# Complex Entropy and Its Application in Decision-Making for Medical Diagnosis

Authors: Fuyuan Xiao; Xiao-Guang Yue
Journal: Journal of Healthcare Engineering
 (2021)

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## Abstract

In decision-making systems, how to measure uncertain information remains an open issue, especially for information processing modeled on complex planes. In this paper, a new complex entropy is proposed to measure the uncertainty of a complex-valued distribution (CvD). The proposed complex entropy is a generalization of Gini entropy that has a powerful capability to measure uncertainty. In particular, when a CvD reduces to a probability distribution, the complex entropy will degrade into Gini entropy. In addition, the properties of complex entropy, including the nonnegativity, maximum and minimum entropies, and boundedness, are analyzed and discussed. Several numerical examples illuminate the superiority of the newly defined complex entropy. Based on the newly defined complex entropy, a multisource information fusion algorithm for decision-making is developed. Finally, we apply the decision-making algorithm in a medical diagnosis problem to validate its practicability.


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## Body

## 1. Introduction

Uncertainty is inevitable in the applications of decision-making systems [1–4]. Considerable attention has addressed uncertainty in the past few decades [5, 6]. How to express the knowledge involved in sources of uncertain information still remains an open issue [7, 8]. Hence, researchers have attempted to model and measure uncertain information using extended soft sets [9], evidence theory [10], reasoning [11–13], belief structures [14, 15], D numbers [16, 17], Z numbers [18, 19], and other hybrid methods [20–23].One successful alternative uncertain information measure is Gini entropy [24], which is simple to implement and has received a substantial amount of attention from researchers. Inspired by Gini entropy [24], Yager and Petry [25] recently devise an intelligent quality-based approach for fusing multisource information [26]. Bouhamed et al.[27] extend it to combine multisource possibilistic information. Later, researchers generalized the Gini entropy-based information quality to belief functions to measure uncertainty. The method of Li et al.[28, 29] is an example that has been well applied in various fields. Although Gini entropy [24] can be used to measure uncertainty, it can only be used for probability distributions.The complex-valued model has potential expressional properties, especially for the modeling of uncertainty [30, 31]. Therefore, the complex-valued model was widely investigated and applied in various fields, such as medical diagnosis [32], decision-making [33, 34], and predicting interference effects [35, 36]. Given that the complex-valued representation model is well suited for certain applications, how can Gini entropy be generalized to complex planes to provide a more powerful capability to measure uncertainty?In this paper, to address the abovementioned issue, a generalized entropy is proposed for measuring the uncertainty of CvDs. When CvDs reduce to probability distributions, the newly defined entropy degrades into Gini entropy. Specifically, vector expressions of CvDs are first proposed to model knowledge in complex planes. After that, a novel complex entropy called Xiao entropy is defined to measure uncertainties of CvDs. Then, the properties of complex entropy, including nonnegativity, maximum and minimum entropies, and boundedness, are analyzed and discussed. Based on the newly defined complex entropy, a multisource information fusion algorithm for decision-making is devised. Finally, we apply the decision-making algorithm in a medical diagnosis problem to verify its practicability.The contributions of this work are summarized as follows:A novel complex entropy, called Xiao entropy, which has the properties of nonnegativity, maximum and minimum entropies, and boundedness, is defined for the CvDThe multisource information fusion algorithm based on the newly defined entropy can be well applied to support decision-makingThis study provides a new perspective of complex-valued representation for uncertain information and offers a promising and generalized solution in terms of uncertainty measurementsThe preliminaries are introduced in Section2. In Section 3, CvD vectors are defined. In Section 4, a complex entropy is defined to measure the uncertainty of CvDs. In Section 5, several numerical examples illustrate the properties of complex entropy. In Section 6, an algorithm for decision-making is designed on the basis of the newly defined entropy. Then, the decision-making algorithm is used in a medical diagnosis. Section 7 concludes this work.


