An interpretable model for stock trading volume prediction based on interval belief rule base with adaptive rule

Abstract

Stock trading volume forecasting is important for maintaining stock market stability. In real stock market trading volume prediction systems, three problems exist: high system complexity, instability, and the need for interpretable prediction results. Thus, an interpretable model for stock trading volume prediction based on the interval belief rule base with adaptive rule (IBRB-AI) is proposed in this paper. First, an interpretable interval belief rule base (IBRB) prediction model is constructed, and an adaptive rule mechanism is defined for the model. Then, a projection covariance matrix adaptation evolution strategy (P-CMA-ES) algorithm with interpretability constraints is used to ensure the interpretability of the optimization process. The final mean square error of the model is 0.0037, and the R-squared value is 0.8812. Compared with black-box models such as the back propagation neural network (BPNN), the IBRB-AI model achieves the accuracy of black-box models, with a complexity reduction of 38.46% compared with the original IBRB, and the interpretability is also improved. The validity of the IBRB-AI model is proven through a case study of the Tesla stock market, and a general analysis is conducted through Apple, Inc. (AAPL).

Keywords

Stock trading volume adaptive rule interpretability belief rule base

1 Introduction

Stock market movements significantly impact the economies of countries and individual consumers. The volume of stock trading is a key indicator of stock market liquidity.¹ The prediction of future stock trading volume is critical for investors because it helps them understand market conditions, identify trends, analyze major forces, and make informed investment decisions.² However, unexplained prediction results can increase investment risk and fail to gain investors’ trust. Investors tend to prefer interpretable models that promote informed decision making.³ Hence, stock volume prediction with interpretability has important research value.

In the present research, there are three types of stock trading volume prediction models: black-box, white-box, and gray-box models.⁴

First, black-box models make predictions through a data-driven approach that relies on many samples to improve accuracy.⁵ These methods include neural networks and gradient boosting methods. Chen et al. used a stock trading volume prediction model based on two-process meta-learning and verified the effectiveness of the model in stock trading volume prediction.⁶ Xu et al. used neural networks for stock trading volume prediction and obtained an overall accurate and stable performance.⁷ While black-box models demonstrate high accuracy and a degree of adaptability through data-driven learning, their adaptability is often confined to interpolation within the range of trained data distributions. Moreover, they struggle to cope with structural changes that frequently occur in volatile markets. In addition, the lack of interpretability undermines their credibility in practical decision-making. Therefore, in the context of financial forecasting, there is a clear need for a model that balances accuracy, adaptability, and interpretability.

The other category is white-box models, where the internal structure and workings of the model are transparent to the outside world.⁸ These include linear regression and decision trees. A Bayesian autoregressive conditional volume (ACV) model was used to predict intraday trading volume in the Polish stock market. It has been demonstrated that the ACV model considering linearity significantly outperforms other competing structures specifying the Burr distribution or the generalized gamma score.⁹ ME introduces a straightforward data-driven decision support system for stock market trading that combines several technologies.¹⁰ However, white-box models, such as linear regression and decision trees, often struggle in dynamic financial markets due to their limited capacity to model nonlinearity, sudden structural changes, and high levels of noise. These models tend to rely on fixed functional forms or rule sets, which do not adapt well to evolving data distributions. As a result, their predictive accuracy may deteriorate over time or under unusual market conditions. Their effectiveness in real-time or high-frequency trading situations is affected by this lack of adaptability. As a result, the application of white-box modeling to the financial market is relatively difficult.

Gray-box models, as a hybrid approach of black-box and white-box models, take full advantage of both, providing greater interpretability, accuracy and adaptability in stock trading volume forecasting. By combining theoretical and data-driven approaches, the gray-box model is able to effectively address market complexity and dynamic changes, making it a powerful forecasting tool.¹¹ These include the belief rule base (BRB) and evidence reasoning (ER). Dymova et al. proposed a new rule-based evidential inference approach for forex trading expert systems, and the ER rules proved effective for financial decision-making.¹² Hossain et al. employed a BRB expert system for technical analysis, integrating it with the bollinger band concept to predict stock prices for the next five days.¹³ The results show that the BRB model has good performance in predicting financial stock movements. In conclusion, an examination of the advantages and disadvantages of the three models reveals that the gray box model is better suited for predicting stock trading volume.

Since investors tend to favor interpretability of the prediction results and a BRB based on expert knowledge and IF-THEN rules is interpretable, it can be applied to financial markets. However, since Cartesian products are used by the BRB to form rules, more stock market indicators are prone to the rule explosion problem. In the context of financial markets, where real-time decision-making and transparency are critical, significant challenges can be posed by the rule explosion problem in BRB. This issue is especially observed in domains characterized by numerous input attributes and highly granular reference values. For example, a model with six attributes, each assigned five reference points, would result in $5^{6} = 15625$ rules being generated. Such exponential growth is found to severely impact inference time and interpretability, rendering these models impractical for real-time financial application^.¹⁴ Additionally, it has been pointed out by Cao et al. that overly complex rule structures can lead to reduced stability and interpretability in belief-based rule systems, thereby limiting their applicability in high-frequency decision-making scenarios.¹⁵ Therefore, the IBRB was proposed by Cheng et al..¹⁴

Using the IBRB to construct the stock trading volume prediction model has the following three problems:

Owing to the excessive number of stock indicators, the accuracy of the IBRB model is closely related to the number of intervals. The higher the precision is, the more complex the model.

Stock markets are characterized by nonlinearity and instability. To ensure the accuracy of the model results, the issue of rationality needs to be considered when constructing the stock trading volume prediction model.

In the stock market, the confidence of investors can increase, and the transparency of decision-making can be enhanced by interpretable forecasting results. How the interpretability of the model can be maintained when the model is being optimized is a problem.

With respect to practical systems, stock trading volume prediction is more difficult because of these three issues. Therefore, all three problems should be handled simultaneously by the model. First, the IBRB is used. The problem of rule explosion is avoided by creating a belief table through interval addition. Then, the IBRB-AI model is proposed. The complexity and interpretability of the model are further considered. In the IBRB-AI model, the complexity problem is addressed by the adaptive rule mechanism. Interpretability is addressed by modeling guidelines and constraints. Therefore, an effective way to solve the above problems is provided by the IBRB-AI model. In this paper, a stock trading volume prediction model is developed via IBRB-AI. Its primary contributions are as follows:

The IBRB-AI model is proposed on the basis of the IBRB model. The number of rules in the modeling process is reduced by the adaptive rule mechanism; thus, the complexity of the model is reduced.

The IBRB-AI model proposed in this paper is effectively applied to stock trading volume prediction. High accuracy and interpretability are achieved by using the model, which effectively supports subsequent decision-making.

The remainder of this paper is organized as follows: In Section 2, the problems faced by the IBRB in stock trading volume prediction are discussed. In Section 3, a detailed introduction to the new IBRB-AI model is provided from the perspectives of modeling, inference, and optimization. In Section 4, the proposed IBRB-AI model is verified by using a stock market case. This thesis is summarized, and some directions for further investigation are suggested in the conclusion section.

2 Problem formulation

In stock market decision-making, investors prefer prediction results that are interpretable, and the IF-THEN rule is used by BRB, which provides interpretability. Thus, it is widely used in the financial stock field, but it suffers from the problem of rule explosion. Its application in the financial field is limited. The rule explosion problem is solved by the IBRB proposed by Cheng et al..¹⁴ However, the IBRB still faces the following three problems in stock trading volume prediction:

Problem 1: How to overcome the problem of excessive model complexity. As there are an excessive number of stock indicators, the accuracy of the IBRB is strongly related to the number of intervals.¹⁴ As the intervals of the model are divided more densely, the accuracy of the model increases; however, this also results in a larger number of rules, making the model more complex.

Problem 2: How to solve the rationality problem of constructing the stock trading volume prediction model. High accuracy and strong interpretability of the results are required for stock trading volume prediction models. The interpretability of the traditional IBRB still needs to be improved.

