Investigations of Symmetrical Incomplete Information Spreading in the Evidential Reasoning Algorithm and the Evidential Reasoning Rule via Partial Derivative Analysis

Abstract

Incomplete information causes great uncertainty in decision making. It is a critical task to understand how incomplete information spreads symmetrically in order to make comprehensive and balanced decisions. A better understanding of the spreading of incomplete information can also be used for accurately locating limited resources to reduce incomplete information in the input for the final purpose of reducing incomplete information in the result. In this study, the way in which incomplete information spreads is studied via the evidential reasoning (ER) algorithm and the evidential reasoning rule (the ER rule), which are known for their transparent analytical procedures. Specifically, the partial derivative analysis is conducted using the steps of ER and the ER rule for calculating the contributions made by the beliefs, weights, and reliability to the incomplete information in the result. The major theoretical contribution of this study is the calculation of the contribution of the input to the incomplete information in the output based on partial derivative analysis. A numerical case is studied to demonstrate the proposed derivative analysis, the contribution calculation, and the consequential results.

1. Introduction

Incomplete information brings uncertainty to traditional decision making, which however, is almost unavoidable under many conditions [1,2]. There are many causes for incomplete information, e.g., degraded sensors [3], inaccurate recording and transferring of data [4], or human knowledge ignorance [5]. Generally existent incomplete information produces challenges in complex systems modeling.

Many traditional machine learning approaches, e.g., the neural network (NN), the support vector machine (SVM), etc., are effective in complex systems modeling, yet they cannot effectively handle incomplete information [1,6,7]. There are several methods for handling incomplete information [8]. When there is a large quantity of data, the data with incomplete information would be deleted or omitted from the later modeling procedure. Alternatively, incomplete information could be imputed using prior information or according to statistical data. Furthermore, the imputed information could be used either in a single value or as a distribution to provide a holistic picture of the problem, if required. Recently, machine learning approaches have been used for inferring the missing information, leading to superior performance. However, those endeavors are all indirect means, i.e., the missing information is either deleted or filled in, becoming complete information which traditional machine learning approaches can handle.

Therefore, the presence of incomplete information remains a challenge in decision-making, leading to the requirement of eliminating or at least reducing it. [9]. For example, there may be ambiguity regarding a patient’s assessment of pain, or the X-rays or CT scans may be blurred [10,11], all of which are sources of incomplete information in the input, thus corresponding to incomplete information in the result. It is understandable that the incomplete information in the result could cause difficulty for the doctors in reaching an accurate final diagnosis for the patients. Therefore, there is a significant incentive for reducing incomplete information in order to reach an accurate diagnosis, which benefits both the doctors and patients [12].

With sufficient or unlimited resources, the incomplete information in the result could be eliminated by eliminating all incomplete information in the input; however, the costs involved would be very high [13]. For example, we can do as many medical examinations as possible to eliminate as much incomplete information as possible [14], e.g., to obtain the clearest X-rays or CT scans possible, but the cost would then be very high, and the human body cannot withstand so many radiative scans. Moreover, some degree of incompleteness may never be eliminated because it is caused by epistemic uncertainty, e.g., human ignorance, which can never be completely avoided.

In more practical conditions, resources are always limited. For instance, unlimited time, funds, or human resources are never available for finding a cure for a certain disease or achieving any goal. As a result, we should only devote limited resources to the key causes of incomplete information in the result. Thus, the task becomes determining how to identify the “key causes” contributing to the final incomplete information.

Building upon the D-S evidence theory, the evidential reasoning algorithm (ER) [15,16] and its later variant, the evidential reasoning rule (ER rule) [17], can directly handle incomplete information. Specifically, the ER and ER rule considers that incomplete information in the results is affected by two factors, namely the rule weight that denotes the relative importance of the rules and the incomplete information in the respective rule as the input. Multiple successful studies on how ER and ER rule can handle incomplete information have been conducted using practical and theoretical cases [15,17,18]. Nonetheless, those studies have not discussed how the input rules could affect the final incomplete information. This is partly owing to the nonlinear modeling ability of the ER algorithm and the ER rule.

The major tasks and theoretical contributions of this study are the mathematical deductions of the contribution made by each factor to the incomplete information in the result via the ER algorithm and the ER rule, respectively. Consequently, the results of this study can be used in studying how the incomplete information spreads in the decision-making process [19], and to guide which adjustments should be made for reducing the incomplete information in the final result in areas such as economics [20], management [21], and medical diagnosis [22]. For example, we can calculate how different tests could affect the diagnosis and then adopt the most appropriate, least expensive tests which are likely to produce the most accurate diagnosis.

The remainder of this study is organized as follows. Section 2 gives a preliminary overview of the ER algorithm and the ER rule. Section 3 provides the detailed steps of how incomplete information spreads via the ER algorithm and the ER rule via partial derivative analysis. Section 4 presents two numerical case studies. The study is concluded in Section 5.

2. The Spreading of Incomplete Information via ER and ER Rule

2.1. The Spreading of Incomplete Information via ER

The kth belief rule is given as follows [15,16],

$$\{(D_1,{\beta }_{1,k}),\cdots ,(D_N,{\beta }_{N,k})\} \mathrm{with} \mathrm{a} \mathrm{rule} \mathrm{weight} \mathrm{of} w_k$$

(1)

where there are N scales and the nth scale contains a belief of β_n,k, and the rule weight w_k denotes the relative importance of the kth belief rule with w_k ∈ [0, 1]. When ${\displaystyle \sum _{n=1}^N{\beta }_{n,k}}<1$, it denotes that there is incomplete information, with a belief of ${\beta }_{D,k}=1-{\displaystyle \sum _{n=1}^N{\beta }_{n,k}}$. Otherwise, when there is no incomplete information, i.e., β_D,k = 0, there is ${\displaystyle \sum _{n=1}^N{\beta }_{n,k}}=1$.