## 2. Preliminaries

In this section, some essential concepts of uncertainty measures related to this work are introduced.Definition 1.
(Gini entropy). LetP=p1,…,pj,…,pn be a probability distribution vector. The Gini entropy of P is defined by [24](1)GP=1−∑j=1npj2.Definition 2.
(Pennecchi and Oberto’s uncertainty measures). Letℂ=c1,…,cj,…,cn be a CvD vector, where cj=aj+bji. Pennecchi and Oberto’s modulus estimations of ℂ are defined by [37](2)c⟶a=1n∑j=1nxj2+yj2,c⟶b=1n∑j=1naj2+1n∑j=1nbj2,where 1−c⟶a and 1−c⟶b can be used as uncertainty measures.


## 3. Vector Representation of CvD

Modeling uncertainty has attracted a substantial amount of attention in a variety of areas [38]. Many methods have been proposed and applied in various fields, such as failure and risk analysis [39], classification [40, 41], information fusion [42], and decision-making [43, 44]. Here, a vector representation of CvD is presented for expressing uncertainty in a complex plane. In addition, the norm of CvD is also defined and analyzed.Definition 3.
(CvD vector). Letℂk be a CvD vector on the frame of discernment (FOD) Ψ=ψ1,…,ψj,…,ψn, denoted by(3)ℂk=ck1,…,ckj,…,ckn,where ckj is the complex value with regard to the occurrence of ψj:(4)ckj=akj+bkji,where akj and bkj are real numbers and i is the imaginary unit, satisfying i2=−1.
ckj in equation (4) satisfies(5)akj2+bkj2∈0,1,∑j=1nckj=1,where ckj is the modulus of ckj.
Equation (4) is also expressed as follows:(6)ckj=rkjeiθkj,with(7)rkj=akj2+bkj2,where rkj=ckj≥0 and θkj∈−π,π denotes an angle (phase) of ckj.Definition 4.
(norm of CvD). Letℂk be a CvD vector on FOD Ψ. Norm of CvD vector, ℂk, is defined by(8)ℂk=∑j=1nckj2=∑j=1nakj2+bkj2.Consider properties of CvD vector in Definition3, where for each ckj, akj2+bkj2∈0,1 and ∑j=1nckj=1, we observe the following:Case 1.
The maximal value ofℂk, denoted by maxℂk, is generated, when(9)ckj=1,for onej,ckj=0,for othersj,such that(10)maxℂk=1.Case 2.
Whenckj degrades into real numbers, i.e., ckj=akjbkj=0, the minimum value of ℂk, denoted by minℂk, is generated, when(11)ckj=1n,1≤j≤n,where(12)minℂk=∑j=1n1n2=1n=1n.
In summary,(13)1n≤ℂk≤1,where ℂk has a maximum value of 1 with ckj=1 for one ψj and others ckj=0; ℂk has a minimal value of 1/n with all ckj=1/n.


