A medical assistant decision-making method based on interval belief rule base with explainability

Abstract

Medical assisted decision-making plays a key role in providing accurate and reliable medical advice. But in medical decision-making, various uncertainties are often accompanied. The belief rule base (BRB) has a strong nonlinear modeling capability and can handle uncertainties well. However, BRB suffers from combinatorial explosion and tends to influence explainability during the optimization process. Therefore, an interval belief rule base with explainability (IBRB-e) is explored in this paper. Firstly, pre-processing using extreme gradient boosting (XGBoost) is performed to filter out features with lower importance. Secondly, based on the filtered features, explainability criterion is defined. Thirdly, evidence reasoning (ER) rule is chosen as an inference tool, while projection covariance matrix adaptive evolutionary strategy (P-CMA-ES) algorithm with explainability constraints is chosen as an optimization algorithm. Lastly, the validation of the model is performed through a breast cancer case. The experimental results show that IBRB-e has good explainability while maintaining high accuracy.

Keywords

Belief rule base decision-making medical assistant explainability evidence reasoning

1. Introduction

Medical assisted decision-making plays a pivotal role in ensuring patients’ health management and subsequent treatment [1]. However, in actual clinical decision-making, a common challenge often arises, that is, the scarcity of an adequate number of human medical decision-making experts with rich experience [2]. Additionally, with the explosive growth of medical data, especially when dealing with a substantial volume of intricate medical information within a short timeframe, various unpredictable challenges and errors may arise during the medical decision-making process [3]. In this context, accurate medical assisted decision-making methods become crucial to ensure that patients receive the best possible medical care and treatment plans.

In recent years, the introduction of artificial intelligence into the medical field has brought about entirely new methods and avenues for medical assisted decision-making [4]. These primarily encompass the following three methods: data-driven methods, model-driven methods, and hybrid information methods. The data-driven methods depend on extensive training data during modeling, utilizing patterns learned from the data to construct models [5]. For example, Haddi et al. [6] proposed an automatic medical decision-making method based on the relevance vector machine (RVM), and optimized the RVM by configuring various kernel functions and parameters. This approach demonstrated excellent performance in addressing atrial fibrillation issues. Andre et al. [7] used a single convolutional neural network (CNN) to classify skin lesions, which achieved performance comparable to all test experts on the task, proving that CNN is capable of classifying skin cancers with the ability of dermatologists. Huang et al. [8] proposed an intelligent data-driven model using support vector machine (SVM) and random forest (RF), which applied to a diagnostic case of cough-variant asthma (CVA). The results demonstrated that the model outperformed individual SVM and RF models, showcasing improved predictive performance. Nicolas et al. [9] introduced a data-driven approach to diagnose Parkinson’s disease (PD), leveraging vertical Ground Reaction Forces (vGRFs) data obtained during the gait cycle. This method effectively distinguishes PD subjects from those with other neurodegenerative disorders. However, in situations involving a small volume of data or high model complexity, the data-driven methods may run the risk of overfitting. This refers to the tendency of the model to excessively adapt to the training data, resulting in poorer performance when applied to new, unseen data.

The model-driven methods describe and solve problems by constructing mathematical models based on prior knowledge, rules, and theories in the modeling process [10]. For instance, Wu et al. [11] applied a “white-box” modeling method. The molecular subtypes of breast cancer are predicted in a multimodal environment using the BI-RADS function of mammography and MRI. Tolks et al. [12] proposed a classification method for a dynamic model of diabetes, which employed principal component analysis (PCA) and support vector machine (SVM) to identify an appropriate classifier. The classification results demonstrated a strong discriminative capability among these categories. Uzair et al. [13] introduced a novel model-driven deep deterministic learning (MDDDL) approach, and defined the system behavior of MDDDL based on pattern recognition of extracted features and subsequently employed empirical methods to effectively evaluate the performance of MDDDL. However, building and maintaining a model can require significant time, resources, and expertise. Especially when complex problems and large-scale data are involved, the design and adjustment of models can become very complex.

The hybrid information methods combine the above two models by combining multiple different types of data to jointly model the data to obtain more comprehensive information and more accurate predictions. For example, Thosini et al. [14] created a deep fuzzy neural network to manage uncertain genetic data for disease diagnosis. Results on six datasets showcased the superior performance of the novel approach over traditional methods using different gene selection techniques. Matteo et al. [15] developed a hybrid machine deep learning model to classify patients into two outcome categories, the CNN classifier is used as a feature extractor, and experiments proved that the model provided doctors with very reliable clinical decision support. Diogo et al. [16] employed a hybrid modeling approach to patients with chronic obstructive pulmonary disease (COPD). By combining hierarchical clustering and decision tree classification techniques, they generated a classification model with a predictive accuracy of 90.4%, thereby offering patients a reliable and affordable alternative for disease management. The hybrid information method combines the advantages of both approaches and allows for faster modeling of data.

BRB is originally proposed by Yang et al. [17], which is a typical hybrid method [18]. As an evidence-based reasoning knowledge representation, it is essentially an expert system [19]. BRB performs well in dealing with various uncertainties [20]. It deals with uncertainty by synthesizing input information and modeling correlations between antecedent properties and outcomes. In addition, BRB can not only maintain a high accuracy rate under small sample data, but also has a certain degree of explainability [21], which is of great significance for medical assistant decision-making. Therefore, BRB has been widely used in many fields [22, 23, 24, 25, 26, 27, 28, 29, 30], in addition, BRB has been widely used in the medical field, for example, Han et al. [31] proposed a hierarchical BRB disease diagnosis model based on power set (HBRBp), which solves the problem of combination rule explosion and inability to effectively deal with local ignorance. It proves the effectiveness of BRB in the field of medical decision-making.

However, the BRB needs to address two issues in medical assisted decision-making: (1) The Cartesian product is employed by BRB for rule combination, which can lead to the problem of rule explosion when there are too many attributes or reference values are considered [32], thereby reducing the efficiency of the model. (2) While optimization algorithms are employed to enhance the BRB model, it is important to note that these algorithms often come with inherent randomness. The explainability of BRB can potentially be impacted by this randomness, consequently leading to a reduction in overall credibility [33]. Therefore, an IBRB-e medical assisted decision-making model with explainable constraints is proposed. Firstly, the reference values for premise attributes are set in interval form, and a new rule combination method is employed, avoiding the issue of rule explosion. Secondly, ER rule is used as the inference engine, taking into account the weights and reliability of the rules. Finally, the optimization process utilizes the P-CMA-ES optimization algorithm with explainability constraints, enhancing the explainability of the model.

This paper offers the following key contributions: (1) The IBRB-e model is constructed to ensure both the accuracy and explainability of the model. (2) Applying IBRB-e to the medical field to help doctors make more reliable medical decisions.

The rest of this paper is as follows: In Section 2, problems in aiding decision-making by disease are addressed. In Section 3, the construction of IBRB-e is introduced, the step to derive feature importance using XGBoost is described, and the process of modeling, inference, and optimization is described. In Section 4, the effectiveness is illustrated by the example of breast cancer. This paper is summarized in Section 5.

2. BRB and problem formulation

This section focuses on the problems faced in the modeling process of the IBRB-e medical assisted decision-making model. In Subsection 2.1, the BRB (rule building and reasoning optimization) is briefly introduced. In Subsection 2.2, various problems suffered by the IBRB-e medical assisted decision-making model during the construction process are discussed.

2.1 Description of BRB

BRB is modeled as a rule-based model with attributes that allow for the incorporation of qualitative knowledge, and uncertainty can be expressed through belief distribution. The description of $k$ -th rule of BRB is as follows:

\begin{aligned} \begin{aligned} {R u l e}_{k} : If x_{1} is A_{1}^{k} \land x_{2} is A_{2}^{k} \land \dots x_{M} is A_{M}^{k} \\ Then consequence is {(D_{1}, β_{1, k}), (D_{2}, β_{2, k}), \dots, (D_{N}, β_{N, k})} \\ And attribute weight δ_{1, k}, δ_{2, k}, \dots, δ_{M, k} \\ \sum_{k = 1}^{N} β_{i, k} ⩽ 1 \end{aligned} \end{aligned}

(1)

where

M

is the count of the premise attribute,

x_{i}

represents the

i

-th of

M

premise attributes.

