Abstract
For medical diagnosis based on fuzzy relations of diseases and symptoms, max-min composition method and distance-based method are frequently used. However, they often lead to diagnostic confusion and puzzle due to their different results. In this paper, we suggest a new method for medical diagnosis using the quantiles of the measures used in both methods to avoid such problems. As an illustration of usefulness, the proposed method is applied to medical diagnosis of headache. The results of the simulation show that the two methods with the normal measures have significant difference in headache diagnosis, while the difference decreases when the quantiles of the normal measures are used as a diagnostic measure.
Introduction
Zadeh [13] first proposed to apply fuzzy set (FS) to the medical diagnosis area. Since then FS theory has been used in various methods to model the diagnostic process. Sanchez [3] expressed the physician’s expertise as fuzzy relation of diseases and symptoms. This approach was further developed by Adlassnig [12] and widely used in researches [1, 15] for medical diagnosis.
There are two popular methods for medical diagnosis based on fuzzy relations. They are the max-min composition method [17] and distance-based method [4]. The max-min composition method is intuitive. The distance-based method can decrease the loss of data information and assign weights for patient’s symptoms.
However, there are some common drawbacks in both methods as follows: (1) The diagnostic measures, max-min composition indicator and distance value (hereafter “normal measures”) generally don’t have the same distribution for each disease in the methods. For example, the composition indicator or distance value 0.6 for disease A does not mean the same degree as that for disease B. (2) The diagnosis results of the methods might be different. It can therefore be inferred that there would be no definite diagnostic criteria.
To overcome those drawbacks, we suggest methods for diagnosis based on max-min composition and based on distance. For disease diagnosis of a disease, we suggest using the quantiles of the normal measures. In this article the quantile of a value when n values are arranged in order is used to call its rank divided by n. For an application of our proposed approach, two methods for disease diagnosis using the interview chart [9, 21] with intuitionistic fuzzy degrees are introduced and the simulation to compare the diagnosis outcomes of the two diagnosis methods has been carried out. The simulation study shows that the two methods provide significantly different diagnostic results. However, the difference decreases when the quantiles of the normal measures are used as a diagnostic measure.
Section 2 introduces two frequently used methods for medical diagnosis. The fuzzy relations between symptoms and diseases based on Sanchez’s approach are presented and then the diagnostic processes using both methods are summarized. In Section 3, the diagnosis results of the methods are illustrated by a numerical example, which is for the differentiation of patients. The two methods give different diagnostic results in this example. In Section 4, a new approach using the quantiles of the normal measures as a diagnostic measure is proposed to avoid the conflicting results of the two methods, which is applied to the same example as in Section 3. The simulation results are summarized and compared with those in Section 3. A brief discussion is given in Section 5.
Two diagnosis methods
Zadeh [14] introduced fuzzy sets which have been applied in various areas [2, 19]. In FS theory, the degrees of membership and non-membership are given by values between 0 and 1 but in reality, the sum of the degrees might not always be exactly 1 [20]. Hence FS has been generalized to intuitionistic fuzzy set (IFS) [11] which are popularly investigated [7, 16].
In this section two methods based on IFS theory and Sanchez’s fuzzy relations are introduced. The relations are summarized in the following outline. Let S = {S1, . . . , S n }, D = {D1, . . . , D m }, and P = {P1, . . . , P q } denote the sets of symptoms, diseases, and patients, respectively. Two fuzzy relations, Q and R, are defined as follows:
Q = { < (p, s) , μ Q (p, s) , τ Q (p, s) > ∣ (p, s) ∈ P × S}
R = {< (s, d) , μ R (s, d) , τ R (s, d) > ∣ (s, d) ∈ S × D}
where μ
Q
(p, s) denotes the degree to which the symptom s is observed in patient p, and τ
Q
(p, s) the degree to which the symptom s is not observed in patient p. Thus μ
Q
(p, s) and τ
Q
(p, s) represent the relationship between symptom and patient, i.e. the degrees of the patient’s symptoms, which are called as (
Max-min composition method
The headache diagnosis based on max-min composition consists of four steps: Step 1: Collect patient data. The data collected are the patient’s degrees and confirmability degrees of each patient’s symptoms. Patient’s degrees, that is the relationship between patient and symptom, are assessed by an expert such as a physician. Confirmability degrees, that is the relationship between disease and symptom, are indicated in an interview chart. The interview chart which was developed in our previous works [9, 21] consists of total 55 items among which 23 (M1 ∼ M23) items are for migraine, 17 (T1 ∼ T17) for tension headache and 15 (C1 ∼ C15) for clusterheadache. Step 2: Calculate the IFWAA (intuitionistic fuzzy weighted arithmetic average) of the patient’s degrees and confirmability degrees, respectively, using the aggregate operator of Def. 1 [22]. Step 3: Calculate the max-min composition indicators using the compositions of Def.2 [5]. Step 4: Idenitify the disease based on the max-min composition indicators. The largest μ
T
in general indicates the most appropriate diagnosis such that both inequalities 0.5 < μ
T
and τ
T
< 0.5 are satisfied.
where ω = (ω1, ω2, . . . , ω n ) T is the weight vector for A with ω i > 0 and .
for all p ∈ P and d ∈ D, where the composition T of two fuzzy relations R and Q in medical diagnosis relates the state of patients to disease as a fuzzy relation between P and D.