## 4. Entropy for CvD

Entropy is useful for measuring uncertainty [45–47], where many kinds of entropies, such as Tsallis entropy [48], fuzzy entropy [49, 50], Deng entropy [51–53], and cross-entropy [54], are presented for different aspects [55–59]. Among them, Shannon and Gini entropies are very popular. The greater the uncertainty is, the greater the entropy is; the lesser the uncertainty is, the lesser the entropy is [60]. We make use of the concept of Gini entropy [24] to measure the uncertainty of CvD.Definition 5.
(complex entropy). Letℂk be a CvD vector on FOD Ψ. The complex entropy of ℂk, denoted as EXℂk, is defined as(14)EXℂk=1−ℂk2=1−∑j=1nckj2.When a CvD reduces to a probability distribution, wherebkj=0 and ckj=akj, then EXℂk can be expressed as follows:(15)EXℂk=1−∑j=1nckj2=1−∑j=1nakj2,which is equal to equation (1).Property 1.
EX is a generalized model of Gini entropy [24]. Specifically, when a CvD becomes a probability distribution, EX degrades into Gini entropy [24].According to equation (13), because 1/n≤ℂk≤1, we have(16)1n≤ℂk2≤1,such that(17)0≤1−ℂk2≤1−1n.Thus,EXℂk has the boundedness of 0,1/n.It is inferred thatEXℂk reaches its maximal value EXℂk=1−1/n when ckj=1/n for 1≤j≤n. When n⟶+∞ and 1/n⟶0, Xℂk reaches the maximum value 1.EXℂk reaches its minimal value EXℂk=0 when one ckj=1 and others ckj=0.Remark 1.
Notably, the largerEXℂk is, the larger the uncertainty in CvD ℂk is, which results in lower certainty.Definition 6.
(the completely certain CvD). CvDℂk is completely certain when EXℂk=0.Definition 7.
(the completely uncertain CvD).
CvDℂk is completely uncertain when EXℂk=1.Theorem 1.
EX has the desired properties of the entropy of the CvD, including nonnegativity, maximum and minimum entropies, and boundedness.Property 2.
Letℂk be an arbitrary CvD:P 2.1
Nonnegativity:EXℂk≥0P 2.2
Maximum entropy:EXℂk≤maxEXℂkP 2.3
Minimum entropy:EXℂk≥minEXℂkP 2.4
Boundedness:0≤EXℂk≤1Proof.
The proofs are trivial.


## 5. Numerical Examples

In this section, several examples are presented to illustrate the entropy for CvD.Example 1.
Consider a CvDℂ in the FOD Ψ=ψ1,ψ2:(18)ℂ=x,1−x.In Example1, ℂ changes as parameter x varies, where x is set within [0,1], such that ℂ reduces to a probability distribution.By leveraging the Gini entropyG and Xiao entropy EX, the corresponding entropy measures are shown in Figure 1. Clearly, EX is the same as G entropy, which verifies that when a CvD reduces to a probability distribution, EX degrades into Gini entropy. Additionally, when x=0 or x=1, such that ℂ=0,1 or ℂ=1,0, Gℂ and EXℂ achieve the minimum entropy of 0, because in this case, ℂ is the completely certain CvD. By contrast, only when x=0.5, such that ℂ=0.5,0.5, can Gℂ and EXℂ achieve a maximum entropy of 0.5.Figure 1
G and EX in Example 1.Example 2.