A_{M}^{k}

is the reference value in the

k

-th rule,

D_{i} (1 ⩽ i ⩽ N)

is the

i

-th consequence,

β_{N, k}

is the belief degree corresponding to the

N

-th consequence, and

δ_{i, k} (1 ⩽ i ⩽ M)

is the attribute weight in the

k

-th rule. Due to the existence of

\land

, a rule may contain multiple premise attributes in the rule composition of BRB.

The reasoning process of BRB is roughly divided into three steps: Step 1: calculating the rule matching degree. Step 2: calculating the activation weights of the rules. Step 3: using ER parsing algorithm for rule synthesis, but ER parsing algorithm does not consider rule reliability, which leads to the inability to clearly understand the contribution of each rule to the result. The P-CMA-ES algorithm is the optimization algorithm of BRB, but P-CMA-ES algorithm without explainable constraints is prone to lose the explainability in the optimization process. In addition, BRB uses the form of Cartesian product to construct the belief table, which is prone to explode the combination rules when there are too many premise attributes. The IBRB-e model presented in this paper successfully solves the above problems.

2.2 Questions with IBRB-e model

Based on the special needs of the medical field, the application of IBRB-e model in this context requires consideration of the following three important questions:

Question 1: How to solve the combinatorial rule explosion problem that exists in BRB? To solve this problem, the IBRB-e model uses interval addition to reduce rule generation, which avoiding exponential growth of rules.

Question 2: How can the problem of belief rule bases losing explainability during optimization be solved? To solve this problem, this paper adds explainability constraints on the basis of the consistent P-CMA-ES to improve the explainability. The explainability constraints can be described as follows:

\begin{aligned} c = {c_{1}, c_{2}, c_{3}, \dots c_{f}} \end{aligned}

(2)

where $f$ represents the number of explainable constraints, on this basis, the initial parameter set is described as follows.

\begin{aligned} ℧ = Z (x, y, c) \end{aligned}

(3)

where $Z (\cdot)$ indicates the optimization function used in the optimization process.

Question 3: How to construct an IBRB-e model for considering rule reliability? This paper establishes a reasonable nonlinear mapping relationship $α (\cdot)$ to solve this problem:

\begin{aligned} y = α (x, c, r, ℧^{0}) \end{aligned}

(4)

where $y$ represents the decision-making result of the model, $x = {x_{1}, x_{2}, \dots, x_{M}}$ depicts the collection of premise attributes, $r$ is the rule reliability, and $c$ is the explainable constraint added to the model. $℧^{0}$ depicts the collection of initial parameters that are not optimized.

3. Modeling of the IBRB-e

This section mainly describes the modeling process of IBRB-e model. Firstly, in Section 3.1, the XGBoost algorithm and its application in feature importance analysis are described in detail. Secondly, in Section 3.2, the rule construction for the IBRB-e model is explained. Thirdly, the Section 3.3 to Section 3.5 describe the modeling, reasoning, and optimization of the IBRB-e model respectively

3.1 Feature importance analysis

In recent years, experts have been exploring techniques for solving problems with high-dimensional datasets [34]. Feature importance analysis plays a very important role in dimensionality reduction [35]. Through this analysis, it can help to determine which features have the most significant impact on the prediction results. By calculating the importance score of the features, the contribution of each feature to the model can be calculated to select and optimize the model.

As an algorithm in the field of machine learning, XGBoost has been widely used in many fields [36]. Therefore, this paper chooses to use XGBoost to analyze the importance of data features. The step of determining feature importance in XGBoost involves training multiple decision trees. During training, feature coverage and gain will be calculated for each decision tree construct. Coverage-based feature importance entails taking the coverage of each feature in the cumulative decision tree and then ranking the features according to their importance. The gain-based feature importance will accumulate the gain of each feature and rank by importance.

Decision tree updates are derived from the objective function given by the formula.

\begin{aligned} Obj = \sum_{i} l ({\hat{o}}_{i}, o_{i}) + \sum_{k} Ω (f_{k}) \end{aligned}

(5)

where $l$ represents a loss function utilized to describe the fitting of the predicted value ${\hat{o}}_{i}$ and the true value $o_{i}$ . $Ω$ is the complexity penalty term for the tree $(f_{k})$ structure. The XGBoost computes feature importance scores derived from the gain score in the training. The equation uncovers the gain score for each tree split, $I_{R}$ denotes the right nodes, $I_{L}$ denotes the left nodes, $g_{i}$ denotes the firstorder gradient related to the loss function, $h_{i}$ denotes the secondorder gradient, $γ$ and $λ$ constitute regularization parameters.

\begin{aligned} gain = \frac{1}{2} [\frac{{(\sum_{i \in I_{L}} g_{i})}^{2}}{\sum_{i \in I_{L}} h_{i} + λ} + \frac{{(\sum_{i \in I_{R}} g_{i})}^{2}}{\sum_{i \in I_{R}} h_{i} + λ} - \frac{{(\sum_{i \in I} g_{i})}^{2}}{\sum_{i \in I} h_{i} + λ}] - γ \end{aligned}

(6)

3.2 Description of IBRB-e

According to the literature [37], the explainability standard can be described as follows: (1) The number of activated rules is the same as the premise attribute. It should not be the case that there is only one input but no corresponding spacing rule (2) The division of reference intervals should be more reasonable, and the division of reference intervals should not overlap in principle. (3) The model should have sufficient meaning. To ensure that it can be adapted to the actual applications. (4) The conversion from input to output should take into account the belief of the original sample information. (5) The reasoning process must be explainable and the results must be traceable. (6) The number of setting interval rules should be appropriate to avoid affecting the performance.

Figure 1.

Overall structure of IBRB-e.

Based on the BRB and inspired by the paper, this paper proposes IBRB-e model. The reference value is set as an interval in this model, and the interval addition is employed to construct a belief table for addressing the issue of rule explosion in BRB. Furthermore, explainability constraints are introduced in the optimization process, which makes the process more reasonable, and the $k$ -th rule can be expressed as:

\begin{aligned} \begin{aligned} {Rule}_{k} : If x_{1} \in [d_{1}, u_{1}] \lor x_{2} \in [d_{2}, u_{2}] \lor \dots x_{M} \in [d_{M}, u_{M}] \\ Then consequence is {(D_{1}, β_{1, k}), (D_{2}, β_{2, k}), \dots, (D_{N}, β_{N, k})} \\ With rule reliability r_{k} \\ And rule weight ϖ_{k} \\ In explainability constraint c_{1}, c_{2}, \dots, c_{f} \end{aligned} \end{aligned}

(7)

where $x_{i} (i = 1, \dots, M)$ represents the premise attributes, $[d_{i}, u_{i}], i \in [1, M]$ represents the reference value interval of the $i$ -th attribute. $k$ satisfies $1 ⩽ k ⩽ L$ , $L$ denotes the number of rules, $J$ represents the number of reference value intervals, $D_{n} (1 ⩽ n ⩽ N)$ represents the model results, $β_{n, k}$ indicates the belief level associated with the $n$ -th outcome in rule $k$ , $r_{k}$ represents the rule reliability, $ϖ_{k}$ represents the rule weight. It can be seen that only one premise attribute exists in a rule of IBRB-e, while multiple premise attributes can exist in a rule of BRB. The framework of IBRB-e is presented in Figure 1.

3.3 Modeling of the IBRB-e

Generally, the modeling is as follows:

Step 1: Analyzing the problem, that is, determine why the problem arose and the effect of the problem when it is solved. The premise attribute is generally selected as the antecedent of the problem, and the output is selected as the effect after the problem is solved.

Step 2: Relevant reference points and reference values are established. Unlike the BRB, the reference values of this model are set in the form of intervals. In the IBRB-e model, when a data sample of a premise attribute falls in an interval, the rule corresponding to that interval is activated. This means that one rule is activated for each data sample of the premise attribute. Subsequently, all the rules activated by the premise attribute are jointly involved in the inference process of the model. For instance, assuming two premise attributes, each with four reference points, the reference points and reference value Settings for BRB and IBRB-e are shown in Tables 1–4. The four tables show the difference between the reference value settings of IBRB-e and BRB.

Table 1
Reference value of premise attribute 1 in BRB.

Reference point P M G E

Reference value 1 1.5 2 2.5

Reference point	P	M	G	E
Reference value	1	1.5	2	2.5

Table 2

Reference value of premise attribute 2 in BRB.

Reference point	VL	L	H	VH
Reference value	3	3.5	4	4.5

Table 3

Reference value of premise attribute 1 in IBRB-e.

