Prediction systems are an important aspect of intelligent decisions. In engineering practice, the complex system structure and the external environment cause many uncertain factors in the model, which influence the modeling accuracy of the model. The belief rule base (BRB) can implement nonlinear modeling and express a variety of uncertain information, including fuzziness, ignorance, randomness, etc. However, the BRB system also has two main problems: Firstly, modeling methods based on expert knowledge make it difficult to guarantee the model’s accuracy. Secondly, interpretability is not considered in the optimization process of current research, resulting in the destruction of the interpretability of BRB. To balance the accuracy and interpretability of the model, a self-growth belief rule base with interpretability constraints (SBRB-I) is proposed. The reasoning process of the SBRB-I model is based on the evidence reasoning (ER) approach. Moreover, the self-growth learning strategy ensures effective cooperation between the data-driven model and the expert system. A case study showed that the accuracy and interpretability of the model could be guaranteed. The SBRB-I model has good application prospects in prediction systems.
As the premise of intelligent decision making, the prediction system can make predictions and judgments about the future development trend and level of things. With the further development of industrialization, a series of environmental pollution problems, such as hazy weather and sandstorms, have caused harm to human physical and mental health. Air quality prediction can scientifically guide people’s daily activities and behaviors and improve their quality of life, which is of great significance to environmental monitoring and governance [
In the current research, the methods of prediction systems can be roughly divided into the following three categories: physical knowledge methods [
In a prediction system for practical engineering, there are two common problems. First, the problem of uncertain information coexistence cannot be effectively handled, such as coexisting fuzzy information and ignorance, reducing the model accuracy. Second, the data-driven model is built on a large amount of data; because the modeling process is not transparent, the rationality of the output results is difficult to convince. The belief rule base (BRB) can effectively solve the above problems [
However, three problems that exist in the BRB model need to be solved. First, expert knowledge can provide a roughly correct direction in the prediction system. However, in current research [
The contributions of this paper are as follows: 1) A new self-growth BRB model with interpretability constraints is proposed. 2) To ensure the interpretability of the SBRB-I model, an optimization algorithm with interpretability constraints is proposed.
The structure of the rest of the paper is as follows. In Section 2, the problems of the prediction model are formulated, and a new prediction model SBRB-I is introduced. The interpretability of the SBRB-I model, including inference interpretability and optimization interpretability, is introduced in Section 3. In Section 4, the reasoning process and optimization process of the SBRB-I model are given. Then, a case study is implemented to verify the effectiveness of the proposed model in Section 5. This paper is concluded in Section 6.
In prediction systems, model construction and optimization have a great impact on prediction accuracy and interpretability. In this section, the problem of the prediction system is formulated.
In a prediction model, the belief rule base is composed of a set of belief rules. The kth belief rule is as follows:
In Section 3.1, the interpretability of the reasoning process of the SBRB-I model is described. Then, in Section 3.2, the optimized interpretability of the SBRB-I model is described.
In the SBRB-I model, the ER algorithm is used as the inference engine; it is a reasoning calculation process based on evidence fusion, and the interpretability is reflected in the causal relationship between the processes. In building the model, the interpretability of the reasoning process refers to sufficient and clear explanations when dealing with uncertain information [
In the current study, Cao et al. established eight general interpretability criteria for the BRB in
Belief rules can give a clear input-output relationship of the prediction system, and it is the main interpretable aspect of BRB [
No. | Rule weight | FlowDiff AND PressureDiff | LeakSize distribution |
---|---|---|---|
{zero, very small, medium, high, very high} | |||
28 | 0.96 | NVS AND PL | {0.53, 0.02, 0, 0.12, 0.33} |
In recent years, BRB has been widely used due to its interpretability [
For interpretable BRB models, the feasible region of optimization is a local judgment based on experts [
BRB is a new intelligent expert system that combines an expert system and a data-driven model. However, because of the complexity of the prediction system, experts face difficulties in developing a deep understanding of the system. Moreover, BRB is typically optimized as a data-driven model, ignoring the characteristics of expert systems [
As shown in
In this strategy, adding constraints can effectively prevent the parameters from being overoptimized.
The interpretability of the SBRB-I model is as follows [
In the SBRB-I prediction model, the ER algorithm is used as the inference engine. As shown in
In the current research, BRB is optimized by many algorithms, including the projection covariance matrix adaptation evolution strategy (P-CMA-ES) [
To obtain higher accuracy, a self-growth learning strategy is proposed. At the same time, interpreted constraints are designed.
The value range of
To exploit the individual capabilities of BRB expert systems and data-driven models, expert knowledge is integrated into the initial population of algorithms, which increases the convergence speed of the algorithm [
Then, to make the initial population carry more expert knowledge information, whales move closer to whales with expert knowledge, forming a solution space centered on expert knowledge.
Specific operation: Turn off the random searchability of the WOA algorithm. Only spiral contraction and contraction surrounding mechanisms are retained. The formulas are
When
When
When
An example of predicting the air quality index (AQI) is given to demonstrate the effectiveness of the proposed model. The experimental data set is from China’s air quality online monitoring and analysis platform from January 2020 to October 2020 in Beijing. Serious air pollution endangers people’s physical and mental health and has become a common threat to the world. Predicting air quality in an interpretable way can help guide future environmental governance. Therefore, the establishment of an accurate and reliable air quality prediction system is of great significance. Thus, SBRB-I is a good choice for predicting air quality.