Consider a CvDℂ in the FOD Ψ=ψ1,ψ2:(19)ℂ=reπ/2i,1−reπ/9i.In Example2, ℂ changes as modulus r varies, where r is set within [0.01,0.99].Becauseℂ consists of complex numbers, Gini entropy is not applicable. The result of EX entropy is shown in Figure 2. As r increases from 0.01 to 0.5, EX entropy increases from 0.0198 to 0.5, while as r increases from 0.5 to 0.99, EX entropy gradually decreases to 0.0198. This result shows a similar trend as the entropy measures in Figure 1.Figure 2
EX in Example 2.A comparison of the results in Examples1 and 2 shows that the proposed EX entropy is a more capable uncertainty measure than Gini entropy.Example 3.
Consider a CvDℂ in the FOD Ψ=ψ1,…,ψj,…,ψx:(20)ℂ=1α,…,1α,…,1α.In Example3, we set six different scales of α, namely, α∈1,10, 1,102, 1,103, 1,104, 1,105, and 1,106, to measure the variation of Gℂ and EXℂ.Figures3(a)–3(f) depict the results of Gℂ and EXℂ with regard to six different cases, respectively. Particularly, as α varies within [1, 10], EXℂ has a maximum value of maxEXℂ=0.9 and a minimum value of minEXℂ=0. When α changes within 1,102, maxEXℂ=0.99 and minEXℂ=0. When α varies within 1,103, maxEXℂ=0.999 and minEXℂ=0. When α changes within 1,104, maxEXℂ=0.9999 and minEXℂ=0. When α varies within 1,105, maxEXℂ=1 and minEXℂ=0. When α changes within 1,106, maxEXℂ=1 and minEXℂ=0. Thus, when a CvD becomes a completely certain distribution, i.e., a probability distribution, in which ckj=akj=1 for one j and other ckj=0, it has a minimum entropy of minEXℂ=0. On the other hand, when α⟶+∞, maxEXℂ is close to 1, because in this case ℂ is completely uncertain.Figure 3
The entropy measures ofGℂ and EXℂ in Example 3. (a)Gℂ and EXℂ: 1≤α≤10. (b)Gℂ and EXℂ: 1≤α≤102. (c)Gℂ and EXℂ: 1≤α≤103. (d)Gℂ and EXℂ: 1≤α≤104. (e)Gℂ and EXℂ: 1 ≤α≤105. (f)Gℂ and EXℂ: 1≤α≤106.
(a)(b)(c)(d)(e)(f)Example 4.
Assume that there is a CvDℂ in the FOD Ψ=ψ1,…,ψj,…,ψx:(21)ℂ=reξπi,1−reξπi.In Example4, ℂ changes as r and ξ vary. Here, we set r within [0,1] and ξ within [-1,1], as shown in Figure 4(a). The entropy measure of EXℂ is presented in Figure 4(b), which shows how the variations in the modulus and angle of the elements in ℂ impact EXℂ.Figure 4
The entropy measure in Example4. (a) The variation of r and ξ. (b)EX
(a)(b)EXℂ changes as r varies, whereas the variation in angle θ=ξπ has no effect on EXℂ. This result is reasonable because rkj2=ckj2=akj2+bkj2 is related to the modulus r rather than θ.Example 5.
Consider Example2.In Example5, r is set within [0,1]. We compare the proposed EX with related works, that is, Pennecchi and Oberto’s uncertainty measures 1−c⟶a and 1−c⟶b.By comparing the results ofEX, 1−μ⟶a, and 1−μ⟶b shown in Figure 5, we can see that 1−c⟶a remains 0.5 and cannot accurately measure the uncertainty. However, 1−μ⟶b provides a better measure of the uncertainty compared to 1−c⟶a because as r increases from 0.01 to 0.5, it increases from 0.2929 to 0.4208, while as r increases from 0.5 to 0.99, it gradually decreases to 0.2929. Nevertheless, the proposed EX has better discrimination as an uncertainty measurement and is superior to other methods.Figure 5
Comparison of different uncertainty measures.