Problem 3: How to ensure that the optimization process does not destroy interpretability. The structure of the initial model defined by the initial expert knowledge may be corrupted by using the global optimization algorithm directly. Therefore, an optimization algorithm that maintains the interpretability of the model is proposed in this paper. The optimization process is shown in Eq. (1):

\begin{aligned} φ_{best} = o p t i m i z e (X_{k}, r, c, τ) \end{aligned}

(1)

where

X_{k} (k = 1, \dots, I_{k})

are the prerequisite properties of the model. r is the prediction result of the stock trading volume prediction model. c is the set of interpretable constraints.

τ

is the set of metrics for the optimization process.

o p t i m i z e (\cdot)

denotes the optimization function.

3 The IBRB-AI model

To address the three problems described above, the IBRB-AI model is proposed in this section. The optimization algorithm P-CMA-ES with interpretability constraints is proposed to ensure that the interpretability of the model remains unaffected.¹⁵ In subsection 3.1, interpretability modeling guidelines for stock trading volume prediction methods are proposed. In this section, subsections 3.2 to 3.4, the IBRB-AI model is described in detail in terms of modeling, inference, and optimization. The overall structure of the IBRB-AI model is shown in Figure 1. Interpretability modeling guidelines are introduced during the modeling and inference process of the IBRB-AI model, and the parameters are optimized via the P-CMA-ES algorithm. To maintain interpretability during optimization, constraints are added. The operational process of the IBRB-AI model is depicted in Figure 2. The workflow of the IBRB-AI model can be composed of the following four steps: first, expert knowledge is used to construct the model on the basis of interpretable modeling guidelines; second, the ER inference algorithm with interpretable guidelines is used for model inference; third, the model is optimized via the P-CMA-ES optimization algorithm with interpretable constraints; and fourth, the expected results are output.

Figure 1.

Overall structure of the IBRB-AI model.

Figure 2.

The operational process of the IBRB-AI model.

3.1 Interpretability of stock trading volume prediction models

Stock trading volume prediction models have high interpretability requirements. Owing to a lack of interpretability, the requirements of some data-driven models for stock trading volume prediction are not fully satisfied. Although interpretability is provided by the traditional BRB model, it is insufficient for the financial domain. On the basis of a study of the interpretability of BRBs by Cao et al. ${C_{general} | C_{1}, C_{2}, \dots, C_{f}}$ , these seven generalized metrics were proposed as construction guidelines for interpretable IBRB-AI models.¹⁵ As shown in Figure 3, according to the interpretability guidelines, the rationality of the BRB should be considered in both modeling and reasoning.

Figure 3.

Modeling interpretability guidelines for IBRB-AI models.

3.1.1 Interpretability guidelines for the modeling process

Guideline 1: Reference values for attributes in the system should be distinguishable.

In BRB, reference values of attributes are used to define different ranges of attribute variables. When constructing a stock trading volume prediction model, each attribute should be assigned at least one reference value with distinct meanings, and the matching functions for each reference value should differ to ensure clear semantics.

Guideline 2: The system should have a complete rule base.

A complete rule base indicates that for every attribute input to the IBRB-AI model, there is at least one corresponding reference value, and at least one rule is activated.¹⁶ The completeness can be determined via Eq. (2). In other words, all the states of the model should be included in the rule base.

\begin{aligned} \begin{matrix} \forall x \in U, \exists 1 \leq v \leq m, a_{v} (x) > 0 \\ \forall x \in U, \exists 1 \leq l \leq L, 0 < w_{k} \leq 1 \end{matrix} \end{aligned}

(2)

where the number of reference values is indicated by m. The matching degree to the vth referential value is indicated by

a_{v} (x)

. L represents the number of rules. The activation weight is indicated by

w_{k}

Guideline 3: The structure and parameters of the system should be physically meaningful.

When a stock trading volume forecasting model is constructed, a reasonable causal relationship is essential. Therefore, each parameter in the prediction process should have real physical meaning. The model parameters of the BRB model include the following: 1)

Rule weights: The importance of different rules is described by it.

Attribute weights: The importance of the input attribute is characterized by it.

Activation weights: The degree to which the rule is activated is indicated by it.

Belief degree: The extent to which the rule matches the stock trading volume forecast is indicated by it.

Guideline 4: The rule base should be simple.

The simplicity of the rule base is the key to the interpretability of the BRB, which helps researchers understand the model's internal logic and achieve better performance.¹⁷ For BRB, a simple rule base is a set of comprehensible, small numbers of rules with simple logical relationships, and the number of rules can be calculated via Eq. (3).

\begin{aligned} L = \prod_{t = 1}^{T} K_{t} \end{aligned}

(3)

where

K_{t}

denotes the reference point number of the t th attribute, T denotes the number of attributes.

Guideline 5: The system rules should be consistent.

Consistency is the basic requirement of interpretability, and conflicting rules are prohibited in BRB.¹⁵ The consistency of the initial rules is ensured by the rules extracted from expert knowledge, preventing the dichotomy of the final result.

3.1.2 Interpretability guidelines for the inference process

Guideline 6: Systematic information transformation must be rational and equivalent.

During the inference process, the integrity of the initial information must be maintained by the system. An approach proposed by Yang et al. combines rules and utilities to effectively implement the transformation of information and ensure that the initial information is effectively associated with the belief rules.^18,19 This approach allows the input information to be transformed into a belief distribution while preserving the characteristics of the output information.

Guideline 7: The system should have a transparent inference engine.

To ensure that the inference process of the stock trading volume prediction model is interpretable, each step of the system should be clearly described, and each inference process needs to be reasonably calculated and have a clear causal relationship. The ER algorithm, which has a transparent and interpretable inference process, is used in the stock volume prediction model based on the IBRB-AI model.²⁰

3.2 Construction of the IBRB-AI model

When the IBRB-AI model is constructed, the first step is that the problem is analyzed, then the reference points and reference intervals are set, and finally, the belief table is constructed. The specific steps are shown below:

1)
The research problem is analyzed mechanistically to derive the input attributes and the model's output as the modeling begins.
2)
The method of setting reference points and reference intervals differs from the traditional BRB method. In the IBRB-AI model, the reference values of the premise attributes are specified in the form of intervals.¹⁴ The reference point and reference interval for one attribute are shown in Table 1.

Table 1.
Reference points and reference intervals for one of the attributes.

Referential point VL L MH H VH EH

Reference interval $[a_{1}, b_{1}]$ $[a_{2}, b_{2}]$ $[a_{3}, b_{3}]$ $[a_{4}, b_{4}]$ $[a_{5}, b_{5}]$ $[a_{6}, b_{6}]$

While the rules of traditional BRB are composed of multiple discrete points, in IBRB-AI, multiple discrete rules are combined into a single rule with wider coverage by means of an interval form. Specific as shown in Table 2: 3)
The approach taken in constructing the belief table is different from the traditional BRB rule combination approach. While the Cartesian products method relies upon rule combination in traditional BRB, interval addition is utilized by the IBRB-AI model. The modeling process of the IBRB-AI model is illustrated in Figure 4. The $I_{k}$ prerequisite attributes are spread across various intervals, aligning with the activation of different rules. The ER inference algorithm is subsequently employed for model inference to generate predictions.

Figure 4.
Diagram of the modeling process of the IBRB-AI model.

Table 2.
Comparison of BRB and IBRB-AI rule structures.