Within the framework of the ER algorithm, the basic probability mass (BPM) is first constructed by the following Equations (2)–(5),

$$m_{n,k}=w_k{\beta }_{n,k}$$

(2)

$$m_{D,k}=1-w_k{\displaystyle \sum _{n=1}^N{\beta }_{n,k}}$$

(3)

$${\overline{m}}_{D,k}=1-w_k$$

(4)

$${\tilde{m}}_{D,k}=w_k(1-{\displaystyle \sum _{n=1}^N{\beta }_{n,k}})$$

(5)

where m_n,k denotes the belief distribution of the nth scale in the kth rule, and m_D,k denotes the belief which is unassigned to any scale. Moreover, there is $m_{D,k}={\overline{m}}_{D,k}+{\tilde{m}}_{D,k}$ where ${\overline{m}}_{D,k}$ denotes the relative importance of the kth rule that is affected by the activation weight, and ${\tilde{m}}_{D,k}$ is caused by the incomplete information in the kth rule.

Then, the following yields the iterative forms of rule integration using the ER algorithm with K rules,

$$m_{n,E(k+1)}={\mu }_{E(k+1)}(m_{n,E(k)}{m}_{n,k+1}+m_{n,E(k)}{m}_{D,k+1}+m_{D,E(k)}{m}_{n,k+1})$$

(6)

$${\overline{m}}_{D,E(k+1)}={\mu }_{E(k+1)}({\overline{m}}_{D,E(k)}{\overline{m}}_{D,k+1})$$

(7)

$${\tilde{m}}_{D,E(k+1)}={\mu }_{E(k+1)}({\tilde{m}}_{D,E(k)}{\tilde{m}}_{D,k+1}+{\tilde{m}}_{D,E(k)}{\overline{m}}_{D,E(k)}+{\overline{m}}_{D,E(k)}{\tilde{m}}_{D,k+1})$$

(8)

$${\mu }_{E(k+1)}={\left(1-{\displaystyle \sum _{n=1}^N{\displaystyle \sum _{t=1,t\ne k}^N{m}_{n,E(k)}{m}_{t,k+1}}}\right)}^{-1}$$

(9)

$${\beta }_n=\frac{m_{n,E(K)}}{1-{\overline{m}}_{D,E(K+1)}}$$

(10)

$${\beta }_D=\frac{{\tilde{m}}_{D,E(K)}}{1-{\overline{m}}_{D,E(K)}}$$

(11)

where β_n denotes the belief of the nth scale in the final result, and β_D denotes the belief of the incomplete information.

Later, the iterative form of the ER algorithm for integrating K rules is deduced

$$m_n=\mu [{\displaystyle \prod _{k=1}^K(m_{n,k}+{\overline{m}}_{D,k}+{\tilde{m}}_{D,k})}-{\displaystyle \prod _{k=1}^K({\overline{m}}_{D,k}+{\tilde{m}}_{D,k})}],\begin{array}{cc}& n=1,\dots ,N\end{array}$$

(12)

$${\tilde{m}}_D=\mu [{\displaystyle \prod _{k=1}^K({\overline{m}}_{D,k}+{\tilde{m}}_{D,k})}-{\displaystyle \prod _{k=1}^K({\overline{m}}_{D,k})}]$$

(13)

$${\overline{m}}_D=\mu [{\displaystyle \prod _{k=1}^K({\overline{m}}_{D,k})}]$$

(14)

$$\mu ={[{\displaystyle \sum _{n=1}^N{\displaystyle \prod _{k=1}^K(m_{n,k}+{\overline{m}}_{D,k}+{\tilde{m}}_{D,k})}}-(N-1){\displaystyle \prod _{k=1}^K({\overline{m}}_{D,k}+{\tilde{m}}_{D,k})}]}^{-1}$$

(15)

$${\beta }_n=\frac{m_n}{1-{\overline{m}}_D}$$

(16)

$${\beta }_D=1-{\displaystyle \sum _{n=1}^N{\beta }_n}$$

(17)

As this study mainly focuses on the incomplete information β_D, the following gives a detailed deduction of β_D.