## 6. Algorithm and Application

How to deal with decision-making problems has attracted much attention [61–65], especially for complex-valued expressed information [66, 67]. In this section, we first design a multisource information fusion algorithm for decision-making based on the proposed entropy. Then, we apply the decision-making algorithm in medical diagnosis to validate its practicability.


### 6.1. A Multisource Information Fusion Algorithm for Decision-Making

Problem statement: letΨ be a FOD with a set of objectives ψ1,…,ψj,…,ψn to be recognized. Suppose there are t CvDs: ℂ=ℂ1,…,ℂk,…,ℂt where ℂk=ck1,…,ckj,…,ckn and ckj=akj+bkji. The decision-making algorithm is to identify the target from ψ1,…,ψj,…,ψn by combining multiple CvDs ℂ1,…,ℂk,…,ℂt.The specific steps are given as follows:Step 1: For1≤k≤t, its corresponding entropy of CvD ℂk, denoted by EXℂk, can be generated as follows:(22)EXℂk=1−ℂk2.Step 2: For1≤k≤t, its corresponding information volume of CvD ℂk, denoted by IVℂk, can be measured by(23)IVℂk=eEXℂk.Step 3: The information volumeIVℂk is normalized by(24)IV¯ℂk=IVℂk∑h=1tIVℂh,1≤k≤t.Step 4: According to the normalized information volumes, the weighted average CvD, denoted asℂ˜, is defined by(25)ℂ˜=∑i=1kIV¯ℂk×ℂk,1≤k≤t,whereℂk=ck1,…,ckj,…,ckn and ckj=xkj2+ykj2.Step 5:ℂ˜ is fused via the complex Dempster’s combination rule [68] by t−1 times:(26)ℂ˜^t−1=ℂ˜⊕ℂ˜1⊕⋯⊕ℂ˜t−1.Step 6: Forℂ˜^t−1ψj, the ψδ with the maximum absolute value is chosen:(27)ℂ˜^t−1ψδ=max1≤j≤nℂ˜^t−1ψj,1≤j≤n.Step 7: Letλ be a threshold value for decision-making, which can be set in advance according to specific applications. If ℂ˜^t−1ψδ≥λ, the ψδ can be identified as the target by(28)δ=argmax1≤j≤nℂ˜^t−1ψj,Target←ψδ.Ifℂ˜^t−1ψδ<λ, it cannot be determined.The corresponding pseudocode is given in Algorithm1.Algorithm 1:Complex entropy-based multisource information fusion algorithm for decision-making.
Input: A FOD Ψ=ψ1,…,ψj,…,ψn;A set of CvDs:ℂ=ℂ1,…,ℂk,…,ℂt(1)
for1≤k≤tdo(2)
Calculate the CvD entropyEXℂk by equation (22)(3)
Measure the CvD information volumeIVℂk by equation (23)(4)
end(5)
Calculate the normalized information volumeIV¯ℂk by equation (24)(6)
Generate the weighted average CvDℂ˜ by equation (25)(7)
Obtain the fusedℂ˜^t−1 via the complex Dempster’s combination rule by equation (26)(8)
Choose the maximum absolute valueℂ˜^t−1ψδ by equation (27)(9)
ifℂ˜^t−1ψδ≥λthen(10)
δ=argmax1≤j≤nℂ˜^t−1ψj(11)
Target←ψδ.(12)
else(13)
Cannot be determined.(14)
end


### 6.2. Application in Medical Diagnosis

In this section, the proposed decision-making method is applied in medical diagnosis to demonstrate its practicability. The scenario and data of the application are based on [32].Considering a medical diagnosis problem, where for a patientP, P suffers with the most possible disease from D  =  D1: viral fever, D2: malaria, D3: typhoid, D4: stomach problem. To clarify which disease the patient may suffer, five experts diagnose the patient’s condition, in which the evaluation data are modeled as CvDs in Table 1. The threshold λ is set as 0.80 for this application to make a decision. We try to diagnose the patient P by integrating the evaluations from the five experts.Table 1
The evaluated data for patient modeled as CvDs.
ExpertsCvDsDiseasesViral fever:D1Malaria:D2Typhoid:D3Stomach problem:D4E1ℂE10.65e0.2i0.10e0.3i0.10e0.3i0.15e0.2iE2ℂE20.10e0.3i0.60e0.2i0.10e0.3i0.20e0.2iE3ℂE30.40e0.3i0.10e0.3i0.30e0.4i0.20e0.4iE4ℂE40.50e0.2i0.20e0.2i0.10e0.3i0.20e0.3iE5ℂE50.55e0.2i0.10e0.3i0.15e0.2i0.20e0.3iThen, the decision-making algorithm is applied to medical diagnosis by the following steps:Step 1: The entropy values of CvDℂEk1≤k≤5 are calculated by equation (22), as shown in Table 2.Step 2: The information volumes of CvDℂEk1≤k≤5 are calculated by equation (23), as shown in Table 2.Step 3: The information volumeIVℂEk1≤k≤5 is normalized by equation (24), as shown in Table 2.Step 4: The weighted average CvDℂ˜ is generated by equation (25), as shown in Table 3.Step 5: By gradually fusing the weighted average CvD with 4 times, their corresponding results are generated by equation (26), as shown in Table 3.Step 6: The maximal absolute value ofℂ˜^t−1Dj is marked with the correct color in Table 3.Step 7: PatientP is diagnosed as most likely to suffer the disease D1:(29)ℂ˜^4Dδ=max1≤j≤4ℂ˜^4Dj≥θ,θ=0.80δ=argmax1≤j≤4ℂ˜^4Dj=1,P←D1.Table 2
The results in terms of entropy, information volume, and normalized information volume.
ResultsCvDsℂE1ℂE2ℂE3ℂE4ℂE5EXℂEk0.53500.58000.70000.66000.6250IVℂEk1.70741.78602.01381.93481.8682IV¯ℂEk0.18340.19180.21630.20780.2007Table 3
The weighted average CvD and fused results obtained by the complex Dempster’s combination rule.
ResultsDiseasesDiagnosis resultsViral fever:D1Malaria:D2Typhoid:D3Stomach problem:D4ℂ˜0.43920.21670.15330.1908Cannot be determinedℂ˜^10.64350.15670.07840.1215Cannot be determinedℂ˜^20.80340.09650.03420.0659Viral feverℂ˜^30.90110.05340.01340.0321Viral feverℂ˜^40.95250.02790.00490.0148Viral fever