Model Structure of rules for input attribute A Result

Traditional BRB Rule1: $A = a_{1}$ Two rules

Rule2: $A = b_{1}$

IBRB-AI Rule1: $A \in [a_{1}, b_{1}]$ One rule

Therefore, the rule explosion problem of the traditional BRB model is solved by the IBRB model through the use of interval summation to form rules. The accuracy of the IBRB model is closely related to the delineation of its reference points. The more densely the model is divided into reference points, the more accurate the model. The denser the reference points are, the greater the number of rules and the more complex the model. IBRB-AI can adapt better to situations with fewer rules. The accuracy of the model is maintained despite a smaller number of rules. By avoiding the setting of denser reference points, the number of rules is reduced, and thus, the complexity of the model decreases. The kth rule of the IBRB-AI model is shown in Eq. (4):
$\begin{aligned} \begin{array}{l} \begin{matrix} R_{k} :if X_{1} \in [a_{1}, b_{1}] \lor X_{2} \in [a_{2}, b_{2}] \lor \dots \lor X_{I_{k}} \in [a_{I_{k}}, b_{I_{k}}] \\ \begin{array}{l} Then result is {(D_{1}, β_{1}^{k}), \dots, (D_{N}, β_{N}^{k})}, \\ \sum_{n = 1}^{N} β_{n}^{k} \leq 1 (k = 1, 2, \dots, L) \end{array} \end{matrix} \\ with a rule weight θ_{k} and rule reliability δ_{k} \\ in interpretable guidelines c_{1}, c_{2}, \dots, c_{f} \end{array} \end{aligned}$
(4)
where $X_{k} (k = 1, \dots, I_{k})$ are the prerequisite attributes of the system. $I_{k}$ is the number of prerequisite attributes. $[a_{I_{k}}, b_{I_{k}}]$ is the reference interval. $D_{N}$ denotes the result of the output. $N$ is the number of results. $β_{N}^{k}$ denotes the belief degree. The sum of the belief degrees for each rule is not greater than 1. $θ_{k}$ is the rule weight. $δ_{k}$ denotes the rule reliability.

For the rule explosion problem, assume that premise attribute 1 has 5 reference points and that premise attribute 2 has 10 reference points; then, $5 \times 10 = 50$ rules can be combined via BRB. Assuming that premise attribute 1 has 6 reference points, premise attribute 2 has 4 reference points, and premise attribute 3 has 5 reference points, $6 \times 4 \times 5 = 120$ rules can be combined via BRB. The number of rules is positively correlated with the number of reference points. Assuming that premise attribute 1 has 5 reference points $A,B,C,D,E$ , then there are 4 intervals $[A,B], [B,C], [C,D], [D,E]$ . Similarly, if premise attribute 2 has 10 reference points, there are 9 intervals, and then $4 + 9 = 13$ rules can be combined by the IBRB-AI. Assuming that premise attribute 1 has 6 reference points, there are 5 intervals, premise attribute 2 has 4 reference points, there are 3 intervals, and premise attribute 3 has 5 reference points, there are 4 intervals; then, $5 + 3 + 4 = 12$ rules can be combined via the IBRB-AI. Compared with BRB, the number of rules is greatly reduced, solving the rule explosion problem while the inference burden is greatly reduced, making the model more suitable for time-sensitive financial forecasting tasks.

When we have four attributes, each with four reference values, the number of rules formed by traditional BRB and IBRB-AI are compared as shown in Figure 5:

Figure 5.
Comparison of rule numbers in BRB and IBRB-AI.

As illustrated in Figure 5, the number of rules in traditional BRB is shown to increase exponentially with the number of attributes, revealing the issue of rule explosion. In contrast, a more gradual increase is observed in IBRB-AI, where a simple structure is maintained even with a large number of attributes.
3.3 Inference process of the IBRB-AI model

Referential point	VL	L	MH	H	VH	EH
Reference interval	$[a_{1}, b_{1}]$	$[a_{2}, b_{2}]$	$[a_{3}, b_{3}]$	$[a_{4}, b_{4}]$	$[a_{5}, b_{5}]$	$[a_{6}, b_{6}]$

Model	Structure of rules for input attribute A	Result
Traditional BRB	Rule1: $A = a_{1}$	Two rules
Rule2: $A = b_{1}$
IBRB-AI	Rule1: $A \in [a_{1}, b_{1}]$	One rule

Step 1: Activate belief rules

In the traditional BRB model, whether a rule is activated depends on the matching of the premise attributes. A rule is only activated when the matching degree of all relevant attributes is non-zero. The input attribute calculation formula for the traditional BRB is shown in Eq. (5).

\begin{aligned} α_{i}^{j} = {\begin{array}{l} \frac{x_{i (k + 1)} - x_{i}^{*}}{x_{i (k + 1)} - x_{i k}}, j = k, x_{i k} \leq x_{i}^{*} \leq x_{i (k + 1)} \\ 1 - α_{i}^{j}, j = k + 1 \\ 0, j = 1, 2, \dots, J, j \neq 1, k + 1 \end{array} \end{aligned}

(5)

where

x_{i}^{*}

represents the input of the i th feature and where

x_{i (k + 1)}

and

x_{i k}

are the reference values of the features in the

(k + 1)

th and k th belief rules, respectively.

Unlike traditional BRB models, IBRB-AI needs to consider the activation interval of rules. When the values of premise attributes fall within the preset interval, the corresponding rules are activated. For example, in Table 3, if the two premise attributes are located in the intervals $[a_{2}, b_{2}]$ and $[a_{5}, b_{5}]$ , then rules 2 and 5 will be activated accordingly.

Table 3.

Belief table.

Number	Point	Output
1	$[a_{1}, b_{1}]$	${(V L, 0.7), (L, 0.3), (M H, 0), (H, 0), (V H, 0), (E H, 0)}$
2	$[a_{2}, b_{2}]$	${(V L, 0.6), (L, 0.2), (M H, 0.2), (H, 0), (V H, 0), (E H, 0)}$
3	$[a_{3}, b_{3}]$	${(V L, 0.6), (L, 0.3), (M H, 0.1), (H, 0), (V H, 0), (E H, 0)}$
4	$[a_{4}, b_{4}]$	${(V L, 0.5), (L, 0.2), (M H, 0.2), (H, 0.1), (V H, 0), (E H, 0)}$
5	$[a_{5}, b_{5}]$	${(V L, 0.3), (L, 0.2), (M H, 0.2), (H, 0.3), (V H, 0), (E H, 0)}$
6	$[a_{6}, b_{6}]$	${(V L, 0), (L, 0.1), (M H, 0.2), (H, 0.2), (V H, 0.2), (E H, 0.3)}$

Step 2: The activation of rules in the BRB is influenced by the observed data inputs related to system characteristics, and the weights of activation can be determined via Eqs. (6) and (7):

\begin{aligned} w_{k} & = \frac{θ_{m} \prod_{i = 1}^{m} {(α_{k}^{i})}^{{\bar{δ}}_{i}}}{\sum_{l = 1}^{L} θ_{l} \prod_{i = 1}^{m} {(α_{k}^{i})}^{{\bar{δ}}_{i}}}, \end{aligned}

(6)

\begin{aligned} \bar{δ_{i}} & = δ_{i} / max_{i = 1, 2, \dots I_{k}} {w_{i}} \end{aligned}

(7)

where the normalized weight is denoted by

\bar{δ_{i}}

. The activation weight is denoted by

w_{k}

Step 3: The belief degree of its output is generated by the activation rule. This rule is then integrated via an ER algorithm to calculate the final output belief degree, as shown in Eq. (8):

\begin{aligned} β_{n} = \frac{μ [\prod_{k = 1}^{L} (w_{k} β_{n}^{k} + 1 - w_{k} \sum_{n = 1}^{N} β_{n}^{k}) - \prod_{k = 1}^{L} (1 - w_{k} \sum_{n = 1}^{N} β_{n}^{k})]}{1 - [\sum_{n = 1}^{N} \prod_{k = 1}^{L} (w_{k} β_{n}^{k} + 1 - w_{k} \sum_{n = 1}^{N} β_{n}^{k}) {- (N - 1) \prod_{k = 1}^{L} (1 - w_{k} \sum_{n = 1}^{N} β_{n}^{k})]}^{- 1} [\prod_{k = 1}^{L} (1 - w_{k})]} \end{aligned}

(8)

where

β_{n}

is the belief degree of the prediction results. `

Step 4: The final distribution of beliefs is shown in Eq. (9):

\begin{aligned} S (x_{n}^{*}) = {(D_{n}, β_{n}); n = 1, 2, \dots, N}, \end{aligned}

(9)

where

x_{n}^{*}

represents the input data for the stock trading volume prediction model.

Step 5: The final utility value can be calculated via Eq. (10):

\begin{aligned} u (S (x_{n}^{*})) = \sum_{n = 1}^{N} u (D_{n}) β_{n} \end{aligned}

(10)

where

u (S (x_{n}^{*}))

denotes the final utility value.