$$\begin{array}{l}{\beta }_D\\ =1-{\displaystyle \sum _{n=1}^N{\beta }_n}\\ =1-{\displaystyle \sum _{n=1}^N(\frac{[{\displaystyle \prod _{k=1}^K(w_k{\beta }_{n,k}+1-}{w}_k{\displaystyle \sum _{n=1}^N{\beta }_{n,k}})-{\displaystyle \prod _{k=1}^K(1-w_k{\displaystyle \sum _{n=1}^N{\beta }_{n,k}})}]}{{\displaystyle \sum _{n=1}^N{\displaystyle \prod _{k=1}^K(w_k{\beta }_{n,k}+1-}{w}_k{\displaystyle \sum _{n=1}^N{\beta }_{n,k}})-(N-1){\displaystyle \prod _{k=1}^K(1-w_k{\displaystyle \sum _{n=1}^N{\beta }_{n,k}})}}-[{\displaystyle \prod _{k=1}^K(1-w_k)}]})}\\ =1-\frac{{\displaystyle \sum _{n=1}^N[{\displaystyle \prod _{k=1}^K(w_k{\beta }_{n,k}+1-}{w}_k{\displaystyle \sum _{n=1}^N{\beta }_{n,k}})-{\displaystyle \prod _{k=1}^K(1-w_k{\displaystyle \sum _{n=1}^N{\beta }_{n,k}})]}}}{{\displaystyle \sum _{n=1}^N{\displaystyle \prod _{k=1}^K(w_k{\beta }_{n,k}+1-}{w}_k{\displaystyle \sum _{n=1}^N{\beta }_{n,k}})-(N-1){\displaystyle \prod _{k=1}^K(1-w_k{\displaystyle \sum _{n=1}^N{\beta }_{n,k}})}}-[{\displaystyle \prod _{k=1}^K(1-w_k)}]}\\ =\frac{\begin{array}{l}\{{\displaystyle \sum _{n=1}^N{\displaystyle \prod _{k=1}^K(w_k{\beta }_{n,k}+1-}{w}_k{\displaystyle \sum _{n=1}^N{\beta }_{n,k}})-(N-1){\displaystyle \prod _{k=1}^K(1-w_k{\displaystyle \sum _{n=1}^N{\beta }_{n,k}})}}\\ -[{\displaystyle \prod _{k=1}^K(1-w_k)}]\}-\{{\displaystyle \sum _{n=1}^N[{\displaystyle \prod _{k=1}^K(w_k{\beta }_{n,k}+1-}{w}_k{\displaystyle \sum _{n=1}^N{\beta }_{n,k}})-{\displaystyle \prod _{k=1}^K(1-w_k{\displaystyle \sum _{n=1}^N{\beta }_{n,k}})]\}}}\end{array}}{{\displaystyle \sum _{n=1}^N{\displaystyle \prod _{k=1}^K(w_k{\beta }_{n,k}+1-}{w}_k{\displaystyle \sum _{n=1}^N{\beta }_{n,k}})-(N-1){\displaystyle \prod _{k=1}^K(1-w_k{\displaystyle \sum _{n=1}^N{\beta }_{n,k}})}}-[{\displaystyle \prod _{k=1}^K(1-w_k)}]}\\ =\frac{N{\displaystyle \prod _{k=1}^K(1-w_k{\displaystyle \sum _{n=1}^N{\beta }_{n,k}})}-(N-1){\displaystyle \prod _{k=1}^K(1-w_k{\displaystyle \sum _{n=1}^N{\beta }_{n,k}})}-{\displaystyle \prod _{k=1}^K(1-w_k)}}{{\displaystyle \sum _{n=1}^N{\displaystyle \prod _{k=1}^K(w_k{\beta }_{n,k}+1-}{w}_k{\displaystyle \sum _{n=1}^N{\beta }_{n,k}})-(N-1){\displaystyle \prod _{k=1}^K(1-w_k{\displaystyle \sum _{n=1}^N{\beta }_{n,k}})}}-{\displaystyle \prod _{k=1}^K(1-w_k)}}\\ =\frac{{\displaystyle \prod _{k=1}^K(1-w_k{\displaystyle \sum _{n=1}^N{\beta }_{n,k}})}-{\displaystyle \prod _{k=1}^K(1-w_k)}}{{\displaystyle \sum _{n=1}^N{\displaystyle \prod _{k=1}^K(w_k{\beta }_{n,k}+1-}{w}_k{\displaystyle \sum _{n=1}^N{\beta }_{n,k}})-(N-1){\displaystyle \prod _{k=1}^K(1-w_k{\displaystyle \sum _{n=1}^N{\beta }_{n,k}})}}-{\displaystyle \prod _{k=1}^K(1-w_k)}}\end{array}$$

(18)

The summary of Equation (18) is Equation (19),

$${\beta }_D=\frac{{\displaystyle \prod _{k=1}^K(1-w_k+w_k{\beta }_{D,k})}-{\displaystyle \prod _{k=1}^K(1-w_k)}}{{\displaystyle \sum _{n=1}^N{\displaystyle \prod _{k=1}^K(w_k{\beta }_{n,k}+1-}{w}_k+w_k{\beta }_{D,k})-(N-1){\displaystyle \prod _{k=1}^K(1-w_k+w_k{\beta }_{D,k})}}-{\displaystyle \prod _{k=1}^K(1-w_k)}}$$

(19)

where it denotes that the incomplete information in the result via the ER algorithm is affected by two factors, namely the relative importance of the rules w_k and the beliefs of the incomplete information in the rules β_D,k.

2.2. The Spreading of Incomplete Information via the ER Rule

Yang and Xu further proposed the evidential reasoning rule (ER rule) by including the reliability r of the rules in the traditional ER algorithm [17]. Thus, the weight of the kth rule within the framework of the ER rule w_k is updated by the following Equation (20),

$${\omega }_k=\frac{w_k}{1+w_k-r_k}$$

(20)

where w_k and r_k denote the original weight (which is the same as in the weight in the ER algorithm) and the reliability of the kth rule, respectively. The weight w and reliability r of a rule can be determined based either on experts’ knowledge or statistical data. As the determination of w and r is not the primary focus of this study, readers can consult the works of [19,20] and references therein.