### 6.3. Discussion

As shown in Table1, we see that ℂE1D1=0.65, ℂE3D1=0.4, ℂE4D1=0.5, and ℂE5D1=0.55, which all support viral fever: D1 disease. However, ℂE2D3=0.6 supports malaria: D2 disease. Hence, ℂE2 conflicts with ℂE1, ℂE3, ℂE4, and ℂE5. By only using Table 1, it is difficult to make an accurate decision because a conflict exists among the experts. It is necessary to fuse the data collected from different experts to better support decision-making. There are five evaluations from five experts. To illuminate the effectiveness of the proposed decision-making algorithm, we gradually fuse the weighted average CvD, and the results are given in Table 3.When the weighted average CvD is fused by 1 time, we obtain the result thatℂ˜^1D1 has the largest value of 0.6435. Because 0.6435 is smaller than the threshold λ=0.80, the patient’s disease cannot be determined. When the weighted average CvD is fused by 2 times, it is calculated that ℂ˜^2D1 has the largest value of 0.8034. Because 0.8034 is larger than the threshold λ=0.80, the patient is diagnosed with viral fever: D1. When the weighted average CvD is fused by 3 and 4 times, it is easy to see that ℂ˜^3D1 and ℂ˜^4D1 have increasingly large values of 0.9011 and 0.9525 to better support decision-making. Finally, the patient is diagnosed as most likely to suffer viral fever: D1. Consequently, the value in terms of disease D1 is increased for decision-making from 0.6435 to 0.8034 to 0.9011 and then to 0.9525 as shown in Figure 6. As a result, the proposed decision-making algorithm is effective to address medical diagnosis problem.Figure 6
The fusion results of the weighted average CvD.