3.4 Optimization process of the IBRB-AI model

When global optimization algorithms are used in the optimization process, complex system structures are generated, making the decision-making process of the model difficult to understand. In addition, global optimization algorithms may sacrifice local interpretability to find the optimal solution. This means that even if an optimal solution is found, the decision path of the model may become opaque. The parameters of the model may become difficult to explain during the optimization process, potentially losing their physical meaning,²¹ leading to the destruction of their interpretability. Therefore, to ensure that the IBRB-AI model remains interpretable after optimization, interpretability constraints are introduced into the original P-CMA-ES optimization algorithm,¹⁵ as shown in Figure 6.

Figure 6.

Steps of the P-CMA-ES optimization algorithm.

The P-CMA-ES optimization algorithm is widely used in practical engineering. The P-CMA-ES optimization algorithm has the following advantages: 1) The model parameters can be optimized, and the model interpretability can be enhanced. 2) The reasonableness of the model outputs can be ensured by setting constraints. 3) The risk of falling into local optima can be avoided.

An adaptive rule mechanism is introduced in the optimization process to improve the accuracy, robustness, and structural adaptability of the model. The mechanism is adapted to different stages of the optimization process through the dynamic adjustment of strategy parameters, such as the covariance matrix and the step size. Specifically, the covariance matrix and step size are adjusted using the cumulative time constant and sigma control time constant, while the learning rate and rank update coefficients are also adaptively modified according to changes in the search environment. In addition, the step-size update formula is dynamically calculated based on the current fitness variation, allowing it to be automatically adjusted according to the search stage in order to fit various target characteristics and search environments.

In the IBRB-AI model, this adaptive rule mechanism is effectively integrated into the P-CMA-ES optimizer, enabling the rules to be automatically compressed or refined in different search phases based on input sensitivity and the characteristics of the objective function. Through this mechanism, dynamic structural adjustment and rule pruning are realized without the need for manual intervention by domain experts. As a result, the number of rules is significantly reduced, and the flexibility and computational efficiency of the model are improved, making it particularly suitable for complex environments such as highly dynamic financial markets.

Before an optimization model is constructed, the optimization objective needs to be clearly defined. In particular, the mean square error (MSE) is used to quantify the discrepancy between the actual and projected values in the IBRB-AI model. Therefore, the optimized function can be expressed as Eq. (11):

\begin{aligned} \begin{array}{l} m i n M S E (β, r, δ), \sum_{n = 1}^{N} β_{n}^{k} = 1, i = 1, \dots, L \\ {\begin{array}{l} 0 \leq β_{n}^{k} \leq 1, n = 1, \dots, N \\ 0 \leq r_{k} \leq 1, k = 1, \dots, L \\ 0 \leq δ_{i} \leq 1, i = 1, 2, \dots L \end{array} \end{array} \end{aligned}

(11)

The formula for the MSE is as follows:

\begin{aligned} M S E (β, r, δ) = \frac{1}{S_{t r}} \sum_{t r = 1}^{S_{t r}} {(v a l u e_{p r e d i c t i o n} - v a l u e_{t r u e})}^{2} \end{aligned}

(12)

where the training sample total is

S_{t r}

The steps involved in the optimization algorithm are as follows:

Step 1 (initialization): Setting the population size to O, the size of the offspring population is $λ$ , the covariance matrix is $μ^{0}$ , the population is $m^{0} = Ω^{0} (β, r, δ)$ , and the step size is $ϵ^{0}$ .

In accordance with guideline 3, the step size $ϵ^{0}$ can be given by expert knowledge. When expert knowledge is more trustworthy, a smaller search is conducted; when expert knowledge is less reliable, a larger search is conducted.

Step 2 (generate population): Generate the population through Eq. (13):

\begin{aligned} Ω_{k}^{f + 1} = m^{f} + ϵ^{(f)} N (0, μ^{(f)}), k = 1, 2, \dots, O \end{aligned}

(13)

where N denotes the normal distribution. The kth population of the

f + 1

th generation is denoted by

Ω_{k}^{f + 1}

Constraint 1: The optimal reference value needs to be within the feasible interval given by the expert, which can be expressed as Eq. (14):

\begin{aligned} a_{u - low}^{k} \leq A_{u}^{k} \leq a_{u - high}^{k}, (k = 1, \dots, L, u = 1, \dots, m) \end{aligned}

(14)

where

A_{u}^{k}

is the u th reference value of the s th rule. The upper and lower bounds of the feasible interval are denoted by

a_{u - l o w}^{k}

and

a_{u - h i g h}^{k}

, respectively.

Constraint 2: The belief distribution should be reasonable and acceptable and can be expressed as Eq. (15):

\begin{aligned} β_{n}^{k} \sim I B D (n = 1, \dots, N, k = 1, \dots, L) \end{aligned}

(15)

where

I B D

is the interpretable belief distribution.

Constraint 3: The reliability of expert knowledge needs to be considered in the optimization process.

\begin{aligned} 0 \leq E_{belief} \leq 1 \end{aligned}

(16)

where the reliability of expert knowledge is indicated by

E_{belief}

, which is subjectively determined from an expert perspective. 1 and 0 indicate that the model is completely trustworthy and completely untrustworthy, respectively; completely untrustworthy models are transformed into data-driven models.

Constraint 4: The kth activation weight should satisfy Eq. (17):

\begin{aligned} w_{k} = {\begin{array}{l} w_{k}, i f w_{k} \geq ℓ \\ 0, i f w_{k} < ℓ \end{array} \end{aligned}

(17)

where

w_{k}

denotes the kth activation weight;

ℓ

denotes the fault tolerance threshold during activation; in the financial stock market, the threshold

ℓ

in the current research does not meet the same standard and is usually given by experts in the field.

Step 3 (modify the belief distribution): Find the wrong belief distribution according to constraint 2 and reset it until the distribution condition is satisfied, as shown in Eq.(18):

\begin{aligned} \begin{matrix} Ω_{k}^{f + 1} \leftarrow β_{k}^{f + 1} = m^{f} + ϵ^{(f)} N (0, μ^{(f)}) \\ β_{k}^{f + 1} \sim I B D, k = 1, \dots, L \end{matrix} \end{aligned}

(18)

where

Ω_{k}^{f + 1}

is the k th solution of the

f + 1

st generation and where

β_{k}^{f + 1}

is the updated belief distribution that satisfies

I B D

\leftarrow

This symbol indicates that the erroneous belief distribution can be reset.

Step 4 (projection operation): Project the solution into the hyperplane via Eq. (19):

\begin{aligned} A_{e} Ω_{k}^{f + 1} (1 + n_{e} (j - 1) : n_{e} j) = 1, j = 1, 2, \dots, N + 1 \end{aligned}

(19)

where

A_{e}

denotes the set of parameters.

n_{e}

and j denote the number of constraints and constraint variables, respectively, in

Ω_{k}^{f}

The projection operation is realized as shown in Eq. (20):

\begin{aligned} \begin{matrix} Ω_{k}^{f + 1} (1 + n_{e} (j - 1) : n_{e} j) = Ω_{k}^{f + 1} (1 + n_{e} (j - 1) : n_{e} j) \\ - A_{e}^{T} {(A_{e} A_{e}^{T})}^{- 1} \times Ω_{k}^{f + 1} (1 + n_{e} (j - 1) : n_{e} j) A_{e} \end{matrix} \end{aligned}

(20)

Step 5 (Selection of the optimal value): Select the optimal solution and update the mean.

\begin{aligned} M S E (Ω_{1 : O}^{f + 1}) \leq M S E (Ω_{2 : O}^{f + 1}) \leq \dots \leq M S E (Ω_{O : O}^{f + 1}) \end{aligned}

(21)

where

(Ω_{a : O}^{f + 1})

denotes the a th solution in the O solutions.

