25] represented an intelligent state monitor that makes an abstraction of a patient’s current status by using fuzzy state transitions. The reasoning as well as the inverse approximate reasoning method [15–17, 19] play an important role in clinical monitoring. The application of the fuzzy set theory in gynaecology has not been researched to the extent that it should be.

Gynaecological diseases involve multiple concomitant causal factors that are difficult to represent using conventional statistical methods. Many times, the diagnostic process constitutes one type of imprecision and uncertainty, and the knowledge related to a patient’s state constitute another type. The physician generally gathers knowledge about the patient from history taking, physical examination, laboratory test results, and other investigative procedures such as ultrasonic diagnosis, X-ray, etc. The knowledge provided by each of these sources carries varying degrees of uncertainty. The history furnished by the patient may be subjective, exaggerated, underestimated, or incomplete. Also, mistakes may be committed in the physical examination and symptoms may sometimes be overlooked. Nonetheless, measurements provided by laboratory tests are often of limited precision, and the exact borderline between normal and abnormal pathological results are often unclear. Diagnostic tests require a correct interpretation of the results [8, 9]. Thus, the state and symptoms of the patient can be known by the physician with only a limited degree of precision. The desire to better understand and teach this difficult and important technique of medical diagnosis has prompted attempts to model the process with the use of the fuzzy set theory [10].

The remaining parts of the paper are organized as follows. Section 2 describes the objectives and explains three stages in the gynaecological framework; the preliminaries are covered in Section 3. The study results and discussion are presented in Section 4.

2 Objective of the Study

“Words can mean different things to different people” [14]. This fact leads to classifying the gynaecologists, as their perceptions could be different in assigning linguistic hedges in defining symptoms, for example, severe pain in the abdomen. Similarly, patients can also be classified depending on the narration of the complaints. Three approaches, viz., similarity measures-based classifications (cosine amplitude and max-min methods) [6, 7, 13] and two-valued binary logic based classification (Gower’s coefficient method), are studied for their efficacy of fuzzy-stat modeling in the classification of gynaecologists and patients. The final objective of the proposed research is to develop a complete application software to be helpful to general physicians.

3 Preliminaries

This section briefly describes fuzzy set theoretic operations and fuzzy relational calculus [10, 20] and Gower’s coefficient used in the paper:

Definition 1: Let U be a universe set. A fuzzy set A of U is defined with a membership μ_A(x) → [0,1], where μ_A(x), ∀x ∈ U indicates the degree of x in A [20, 28, 29].

Definition 2: Let R be a fuzzy relation on X × Y, i.e., R = {((x, y), f_R(x, y))|(x, y) ∈ X × Y}, the α-cut matrix R_α is denoted by [20, 28, 29]

Definition 3: Let R ⊂ X × Y and S ⊂ Y × Z be fuzzy relations; the max-min composition R o S is defined by [20, 28, 29]

Definition 4: A fuzzy relation R on X × Y is called a fuzzy equivalence relation if the following three conditions held [20, 28, 29],

1. (3)R is reflexive, if fR(x, x) = 1, ∀x ∈ X. (3)
2. (4)R is symmetric, if fR(x, y) = fR(y, x) ∀x, y ∈ X. (4)

3. R is transitive, if R⁽²⁾ = (R o R) ⊂ R, or more explicitly

   (5)fR(x, z) = max{min {fR(x, y), fR(y, z)}}, ∀x, y, z ∈ X. (5)

Definition 5: A fuzzy relation R on X × Y is called is a fuzzy compatible or tolerance or proximity relation if it satisfies reflexive and symmetric conditions [20, 28, 29].

Definition 6: The transitive closure, R_T, of a fuzzy relation R is defined as the relation that is transitive, contain R, and has the smallest possible membership grades [20, 28, 29].

It can be reformed into a fuzzy equivalence relation by at most (n − 1) compositions, just as a crisp tolerance relation can be reformed into a crisp equivalence relation. That is,

Definition 7: The cosine amplitude method [28] is manipulated on a collection of n data samples. The position of each datum in space is represented by m feature values. Relation value r_ij reflects a similarity relationship between x_i and x_j data, calculated through the following equation [20, 28, 29]:

If x_i and x_j are very similar to each other, r_ij becomes close to 1. In contrast, if they are very dissimilar to each other, r_ij becomes close to 0.

4 The Study

To study the efficacy of fuzzy-stat modeling in classification of gynaecologists and patients, the domain experts and the patients were identified. The domain experts, based on their over two decades of experience, confirmed that there are 31 commonly observed gynaecological diseases and 123 related symptoms. The case study is divided into two parts.

The first part focuses on classifying the experts depending on their perceptions on the symptom-disease relationship. This classification uses fuzzy similarity measures methods [1], while in the second part, classification of the patients based on their symptoms narration is carried out. The techniques used in the second part are the fuzzy similarity-based classification method and the statistical approach; Gower’s coefficient is used in classifying measurable (say, age) and non-measureable (say pain in abdomen) parameters.

The gynaecologists recorded their perceptions linguistically with corresponding membership values (mentioned in brackets) as A – always (1), VO – very often (0.5625), O – often (0.75), NS – not specific (0.5), S – seldom (0.25), VS – very seldom (0.0625), and N – never (0). The concept of Zadeh’s concentration operator was used in assigning the membership value [20].