Reference point	P	M	G	E
Reference value	$[d_{1}, u_{1}]$	$[d_{2}, u_{2}]$	$[d_{3}, u_{3}]$	$[d_{4}, u_{4}]$

Table 4

Reference value of premise attribute 2 in IBRB-e.

Reference point	VL	L	H	VH
Reference value	$[l_{1}, r_{1}]$	$[l_{2}, r_{2}]$	$[l_{3}, r_{3}]$	$[l_{4}, r_{4}]$

Step 3: The corresponding belief table is set up. The BRB creates the belief table using a Cartesian product approach, and the method of interval addition and combination is employed by IBRB-e to create the belief rule table [38]. The difference between the generation rules is presented in Figure 2, and the difference in rule generation is illustrated in Figure 3. Specifically, when BRB and IBRB-e in Step 2 create their own belief tables, the generated rules are shown in Tables 5 and 6.

Table 5

The belief table of BRB.

Number	Reference values of attribute		Output
	Attribute 1	Attribute 2
1	P	VL	${$ (M, 1), (B, 0) $}$
2	P	L	${$ (M, 0.9), (B, 0.1) $}$
3	P	H	${$ (M, 0.8), (B, 0.2) $}$
4	P	VH	${$ (M, 0.75), (B, 0.25) $}$
5	M	VL	${$ (M, 0.7), (B, 0.3) $}$
6	M	L	${$ (M, 0.65), (B, 0.35) $}$
7	M	H	${$ (M, 0.6), (B, 0.4) $}$
8	M	VH	${$ (M, 0.55), (B, 0.45) $}$
9	G	VL	${$ (M, 0.5), (B, 0.5) $}$
10	G	L	${$ (M, 0.45), (B, 0.55) $}$
11	G	H	${$ (M, 0.4), (B, 0.6) $}$
12	G	VH	${$ (M, 0.35), (B, 0.65) $}$
13	E	VL	${$ (M, 0.3), (B, 0.7) $}$
14	E	L	${$ (M, 0.2), (B, 0.8) $}$
15	E	H	${$ (M, 0.1), (B, 0.9) $}$
16	E	VH	${$ (M, 0), (B, 1) $}$

Table 6

The belief table of IBRB-e.

Number	Reference intervals	Output
1	$[d_{1}, u_{1}]$	${$ (M, 0.9), (B, 0.1) $}$
2	$[d_{2}, u_{2}]$	${$ (M, 0.8), (B, 0.2) $}$
3	$[d_{3}, u_{3}]$	${$ (M, 0.7), (B, 0.3) $}$
4	$[d_{4}, u_{4}]$	${$ (M, 0.6), (B, 0.4) $}$
5	$[l_{1}, r_{1}]$	${$ (M, 0.5), (B, 0.5) $}$
6	$[l_{2}, r_{2}]$	${$ (M, 0.4), (B, 0.6) $}$
7	$[l_{3}, r_{3}]$	${$ (M, 0.3), (B, 0.7) $}$
8	$[l_{4}, r_{4}]$	${$ (M, 0.4), (B, 0.6) $}$

From the Tables 5 and 6, when BRB and IBRB-e each have four reference points for a single premise attribute, BRB generates 16 rules, while IBRB-e only generates 8 rules. This effect is more obvious when the reference value is large. Therefore, IBRB-e does not require too many premise properties to maintain model efficiency while improving accuracy.

Figure 2.

The distinction between IBRB-e and BRB generation belief rules.

Figure 3.

The difference in rule generation.

Figure 4.

Reasoning process.

According to Figure 3, the reference points are set for each of the two premise attributes. The reference points for the first premise attribute are P, M, G, and E, and the reference points for the second premise attribute are VL, L, H, and VH, respectively. in the BRB model, since the rule combination is performed by Cartesian product, the number of generated rules is 4*4 $=$ 16. while in the IBRB-e model, the reference values are set in the form of intervals, and the rules are generated as the sum of intervals, which generates a number of rules of 4 $+$ 4 $=$ 8, each interval corresponds to one rule, when the input value falls into this interval, the corresponding rule is activated.

3.4 Inference for ER rules

The ER parsing algorithm is not utilized by the IBRB-e medical assistant decision-making model. Instead, IBRB-e emphasizes the weight of the evidence and the reliability of the evidence, using ER rule as the inference machine. The process of inference is depicted in Figure 4.

Step 1: In ER rule, evidence is acquired through varying approaches, it may be affected by various interferences during the acquisition process, and the evidence might not be entirely dependable. $L$ pieces of independent evidence $e_{i} (i = 1 \dots, L)$ , the identification framework $Θ$ consists of $N$ evaluation levels $D_{n} (n = 1, \dots, N)$ , namely, $Θ = {D_{1}, \dots D_{N}}$ . An item of evidence can be expressed as follows:

\begin{aligned} e_{i} = {(D_{n}, β_{n, i}), n = 1, \dots, N; (Θ, β_{Θ, i})} \end{aligned}

(8)

where $β_{n, i}$ represents the belief degree that the evaluation plan being assessed as $D_{n}$ under $e_{i}$ , and $β_{Θ, i}$ represents the global ignorance.

Step 2: If the weight $ϖ_{i} (i = 1, \dots, L)$ and the reliability $r_{i} (i = 1, \dots, L)$ are considered, both belong to the interval $[0, 1]$ at the same time. Then, the belief distribution calculated by the mixed weight of $ϖ_{i}$ and $r_{i}$ is given by:

\begin{aligned} m_{i} = {(D_{n}, {\tilde{m}}_{n, i}); (β (Θ), {\tilde{m}}_{β (Θ), i})} \end{aligned}

(9)

where any $D_{n} \subseteq Θ$ is satisfied, the power collection is represented by $β (Θ)$ .

Step 3: The $i$ -th mixed probability mass in any outcome $D_{n}$ is as follows:

\begin{aligned} {\tilde{m}}_{n, i} & = {\begin{aligned} 0, D_{n} = \emptyset \\ d_{r ϖ, i} m_{n, i}, D_{n} \subseteq Θ, D_{n} \neq \emptyset \\ d_{r ϖ, i} (1 - r_{i}), D_{n} = β (Θ) \end{aligned} \end{aligned}

(10)

\begin{aligned} d_{r ω, i} & = \frac{1}{(1 + ϖ_{i} - r_{i})} \end{aligned}

(11)

\begin{aligned} m_{n, i} & = ϖ_{i} β_{n, i} \end{aligned}

(12)

where $d_{r ω, i}$ represents the normalization coefficient, ${\tilde{m}}_{n, i}$ and ${\tilde{m}}_{β (Θ), i}$ satisfy $\sum_{n = 1}^{N} ({\tilde{m}}_{n, i} + {\tilde{m}}_{β (Θ), i}) = 1$ .

Step 4: The collective support degree $β_{N, e (L)}$ of $L$ independent evidence can be computed as follows:

\begin{aligned} {\hat{m}}_{n, e (k)} & = [(1 - r_{k}) m_{n, e (k - 1)} + m_{P (Θ), e (k - 1)} m_{n, k}] + \sum_{Q \cap E = D_{n}} m_{Q, e (k - 1)} m_{E, k} \end{aligned}

(13)

\begin{aligned} {\hat{m}}_{β (Θ), e (k)} & = (1 - r_{k}) m_{β (Θ), e (k - 1)} \end{aligned}

(14)

\begin{aligned} m_{n, e (k)} & = {\begin{matrix} \frac{\begin{matrix} 0, D_{n} = \emptyset \\ {\hat{m}}_{n, e (k)} \end{matrix}}{(\sum_{Q \subseteq Θ} {\hat{m}}_{Q, e (k)} + {\hat{m}}_{β (Θ), e (k)})} \end{matrix}, else \end{aligned}

(15)

\begin{aligned} β_{n, e (k)} & = {\begin{matrix} \frac{\begin{matrix} 0, D_{n} = \emptyset \\ {\hat{m}}_{n, e (k)} \end{matrix}}{(\sum_{Q \subseteq Θ} {\hat{m}}_{Q, e (k)})} \end{matrix}, else, D_{n} \subseteq Θ \end{aligned}

(16)

where $k$ represents one of the $L$ rules, the belief result of $D_{n}$ after fusing randomly selected $k$ rules is expressed as $P_{n, e (k)}$ , and the formula satisfies $e (1) = 1$ .