Distance-based method
The medical diagnosis based on distance is also divided into four steps, and the first two steps are identical to Steps 1 and 2 of max-min composition. Step 3: Calculate distance with the measure in Def. 3 [6]. Step 4: Identify the disease based on the distance. The shortest distance indicates the most appropriate diagnosis.
l h (A, B) = (1/2n)∑ [ |μ A (x i ) - μ B (x i ) | +
|τ A (x i ) - τ B (x i ) | + |π A (x i ) - π B (x i ) | ]
where π is the degree of hesitation part, i.e., π A (x i ) =1 - [μ A (x i ) + τ A (x i )] and π B (x i ) =1 - [μ B (x i ) + τ B (x i )].
A headache example: max-min composition method and distance-based method
We now illustrate our proposed approach on an example for diagnosis of headaches. The three primary types of headaches are migraine, tension, cluster. The approach here uses the confirmability degrees <μ R (s, d) , τ R (s, d)> indicated in the interview chart and the patient’s degrees<μ Q (p, s) , τ Q (p, s)> assessed by a physician. In this example, we have seen that the diagnostic results depend on which methods were applied.
Let us consider patient P1. P1’s symptoms are (M11, M20) of migraine, (T2, T3, T9, T17) of tension headache, and (C2, C8) of cluster headache. Tables 1 and 2 report the confirmability degrees presented in the interview chart and the degrees for P1’s symptoms assessed by a physician, respectively. In general, values above 0.3 will be assessed by a physician as a patient’s membership value μ Q for each symptom. Based on membership value, the patient degrees are randomly generated. We present in Tables 3 and 4 the IFWAAs of the patient’s degree and confirmability degree for patient P1, which are calculated by IFWAA operator of Def. 1 using Tables 1 and 2.
•Results from max-min composition method
Table 5 shows the max-min composition indicators for patient P1 using the patient’s degree and confirmability degree given in Tables 3 and 4. We observe that μ T for tension is the largest so that we can make preliminary diagnosis that patient P1 suffers most likely from tension headache.
•Results from distance-based method
The distances between IFS’s are calculated by applying the distance measure defined in Def. 2.2 to the data in Tables 3 and 4, which are presented in Table 5. Since the distance value of migraine is the smallest in this case, our preliminary diagnosis could be that patient P1 suffers most likely from migraine headache.
We note that Tables 5 leads to a different diagnosis and hence there would be no definite diagnostic criteria. This problem is partly caused by the fact that the normal measures for each disease generally don’t have the same distribution, and we propose to use their quantiles of the normal measures to avoid this drawback.
The proposed approach and its application
We investigate the distributional characteristics and calculate the quantile of each normal measure by simulation, which is conducted by varying the number of symptoms k in headache from 3 to 20. The simulation study has been conducted in IBM-PC with 2.80GHz Pentium CPU and 4GB memory on Windows 2000. We use Java language (JDK 7) and Eclipse integrated development environment. First, we select k symptoms out of 50 symptoms of headache. Then we assign the patient’s degrees μ Q and τ Q to the selected symptoms, where the values of μ Q and τ Q are randomly chosen between 0.3 and 0.7 and between 0.0 to 0.3 respectively and their sum is less than 1. Based on these patient’s degrees, the normal measures are obtained by following the steps given in Section 2. We repeat these calculations 10,000 times foreach k.
Tables 6 and 7 show the normal measures and their quantiles of Patient P1 who have 8 specific symptoms of headache. The diagnosis results of max-min composition methods are the same both in using the normal measures and using their quantiles as given in Table 7. For the case of distance-based method, the diagnosis result using the normal measures is different from the one with quantiles as in Table 7 migraine has the lowest distance but the lowest quantile is for tension headache. Although the max-mincomposition method and distance-based method gives the different results, we can diagnose preferentially that patient P1 suffers from tension headache using the quantiles in both Tables 6 and 7.
In the remainder of this section, the simulation results are presented to compare the outcomes of the diagnostic measures, the normal measures and quantiles. A part of the simulation results are reported in Table 8, where the number in each cells represents the coincidence rate of the diagnosis results of max-min composition method and distance-based method. The value 0.542 in the top-leftmost cell of Table 8 represents the coincidence rate of the diagnosis results when the max-min composition indicator and distance value as a diagnostic measure for a patient with three symptoms are used. Similarly, the value 0.746 is the coincidence rate of the diagnosis results using the quantiles of the normal measures. The averages of coincidence rates are about 0.470 and 0.587 for each case, respectively, which seems to imply that the two methods give significant difference in the diagnosis results of the normal measures and quantiles. Figure 1 shows an overall tendency of the diagnostic measures, where we could see that the diagnosis based on quantiles gives more efficient results.
Conclusion
Recently, FS theory and its concept have been applied successfully in medical diagnosis processes, and various methods have been proposed. However, they often lead to diagnostic confusion and puzzle because the methods produce different results.
In this study, we made a comparison between the max-min composition method and the distance-based method. The important part of this study is the comparison of the simulation result with the diagnosis outcome. The results of the simulation indicate that the two methods show significant difference in diagnosis results when the normal measures, the max-min composition indicator and distance value are used. However, the difference decreases when we use the quantile of the normal measures as a diagnostic measure. The proposed method may be applied to diagnose the type of diseases based on the patient’s answer and doctor’s confirmation to the interview chart.
Acknowledgment
This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) (NRF-2016R1A2B1010253).