5 Discussion

The first part of the procedures section refers to application of similarity measures (cosine amplitude and max-min) to the perception of different domain experts (Tables 1–3). The classification result and classification diagram for both methods are shown in Table 8. It is observed that experts E₄ and E₆ are in one group with their agreement having a possibility of 0.981 using the cosine amplitude method, while the possibility is 0.95 using the max-min method. The degree of possibility to arrive at the same result in the cosine amplitude method, in this particular case, is better than that of the max-min method; thus, the results of the cosine amplitude method are considered for further analysis. Furthermore, it can be inferred that for the possibility value of 0.973 in the cosine amplitude method, the experts get classified into three groups {E₁}, {E₈}, and {E₂, E₃, E₄, E₅, E₆, E₇}, while for the possibility value of 0.937 the same groups get classified in the max-min method. We can further state that the experts {E₂, E₃, E₄, E₅, E₆, E₇} agree with one another in their diagnostic labels. Therefore, their perceptions in linguistic are considered in further analysis in our subsequent research. The reason two experts, E₁ and E₈, are not in the classification is because their perceptions vary significantly while the perceptions of the remaining six experts are very close to each other. The authors cross-verified the diagnosis made using each expert’s perceptions with the actual diagnosis made by the experts. We observed that the diagnosis is similar for those experts who are in one group, indicating that the mathematical classification of experts is in complete agreement with the actual diagnostic process.

Table 8

Classification of Experts: Classification Result and Classification Diagram Using the Cosine Amplitude and Max-Min Methods (Eight Experts, 226 Patients).

	Cosine Amplitude Method		Max-Min Method	Classification Diagram
α1	{E1, E2, E3, E4, E5, E6, E7, E8}	α1	{E1, E2, E3, E4, E5, E6, E7, E8}
α0.981	{(E4, E6), E1, E2, E3, E5, E7, E8}	α0.95	{(E4, E6), E1, E2, E3, E5, E7, E8}
α0.976	{(E4, E5, E6), E1, E2, E3, E7, E8}	α0.938	{(E4, E5, E6, E7), E1, E2, E3, E8}
α0.975	{(E2, E3, E4, E5, E6), E1, E7, E8}	α0.937	{(E2, E3, E4, E5, E6, E7), E1, E8}
α0.973	{(E2, E3, E4, E5, E6, E7), E1, E8}	α0.795	{(E1, E8, E2, E3, E4, E5, E6, E7)}
α0.935	{(E1, E8, E2, E3, E4, E5, E6, E7)}

The second part of the method relates to the application of fuzzy similarity measures method (cosine amplitude and max-min) and Gower’s coefficient method to classify patients, which show a confirmation of the classification of different patients after the application of the three methods (Figure 1). Table 9 shows the result of the three methods – cosine amplitude, max-min, and Gower’s coefficient method – applied to all 226 patients and 123 symptoms (124 symptoms for Gower’s coefficient). The patients not listed in Table 9 are those that do not get classified in any group. They are singleton sets.

Figure 1:

Classification of Patients.

Classification diagram using the cosine amplitude, max-min, and Gower’s coefficient methods (case study: nine patients, nine symptoms).

The patients are classified into various groups for various α-cut values, which infer the possibility of their similarity:

• Patients P₁₅₇, P₁₉₈, and P₂₂₀ get classified into one group using fuzzy similarity measures methods with a degree of confidence of 1, indicating that the symptoms in these three patients exactly match.

• Patient P₁₅₇ is not classified with P₁₉₈ and P₂₂₀ using Gower’s coefficient method. The measureable parameter “age” is used in this method, which is not considered in other two fuzzy set theoretic-based similarity measures. More important, partial belief is the strength of fuzzy set theory approaches, while in Gower’s coefficient method, the symptoms described linguistically are in dichotomous terms (yes/no). These could be the possible reasons why the output using statistical and fuzzy set theory methods are different. For example, the age of P₅₇ is 30 years and that of P₁₉₈ and P₂₂₀ is 32 years, so P₅₇ does not get classified in the group using Gower’s coefficient.

• Similar is the case for P₅ and P₅₀ and other groups.

• The other α-cut value that matches all three methods is 0.88, which classifies some more patients in different groups. This indicates that the patients who get classified at this cut-off value also have the same symptoms.

• The classification is based on the symptom of the patient; we decided the cut-off at a degree of confidence value 1 (i.e., α-cut = 1) so as to match the symptoms with a high degree of confidence.

The specific symptom that occurs more frequently in patients is observed. For example, for patients P₁₅₇ and P₁₉₈, the symptom is “pain in the abdomen.” This information can be helpful for health experts to find the cause of the disease related to these symptoms in a specific area. We can infer that the symptoms of patients classified into one group being found to be exactly the same at α-cut level 1 indicates that the patients get classified correctly using the fuzzy similarity measures and two-valued binary logic-based Gower’s coefficient. In gynaecology, the diagnosis depends heavily on the parameter “age” for certain diseases. For such diseases, the Gower’s coefficient method is more efficient.

6 Concluding Remarks

• The basis of fuzzy logic-based inference systems is strongly dependent on the perception of the domain experts. The authors firmly believe that the similarity should be first evaluated before using the knowledgebase of the experts in decision analysis. In this regard, fuzzy similarity measures-based methods can be a preferred armamentarium in experts classification. It can be concluded that the cosine amplitude method is better and faster in achieving the results in experts classification than the max-min method.

• Fuzzy similarity measures and similarity measures using mixed variables (Gower’s coefficient) can be implemented in estimating the possibility of agreement in symptom-patient among patients.

• The study concludes with a note that the efficacy of two-valued binary logic-based Gower’s coefficient is low as compared to the cosine amplitude method.

• In summary, the classification of patients can be used in policy issues to find the prevalence and severity of a certain disease in specific areas/locations. The classification of experts and patients using fuzzy similarity measures is an effective formalism and could be effectively used in overall medical decision support systems.