Then, there is still

$${\beta }_n=\frac{\mu \left[{\displaystyle \prod _{k=1}^K\left({\omega }_k{\beta }_{n,k}+1-{\omega }_k{\displaystyle \sum _{n=1}^N{\beta }_{n,k}}\right)}-{\displaystyle \prod _{k=1}^K\left(1-{\omega }_k{\displaystyle \sum _{n=1}^N{\beta }_{n,k}}\right)}\right]}{1-\mu \left[{\displaystyle \prod _{k=1}^K(1-{\omega }_k)}\right]}$$

(21)

$$\mu ={\left[{\displaystyle \sum _{n=1}^N{\displaystyle \prod _{k=1}^K\left({\omega }_k{\beta }_{n,k}+1-{\omega }_k{\displaystyle \sum _{n=1}^N{\beta }_{n,k}}\right)}}-(N-1){\displaystyle \prod _{k=1}^K\left(1-{\omega }_k{\displaystyle \sum _{n=1}^N{\beta }_{n,k}}\right)}\right]}^{-1}$$

(22)

Correspondingly, the incomplete information is deduced as follows,

$$\begin{array}{l}{\beta }_D\\ =1-{\displaystyle \sum _{n=1}^N{\beta }_n}\\ =1-{\displaystyle \sum _{n=1}^N\frac{\mu \left[{\displaystyle \prod _{k=1}^K\left({\omega }_k{\beta }_{n,k}+1-{\omega }_k{\displaystyle \sum _{n=1}^N{\beta }_{n,k}}\right)}-{\displaystyle \prod _{k=1}^K\left(1-{\omega }_k{\displaystyle \sum _{n=1}^N{\beta }_{n,k}}\right)}\right]}{1-\mu \left[{\displaystyle \prod _{k=1}^K(1-{\omega }_k)}\right]}}\\ =1-{\displaystyle \sum _{n=1}^N\frac{\mu \left[{\displaystyle \prod _{k=1}^K\left(w_k{\beta }_{n,k}+1+w{}_k-r_k-w_k{\displaystyle \sum _{n=1}^N{\beta }_{n,k}}\right)}-{\displaystyle \prod _{k=1}^K\left(1+w_k-r_k-w_k{\displaystyle \sum _{n=1}^N{\beta }_{n,k}}\right)}\right]}{1-\mu \left[{\displaystyle \prod _{k=1}^K(1-w_k)}\right]}}\\ =\frac{{\displaystyle \prod _{k=1}^K(1-r_k+w_k{\beta }_{D,k})}-{\displaystyle \prod _{k=1}^K(1-r_k)}}{{\displaystyle \sum _{n=1}^N{\displaystyle \prod _{k=1}^K(w_k{\beta }_{n,k}+1-}{r}_k+w_k{\beta }_{D,k})-(N-1){\displaystyle \prod _{k=1}^K(1-r_k+w_k{\beta }_{D,k})}}-{\displaystyle \prod _{k=1}^K(1-r_k)}}\end{array}$$

(23)

3. Contribution Calculation within the Framework of ER and the ER Rule

3.1. Contribution Calculation within the Framework of ER

By introducing two functions, f and g, Equation (19) is transformed into the following Equation (24),

$${\beta }_D=\frac{f(w_k,{\beta }_{D,k})}{g(w_k,{\beta }_{D,k})}$$

(24)

where

$$\{\begin{array}{l}f(w_k,{\beta }_{D,k})={\displaystyle \prod _{k=1}^K(1-w_k+w_k{\beta }_{D,k})}-{\displaystyle \prod _{k=1}^K(1-w_k)}\\ g(w_k,{\beta }_{D,k})={\displaystyle \sum _{n=1}^N{\displaystyle \prod _{k=1}^K(w_k{\beta }_{n,k}+1-}{w}_k+w_k{\beta }_{D,k})-(N-1){\displaystyle \prod _{k=1}^K(1-w_k+w_k{\beta }_{D,k})}}-{\displaystyle \prod _{k=1}^K(1-w_k)}\end{array}$$

(25)

Then, calculate the partial derivatives of Equation (25), and the following can be derived,

$$\{\begin{array}{l}\frac{\partial {\beta }_D}{\partial w_k}=\frac{\partial f}{\partial w_k}•\frac{1}{g}+f•(-\frac{1}{g^2})•\frac{\partial g}{\partial w_k}\\ \frac{\partial {\beta }_D}{\partial {\beta }_{D,k}}=\frac{\partial f}{\partial {\beta }_{D,k}}•\frac{1}{g}+f•(-\frac{1}{g^2})•\frac{\partial g}{\partial {\beta }_{D,k}}\end{array}$$

(26)

where

$$\{\begin{array}{l}\frac{\partial f}{\partial w_k}=({\beta }_{D,k}-1){\displaystyle \prod _{k^{\prime }=1,k^{\prime }\ne k}^K(1-w_{k^{\prime }}+w_{k^{\prime }}{\beta }_{D,k^{\prime }})}+{\displaystyle \prod _{k^{\prime }=1,k^{\prime }\ne k}^K(1-w_{k^{\prime }})}\\ \frac{\partial g}{\partial w_k}={\displaystyle \sum _{n=1}^N[({\beta }_{n,k}+{\beta }_{D,k}-1){\displaystyle \prod _{k^{\prime }=1,k^{\prime }\ne k}^K(w_{k^{\prime }}{\beta }_{n,k^{\prime }}+1-w_{k^{\prime }}+}{w}_{k^{\prime }}{\beta }_{D,k^{\prime }})]}\\ -(N-1)({\beta }_{D,k}-1){\displaystyle \prod _{k^{\prime }=1,k^{\prime }\ne k}^K(1-w_{k^{\prime }}+w_{k^{\prime }}{\beta }_{D,k^{\prime }})}+{\displaystyle \prod _{k^{\prime }=1,k^{\prime }\ne k}^K(1-w_{k^{\prime }})}\\ \frac{\partial f}{\partial {\beta }_{D,k}}=w_k{\displaystyle \prod _{k^{\prime }=1,k^{\prime }\ne k}^K(1-w_{k^{\prime }}+w_{k^{\prime }}{\beta }_{D,k^{\prime }})}\\ \frac{\partial g}{\partial {\beta }_{D,k}}={\displaystyle \sum _{n=1}^N[w_k{\displaystyle \prod _{k^{\prime }=1,k^{\prime }\ne k}^K(w_{k^{\prime }}{\beta }_{n,k^{\prime }}+1-}{w}_{k^{\prime }}+w_{k^{\prime }}{\beta }_{D,k^{\prime }})]}\\ -(N-1)w_k{\displaystyle \prod _{k^{\prime }=1,k^{\prime }\ne k}^K(1-w_{k^{\prime }}+w_{k^{\prime }}{\beta }_{D,k^{\prime }})}\end{array}$$