The subgroup can be derived from Eq. (22):

\begin{aligned} m^{f + 1} = \sum_{i = 1}^{λ} σ_{i} Ω_{i : O}^{f + 1} \end{aligned}

(22)

where the weights set on the basis of the adaptation values are denoted by

σ_{i}

Step 6 (adaptive operation): Update the covariance matrix of the population via Eq.(23):

\begin{aligned} μ^{f + 1} & = (1 - ε_{1} - ε_{2}) μ^{f} + ε_{1} y_{ε}^{f + 1} {(y_{ε}^{f + 1})}^{T} {+ ε_{2} \sum_{i = 1}^{λ} σ_{i} (\frac{Ω_{i : O}^{f + 1} - m^{f}}{ϵ^{f}})) (\frac{Ω_{i : O}^{f + 1} - m^{f}}{ϵ^{f}}))}^{T} \\ y_{ε}^{f + 1} & = 1 - (ε_{ε}) y_{ε}^{f} + \sqrt{ε_{ε} (2 - ε_{ε}) {(\sum_{i = 1}^{λ} σ_{i}^{2})}^{- 1}} (m^{f + 1} - m^{f}) / ϵ^{f} \end{aligned}

(23)

where the step size

ϵ^{f + 1}

is given by Eq. (24):

\begin{aligned} ϵ^{f + 1} & = ϵ^{f} \exp (\frac{ε_{c}}{b_{c}} (\frac{| | y_{c}^{f + 1} | |}{E | | N (0, 1) | |} - 1)) \\ y_{c}^{f + 1} & = (1 - ε_{ε}) y_{c}^{f} + \sqrt{ε_{ε} (2 - ε_{ε}) {(\sum_{i = 1}^{λ} σ_{i}^{2})}^{- 1}} {(μ^{f})}^{- \frac{1}{2}} (m^{f + 1} - m^{f}) / ϵ^{f} \end{aligned}

(24)

where

ε_{1}

and

ε_{2}

denote the learning rate,

y_{ε}

denotes the evolutionary path,

ε_{ε}

denotes the evolutionary path parameter, and

b_{c}

denotes the damping parameter.

Based on the above discussion, the steps of the P-CMA-ES optimization algorithm are illustrated in Figure 6. Four additional interpretability constraints are incorporated into the original P-CMA-ES optimization algorithm: 1)

Reference value constraint: Ensures that the optimized reference values are kept within the range defined by experts.

Belief normalization: Ensures that a valid belief distribution is maintained and the probabilistic interpretation is preserved.

Belief interval: Restricts the update of belief weights to within the belief boundaries provided by experts.

Activation rule: Ensures that rule activation weights are not allowed to conflict, thereby preserving behavioral consistency.

Through these constraints, the solution space is restricted to a region where performance is improved without sacrificing model transparency.

4 Case study

The experimental steps are described in subsection 4.1. The dataset setup and experimental parameters are outlined in subsection 4.2. Finally, the experimental results are analyzed in subsection 4.3.

4.1 Experimental steps

Data acquisition, data preprocessing, model construction, model training, and comparative experimental analysis are included in the experiments. The steps of stock trading volume prediction are shown in Figure 7.

4.2 Dataset and initial parameter setup

Fluctuations in stock data are characterized by irregularity, a nonnormal distribution, and nonlinearity. Therefore, the ability to address uncertain information is required when constructing a stock trading volume prediction model, while the interpretability of the model is also key to ensuring the credibility of the results.

Tesla stock data were chosen for this experiment to predict the results. In the stock market, all the technical indicators involved are calculated on the basis of opening, closing, high, and low prices. In this case, the market trading price at the beginning of a new trading cycle is reflected by the opening price, and the market trading price at the end of the trading cycle is reflected by the closing price. Trading decisions can be made by investors on the basis of the upward or downward trend of opening and closing prices. When the closing price is higher than the opening price, there is an upward trend in the stock. Investors may consider buying, and thus, trading volume is affected. The high and low prices are represented by the highest and lowest prices traded on the day, respectively, and the volatility of the market and the willingness of investors to trade are reflected by these prices. When high volatility is observed, it indicates that market activity is elevated, suggesting an increase in potential investment opportunities. In such cases, an increase in trading volume may also be observed in the stock market. In addition, the causal “news” effect and the strong correlation of the opening price can be utilized to better predict high, low, and closing prices.^22,23 Therefore, in this paper, the opening, closing, high, and low prices are chosen to predict stock trading volume. The stock trading data covered the period from June 29, 2010, to February 3, 2020 (the stock market was closed on Saturdays and Sundays), with a final dataset of 2417 data points. In this work, the proportion of training datasets to test datasets is set at 7:3. The development environment for the experiment was Windows 11 and an AMD Ryzen 7 5800U with Radeon Graphics. The experiments were performed in MATLAB, and no other library functions were used.

Figure 7.

Experimental steps of stock trading volume prediction.

To eliminate the quantitative effects between the indicators of the stock market, the data need to be normalized via Eq. (25):

\begin{aligned} x^{'} = \frac{x - x_{min}}{x_{max} - x_{min}} \end{aligned}

(25)

where x is the original input value to the predictive model,

x^{'}

is the normalized input value, and

x_{max}

and

x_{min}

are the maximum and minimum values, respectively.

However, due to missing samples and incomplete statistics, specific treatments need to be developed for each patient. In situations of widespread missing historical data, direct elimination can be used to exclude missing data. In addition, for the factors with less missing historical data, the mean padding method was employed for data padding, and data integrity was ensured. Owing to the small amount of missing historical data in this experiment, the mean padding method was used to populate the data.

The distribution of the processed data is shown in Figure 8.

Figure 8.

Distribution of the processed data.

Six semantic values—"very low” (VL), “low” (L), “medium” (M), “high” (H), “very high” (VH), and “extremely high” (EH)—were chosen to characterize the results of the expert knowledge-based prediction.

Therefore, the first rule of the stock trading volume prediction model of IBRB-AI is shown in Eq. (26):

\begin{aligned} \begin{matrix} \begin{array}{l} Rul e_{k} : \\ i f X_{1} \in [a_{1}, b_{1}] \lor X_{2} \in [a_{2}, b_{2}] \lor X_{3} \in [a_{3}, b_{3}] \lor X_{4} \in [a_{4}, b_{4}] \end{array} \\ T h e n {(V L, β_{1}), (L, β_{2}), (M, β_{3}), (H, β_{4}), (V H, β_{5}), (E H, β_{6})} \\ \begin{array}{l} with θ_{k} = 1 and δ_{k} = 1 \\ in interpretable guidelines c_{1}, c_{2}, \dots, c_{7} \end{array} \end{matrix} \end{aligned}

(26)

4.3 Analysis of the experimental results

The initial attribute weights and initial reference values of IBRB-AI are shown in Table 4, and the initial rule weights and belief distributions are shown in Table 5. An interval of values is given by expert knowledge for optimization, and the optimized model parameters are shown in Table 14 .

Table 4.
Initial weights of the attributes and reference values.

Referential values

Attribute Attribute weight $w_{I_{k}}$ Attribute weight constraint $w_{I_{k}} \sim c_{3}$ VL L M H VH EH

Open 1 0.6∼1 −1 −0.2 0 0.2 0.5 1

High 1 0.6∼1 −1 −0.2 0 0.2 0.5 1

Low 1 0.6∼1 −1 −0.2 0 0.2 0.5 1

Close 1 0.6∼1 −1 −0.2 0 0.2 0.5 1

Output −1 −0.2 0 0.2 0.5 1

			Referential values
Open	1	0.6∼1	−1	−0.2	0.2	0.5	1
High	1	0.6∼1	−1	−0.2	0.2	0.5	1
Low	1	0.6∼1	−1	−0.2	0.2	0.5	1
Close	1	0.6∼1	−1	−0.2	0.2	0.5	1
Output			−1	−0.2	0.2	0.5	1

Table 5.

Initial rule weights and belief distributions.