## 6.1. A Multisource Information Fusion Algorithm for Decision-Making

Problem statement: letΨ be a FOD with a set of objectives ψ1,…,ψj,…,ψn to be recognized. Suppose there are t CvDs: ℂ=ℂ1,…,ℂk,…,ℂt where ℂk=ck1,…,ckj,…,ckn and ckj=akj+bkji. The decision-making algorithm is to identify the target from ψ1,…,ψj,…,ψn by combining multiple CvDs ℂ1,…,ℂk,…,ℂt.The specific steps are given as follows:Step 1: For1≤k≤t, its corresponding entropy of CvD ℂk, denoted by EXℂk, can be generated as follows:(22)EXℂk=1−ℂk2.Step 2: For1≤k≤t, its corresponding information volume of CvD ℂk, denoted by IVℂk, can be measured by(23)IVℂk=eEXℂk.Step 3: The information volumeIVℂk is normalized by(24)IV¯ℂk=IVℂk∑h=1tIVℂh,1≤k≤t.Step 4: According to the normalized information volumes, the weighted average CvD, denoted asℂ˜, is defined by(25)ℂ˜=∑i=1kIV¯ℂk×ℂk,1≤k≤t,whereℂk=ck1,…,ckj,…,ckn and ckj=xkj2+ykj2.Step 5:ℂ˜ is fused via the complex Dempster’s combination rule [68] by t−1 times:(26)ℂ˜^t−1=ℂ˜⊕ℂ˜1⊕⋯⊕ℂ˜t−1.Step 6: Forℂ˜^t−1ψj, the ψδ with the maximum absolute value is chosen:(27)ℂ˜^t−1ψδ=max1≤j≤nℂ˜^t−1ψj,1≤j≤n.Step 7: Letλ be a threshold value for decision-making, which can be set in advance according to specific applications. If ℂ˜^t−1ψδ≥λ, the ψδ can be identified as the target by(28)δ=argmax1≤j≤nℂ˜^t−1ψj,Target←ψδ.Ifℂ˜^t−1ψδ<λ, it cannot be determined.The corresponding pseudocode is given in Algorithm1.Algorithm 1:Complex entropy-based multisource information fusion algorithm for decision-making.
Input: A FOD Ψ=ψ1,…,ψj,…,ψn;A set of CvDs:ℂ=ℂ1,…,ℂk,…,ℂt(1)
for1≤k≤tdo(2)
Calculate the CvD entropyEXℂk by equation (22)(3)
Measure the CvD information volumeIVℂk by equation (23)(4)
end(5)
Calculate the normalized information volumeIV¯ℂk by equation (24)(6)
Generate the weighted average CvDℂ˜ by equation (25)(7)
Obtain the fusedℂ˜^t−1 via the complex Dempster’s combination rule by equation (26)(8)
Choose the maximum absolute valueℂ˜^t−1ψδ by equation (27)(9)
ifℂ˜^t−1ψδ≥λthen(10)
δ=argmax1≤j≤nℂ˜^t−1ψj(11)
Target←ψδ.(12)
else(13)
Cannot be determined.(14)
end


## 6.2. Application in Medical Diagnosis

In this section, the proposed decision-making method is applied in medical diagnosis to demonstrate its practicability. The scenario and data of the application are based on [32].Considering a medical diagnosis problem, where for a patientP, P suffers with the most possible disease from D  =  D1: viral fever, D2: malaria, D3: typhoid, D4: stomach problem. To clarify which disease the patient may suffer, five experts diagnose the patient’s condition, in which the evaluation data are modeled as CvDs in Table 1. The threshold λ is set as 0.80 for this application to make a decision. We try to diagnose the patient P by integrating the evaluations from the five experts.Table 1
The evaluated data for patient modeled as CvDs.
ExpertsCvDsDiseasesViral fever:D1Malaria:D2Typhoid:D3Stomach problem:D4E1ℂE10.65e0.2i0.10e0.3i0.10e0.3i0.15e0.2iE2ℂE20.10e0.3i0.60e0.2i0.10e0.3i0.20e0.2iE3ℂE30.40e0.3i0.10e0.3i0.30e0.4i0.20e0.4iE4ℂE40.50e0.2i0.20e0.2i0.10e0.3i0.20e0.3iE5ℂE50.55e0.2i0.10e0.3i0.15e0.2i0.20e0.3iThen, the decision-making algorithm is applied to medical diagnosis by the following steps:Step 1: The entropy values of CvDℂEk1≤k≤5 are calculated by equation (22), as shown in Table 2.Step 2: The information volumes of CvDℂEk1≤k≤5 are calculated by equation (23), as shown in Table 2.Step 3: The information volumeIVℂEk1≤k≤5 is normalized by equation (24), as shown in Table 2.Step 4: The weighted average CvDℂ˜ is generated by equation (25), as shown in Table 3.Step 5: By gradually fusing the weighted average CvD with 4 times, their corresponding results are generated by equation (26), as shown in Table 3.Step 6: The maximal absolute value ofℂ˜^t−1Dj is marked with the correct color in Table 3.Step 7: PatientP is diagnosed as most likely to suffer the disease D1:(29)ℂ˜^4Dδ=max1≤j≤4ℂ˜^4Dj≥θ,θ=0.80δ=argmax1≤j≤4ℂ˜^4Dj=1,P←D1.Table 2
The results in terms of entropy, information volume, and normalized information volume.
ResultsCvDsℂE1ℂE2ℂE3ℂE4ℂE5EXℂEk0.53500.58000.70000.66000.6250IVℂEk1.70741.78602.01381.93481.8682IV¯ℂEk0.18340.19180.21630.20780.2007Table 3
The weighted average CvD and fused results obtained by the complex Dempster’s combination rule.
ResultsDiseasesDiagnosis resultsViral fever:D1Malaria:D2Typhoid:D3Stomach problem:D4ℂ˜0.43920.21670.15330.1908Cannot be determinedℂ˜^10.64350.15670.07840.1215Cannot be determinedℂ˜^20.80340.09650.03420.0659Viral feverℂ˜^30.90110.05340.01340.0321Viral feverℂ˜^40.95250.02790.00490.0148Viral fever