(27)

Based on the above derivative analysis, the contribution made by the kth rule to the incomplete information is calculated as in Equation (28),

$$ctr(R_k,{\beta }_D)=\frac{\partial {\beta }_D}{\partial w_k}+\frac{\partial {\beta }_D}{\partial {\beta }_{D,k}}-\frac{\partial {\beta }_D}{\partial w_k}•\frac{\partial {\beta }_D}{\partial {\beta }_{D,k}}$$

(28)

where $\frac{\partial {\beta }_D}{\partial w_k}$ and $\frac{\partial {\beta }_D}{\partial {\beta }_{D,k}}$ denote the contributions made by the relative weight w_k and β_D,k, respectively, and $\frac{\partial {\beta }_D}{\partial w_k}•\frac{\partial {\beta }_D}{\partial {\beta }_{D,k}}$ denotes their joint contribution, which should then be deducted, as it would have been double calculated in $\frac{\partial {\beta }_D}{\partial w_k}+\frac{\partial {\beta }_D}{\partial {\beta }_{D,k}}$.

Figure 1 presents the mathematical logic for calculating ctr(R_k, β_D). Specifically, Equation (28) sees the contribution made by the rule weight and the rule belief as two independent factors. Thus, their integrated contribution follows the addition law of probability [23].

3.2. Contribution Calculation within the Framework of the ER Rule

By introducing two functions f and g, Equation (23) is transformed into the following Equation (29),

$${\beta }_D=\frac{f(w_k,r_k,{\beta }_{D,k})}{g(w_k,r_k,{\beta }_{D,k})}$$

(29)

where

$$\{\begin{array}{l}f(w_k,r_k,{\beta }_{D,k})={\displaystyle \prod _{k=1}^K(1-r_k+w_k{\beta }_{D,k})}-{\displaystyle \prod _{k=1}^K(1-r_k)}\\ g(w_k,r_k,{\beta }_{D,k})={\displaystyle \sum _{n=1}^N{\displaystyle \prod _{k=1}^K(w_k{\beta }_{n,k}+1-}{r}_k+w_k{\beta }_{D,k})}\\ -(N-1){\displaystyle \prod _{k=1}^K(1-r_k+w_k{\beta }_{D,k})}-{\displaystyle \prod _{k=1}^K(1-r_k)}\end{array}$$

(30)

Then, calculate the partial derivative in Equation (30), and the following can be derived,

$$\{\begin{array}{l}\frac{\partial {\beta }_D}{\partial w_k}=\frac{\partial f}{\partial w_k}•\frac{1}{g}-\frac{f}{g^2}•\frac{\partial g}{\partial w_k}\\ \frac{\partial {\beta }_D}{\partial r_k}=\frac{\partial f}{\partial r_k}•\frac{1}{g}-\frac{f}{g^2}•\frac{\partial g}{\partial r_k}\\ \frac{\partial {\beta }_D}{\partial {\beta }_{D,k}}=\frac{\partial f}{\partial {\beta }_{D,k}}•\frac{1}{g}-\frac{f}{g^2}•\frac{\partial g}{\partial {\beta }_{D,k}}\end{array}$$

(31)

where

$$\{\begin{array}{l}\frac{\partial f}{\partial w_k}={\beta }_{D,k}{\displaystyle \prod _{k^{\prime }=1,k^{\prime }\ne k}^K(w_{k^{\prime }}{\beta }_{D,k^{\prime }}+1-r_{k^{\prime }})}\\ \frac{\partial f}{\partial r_k}=-{\displaystyle \prod _{k^{\prime }=1,k^{\prime }\ne k}^K(w_{k^{\prime }}{\beta }_{D,k^{\prime }}+1-r_{k^{\prime }})}+{\displaystyle \prod _{k^{\prime }=1,k^{\prime }\ne k}^K(1-r_{k^{\prime }})}\\ \frac{\partial f}{\partial {\beta }_{D,k}}=w_k{\displaystyle \prod _{k=1,k^{\prime }\ne k}^K(w_{k^{\prime }}{\beta }_{D,k^{\prime }}+1-r_{k^{\prime }})}\\ \frac{\partial g}{\partial w_k}={\displaystyle \sum _{n=1}^N[({\beta }_{n,k}+{\beta }_{D,k^{\prime }}){\displaystyle \prod _{k=1,k^{\prime }\ne k}^K(w_{k^{\prime }}{\beta }_{n,k^{\prime }}+1-r_{k^{\prime }}+w_{k^{\prime }}{\beta }_{D,k^{\prime }})]}}\\ -(N-1){\beta }_{D,k}{\displaystyle \prod _{k^{\prime }=1,k^{\prime }\ne k}^K(w_{k^{\prime }}{\beta }_{D,k^{\prime }}+1-r_{k^{\prime }})}\\ \frac{\partial g}{\partial r_k}=-{\displaystyle \sum _{n=1}^N[{\displaystyle \prod _{k=1,k^{\prime }\ne k}^K(w_{k^{\prime }}{\beta }_{n,k^{\prime }}+1-r_{k^{\prime }}+w_{k^{\prime }}{\beta }_{D,k^{\prime }})]}}\\ +(N-1){\displaystyle \prod _{k^{\prime }=1,k^{\prime }\ne k}^K(1-r_{k^{\prime }}+w_{k^{\prime }}{\beta }_{D,k^{\prime }})}+{\displaystyle \prod _{k^{\prime }=1,k^{\prime }\ne k}^K(1-r_{k^{\prime }})}\\ \frac{\partial g}{\partial {\beta }_{D,k}}={\displaystyle \sum _{n=1}^N[w_k{\displaystyle \prod _{k=1,k^{\prime }\ne k}^K(w_{k^{\prime }}{\beta }_{n,k^{\prime }}+1-r_{k^{\prime }}+w_{k^{\prime }}{\beta }_{D,k^{\prime }})]}}\\ -(N-1)w_k{\displaystyle \prod _{k^{\prime }=1,k^{\prime }\ne k}^K(1-r_{k^{\prime }}+w_{k^{\prime }}{\beta }_{D,k^{\prime }})}\end{array}$$