In the actual air quality prediction system, the AQI is based on air quality standards and the impact of various pollution factors on the ecological environment, which comprehensively reflects the pollution degree of
Second, these three groups are put into the sub-BRB model. Expert knowledge can be obtained through mechanism analysis of actual systems and long-term practice accumulation [
Attribute | P | M | G | E | |
---|---|---|---|---|---|
1 | 207 | 100 | 50 | 3 | |
1 | 222 | 100 | 50 | 8 | |
1 | 13 | 8 | 4 | 2 | |
1 | 77 | 50 | 30 | 5 | |
1 | 2.4 | 1 | 0.5 | 0 | |
1 | 283 | 200 | 100 | 12 |
Model | H | M | L | VL |
---|---|---|---|---|
Sub-BRB model 1 | 257 | 100 | 50 | 8 |
Sub-BRB model 2 | 97 | 60 | 40 | 7 |
Sub-BRB model 3 | 204 | 150 | 50 | 18.0 |
Third,
Attribute | H | M | L | |
---|---|---|---|---|
1 | 1 | 0.5 | 0 | |
1 | 1 | 0.5 | 0 | |
1 | 1 | 0.5 | 0 |
Reference points | S | B | M | G | E |
---|---|---|---|---|---|
Reference value | 1 | 0.6 | 0.4 | 0.2 | 0 |
The initial parameters of the WOA are as follows: population size
As shown in
The Euclidean distance shows the similarity between two vectors, as shown in
As shown in
In
Models | WOA-BRB | SBRB(1) | SBRB-I(42) |
---|---|---|---|
Minimum MSE | 0.0032 | 0.0030 | 0.0044 |
Maximum MSE | 0.0186 | 0.0050 | 0.0058 |
Average MSE | 0.0094 | 0.0038 | 0.0050 |
The standard deviation of MSE | 0.0059 | 5.51e−04 | 3.59e−04 |
The projection covariance matrix adaptation evolution strategy (P-CMA-ES), gray wolf optimization algorithm (GWO), differential evolution algorithm (DE), backpropagation neural network (BPNN), radial basis function (RBF), deep belief networks (DBN), long short-term memory (LSTM) and decision tree are used for experimental comparison. In
Model | MSE | Model | MSE | ||
---|---|---|---|---|---|
WOA-BRB | 0.0032 | DBN | 0.0043 | ||
P-CMA-ES-BRB | 0.0033 | LSTM | 0.0038 | ||
Part 1 | DE-BRB | 0.0060 | Part 2 | Decision tree | 0.0039 |
GWO-BRB | 0.0037 | BPNN | 0.0036 | ||
SBRB-I | 0.0046 | SBRB-I | 0.0046 |
Compared with the BPNN, RBF, DBN, LSTM prediction models, the interpretation of SBRB-I can be described as follows: 1. The SBRB-I model is a modeling method based on IF-THEN rules, and its output can be traced back. However, the prediction models of BPNN and RBF are essentially black-box models with few parameter meanings, and their input–output relationships are difficult to interpret. 2. SBRB-I has a clear and transparent reasoning calculation process, while the internal structure of prediction models such as BPNN is invisible. 3. The expert knowledge and system mechanism can be integrated into the SBRB-I model, so the SBRB-I model is much more easily understood by users.
The SBRB-I model is an interpretable model, and it can provide guidance on air quality governance through its analysis [
Interpretability and accuracy are important requirements to achieve reliable and accurate prediction systems. However, in the current study, three problems need to be solved for interpretability of BRBs: expert knowledge is not used effectively, how to improve model accuracy while maintaining interpretability and how to make expert systems and data-driven models cooperate effectively.
There are two innovations in this paper. For the first problem, one interpretability guideline is designed. Expert knowledge is used to form a local optimization space based on expert judgment. Moreover, expert knowledge is also integrated into the optimization process, which improves the convergence speed when optimizing and enhances the model’s interpretability. For the second and third problems, a new prediction system based on a self-growth BRB with interpretability constraints (SBRB-I) is proposed. The SBRB-I model uses any available knowledge to guide the optimization direction, including domain expert knowledge and knowledge optimized by correlation functions. The SBRB-I model realizes cooperation between BRB’s expert system and the data-driven model. Moreover, the optimization process guided by experts and the limitation of interpretability constraints makes the knowledge after model optimization highly similar to the expert knowledge. Therefore, the knowledge optimized by the data-driven model can be used as a supplementary source of expert systems. Finally, a case study of the prediction system of the air quality index is conducted to verify the effectiveness of the proposed model. SBRB-I can improve prediction accuracy while maintaining interpretability.
SBRB-I proposed in this paper is an exploration. More interpretability constraints have been added to this model, which enhances its interpretability. At the same time, after limiting the constraint space, it is necessary to improve the local search ability of the algorithm, which will obtain better prediction results. Finally, the number of iterations of each self-growth needs to be further studied.
We thank the anonymous reviewers for their valuable comments and suggestions which helped us to improve the content and presentation of this paper.
This work was supported in part by the
The authors declare that they have no conflicts of interest to report regarding the present study.