Step 5: Finally, the final expected utility and output belief distribution is given by:

\begin{aligned} e (L) & = {(D_{n}, β_{n, e (L)}), n = 1, \dots, N, (Θ, β_{Θ, e (L)})} \end{aligned}

(17)

\begin{aligned} U & = \sum_{n = 1}^{N} μ (D_{n}) β_{n, e (L)} + μ (Θ) β_{Θ, e (L)} \end{aligned}

(18)

where $U$ represents the final mathematical expected utility. $μ (D_{n})$ represents the utility value under outcome $D_{n}$ .

3.5 Optimization of the IBRB-e

As an optimization algorithm, the P-CMA-ES exhibits the following benefits [39]: (1) Fast convergence: Using parallel computing, multiple solutions can be evaluated at the same time to speed up the convergence speed. (2) Scalability: It is suitable for high-performance computing systems to fully utilize computing resources. (3) Robustness: It can handle complex, nonlinear and multimodal optimization problems. (4) Global optimization does not rely on gradient information and is suitable for problems where gradients are unavailable or expensive. (5) Adaptive step size: By adaptively adjusting the step size, a balance is found between exploration and utilization. (6) Population strategy: Maintaining the population of candidate solutions helps to maintain diversity and avoid falling into local optimal solutions. (7) Wide application: It can be used for optimization problems with continuous, discrete and mixed variables.

Since the parameters of the model may lose meaning during the optimization process, in this paper, a P-CMA-ES with explainability constraints is represented, as shown in Figure 5, which has a certain degree of explainability and holds significant importance within the medical domain. The optimized objective function can be expressed as follows:

\begin{aligned} \begin{aligned} min ~MSE (β, r, ϖ) \\ s t . \sum_{n = 1}^{N} β_{n, k} = 1, k = 1, \dots, L \\ 0 ⩽ β_{n, k} ⩽ 1, n = 1, \dots, N, k = 1, \dots, L \\ 0 ⩽ r_{k} ⩽ 1, k = 1, \dots, L \\ 0 ⩽ ϖ_{i} ⩽ 1, i = 1, \dots, L \end{aligned} \end{aligned}

(19)

where MSE is a loss function, and the specific formula is given by:

\begin{aligned} MSE (β, r, ϖ) = \frac{1}{T} \sum_{t = 1}^{T} {(y - \hat{y})}^{2} \end{aligned}

(20)

where $T$ represents the count of training samples. The outcome of the IBRB-e is expressed by $y$ , and the data sample label is denoted by $\hat{y}$ , therefore, an optimization algorithm with explainable constraints can be expressed as below:

Step 1: The required parameters is established. The parameters subject to optimization are as follows:

\begin{aligned} ℧^{0} & = ψ^{0} \end{aligned}

(21)

\begin{aligned} ψ^{0} & = {β_{1, 1}, \dots, β_{N, L}; r_{1}, \dots, r_{L}; ϖ_{1}, \dots, ϖ_{L}} \end{aligned}

(22)

where $℧^{g}$ denotes the average value after $g$ round of retrieval distribution, and $ψ^{g}$ denotes the collection of parameters requiring optimization throughout the procedure

Step 2: Data sampling. The update operation of the parameters through the operation can be expressed as follows:

\begin{aligned} ψ_{u}^{g + 1} \sim ℧^{g} + s^{g} N (0, C^{g}), u = 1, \dots, h \end{aligned}

(23)

where $ψ_{u}^{g + 1}$ represents the $u$ -th solution in $g + 1$ round of optimization, $s^{g}$ is the step size, $N (\cdot)$ is the normal distribution function, $h$ is the offspring count, and $C^{g}$ is the covariance matrix.

Step 3: Incorporating explainability constraints. Adding explainability to ensure that IBRB-e is reasonable, the constraints are as follows:

\begin{aligned} \begin{aligned} β_{k} \sim O_{k} (k = 1, \dots, L) \\ O_{k} \in {{β_{1} ⩽ β_{2} ⩽ β_{3} ⩽ \dots ⩽ β_{N}}, \\ {β_{1} ⩽ \dots ⩽ max (β_{1}, β_{2}, \dots, β_{N}) ⩾ \dots ⩾ β_{N}}, \\ {β_{1} ⩾ β_{2} ⩾ \dots ⩾ β_{N}}} \end{aligned} \end{aligned}

(24)

where $O_{k}$ denotes the explainability constraint under rule $k$ , the constrained belief distribution should satisfy one of ${β_{1} ⩽ β_{2} ⩽ β_{3} ⩽ \dots ⩽ β_{N}}$ , ${β_{1} ⩽ \dots ⩽ max (β_{1}, β_{2}, \dots, β_{N}) ⩾ \dots ⩾ β_{N}}$ , and ${β_{1} ⩾ β_{2} ⩾ \dots ⩾ β_{N}}$ and the explainability limitation factor of the belief distribution is flexible and variable, depending on expert knowledge.

Step 4: Projection. Through projection, the solution vector can conform the explainable conditions added to the optimization, and the equation is given by:

\begin{aligned} \begin{aligned} ψ_{u}^{g + 1} (1 + ε \times (κ - 1) : ε \times κ) \\ = ψ_{u}^{g + 1} (1 + ε \times (κ - 1) : ε \times κ) - W^{T} \times (W \times W^{T})^{- 1} \\ \times ψ_{u}^{g + 1} (1 + ε \times (κ - 1) : ε \times κ) \times W \end{aligned} \end{aligned}

(25)

where $W$ represents a vector whose dimension is $1 \times N$ and its value is 1, $ε = 1, \dots, N$ represents the variables count with constraints, and $κ = 1, \dots, N + 1$ represents the count of variables with the same constraints.

Step 5: Updating the parameter. The formula is as follows:

\begin{aligned} ℧^{g + 1} = \sum_{u = 1}^{λ} w_{u} ψ_{u, h}^{g + 1} \end{aligned}

(26)

where $w_{u}$ represents the coefficient of the weight, $λ$ is the descendants count, and $ψ_{u, h}^{g + 1}$ denotes the $u$ -th solution of $h$ solutions in the $g + 1$ round optimization.

Step 6: Calculating the covariance matrix. The specific calculation is as follows:

\begin{aligned} \begin{aligned} C^{g + 1} & = (1 - e_{1} - e_{2}) C^{g} + e_{1} P_{e}^{g + 1} (P_{e}^{g + 1})^{T} + e_{2} \sum_{u = 1}^{τ} w_{u} (\frac{K_{u, h}^{g + 1} - θ^{g}}{ς^{g}}) \\ \times {(\frac{K_{u, h}^{g + 1} - θ^{g}}{ς^{g}})}^{T} \end{aligned} \end{aligned}

(27)

where $ς^{g}$ is the step size under generation $g$ , and $P_{e}^{g + 1}$ is the evolutionary path under $(g + 1)$ -th generation $e_{1}$ and $e_{2}$ are learning rates.

Under generation $g$ , $θ^{g}$ is the number of offspring. $K_{u, h}^{g + 1}$ represents the $u$ -th solution from the $h$ solution within $g + 1$ generation In particular, the process is cyclically iterative.

Figure 5.

The process of P-CMA-ES.

In IBRB-e, it is inevitable that two inputs fall into the same range, thereby activating the same rule. For this situation, setting the range more densely is a very good solution.

4. Case study

The running environment of this experiment is Windows 11 version and the running software is MATLAB R2021b version. The datasets used in the experiment are all opening source datasets. The experimental division of the dataset is done by the leave-out method, where a portion of the data is selected, after which the training and test sets are divided proportionally.

4.1 Description of the dataset

The dataset comes from the Wisconsin Breast Cancer Diagnostic Data Set on the UCI website, which is open source and extensively utilized in the domains of machine learning and data mining tasks for breast cancer diagnosis. The data is a class-balanced dataset, and most commonly used for dichotomous problems, to determine whether a tumor is benign or malignant.

The dataset contains the following characteristics (feature values):

Radius: The average distance from the tumor boundary to the tumor centroid.

Texture: The standard deviation of the gray, which measures the degree of gray change in an image.

Perimeter: The length of the perimeter of the tumor. Area: The size of the area of the tumor.

Smoothness: The local length change of the boundary.

Compactness: Square of perimeter divided by area minus 1.

Concavity: The severity of the concave part of the profile.

Concave points: The count of contour points within the depression.

Symmetry: The symmetry of the tumor.

Fractal dimension: A fractal dimension that uses “coastline estimation”.