ID	Referential interval $[a_{I_{k}}, b_{I_{k}}]$	Rule weight $θ_{k}$	Rule weight constraint $θ_{k} \sim c_{3}$	The initial belief ${β_{1}, β_{2}, β_{3}, β_{4}, β_{5}, β_{6}}$
1	[−1, 0.97]	1	0.5∼1	{0 0 0 0 0.2 0.8}
2	[−0.97, 0.5]	1	0.5∼1	{0 0 0.1 0.1 0 0.8}
3	[−0.5, 0.33]	1	0.5∼1	{0 0 0 0.1 0.5 0.4}
4	[−0.33, 0.17]	1	0.5∼1	{0.1 0.3 0.1 0 0.4 0.1}
5	[−0.17,0.2]	1	0.5∼1	{0.2 0 0 0.2 0.5 0.1}
6	[0.2,0.5]	1	0.5∼1	{0.1 0.1 0.3 0.2 0.2 0.1}
7	[0.5,0.7]	1	0.5∼1	{0.2 0.2 0 0.2 0.2 0.2}
8	[0.7,1]	1	0.5∼1	{0.3 0.2 0.1 0.3 0 0.1}
9	[−1, 0.97]	1	0.5∼1	{0 0 0 0 0.3 0.7}
10	[−0.97, 0.5]	1	0.5∼1	{0 0 0.1 0.1 0.3 0.5}
11	[−0.5, 0.33]	1	0.5∼1	{0 0 0 0.2 0.5 0.3}
12	[−0.33, 0.17]	1	0.5∼1	{0 0 0 0.2 0.4 0.4}
13	[−0.17,0.2]	1	0.5∼1	{0.2 0.1 0.1 0.4 0.1 0.1}
14	[0.2,0.5]	1	0.5∼1	{0.1 0.1 0 0.3 0.2 0.3}
15	[0.5,0.7]	1	0.5∼1	{0 0.2 0.3 0.1 0.1 0.3}
16	[0.7,1]	1	0.5∼1	{0.1 0 0.2 0.3 0.1 0.3}
17	[−1, 0.97]	1	0.5∼1	{0 0.1 0 0.2 0.1 0.6}
18	[−0.97, 0.5]	1	0.5∼1	{0 0 0 0 0.1 0.9}
19	[−0.5, 0.33]	1	0.5∼1	{0 0 0.1 0.2 0.3 0.4}
20	[−0.33, 0.17]	1	0.5∼1	{0 0.1 0.2 0.2 0.3 0.2}
21	[−0.17,0.2]	1	0.5∼1	{0 0 0 0.2 0.7 0.1}
22	[0.2,0.5]	1	0.5∼1	{0 0.2 0.5 0.1 0.2 0}
23	[0.5,0.7]	1	0.5∼1	{0 0.3 0.3 0 0.1 0.3}
24	[0.7,1]	1	0.5∼1	{0.3 0 0.5 0.1 0.1 0}
25	[−1, 0.97]	1	0.5∼1	{0 0 0.2 0.1 0.1 0.6}
26	[−0.97, 0.5]	1	0.5∼1	{0.1 0.1 0.1 0 0.1 0.6}
27	[−0.5, 0.33]	1	0.5∼1	{0.1 0.1 0 0.2 0.1 0.5}
28	[−0.33, 0.17]	1	0.5∼1	{0.1 0.2 0.2 0 0.3 0.2}
29	[−0.17,0.2]	1	0.5∼1	{0 0.2 0 0.1 0.2 0.5}
30	[0.2,0.5]	1	0.5∼1	{0.2 0.1 0.1 0.3 0.2 0.1}
31	[0.5,0.7]	1	0.5∼1	{0.1 0.1 0.1 0.3 0.2 0.2}
32	[0.7,1]	1	0.5∼1	{0.1 0.3 0.3 0.1 0 0.2}

Table 6.

Average accuracy analysis of various models.

Models	Average MSE	Average R-squared
IBRB	0.0025	0.9257
IBRB-A	0.0034	0.8835
IBRB-AI	0.0037	0.8812
Elman	0.0119	0.6190
BPNN	0.0047	0.8551
KNN	0.0105	0.7076
SVM	0.0097	0.6819
BVAR	0.0099	0.6793
LR	0.0081	0.7338

Table 7.

Rule count and training time comparison across models.

Models	Rule count	Avg. training time (s)	Avg. MSE
IBRB	52	24.24	0.0025
IBRB-A	32	16.38	0.0034
IBRB-AI	32	16.85	0.0037

Table 8.

Fluctuation interval of the MSE for each model.

Models	Fluctuation interval
IBRB	0.0013
IBRB-A	0.0019
IBRB-AI	0.0010
Elman	0.0030
BPNN	0.0015
KNN	0.0112
SVM	0.0046
BVAR	0.0068
LR	0.0041

Table 9.

MSEs for each model with different test samples.

Models	10%	20%	30%	70%	Fluctuation interval
IBRB	0.0042	0.0025	0.0019	0.0015	0.0027
IBRB-A	0.0048	0.0036	0.0049	0.0048	0.0013
IBRB-AI	0.0048	0.0039	0.0045	0.0047	0.0008
Elman	0.0132	0.0106	0.0117	0.0123	0.0026
BPNN	0.0032	0.0044	0.0033	0.0048	0.0016
KNN	0.0099	0.0089	0.0098	0.0085	0.0014
SVM	0.0097	0.0086	0.0102	0.0098	0.0016
BVAR	0.0110	0.0091	0.0092	0.0122	0.0031
LR	0.0068	0.0062	0.0081	0.0073	0.0019

Table 10.

Comparison between traditional BRB and IBRB-AI models.

Feature	Traditional BRB	IBRB-AI (Proposed)
Rule Source	Expert-defined only	Expert + Data-driven Optimization
Rule Type	Discret	Interval-based
Rule Explosion Risk	High	Controlled
Interpretability	Full	High(Semantic constraints applied)
Accuracy (MSE)	Not applicable in this task	0.0037

Table 11.

Experimental results of various models for obtaining AAPL data.

Models	Average MSE	Average R-squared
IBRB	0.0068	0.8324
IBRB-A	0.0074	0.8041
IBRB-AI	0.0076	0.7967
BPNN	0.0081	0.8196
Elman	0.0083	0.7354
KNN	0.0078	0.7506
SVM	0.0122	0.6108
BVAR	0.0108	0.7128
LR	0.0079	0.7531

Table 12.

Analysis of the resistance to volatility of each model.

Models	MSE volatility range
IBRB	0.0010
IBRB-A	0.0052
IBRB-AI	0.0004
Elman	0.0017
BPNN	0.0032
KNN	0.0010
SVM	0.0011
BVAR	0.0008
LR	0.0061

Table 13.

Model MSE performance across multiple stocks from different sectors.

Ticker	Company	Sector	MSE
AMZN	Amazon	Consumer Tech	0.0101
FB	Meta (Facebook)	Communication	0.0032
IBM	IBM	Technology	0.0057
GOOGL	Alphabet (Google)	Information Tech	0.0054
GS	Goldman Sachs	Finance	0.0033

Table 14.

Table of optimized parameters.