## 6.3. Discussion

As shown in Table1, we see that ℂE1D1=0.65, ℂE3D1=0.4, ℂE4D1=0.5, and ℂE5D1=0.55, which all support viral fever: D1 disease. However, ℂE2D3=0.6 supports malaria: D2 disease. Hence, ℂE2 conflicts with ℂE1, ℂE3, ℂE4, and ℂE5. By only using Table 1, it is difficult to make an accurate decision because a conflict exists among the experts. It is necessary to fuse the data collected from different experts to better support decision-making. There are five evaluations from five experts. To illuminate the effectiveness of the proposed decision-making algorithm, we gradually fuse the weighted average CvD, and the results are given in Table 3.When the weighted average CvD is fused by 1 time, we obtain the result thatℂ˜^1D1 has the largest value of 0.6435. Because 0.6435 is smaller than the threshold λ=0.80, the patient’s disease cannot be determined. When the weighted average CvD is fused by 2 times, it is calculated that ℂ˜^2D1 has the largest value of 0.8034. Because 0.8034 is larger than the threshold λ=0.80, the patient is diagnosed with viral fever: D1. When the weighted average CvD is fused by 3 and 4 times, it is easy to see that ℂ˜^3D1 and ℂ˜^4D1 have increasingly large values of 0.9011 and 0.9525 to better support decision-making. Finally, the patient is diagnosed as most likely to suffer viral fever: D1. Consequently, the value in terms of disease D1 is increased for decision-making from 0.6435 to 0.8034 to 0.9011 and then to 0.9525 as shown in Figure 6. As a result, the proposed decision-making algorithm is effective to address medical diagnosis problem.Figure 6
The fusion results of the weighted average CvD.


## 7. Conclusions

In this paper, a complex entropy, called Xiao entropy, is proposed to measure the uncertainty of complex-valued distributions (CvDs). The complex entropy is a generalized model of Gini entropy. Specifically, when the CvD turns into a probability distribution, the proposed entropy degrades into Gini entropy. Furthermore, we study the properties of complex entropy, including nonnegativity, maximum and minimum entropies, and boundedness. Several numerical examples compare the proposed complex entropy with related works. The results illuminate the superiority of the proposed complex entropy. Based on the complex entropy, a multisource information fusion algorithm for decision-making is devised. Finally, we apply the decision-making algorithm in a medical diagnosis problem to validate its practicability.The main contributions are that this study provides a new perspective of complex-valued representation for uncertain information; the newly defined complex entropy has a powerful capability to measure uncertainty. Additionally, it offers a promising application in decision theory. In the future work, we intend to apply this complex entropy to handle more complex decision-making problems, such as the analyzing and processing of image and physiological signals.



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