(32)

Based on the above derivative analysis, the contribution made by the kth rule to the incomplete information is calculated as in Equation (33),

$$\begin{array}{l}ctr(R_k,{\beta }_D)=\frac{\partial {\beta }_D}{\partial w_k}+\frac{\partial {\beta }_D}{\partial r_k}+\frac{\partial {\beta }_D}{\partial {\beta }_{D,k}}-\frac{\partial {\beta }_D}{\partial w_k}•\frac{\partial {\beta }_D}{\partial r_k}-\frac{\partial {\beta }_D}{\partial w_k}•\frac{\partial {\beta }_D}{\partial {\beta }_{D,k}}\\ -\frac{\partial {\beta }_D}{\partial r_k}•\frac{\partial {\beta }_D}{\partial {\beta }_{D,k}}+\frac{\partial {\beta }_D}{\partial w_k}•\frac{\partial {\beta }_D}{\partial r_k}•\frac{\partial {\beta }_D}{\partial {\beta }_{D,k}}\end{array}$$

(33)

where $\frac{\partial {\beta }_D}{\partial w_k}$, $\frac{\partial {\beta }_D}{\partial {\beta }_{D,k}}$, and $\frac{\partial {\beta }_D}{\partial r_k}$ are calculated in Equation (32). Furthermore, Figure 2 reveals why Equation (33) is designed as such.

Figure 2 presents the mathematical logic for calculating ctr(R_k, β_D), which is consistent with Figure 1. Specifically, Equation (33) sees the contribution made by the rule weight, rule belief, and rule reliability as three independent factors. Thus, their integrated contribution follows the addition law of probability [23].

3.3. Similarities and Dissimilarities between the ER Algorithm and the ER Rule

Table 1. Comparison between the ER algorithm and the ER rule.

Items	ER Algorithm	ER Rule
Theoretical basis	D-S evidence theory	D-S evidence theory
Ability to handle incomplete information	Yes	Yes
Major factors	Rule weight, rule beliefs	Rule weight, rule beliefs, and rule reliability
Factors that affect the integration results	Rule weight, rule beliefs	Rule weight, rule beliefs, and rule reliability

summarizes the similarities and dissimilarities between the ER algorithm and the ER rule. According to Table 1, both the ER algorithm and the ER rule can handle incomplete information, which is inherited from their shared theoretical basis, i.e., the D-S evidence theory. Among many dissimilarities, the major difference between the ER algorithm and the ER rule is whether the reliability of the rule is considered. To put it simply, the ER rule includes the rule reliability, while the ER algorithm does not. Section 3 separately studies how the ER algorithm and the ER rule affect the spreading of incomplete information.

4. Numerical Case Study

4.1. A Numerical Case via ER: Incomplete Information Calculation

The parameter settings of the first numerical case study are as follows: (1) There are only two rules with equal weights of w₁ = w₂ = 0.5. (2) The belief distributions of the two rules are also assumed to be the same as {(D₁, β_D,1),(D₂, β_D,2)}.

First, the results following the above parameter settings, subject to different conditions of the weights and incomplete information in the rules, are provided below. Figure 3 shows the results using different weights. Specifically, Figure 3a presents the result with w₁ = w₂ = 0.5, and Figure 3b,c shows the result with w₁ = 0.3, w₂ = 0.7, and w₁ = 0.7, w₂ = 0.3, respectively. Accordingly, Figure 3a shows a symmetrical image, as does Figure 3b,c.

4.2. A Numerical Example via ER: Information Spreading Based on Contribution Calculation

First, we will analyze the contributions made by the rules to the incomplete information in the result with fixed rule weights. Figure 4a–f, present the contribution made to the incomplete information in the result β_D by the first and second rule, R₁ and R₂, i.e., ctr(R₁, β_D) and ctr(R₂, β_D), respectively.

According to Figure 4a–f, the following conclusions can be drawn:

(1)

Neither ctr(R₁, β_D) nor ctr(R₂, β_D) is symmetrical. Generally, we can observe that the highest contribution is made when β_D,1 = 1 and β_D,2 = 1, but the lowest contribution is NOT made when β_D,1 = 0 and β_D,2 = 0. Instead, the contribution reaches the lowest value when β_D,1 = 0 and β_D,2 = 1, as shown in Figure 4a, and the contribution reaches the lowest value when β_D,1 = 1 and β_D,2 = 0, as shown in Figure 4b.

(2)

Symmetrical results can be observed by comparing Figure 4c,d with Figure 4e,f, i.e., Figure 4c,f and Figure 4d,e are identical. Nonetheless, it can be observed that they both produce tilted images compared with Figure 4a,b. The contribution reaches the lowest value when β_D,1 = 0 and β_D,2 = 1 (Figure 4c) and when β_D,1 = 1 and β_D,2 = 0 (Figure 4f). The lowest contribution is not achieved at any endpoint in Figure 4d,f.