The decision-making is divided into two types: benign tumors and breast cancer. Benign tumors are evaluated as “0”, and breast cancer tumors are evaluated as “1”. The mean, standard error, and worst values of the above ten features are included in the dataset, resulting in a total of 30 features in the dataset, there are 569 samples for each feature, including 212 samples for malignant tumors and 357 samples for benign tumors. In order to better advance the experimental process, outliers are removed from this dataset and mean interpolation operations are performed on the missing values. After the set-aside method, 96 malignant tumors and 106 benign tumors are selected as test samples, and the remaining samples are used for training.

4.2 Analysis of the importance of characteristics to the results

The purpose of feature importance analysis is to improve the applicability of the data [40], and XGBoost is an efficient algorithm commonly used for classification and regression, therefore, the XGBoost is used in this paper as a tool for importance analysis. When performing importance analysis, the importance of the input features to the results is important, which can be performed by calculating feature importance methods: XGBoost provides a way to directly view feature importance, by calling the properties, the importance score of each feature to the prediction can be obtained. This can help quickly understand which features exert the most significant on the prediction. The feature importance analysis is shown in Figure 6.

Figure 6.

The degree to which the feature affects the outcome.

Since the influence degree of some features on the results is negligible compared with the features with a high degree of influence, only the first eight features are selected and placed in the above table, and the first two features with the greatest influence are opted as the premise attributes, which greatly reduces the complexity of the model.

4.3 Model building

(1) Settings of optimization parameters

Optimizing parameter settings holds importance in enhancing algorithm or model performance, boosting efficiency, mitigating overfitting or underfitting, accommodating diverse datasets, conserving resources, and gaining insights into algorithm behavior. By adjusting the parameters, computer algorithms and machine learning models can perform better in real-world applications.

In this paper, a round of experiments is randomly selected, and the parameters in the experiment are presented in Table 7, in Table 7, the number of rules generated is 31, and the reference interval, rule reliability, rule weight, and output belief distribution (malignant versus benign tumors) are described in detail for each rule.

Table 7
Parameter settings.

No. Referential interval Rule reliability Rule weight Output distribution

1 [50, 55] 0.3631 0.0159 ${$ 0.3329, 0.6671 $}$

2 [55, 57] 0.8741 0.1784 ${$ 0.7497 0.2503 $}$

3 [57, 59] 0.5673 0.0578 ${$ 0.3377 0.6623 $}$

4 [59, 95] 0.6862 0.4246 ${$ 0.1140, 0.8860 $}$

5 [95, 201] 0.7466 0.9342 ${$ 0.9818, 0.0182 $}$

6 [201, 210] 0.8401 0.3886 ${$ 0.9591, 0.0409 $}$

7 [210, 220] 0.4604 0.7961 ${$ 0.9835, 0.0165 $}$

8 [220, 230] 0.7544 0.8908 ${$ 0.9772, 0.0228 $}$

9 [230, 240] 0.3614 0.7698 ${$ 0.4123, 0.5877 $}$

10 [240, 250] 0.6182 0.5021 ${$ 0.6556, 0.3444 $}$

11 [250, 252] 0.5290 0.3939 ${$ 0.4345, 0.5655 $}$

12 [ $-$ 0.001, 0.163] 0.6467 0.7985 ${$ 6.41e-04, 0.9994 $}$

13 [0.163, 0.2] 0.6911 0.4064 ${$ 0.8939, 0.1061 $}$

14 [0.2, 0.267] 0.6044 0.7482 ${$ 0.9744, 0.0256 $}$

15 [0.267, 0.269] 0.1278 0.1431 ${$ 0.9047, 0.0953 $}$

16 [0.269, 0.272] 0.3985 0.8193 ${$ 0.9676, 0.0324 $}$

17 [0.272, 0.274] 0.6574 0.3851 ${$ 0.8628, 0.1372 $}$

18 [0.274, 0.275] 0.3198 0.3554 ${$ 0.2994, 0.7006 $}$

19 [0.275, 0.276] 0.3216 0.8061 ${$ 0.4580, 0.5420 $}$

20 [0.276, 0.277] 0.6371 0.6361 ${$ 0.2348, 0.7652 $}$

21 [0.277, 0.279] 0.6229 0.1444 ${$ 0.2761, 0.7239 $}$

22 [0.279, 0.281] 0.1672 0.0823 ${$ 0.9171, 0.0829 $}$

23 [0.281, 0.283] 0.4590 0.5924 ${$ 0.2858, 0.7142 $}$

24 [0.283, 0.285] 0.2215 0.7188 ${$ 0.8168, 0.1832 $}$

25 [0.285, 0.287] 0.3263 0.1875 ${$ 0.7754, 0.2246 $}$

26 [0.287, 0.289] 0.8339 0.4656 ${$ 0.7082, 0.2918 $}$

27 [0.289, 0.290] 0.8447 0.4066 ${$ 0.9275, 0.0725 $}$

28 [0.290, 0.2901] 0.1651 0.0534 ${$ 0.1782, 0.8218 $}$

29 [0.2901, 0.2903] 0.9672 0.7762 ${$ 0.8709, 0.1291 $}$

30 [0.2903, 0.2907] 0.5065 0.1154 ${$ 0.6891, 0.3109 $}$

31 [0.2907, 0.2911] 0.3817 0.6315 ${$ 0.9665, 0.0335 $}$

No.	Referential interval	Rule reliability	Rule weight	Output distribution
1	[50, 55]	0.3631	0.0159	${$ 0.3329, 0.6671 $}$
2	[55, 57]	0.8741	0.1784	${$ 0.7497 0.2503 $}$
3	[57, 59]	0.5673	0.0578	${$ 0.3377 0.6623 $}$
4	[59, 95]	0.6862	0.4246	${$ 0.1140, 0.8860 $}$
5	[95, 201]	0.7466	0.9342	${$ 0.9818, 0.0182 $}$
6	[201, 210]	0.8401	0.3886	${$ 0.9591, 0.0409 $}$
7	[210, 220]	0.4604	0.7961	${$ 0.9835, 0.0165 $}$
8	[220, 230]	0.7544	0.8908	${$ 0.9772, 0.0228 $}$
9	[230, 240]	0.3614	0.7698	${$ 0.4123, 0.5877 $}$
10	[240, 250]	0.6182	0.5021	${$ 0.6556, 0.3444 $}$
11	[250, 252]	0.5290	0.3939	${$ 0.4345, 0.5655 $}$
12	[ $-$ 0.001, 0.163]	0.6467	0.7985	${$ 6.41e-04, 0.9994 $}$
13	[0.163, 0.2]	0.6911	0.4064	${$ 0.8939, 0.1061 $}$
14	[0.2, 0.267]	0.6044	0.7482	${$ 0.9744, 0.0256 $}$
15	[0.267, 0.269]	0.1278	0.1431	${$ 0.9047, 0.0953 $}$
16	[0.269, 0.272]	0.3985	0.8193	${$ 0.9676, 0.0324 $}$
17	[0.272, 0.274]	0.6574	0.3851	${$ 0.8628, 0.1372 $}$
18	[0.274, 0.275]	0.3198	0.3554	${$ 0.2994, 0.7006 $}$
19	[0.275, 0.276]	0.3216	0.8061	${$ 0.4580, 0.5420 $}$
20	[0.276, 0.277]	0.6371	0.6361	${$ 0.2348, 0.7652 $}$
21	[0.277, 0.279]	0.6229	0.1444	${$ 0.2761, 0.7239 $}$
22	[0.279, 0.281]	0.1672	0.0823	${$ 0.9171, 0.0829 $}$
23	[0.281, 0.283]	0.4590	0.5924	${$ 0.2858, 0.7142 $}$
24	[0.283, 0.285]	0.2215	0.7188	${$ 0.8168, 0.1832 $}$
25	[0.285, 0.287]	0.3263	0.1875	${$ 0.7754, 0.2246 $}$
26	[0.287, 0.289]	0.8339	0.4656	${$ 0.7082, 0.2918 $}$
27	[0.289, 0.290]	0.8447	0.4066	${$ 0.9275, 0.0725 $}$
28	[0.290, 0.2901]	0.1651	0.0534	${$ 0.1782, 0.8218 $}$
29	[0.2901, 0.2903]	0.9672	0.7762	${$ 0.8709, 0.1291 $}$
30	[0.2903, 0.2907]	0.5065	0.1154	${$ 0.6891, 0.3109 $}$
31	[0.2907, 0.2911]	0.3817	0.6315	${$ 0.9665, 0.0335 $}$

It can be seen from the table that the reference interval setting of premise attribute 1 of IBRB-e ranges from 50 to 252, with a total of 12 reference values; the reference interval setting of premise attribute 2 ranges from $-$ 0.001 to 0.2911, with a total of 21 reference values. If BRB is used to deal with this problem, the upper and lower bounds of these intervals are all single reference values. Finally, the generated rules are 12*21 $=$ 252 rules, the comparison shows that the effect of IBRB-e on rule reduction is very obvious.