ID	Referential interval $[a_{I_{k}}, b_{I_{k}}]$	Rule reliability $δ_{k}$	Rule weight $θ_{m}$	Output belief ${β_{1}, β_{2}, β_{3}, β_{4}, β_{5}, β_{6}}$
1	[−1, 0.97]	0.68	0.92	{0.04 0.03 0.34 0.19 0.01 0.39}
2	[−0.97, 0.5]	0.72	0.66	{0 0 0 0.13 0.16 0.72}
3	[−0.5, 0.33]	0.57	0.69	{0.24 0.14 0.16 0.13 0.13 0.20}
4	[−0.33, 0.17]	0.55	0.77	{0.09 0.04 0.11 0.02 0.35 0.39}
5	[−0.17,0.2]	0.54	0.67	{0.06 0.22 0.26 0.29 0.09 0.09}
6	[0.2,0.5]	0.55	1.00	{0.09 0.17 0.08 0.46 0.10 0.11}
7	[0.5,0.7]	0.77	0.94	{0.08 0.04 0.10 0.12 0.16 0.49}
8	[0.7,1]	0.60	0.71	{0.15 0.17 0.21 0.08 0.24 0.15}
9	[−1, 0.97]	0.92	0.86	{0.08 0.33 0.19 0.19 0.18 0.03}
10	[−0.97, 0.5]	0.51	0.75	{0.02 0.20 0.14 0.11 0.36 0.16}
11	[−0.5, 0.33]	0.81	0.91	{0.07 0.04 0.04 0.28 0.04 0.54}
12	[−0.33, 0.17]	0.51	0.80	{0.14 0.05 0.08 0.09 0.56 0.09}
13	[−0.17,0.2]	0.70	0.73	{0.57 0.12 0.08 0.16 0.07 0.01}
14	[0.2,0.5]	0.88	0.75	{0.25 0.15 0.08 0.31 0.02 0.19}
15	[0.5,0.7]	0.71	0.78	{0.02 0.06 0.38 0.33 0.09 0.11}
16	[0.7,1]	0.60	0.77	{0.09 0.21 0.19 0.16 0.24 0.11}
17	[−1, 0.97]	0.53	0.78	{0.11 0.11 0.30 0.07 0.08 0.32}
18	[−0.97, 0.5]	0.58	0.67	{0.03 0.12 0.37 0.12 0.15 0.21}
19	[−0.5, 0.33]	0.61	0.73	{0 0.05 0.05 0.27 0.32 0.33}
20	[−0.33, 0.17]	0.95	0.64	{0.03 0.03 0.07 0.13 0.25 0.49}
21	[−0.17,0.2]	0.82	0.96	{0.02 0.01 0.02 0.12 0.12 0.71}
22	[0.2,0.5]	0.54	0.84	{0.06 0.34 0.08 0.27 0.11 0.13}
23	[0.5,0.7]	0.76	0.98	{0.15 0.07 0.41 0.09 0.25 0.03}
24	[0.7,1]	0.92	0.89	{0.37 0.04 0.17 0.14 0.19 0.09}
25	[−1, 0.97]	0.84	0.91	{0 0 0.03 0 0.05 0.92}
26	[−0.97, 0.5]	0.72	0.87	{0.03 0.05 0.12 0.09 0 0.7}
27	[−0.5, 0.33]	0.88	0.71	{0.04 0.05 0.03 0.05 0.20 0.63}
28	[−0.33, 0.17]	0.73	0.81	{0.09 0.15 0.11 0.09 0.53 0.03}
29	[−0.17,0.2]	0.81	0.96	{0.04 0.05 0.02 0.53 0.02 0.34}
30	[0.2,0.5]	0.91	0.83	{0.04 0.03 0.25 0.10 0.34 0.25}
31	[0.5,0.7]	0.53	0.84	{0.22 0.13 0.04 0.34 0.02 0.25}
32	[0.7,1]	0.91	0.83	{0.17 0.32 0.24 0.12 0.10 0.04}

To better evaluate the performance of the models, BPNN, Elman network (Elman), K-nearest neighbors (KNN), support vector machine (SVM), Bayesian vector autoregression (BVAR), and linear regression (LR) are used for comparison. In addition, the IBRB and the IBRB-A without interpretability constraints are compared. In this work, the MSE and R-squared are selected for model accuracy analysis and comparison of each model. The formulas are shown in Eqs. (27) and (28):

\begin{aligned} M S E & = \frac{1}{n} \sum_{i = 1}^{n} (Q_{i} - {\hat{Q}}_{i})^{2} \end{aligned}

(27)

\begin{aligned} R^{2} & = 1 - \frac{\sum_{i = 1}^{n} {({\hat{Q}}_{i} - Q_{i})}^{2}}{\sum_{i = 1}^{n} {(Q_{i} - \bar{Q})}^{2}} \end{aligned}

(28)

where the actual value is expressed as

Q_{i}

, the predicted value is denoted by

{\hat{Q}}_{i}

, the average value of

Q_{i}

is denoted by

\bar{Q}

, and the number of data points is denoted by n.

4.3.1 Goodness-of-fit analysis

The degree of model fit refers to the relationship between the expected and true values. That is, whether the expected value is close to the real value is determined, and the fitting results are shown in Figure 9. If the absolute error between the predicted value and the actual value is less than a certain threshold value, the model correctly predicts the value. Conversely, if the absolute error exceeds the threshold, the model is not correctly predicted. The fitting accuracy of this experiment is as high as 97.93%. It is clear from Figure 9 that the IBRB-AI model has a good fit for the overall prediction, with deviations in the prediction of individual data points only. This shows that most of the stock trading volume can be accurately predicted by the model with better results.

Figure 9.

Goodness-of-fit for each method.

4.3.2 Accuracy analysis

In this paper, MSE and R-squared are used to assess the accuracy of the model. The average MSE and R-squared obtained by each method through 20 experiments are shown in Table 6. The MSE of the IBRB-AI model is slightly greater than those of the IBRB and IBRB-A models. This decrease in accuracy is caused by the consideration of expert knowledge and interpretability of the IBRB-AI model. Black-box models such as BPNN and SVM are known for their high accuracy but are criticized for lacking interpretability, whereas the IBRB-AI model achieves better performance than the black-box models do, with the interpretability of the model being maintained. As shown in Table 6, traditional white-box models like linear regression (LR) and Bayesian VAR (BVAR) exhibit significantly lower R-squared values and higher MSE compared to IBRB-AI. This reflects their limitations in capturing nonlinear dynamics and adapting to shifting data patterns in real-world stock markets. Such models are more prone to underfitting, which hinders their practical utility in dynamic trading environments.

Comparisons between the IBRB model, the IBRB-A model, and the IBRB-AI model show that model capability under interpretability constraints is improved by combining expert-based modeling with data-driven optimization. While the lowest MSE is achieved by IBRB, its adaptive robustness is limited. Flexibility is improved by IBRB-A at the expense of interpretability, whereas a balance of performance is achieved by IBRB-AI through the integration of the two approaches, allowing strong robustness and accuracy to be maintained along with interpretability.

4.3.3 Complexity analysis

The numbers of rules for the IBRB, IBRB-A, and IBRB-AI models are shown in Table 7. As shown in Table 7, the rule count for the IBRB model is 52, whereas only 32 rules are required by the IBRB-A model, which is a reduction of 38.46% compared with the IBRB model. The IBRB-A model achieves an MSE of 0.0034; compared with the IBRB model, the MSE difference is only 0.0009, and the complexity is significantly reduced. The MSE of the IBRB-AI model is 0.0037, which is slightly greater than that of the IBRB-A model. This is because the interpretability of the IBRB-AI is greater than that of the IBRB-A.

To further evaluate the practical benefits of complexity simplification, ten experiments were conducted to compare the average training time of the IBRB model (52 rules), the IBRB-A model (32 rules), and the IBRB-AI model (32 rules). As shown in Table 7, the training time was reduced from 24.24 s to 16.38 s, representing a 32.4% improvement. This confirms that rule complexity reduction not only leads to a simplified model structure but also yields quantifiable improvements in computational efficiency.

In conclusion, rule explosion can be effectively addressed and training efficiency can be improved by reducing the reference value for each attribute. Given the significant improvement in computational efficiency, the slight increase in MSE is considered acceptable. The trade-off between accuracy and computational efficiency is taken as evidence that IBRB-AI can be practically applied in stock trading volume prediction.

4.3.4 Robustness analysis

To verify the robustness of the model, 20 experiments were conducted for the IBRB-AI, IBRB-A, IBRB, BPNN, KNN, SVM, Elman, BVAR, and LR methods, and then the MSE of the model was analyzed. The results of the 20 experiments are shown in Figure 10. The fluctuation intervals for each method over the 20 experiments are shown in Table 8. As shown in Table 8, the IBRB-AI model has the smallest fluctuation interval, and the robustness of the model is proven.

Figure 10.

MSEs of the 20 experiments.

The robustness of the model can also be verified by comparing the change in MSE for each method for different test sample sizes. In this work, MSEs of 10%, 20%, 30%, and 70% of the test samples are studied. The results of the study are shown in Table 9. The IBRB-AI model is minimized by the fluctuation of the data perturbation, and its robustness is well maintained.

In summary, the robustness of the IBRB-AI model is superior. The reason is that the IBRB-AI model introduces interpretability guidelines to achieve the optimal solution. The optimization search space is limited by these guidelines, and the search fluctuations are reduced. Consequently, the optimal parameters are more closely aligned with expert knowledge since the algorithm prioritizes solutions that are more interpretable and consistent with expert knowledge. The model is thus more robust.