(3)

The nonlinearity is more novel when there is a tilted weight distribution in Figure 4c–f compared with equal weights in Figure 4a,b. This is especially clear in Figure 4d,e.

Next, we further test the contributions of the rules to the incomplete information in the result, with fixed beliefs in the rules as the input, namely β_D,1 and β_D,2. Specifically, Figure 5a–c presents the results when β_D,1 = β_D,2 = 0.5, β_D,1 = 0.3/β_D,2 = 0.7, and β_D,1 = 0.7/β_D,2 = 0.3, respectively.

According to Figure 5, the following conclusions can be drawn:

(1)

There is an increasing nonlinearity presented in Figure 5a–c. Even in Figure 5a, the nonlinearity is much higher compared with Figure 4a,b. The nonlinearity presented in Figure 5b,c is even higher. Specifically, there is a sharp change in β_D when w₂→0 and β_D,1 = 0.3/β_D,2 = 0.7, and also when w₁→0 and β_D,1 = 0.7/β_D,2 = 0.3.

(2)

The minimum β_D in Figure 5a is obtained while w₁ = w₂ = 0, and the highest β_D is NOT obtained when w₁ = w₂ = 1, but when w₁ = 1 and w₂ = 0, or w₁ = 0 and w₂ = 1, which is beyond the expectation of common understanding. As for when β_D,1 = 0.3/β_D,2 = 0.7, or β_D,1 = 0.7/β_D,2 = 0.3, the results are even more unstable, which is consistent with the high nonlinearity, as summarized in (1).

(3)

Even with the high nonlinearity presented in Figure 5a–c, symmetry can still be observed, not only in Figure 5a, but also in Figure 5b,c.

Figure 6 presents the contribution calculation results with different β_D,1 and β_D,2. The following conclusions can be drawn:

(1)

There is a corresponding high nonlinearity presented by Figure 6b,c compared with Figure 5a–c, which is consistent with the enhanced nonlinearity presented in Figure 5b,c vs. Figure 5a.

(2)

A high level of symmetry can still be observed in Figure 6a and also in Figure 6b,c. Moreover, there is a certain level of asymmetry within Figure 6b,c. It seems that there is a drastic change in the contribution when β_D,1 and β_D,2 are imbalanced. Specifically, the image is tilted towards one direction, favoring the rule with a higher β_D,k rather than along the axis of w₁ = w₂.

4.3. A Numerical Case via the ER Rule: Incomplete Information Calculation

Since there are three influential factors regarding β_D in the context of the ER rule, namely the rule weight w, the rule reliability r, and the incomplete information in the input rules β_D,k, this section will study the co-influence of two of the three factors on β_D via the ER rule.

First, the co-influence of β_D,1 and w₁ is studied with a fixed r. The specific parameter settings are as follows. (1) The two rules are assumed with fixed reliabilities of r₁ = r₂ = 0.5. (2) There are only two rules with w₂ = 0.5, and w₁∈[0, 0.5]. (3) The incomplete information in Rule 2 is 0.5, i.e., β_D,2 = 0.5, and β_D,1∈[0, 1]. Figure 7a presents the results, and the following observations can be made:

(1)

β_D reaches its minimum when β_D,1 = 0 and w₁ = 0.5, and β_D reaches its maximum when β_D,1 = 1 and w₁ = 0.5.

(2)

When w₁ is approaching 0, β_D is completely determined by β_D,2, as it is equal to β_D,2, and it does not change along with β_D,1. In other words, β_D,1 does not contribute β_D, and β_D = 0.5.

(3)

When w₁ ≠ 0, β_D increases along with the increase in β_D,1.

(4)

When β_D,1 is fixed and below 0.5, β_D decreases along with the increase in w₁.

(5)

When β_D,1 is fixed and above 0.5, β_D increases along with the increase in w₁.

The co-influence of β_D,1 and r₁ is studied with a fixed w. The specific parameter settings are as follows. (1) The incomplete information in Rule 1 and Rule 2 is 0.5, i.e., β_D,1 = β_D,2 = 0.5. (2) There are only two rules with w₂ = 0.5, and w₁∈[0, 0.5]. (3) The two rules are assumed with equal reliabilities of r₂ = 0.5, and r₁∈[0, 1]. Figure 7c presents the results, and the following observations can be made: (1) β_D reaches its minimum when r₁ = 1 and w₁ = 0.5, and β_D reaches its maximum when w₁ = 0.5. (2) When w₁ is approaching 0, β_D is completely determined by β_D,2, as it is equal to β_D,2 and it does not change along with r₁. In other words, r₁ does not contribute β_D, and β_D = 0.5. (2) When w₁ ≠ 0, β_D decreases along with the increase in r₁. (3) β_D decreases along with the increase in w₁.

Finally, the co-influence on β_D by w₁ and r₁ is studied with a fixed β_D,k. The specific parameter settings are as follows. (1) There are only two rules with w₁ = w₂ = 0.5. (2) The two rules are assumed with reliabilities of r₂ = 0.5, r₁∈[0, 1]. (3) The incomplete information in Rule 2 is 0.5, i.e., β_D,2 = 0.5, and β_D,1∈[0, 1]. Figure 7b presents the results, and the following observations can be made: (1) β_D increases along with the increase in β_D,1. (2) When β_D,1 is fixed and below 0.5, β_D decreases along with the increase in r₁. (3) When β_D,1 is fixed and above 0.5, β_D increases along with the increase in r₁. (4) β_D reaches its minimum when β_D,1 = 0 and r₁ = 1, and β_D reaches its maximum when β_D,1 = 0 and r₁ = 1.