(2) Rules for the diagnosis of breast cancer diseases are established

After determining the indicators needed for diagnosis and the feature importance of the diagnostic indicators, to avoid the problem of too many input attributes of BRB causing the explosion of combination rules, the IBRB-e model is established, at the same time, based on the existing expert knowledge, the reference values for the premise attributes are selected.

\begin{aligned} \begin{aligned} R_{K} : If x_{1} is Perimeter_worst~ \lor x_{2} is Concave_points mean \\ Then consequence is {(D_{1}, β_{1, k}) \dots (D_{m}, β_{m, k})} \\ With rule reliability r_{k} \\ And rule weight ϖ_{k} \\ In explainability constraint c_{1}, c_{2}, \dots c_{f} \end{aligned} \end{aligned}

(28)

4.4 Result analysis

(1) Evaluation indicators

Case analysis usually requires appropriate evaluation indicators to be more reliable, so this paper uses the following indicators to evaluate the case:

(1) Accuracy: This reflects how many of the disease samples are correctly diagnosed. The calculation equation is shown below:

\begin{aligned} accuracy = \frac{T P + T N}{T P + F P + F N + T N} \end{aligned}

(29)

(2) Precision: Precision reflects how many samples of a certain type are correctly diagnosed when they have been diagnosed. The calculation equation is shown below:

\begin{aligned} precision = \frac{T P}{T P + F P} \end{aligned}

(30)

(3) Recall: Recall reflects the number of the initial actual samples correctly diagnosed. The calculation equation is shown below:

\begin{aligned} recall = \frac{T P}{T P + F N} \end{aligned}

(31)

where TP is the True positives (predicted to be positive case and actually positive case), TN is the True Negatives (predicted negative case and actually negative case), FP is the False Positives (predicted positive case and actually negative case), FN is the False Negatives (predicted negative case and actually positive case). In the experiment, positive cases are benign tumors and negative cases are malignant tumors.

(2) Model output

After the preparatory work is completed, the two selected features can be obtained, namely, the maximum circumference of the tumor and the average concave point. Figures 7 and 8 show the data distribution of perimeter worst and concave point worst in benign and malignant tumors respectively

Figure 7.

Data distribution with the worst perimeter.

Figure 8.

Data distribution with the mean concave point.

Figure 9.

The two most important feature directions.

The distribution of experimental samples for these two features is shown in Figure 9, with blue indicating benign and red indicating malignant. The selected premise attributes are input into the IBRB-e model, and the corresponding fitting results are shown in Figure 10. The accuracy rate of the model can reach 99.5%. As seen from the above figure, the IBRB-e has a good fitting effect for the diagnosis of breast cancer.

(3) Robustness of IBRB-e

To evaluate the robustness of the model, different proportions of samples are randomly selected from 212 malignant tumor samples and 357 benign tumor samples each time for the test set, and the rest for the training set. The accuracy of IBRB-e and BRB under the ratio of training set to test set is 2:8, 3:7, 4:6, 5:5, 6:4, 7:3, 8:2, respectively. The evaluation results are shown in the Table 8.

Table 8

Accuracy of IBRB-e and BRB under different data ratios.

Ratio of train set to test set	IBRB-e	BRB
2:8	96.53%	84.16%
3:7	97.02%	84.65%
4:6	97.52%	85.64%
5:5	98.01%	86.13%
6:4	98.51%	86.67%
7:3	99.00%	87.12%
8:2	99.50%	89.11%

Figure 10.

Result fit plot.

As can be seen from the Table 8, the accuracy rate fluctuated with different sample sizes, but remained generally high and much higher than traditional BRB. It indicates that IBRB-e has certain robustness and maintains good performance when the data sample size changes.

To sum up, IBRB-e performs well in this comparative experiment in terms of average accuracy, highest accuracy, lowest accuracy and robustness. The IBRB-e’s superior capabilities in accuracy, stability and reliability make it a very favorable choice for solving specific problems.

(4) the efficiency of IBRB-e

After proving the robustness of IBRB-e, the efficiency of the model needs to be further verified. The model generates rules by adding intervals, which is much more convenient than the traditional brb by Cartesian product, so the running time is shorter.

Figure 11.

Operation efficiency comparison.

Then, an experiment is set up. In order to facilitate recording, the number of reference values of the two premise attributes of the control IBRB-e and BRB model is the same, and then the regular number generated by IBRB-e and BRB is recorded when the number of reference values is 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 respectively. As can be seen in subgraph a) of Figure 11, when the number of reference values of premise attribute 1 and premise attribute 2 are both 10, the number of BRB rules has reached 100, which increases the complexity of the model.

Further, the running time of IBRB-e and BRB is also shown in subgraph b) of Figure 11. When the number of reference values of the premise attribute increases continuously (here, the number of reference values of the two premise attributes is also the same), the running time of IBRB-e does not significantly slow down. However, when the number of reference values of the premise attribute 1 and premise attribute 2 are both 10, the running time of IBRB-e does not significantly slow down. The BRB running time has reached 98 seconds, which greatly affects the efficiency of the model, so the optimization effect of IBRB-e in terms of simplification of rules and running time is very obvious.

(5) Belief distribution

Expert knowledge is the key to optimization. IBRB-e uses the improved P-CMA-ES optimization algorithm, and its optimization process involves the gradual absorption of expert knowledge. In contrast, IBRB uses the P-CMA-ES optimization algorithm without adding explainability constraints in the optimization process, and lacks the ability to absorb expert knowledge. Resulting in an overemphasis on accuracy and a decline in explainability. Figure 12 illustrates a comparison of the belief distribution between expert knowledge, IBRB and IBRB-e.

When the optimized belief is close to the expert knowledge, more key features of the expert knowledge are retained. As can be seen from the figure, the IBRB-e model shows better agreement with expert knowledge and effectively describes the actual benign and malignant tumors of breast cancer. The model retains more features of expert knowledge in the optimization process, thus enhancing explainability, on the contrary, IBRB makes many rules that are inconsistent with expert knowledge, resulting in a loss of explainability.

Figure 12.

Belief distribution of IBRB-e, IBRB, and expert knowledge.

In addition, in assisted decision problems, there is an intermediate transition phase between different decision classes. At this stage, the attribute similarity between two similar categories is very high, resulting in local ignorance information, which may lead to the objective fact error of the model’s decision. Therefore, in order to effectively represent ignorance information, an intermediate state is introduced, and the belief rule is assigned to three states, namely M (malignant), Mid, and B (Benign). This approach better captures local ignorance and enhances representation.

4.5 Comparative experiment

(1) The comparison accuracy

Figure 13.

Comparative experiment fit plot.

Table 9

Accuracy stability analysis of different algorithms.

Machine learning algorithm	The average accuracy	The maximum accuracy	The minimum accuracy
IBRB-e	98.01%	99.50%	9653%
BRB	86.21%	89.11%	84.16%
Decision tree	87.35%	91.30%	78.80%
Support vector machine	87.78%	91.50%	82.40%
K nearest neighbors	88.20%	91.50%	82.90%
Back propagation	86.45%	91.90%	81.80%
Linear discriminant analysis	93.18%	94.60%	92.40%

Table 10

Comparison of recall and precision.

Machine learning algorithm	The average precision	The average recall
IBRB-e	99.05%	97.71%
BRB	92.45%	91.37%
Logistic regression	97.16%	96.36%
Decision tree	86.79%	84.80%
Support vector machine	94.33%	88.33%
K nearest neighbors	89.62%	87.60%
Back propagation	97.16%	89.83%
Linear discriminant analysis	95.28%	94.64%
Artificial neural network	93.39%	93.80%

In the comparative experiment shown in Table 9 and Figure 13, various machine learning algorithms, including IBRB-e, BRB, decision trees, SVM, KNN, and BP, are assessed for the performance in a specific task. Analyzing the experimental outcomes allows the remarkable performance of the IBRB-e model to stand out: a)

Highest Accuracy: Exceptional performance is demonstrated by the IBRB-e, achieving the highest accuracy of 99.5%, surpassing the highest accuracy achieved by other models. This underscores the capability of IBRB-e to achieve a significantly high classification accuracy for this task.