4.3.5 Interpretability analysis

4.3.5.1 Analyzing the differentiability of reference values

Figure 11 shows that the established constraint guidelines are consistent with the rule weights of the IBRB-AI. A more transparent inference process and a more meaningful set of model parameters are possessed by the IBRB-AI model than by the other models.

Figure 11.

Rule weights for IBRB-A and IBRB-AI.

4.3.5.2 Belief distribution analysis

The belief distributions of IBRB-A and IBRB-AI are shown in Figure 12, and IBRB-AI and expert knowledge have more similar belief distributions. Most of the rules in IBRB-A deviate from expert knowledge and even differ from it more, making them more difficult to understand. For example, rules 5, 6, 8, 9, 12, 15, 19, 24, 26, and 31 in Figure 12 are very different from the initial belief distribution, and the interpretability of the model is violated. This result more clearly reflects that the interpretability of the model is strongly influenced by the rational use of optimization algorithms.

Figure 12.

The belief distributions of each rule.

4.3.5.3 Theoretical Comparison between Traditional BRB and IBRB-AI

Although full interpretability is ensured in traditional BRB models due to reliance on discrete rules defined by experts, direct experimental validation of traditional BRB is hindered by the rule explosion problem and the lack of adaptability in handling high-dimensional data.

In contrast, in the IBRB-AI model, expert knowledge is combined with data-driven optimization through an interval structure and semantic constraints. Interpretability is retained in this hybrid gray box model, while model accuracy and robustness are ensured. The main differences between traditional BRB and IBRB-AI are summarized in Table 10.

4.3.6 Applicability analysis

To validate the applicability of the method proposed in this paper, stock data from Apple, Inc. (AAPL) were collected from September 20, 2012, to September 1, 2020, for a total of 2000 pieces of data. The experimental results are shown in Table 11. The experimental results show that this IBRB-AI model can be effectively used in stock prediction

4.3.7 Volatility analysis

To investigate the impact of stock market instability on the effectiveness of model predictions for stock trading volume, Gaussian noise was introduced into the training data to account for the effects of stock market volatility on the model. The model's robustness to volatility can be measured by comparing its performance before and after adding noise. The MSE fluctuation range of the model before and after adding noise is shown in Table 12. Table 12 shows that when the stock market is unstable, the MSE fluctuation range of the IBRB-AI model is only 0.0004, indicating high reliability. White-box models such as LR exhibit higher MSE fluctuation rates under stock market instability.

In this section, Gaussian noise is added to model the volatility of the stock market, and the model's ability to remain stable under non-stationary input conditions is reflected in the results. The adaptive rule mechanism, combined with the constraint-based optimization algorithm, is employed to enable dynamic adjustment to local data perturbations, through which the model's ability to handle structural changes during data generation is demonstrated.

4.3.8 Trade-off between rule complexity and semantic expressiveness

In belief rule-based systems, a reduction in the number of rules is often accompanied by a risk of decreased semantic interpretation capabilities or interpretability. To address this issue, an interval rule representation combined with adaptive rules is employed in IBRB-AI. This enables meaningful coverage of the input space to be provided by a smaller set of rules. Detailed rules are retained in areas where high data density is observed or where importance is defined by experts, while simplification is applied to less active areas. Furthermore, interpretability constraints are imposed during the optimization process to ensure that confidence levels are kept logically consistent.

Despite a 38.46% reduction in the number of rules, high prediction accuracy and consistency between rules and expert semantics are maintained, as demonstrated in Table 6 and Figure 12. This confirms that a balance between reduced complexity and maintained model expressiveness is achieved by IBRB-AI.

4.3.9 Generalization test

To evaluate the model's generalization ability under different industry and market conditions, the scope of the experiment was expanded to include five additional stocks from multiple industries. The selected stocks include companies from the technology (AMZN, FB, GOOGL, IBM), e-commerce (AMZN), and financial services (GS) sectors, covering a diverse set of market behaviors and liquidity profiles.

These stocks were chosen to include not only high-volume assets but also those with varying volatility levels and market dynamics, enabling a more comprehensive assessment of the model's real-world applicability.

The experimental results are shown in Table 13. A low MSE was consistently maintained by the IBRB-AI model across all stock experiments, demonstrating that strong adaptability and robustness were exhibited in different financial markets. The effectiveness of interval-based rule modeling and adaptive rule structures in predicting financial stock trading volumes is further validated by these results.

4.4 Experimental summary

On the basis of the above case study, the proposed IBRB-AI has several advantages in stock trading volume prediction methods.

Expert knowledge and data-driven methods are combined via the IBRB-AI model. The accuracy of the model being closer to the black box model is ensured by this combined approach. The feasibility of the method for stock trading volume prediction is demonstrated.

The number of rules for the IBRB-AI model is 32. Compared with the IBRB model, the number of rules is reduced by 38.46%, with approximate accuracy. A better adaptive ability with a reduced number of rules is demonstrated by the IBRB-AI model. Changes in the data are well adapted by the model, which has better generalization ability, and the complexity of the model is reduced.

IBRB-AI is understood because interpretability guidelines are followed. By following guidelines such as keeping the rule base as simple as possible, having distinguishable attribute references, and having transparent inference machines, the model is also internally easier to understand. This interpretability helps the decision basis of the model to be better understood by investors, thus enabling more informed investment or trading decisions to be made.

The potential of the IBRB-AI model as a stock volume prediction method is highlighted by these three advantages. Accuracy, adaptability, and interpretability were combined to create a reliable system for stock market investment decisions.

5 Conclusion

This paper makes two contributions. First, the number of rules in the modeling process is reduced by the IBRB-AI model through the adaptive rule mechanism, resulting in a complexity reduction of 38.46% compared with the original IBRB model. Second, an accuracy of 0.0037 is achieved by the IBRB-AI model, which is close to that of the black-box model, while interpretability in the stock trading volume prediction results is ensured, effectively supporting subsequent decisions.

The effectiveness of the IBRB-AI model in addressing the problem of stock trading volume prediction has been demonstrated. The rule base of this paper is built on data from the last ten years, which predict the short-term future, but the problem of large fluctuations in the actual stock market is not taken into account, which imposes limitations on the model's performance. We plan to continue our research in the future as follows: 1)

Design a BRB that can update the rule base in real time and can adjust the parameters of the rule base in real time through the fluctuation of data.

It is necessary to design multi-objective optimization algorithms to ensure that the model is maximized in terms of accuracy and interpretability.

Considering the multi-featured nature of the stock market, it is necessary to study other features that affect the trading volume of the stock market.

Footnotes

Acknowledgments

This work was supported in part by the Social Science Foundation of Heilongjiang Province under Grant No. 21GLC189, in part by the Foreign Expert Projects in Heilongjiang Province under Grant No. GZ20220131, in part by the Shandong Provincial Natural Science Foundation under Grant No. ZR2023QF010, in part by the Social Science Planning Foundation of Liaoning Province under Grant No. L23BTQ005, and in part by the Scientific Research Project of Liaoning Provincial Education Department under Grant No. JYTMS20230555.

Author contribution statement

Xiaokang Gao: Conceptualization, Methodology, Writing – original draft, Validation, Formal analysis, Writing – review & editing, Visualization. Jiabing Wu: Writing – original draft, Writing – review, Supervision. Wei He: Supervision, Writing – review. Yanzi Gao: Writing – original draft, Investigation.

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Data availability

The authors do not have permission to share data.

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			Referential values
Attribute	Attribute weight $w_{I_{k}}$	Attribute weight constraint $w_{I_{k}} \sim c_{3}$	VL	L	M	H	VH	EH
Open	1	0.6∼1	−1	−0.2	0	0.2	0.5	1
High	1	0.6∼1	−1	−0.2	0	0.2	0.5	1
Low	1	0.6∼1	−1	−0.2	0	0.2	0.5	1
Close	1	0.6∼1	−1	−0.2	0	0.2	0.5	1
Output			−1	−0.2	0	0.2	0.5	1