4.4. A Numerical Case via ER Rule: Information Spreading Based on Contribution Calculation

As Section 4.3 studies how incomplete information spreads via the ER rule in three separate settings, this section also follows this train of thought by calculating the contribution in three separate settings.

First, the contribution made by (β_D,1, w₁) in Rule 1 and Rule 2 to β_D with w₂ = 0.5, β_D,2 = 0.5, r₁ = r₂ = 0.5 is calculated and presented in Figure 8. According to Figure 8a, (1) ctr(R₁, β_D) reaches its minimum when β_D,1 = 0 and w₁ = 0, and β_D reaches its maximum when β_D,1 = 1 and w₁ = 0. (2) When β_D,1 is fixed and below 0.5, ctr(R₁, β_D) increases along with the increase in w₁. (3) When β_D,1 is fixed and below 0.5, ctr(R₁, β_D) decreases along with the increase in w₁. According to Figure 8b, (1) ctr(R₂, β_D) reaches its minimum when β_D,1 = 1 and w₁ = 0.5, and ctr(R₂, β_D) reaches its maximum when β_D,1 = 0 and w₁ = 0. (2) There is a monotonic decreasing trend from the maximum to all directions of (β_D,1, w₁), although at different decreasing speeds.

Second, the contribution made by (w₁, r₁) in Rule 1 and Rule 2 to β_D with β_D,1 = β_D,2 = 0.5 is calculated and presented in Figure 9. According to Figure 9a, (1) ctr(R₁, β_D) reaches its minimum when w₁ = 0 and r₁ = 1, and ctr(R₁, β_D) reaches its maximum when w₁ = 0.5 and r₁ = 1. (2) Other than the minimum, at (w₁ = 0 and r₁ = 1) ctr(R₁, β_D) stays rather flat for almost all situations. According to Figure 9b, (1) ctr(R₂, β_D) reaches its minimum when r₁ = 1 and w₁ = 0.5, and ctr(R₂, β_D) reaches its maximum when w₁ = 0. (2) When w₁ ≠ 0, ctr(R₂, β_D) decreases along with the increase in r₁. (3) When β_D,1 -> 1, ctr(R₂, β_D) slightly decreases along with the increase in r₁, and when β_D,1 -> 0, ctr(R₂, β_D) slightly increases along with the increase in r₁.

Finally, the contribution made by (β_D,1, r₁) in Rule 1 and Rule 2 to β_D with w₁ = w₂ = 0.5 is calculated and presented in Figure 10. According to Figure 10a, (1) ctr(R₁, β_D) reaches its minimum when β_D,1 = 0 and r₁ = 0, and ctr(R₁, β_D) reaches its maximum when β_D,1 = 0 and r₁ = 1. (2) When r₁ -> 1, ctr(R₁, β_D) first decreases and then increases along with the increase in β_D,1. (3) When r₁ -> 0, ctr(R₁, β_D) decrease along with decrease in β_D,1. According to Figure 10b, (1) ctr(R₂, β_D) reaches its minimum when β_D,1 = 1 and r₁ = 1, and ctr(R₂, β_D) reaches its maximum when β_D,1 = 0 and r₁ = 0. (2) When r₁ -> 0, ctr(R₂, β_D) decreases along with the increase in β_D,1. (3) When β_D,1 -> 0, ctr(R₂, β_D) first increases and decreases along with the increase in r₁.

4.5. Summarization

The following conclusions can be drawn, according to the results from the numerical case studies in prior sections.

(1)

There is a high level of nonlinearity observed both when integrating incomplete information and calculating the contributions made by influential factors. This is mostly owing to the superior nonlinearity modeling ability of the ER algorithm and the ER rule. From a counter perspective, the nonlinearity is expected, as a superior machine learning approach is anticipated to handle the nonlinearity residing in practical problems, or it would be deemed a failed approach.

(2)

Symmetry is observed, even when certain parameters are adjusted. This is especially novel in the context of the ER algorithm when the rule reliability is not considered as compared to when the ER rule is considered (in terms of the rule reliability).

(3)

Asymmetry is observed, especially concerning the contribution calculation results, and is more novel in the context of the ER rule compared with the ER algorithm. For the ER rule, this is likely because there are three factors, namely the rule reliability, the rule weight, and the incomplete information of the rule. Specifically, there is a higher level of asymmetry and nonlinearity between the rule weight and rule reliability with fixed β_D,1 and β_D,2.

Overall, the investigations regarding how incomplete information spreads via the ER algorithm and the ER rule demonstrate high levels of nonlinearity, symmetry, and asymmetry, which is consistent with the superior modeling ability and the ER algorithm and the ER rule, and it is also the reason why ER qualifies as a transparent white-box machine learning approach.

5. Conclusions

An investigative study is conducted on the spreading of incomplete information via the ER algorithm and the ER rule by calculating the contributions made by the rule weight, rule reliability, and the rule beliefs as the input to the incomplete information in the result. The theoretical contribution is calculated based on partial derivative analysis, which is analytical and different from traditional sensitive analysis or qualitative analysis.

The major findings of this study using two numerical cases include the discovery of high levels of nonlinearities among major symmetrical features with the given parameter settings. More importantly and unpredictably, there are also asymmetrical features revealed when the parameter settings are even slightly adjusted.

The limitations are that only numerical investigations, rather than practical cases, are conducted in this study, which presents obstacles for non-academics to better understand this study. Moreover, the mathematical investigation is conducted under the premise of the integration of two rules instead of K rules, which is a more generic condition.

For future studies, the contribution calculation results can be used directly as the mathematical and deterministic proof for further reducing the incomplete information in the result, with consideration of the rule costs. Moreover, the validation should be extended to practical cases to grant specific and physical meanings to rules, reliabilities, weights, costs, etc.