Average Accuracy: An impressive average accuracy of 98.01% is achieved by IBRB-e, which far exceeded the average accuracy of the other models. The robustness and reliability of IBRB-e are well proven.

Lowest Accuracy: Even in terms of the lowest accuracy, the performance of IBRB-e remained outstanding, reaching 96.53%, which is notably higher than the lowest accuracy of other models. This underscores the resilience of IBRB-e, maintaining elevated accuracy even in the face of challenging data instances.

(2) The comparison of precision and recall

In order to evaluate the model more comprehensively. The precision and recall metrics are introduced. The recall rate measures the model’s ability to correctly identify positive cases and therefore provides information about the proportion of cases the model successfully identifies out of all positive cases detected. The experiment is still conducted for ten rounds and two methods, logistic regression as well as artificial neural networks, are added and the results of the experimental evaluation are shown in the Table 10.

As can be seen from the table, IBRB-e has a better performance in Precision as well as Recall, with an average Precision of 99.05% and an average Recall of 97.71%, which indicates that IBRB-e has a more prominent advantage in diagnosing malignant tumors. In addition, Logistic Regression Linear Discriminant Analysis and Artificial Neural Network also have more stable performance Back Propagation and Support Vector Machine have larger differences in Precision and Recall, which may be caused by their poorer effect of predicting malignant tumors and better effect of predicting benign tumors instead. Decision Tree and K Nearest Neighbors perform very generally and do not predict malignancy very accurately.

4.6 Extensibility experiments

To verify the scalability of the IBRB-e disease diagnosis method, 10 rounds of experiments are conducted using three publicly available datasets. The three data sets are Obesity risk, Heart Failure prediction, and Diabetes prediction. The experimental results show that IBRB-e performs well on these datasets. Details of the complete experiment can be found in Table 11 of this paper.

Table 11
Reference value intervals for different disease decisions.

Dataset Premise attributes Reference value interval settings

Obesity risk $x 1$ ${$ [50, 60], [60, 75], [75, 90], [90, 100], [100, 125] $}$

$x 2$ ${$ [ $-$ 0.01, 0.05], [0.05, 0.075], [0.075, 1.001] $}$

Heart failure $x 3$ ${$ [60, 130], [130, 135], [135, 140], [140, 150], [150, 160] $}$

$x 4$ ${$ [50, 75], [75, 94], [94, 120], [120, 140], [140, 175], [175, 260] $}$

Diabetes $x 5$ ${$ [3, 4], [4, 5.9], [5.9, 7.4], [7.4, 10] $}$

$x 6$ ${$ [70, 120], [120, 199], [199, 250], [250, 310] $}$

Dataset	Premise attributes	Reference value interval settings
Obesity risk	$x 1$	${$ [50, 60], [60, 75], [75, 90], [90, 100], [100, 125] $}$
	$x 2$	${$ [ $-$ 0.01, 0.05], [0.05, 0.075], [0.075, 1.001] $}$
Heart failure	$x 3$	${$ [60, 130], [130, 135], [135, 140], [140, 150], [150, 160] $}$
	$x 4$	${$ [50, 75], [75, 94], [94, 120], [120, 140], [140, 175], [175, 260] $}$
Diabetes	$x 5$	${$ [3, 4], [4, 5.9], [5.9, 7.4], [7.4, 10] $}$
	$x 6$	${$ [70, 120], [120, 199], [199, 250], [250, 310] $}$

Figure 14.

Extended experimental results of IBRB-e.

Where $x 1$ represents body weight, $x 2$ represents frequent intake of high-fat foods, $x 3$ represents resting blood pressure, $x 4$ represents cholesterol level, $x 5$ represents hemoglobin concentration, and $x 6$ represents blood glucose level. These features have an important influence on the occurrence of the disease.

Table 12

Average accuracy of IBRB-e extended experiment.

Dataset	Obesity risk	Heart failure	Diabetes
Average accuracy	95.7142%	91.7417%	94.0667%

From Figure 14 and Table 12, it can be seen that the IBRB-e medical decision-making method consistently maintains an accuracy of over 88.75% in scalability experiments. Particularly noteworthy is its robust performance when dealing with the obesity risk dataset, and the average accuracy is 95.7142% for the obesity risk dataset, 91.7417% for the heart failure dataset and 94.0667% for the diabetes dataset, which is closely related to the excellent explainability of the IBRB-e medical decision-making method. Through extended experiments, it can be demonstrated that IBRB-e has potential application value, providing effective clinical decision support for medical decision-makers.

5. Conclusion

To address the problems of BRB in the field of medical assisted decision making, this paper applies the IBRB-e model, which presents the reference values in the form of intervals, and builds the belief rule table by adding the intervals, thus effectively solving the dilemma of combinatorial rule explosion. In addition, while using ER rules as the reasoning machine, the reliability of the rules is considered and the model is optimized using the P-CMA-ES optimization algorithm with explainability constraints, which improves the accuracy and explainability of the model.

However, the applicability of IBRB-e is limited in the case of uneven sample distribution. In this case, an unreasonable activation of rules may occur. And if there are fewer premise attributes and reference values, then the advantage of the IBRB-e is not obvious. Future work can solve these two shortcomings and refine the model to make it more suitable for real-world clinical applications and provide more reliable and transparent decision support for medical decisions.

Footnotes

Funding statement

This work was supported in part by the Postdoctoral Science Foundation of China under Grant No. 2020M683736, in part by the Teaching reform project of higher education in Heilongjiang Province under Grant No. SJGY20210456, in part by the Natural Science Foundation of Heilongjiang Province of China under Grant No. LH2021F038, in part by the Social Science Foundation of Heilongjiang Province of China under Grant No. 21GLC189, in part by the Foreign Expert Projects in Heilongjiang under Grant No. GZ20220131, in part by the Graduate innovation project of Harbin Normal University under Grant No.HSDSSCX2023-3.

Competing interests

The authors declare no competing interests.

Data availability

The datasets generated and analysed during the current study are available on the following website:

http://archive.ics.uci.edu/dataset/17/breast+cancer+wisconsin+diagnostic,

https://www.kaggle.com/datasets/iammustafatz/diabetes-prediction-dataset,

https://www.kaggle.com/datasets/fedesoriano/heart-failure-prediction,

https://www.kaggle.com/datasets/aravindpcoder/obesity-or-cvd-risk-classifyregressorcluster.

References

Milad

Stieglitz

Frick

, Artificial intelligence in disease diagnostics: A critical review and classification on the current state of research guiding future direction, Health and Technology 11(4) (2021), 693–731.

Adeola

Wang

Q.G.

, XGBoost model for chronic kidney disease diagnosis, IEEE/ACM Transactions on Computational Biology and Bioinformatics 17(6) (2019), 2131–2140.

Chinthaka

J.S.M.D.A.

Ganegoda

G.U.

, Involvement of machine learning tools in healthcare decision making, Journal of Healthcare Engineering 2021 (2021).

Jiang

Zhi

Dong

S.F.

Wang

Y.L.

Dong

Shen

H.P.

Wang

Y.J.

, Artificial intelligence in healthcare: Past, present and future, Stroke and Vascular Neurology 2(4) (2017).

Solomatine

Ostfeld

, Data-driven modelling: Some past experiences and new approaches, Journal of Hydroinformatics 10(1) (2008), 3–22.

Haddi

Ananou

Trardi

Ouladsine

Pons

Delliaux

Deharo

, Relevance Vector Machine as Data-Driven Method for Medical Decision Making, in: 2019 18th European Control Conference (ECC), IEEE, 2019.

Esteva

Kuprel

Novoa

Swetter

Blau

Thrun

, Dermatologist-level classification of skin cancer with deep neural networks, Nature 542(7639) (2017), 115–118.

Gao

C.M.

, An intelligent data-driven model for disease diagnosis based on machine learning theory, Journal of Combinatorial Optimization 42 (2021), 884–895.

Khoury

Attal

Amirat

Oukhellou

Mohammed

, Data-driven based approach to aid Parkinson’s disease diagnosis, Sensors 19(2) (2019), 242.

10.

Zhang

Y.Y.

Wang

S.C.

Zhang

Tang

S.W.

Zhou

G.H.

, A new health analysis method for lithium-ion batteries based on the evidential reasoning rule considering perturbation, Batteries 9(2) (2023), 88.

11.

M.X.

Zhong

X.L.

Peng

Q.Z.

Huang

S.L.

Yuan

J.L.

Tan

, Prediction of molecular subtypes of breast cancer using BI-RADS features based on a “white box” machine learning approach in a multi-modal imaging setting, European Journal of Radiology 114 (2019), 175–184.

12.

Tolks

Ament

Eberle

, Model-driven classification of different diabetes types within a personalized diabetes management, in: 2018 40th Annual International Conference of the IEEE Engineering in Medicine and Biology Society (EMBC), IEEE, 2018.

13.

Iqbal

Wah

T.Y.

Rehman

M.H.U.

Shah

J.H.

, Prediction analytics of myocardial infarction through model-driven deep deterministic learning, Neural Computing and Applications 32 (2020), 15909–15928.

14.

Mudiyanselage

T.K.B.

Xiao

X.L.

Zhang

Y.Q.

Pan

, Deep fuzzy neural networks for biomarker selection for accurate cancer detection, IEEE Transactions on Fuzzy Systems 28(12) (2019), 3219–3228.

15.

Chieregato

Frangiamore

Morassi

Baresi

Nici

Bassetti

Bna

Galelli

, A hybrid machine learning/deep learning COVID-19 severity predictive model from CT images and clinical data, Scientific Reports 12(1) (2022), 4329.

16.

Martinho

Freitas

Sousa

Vieira

Meira

Martins

Marreiros

, A hybrid model to classify patients with chronic obstructive respiratory diseases, Journal of Medical Systems 45(3) (2021), 31.

17.

Yang

J.B.

Liu

Wang

Sii

H.S.

Wang

H.W.

, Belief rule-base inference methodology using the evidential reasoning approach-RIMER, IEEE Transactions on Systems, Man, and Cybernetics-part A: Systems and Humans 36(2) (2006), 266–285.

18.

Zhou

Z.J.

G.Y.

C.H.

Wen

C.L.

Chang

L.L.

, A survey of belief rule-base expert system, IEEE Transactions on Systems, Man, and Cybernetics: Systems 51(8) (2019), 4944–4958.

19.

G.Y.

Zhou

Z.J.

Qiao

P.L.

Han

X.X.

Y.Y.

Wei

Shi

, A new hierarchical belief-rule-based method for reliability evaluation of wireless sensor network, Microelectronics Reliability 87 (2018), 33–51.

20.

Zhou

Z.J.

G.Y.

Zhang

B.C.

C.H.

Zhou

Z.G.

Qiao

P.L.

, A model for hidden behavior prediction of complex systems based on belief rule base and power set, IEEE Transactions on Systems, Man, and Cybernetics: Systems 48(9) (2017), 1649–1655.

21.

Cheng

X.Y.

Zhao

Zhou

G.H.

Zhu

H.L.

Zhao

E.K.

Qian

G.Y.

, An interval construction belief rule base with interpretability for complex systems, Expert Systems with Applications 229 (2023), 120485.

22.

Zhao

F.J.

Zhou

Z.J.

C.H.

Chang

L.L.

Zhou

Z.G.

G.L.

, A new evidential reasoning-based method for online safety assessment of complex systems, IEEE Transactions on Systems, Man, and Cybernetics: Systems 48(6) (2016), 954–966.

23.

Zhang

B.C.

Yin

X.J.

Wang

Z.L.

Zhou

Z.J.

Zhang

Y.L.

K.W.

, Fault diagnosis of CNC servo system based on belief rule base, J. Vibrat., Meas. Diagnosis 33(4) (2013), 694–700.

24.

G.L.

Zhou

Z.J.

C.H.

Chang

L.L.

Zhou

Z.G.

Zhao

F.J.

, A new safety assessment model for complex system based on the conditional generalized minimum variance and the belief rule base, Safety Science 93 (2017), 108–120.

25.

Zhou

Z.J.

Chang

L.L.

C.H.

Han

X.X.

Zhou

Z.G.

, A new BRB-ER-based model for assessing the lives of products using both failure data and expert knowledge, IEEE Transactions on Systems, Man, and Cybernetics: Systems 46(11) (2015), 1529–1543.

26.

Zhou

Z.J.

C.H.

G.Y.

Han

X.X.

Zhang

B.C.

Chen

Y.W.

, Hidden behavior prediction of complex systems under testing influence based on semiquantitative information and belief rule base, IEEE Transactions on Fuzzy Systems 23(6) (2015), 2371–2386.

27.

Zhou

Z.J.

C.H.

Zhang

B.C.

D.L.

Chen

Y.W.

, Hidden behavior prediction of complex systems based on hybrid information, IEEE Transactions on Cybernetics 43(2) (2013), 402–411.

28.

Zhou

Z.G.

Liu

Jiao

L.C.

Zhou

Z.J.

Yang

J.B.

Gong

M.G.

Zhang

X.P.

, A bi-level belief rule based decision support system for diagnosis of lymph node metastasis in gastric cancer, Knowledge-based Systems 54 (2013), 128–136.

29.

Zhou

Z.G.

Liu

L.L.

Jiao

L.C.

Zhou

Z.J.

Yang

J.B.

Wang

Z.L.

, A cooperative belief rule based decision support system for lymph node metastasis diagnosis in gastric cancer, Knowledge-based Systems 85 (2015), 62–70.

30.

Liu

Yang

J.B.

Ruan

Martinez

Wang

, Self-tuning of fuzzy belief rule bases for engineering system safety analysis, Annals of Operations Research 163 (2008), 143–168.

31.

Han

W.C.

Xiao

Jiang

H.Y.

, A new method for disease diagnosis based on hierarchical BRB with power set, Heliyon 9(2) (2023).

32.

You

Y.Q.

Sun

J.B.

Chen

Y.W.

Niu

C.Y.

Jiang

, Ensemble belief rule-based model for complex system classification and prediction, Expert Systems with Applications 164 (2021), 113952.

33.

Han

Zhao

B.Y.

Kong

L.K.

Y.M.

Zhou

G.H.

Feng

J.C.

, An interpretable BRB model with interval optimization strategy for lithium battery capacity prediction, Energy Science & Engineering 11(6) (2023), 1945–1959.

34.

Naeimeh

Muniyandi

R.C.

, Membrane computing inspired feature selection model for microarray cancer data, Intelligent Data Analysis 21(S1) (2017), S137–S157.

35.

Anis

B.I.

, Variable selection using support vector regression and random forests: A comparative study, Intelligent Data Analysis 20(1) (2016), 83–104.

36.

Chen

T.Q.

Carlos

, Xgboost: A scalable tree boosting system, in: Proceedings of the 22nd Acm Sigkdd International Conference on Knowledge Discovery and Data Mining, 2016.

37.

Cao

Zhou

Z.J.

C.H.

Tang

S.W.

, On the interpretability of belief rule-based expert systems, IEEE Transactions on Fuzzy Systems 29(11) (2020), 3489–3503.

38.

Cheng

X.Y.

Han

Zhou

G.H.

, A new interval constructed belief rule base with rule reliability, The Journal of Supercomputing 79(14) (2023), 15835–15867.

39.

Zhang

Q.X.

Zhao

B.Y.

Zhu

H.L.

Zhou

G.H.

, A behavior prediction method for complex system based on belief rule base with structural adaptive, Applied Soft Computing 151 (2024), 111118.

40.

Salama

K.M.

Abdelbar

A.M.

Anwar

I.M.

, Data reduction for classification with ant colony algorithms, Intelligent Data Analysis 20(5) (2016), 1021–1059.

A medical assistant decision-making method based on interval belief rule base with explainability

Abstract

Keywords

1. Introduction

2. BRB and problem formulation

2.1 Description of BRB

3.1 Feature importance analysis

Table 1 Reference value of premise attribute 1 in BRB. Reference point P M G E Reference value 1 1.5 2 2.5

4.1 Description of the dataset

4.2 Analysis of the importance of characteristics to the results

(1) Settings of optimization parameters

(2) Rules for the diagnosis of breast cancer diseases are established

(1) Evaluation indicators

(2) Model output

(3) Robustness of IBRB-e

(4) the efficiency of IBRB-e

(5) Belief distribution

(1) The comparison accuracy

(2) The comparison of precision and recall

Footnotes

Funding statement

Competing interests

Data availability

References

Table 1
Reference value of premise attribute 1 in BRB.

Reference point P M G E

Reference value 1 1.5 2 2.5