Combinatorial Explosion Problem Research Articles

Interpretable large-scale belief rule base for complex industrial systems modeling with expert knowledge and limited data

Complex system modeling technology is a hot topic. Nowadays, many complex industrial systems present three characteristics: multiple input indicators, limited data and interpretability requirements. With good interpretability, belief rule base (BRB) serves as an efficient tool for modeling complex systems. However, as the number of input indicators of industrial systems increases, BRB suffers from the combinatorial explosion problem, which makes it hard to generate large-scale BRB and optimize it while maintaining its interpretability. For this purpose, an interpretable large-scale BRB is proposed for complex systems with limited data, where expert knowledge can be utilized effectively. First, a framework for generating an initial large-scale BRB using expert knowledge and limited data is developed, including the determination of attribute weight, basic belief degree and rule weight. Afterwards, a new parameter optimization model is designed to reduce the burden of parameter optimization and maintain the interpretability of BRB, where the Adaptive Moment Estimation (Adam) algorithm is adopted to further improve the efficiency of large-scale parameter optimization. Finally, a health assessment case of an inertial navigation system (INS) verifies the proposed method.

Semantic-Aware Dynamic Generation Networks for Few-Shot Human-Object Interaction Recognition.

Recognizing human-object interaction (HOI) aims at inferring various relationships between actions and objects. Although great progress in HOI has been made, the long-tail problem and combinatorial explosion problem are still practical challenges. To this end, we formulate HOI as a few-shot task to tackle both challenges and design a novel dynamic generation method to address this task. The proposed approach is called semantic-aware dynamic generation networks (SADG-Nets). Specifically, SADG-Net first assigns semantic-aware task representations for different batches of data, which further generates dynamic parameters. It obtains the features that highlight intercategory discriminability and intracategory commonality adaptively. In addition, we also design a dual semantic-aware encoder module (DSAE-Module), that is, verb-aware and noun-aware branches, to yield both action and object prototypes of HOI for each task space, which generalizes to novel combinations by transferring similarities among interactions. Extensive experimental results on two benchmark datasets, that is, humans interacting with common objects (HICO)-FS and trento universal HOI (TUHOI)-FS, illustrate that our SADG-Net achieves superior performance over state-of-the-art approaches, which proves its impressive effectiveness on few-shot HOI recognition.

Pairwise test case generation with harmony search, one-parameter-at-at-time, seeding, and constraint mechanism integration

Pairwise testing is a method for identifying defects through combinatorial analysis. It involves testing all possible combinations of input parameters in pairs within a system, ensuring that each pair is tested at least once. The field of test case generation is highly active in the realm of combinatorial interaction testing. Research in this area is particularly encouraged, as it falls under the category of non-deterministic polynomial-time hardness. A big challenge in this field is the combinatorial explosion problem. It is about finding the best test suite that covers all possible combinations of interaction strength. In this paper, we present the task of discovering a pairwise test set as a search problem and introduce an innovative testing tool referred to as pairwise test case generation in harmony search algorithm with seeding and constraint mechanism (PHOSC). Experimental results show that PHOSC performs better compared to some existing pairwise strategies in terms of test suite size. Additionally, PHOSC provides a comprehensive framework and serves as a research platform for the generation of pairwise test sets employing the harmony search algorithm. It adopts an approach that focuses on one parameter at a time (OPAT) and incorporates seeding and constraint mechanisms at the same time, thereby enhancing the efficiency and effectiveness of the testing process.

A deep belief rule base-based fault diagnosis method for complex systems

Complex systems are prone to faults due to their intricate structures, potentially impacting system stability. Therefore, fault diagnosis has become crucial for maintaining stable operation. In the field of complex systems, the combinatorial explosion problem in belief rule base (BRB) has attracted significant attention. The interdependence among system components leads to numerous variables and the need for rules, heightening model complexity. Regarding the combinatorial explosion problem, an improved belief rule network structure called deep BRB (DBRB) is proposed. First, the extreme gradient boosting (XGBoost) feature selection method is employed to choose the relatively important feature subset. Next, driven by the importance of features, different levels of features are input into the model, forming a complete and progressive network structure. Finally, the model undergoes the reasoning and optimization process. The effectiveness of the model is confirmed with a bearing fault dataset. After a comprehensive evaluation of multiple indicators, this method demonstrates a consistent improvement in classification performance as the depth increased. Moreover, compared to the traditional BRB model, this method notably reduces the number of parameters, improving its efficiency of processing complex data. In short, this method effectively tackles combinatorial explosion while ensuring model performance. The selection and assignment of feature subsets enhance the logic and readability of the model. Through the network structure, various fault features are captured well. This fault diagnosis method, rooted in the DBRB, offers a novel perspective on diagnosing complex system faults.

A belief rule-based classification system using fuzzy unordered rule induction algorithm

A rule-based system is a widely used artificial intelligence system that employs a set of rules to make decisions. The belief rule-based (BRB) classification system is an extension of fuzzy rule-based (FRB) system that handles uncertainty and imprecision in classification tasks by incorporating Dempster-Shafer evidence theory and fuzzy set theory. However, the BRB classification system suffers from the combinatorial explosion and high time complexity problems. To solve these issues, the fuzzy unordered rule induction algorithm (FURIA) is introduced into the BRB classification systems to design a novel BRB classification system in this paper. FURIA is computationally less complex and has superior classification performance. The proposed system outperforms BRB classification systems in terms of classification performance and significantly reduces the number of rules and conditions. Furthermore, the proposed system offers a more effective solution than FURIA. To evaluate the validity and superiority of the proposed system, we conduct two classification experiments, comparing it with five traditional classifiers and seven rule-based systems, respectively, in terms of classification performance and interpretability.

Combinatorial t-way test suite generation using an improved asexual reproduction optimization algorithm

To ensure the correctness and quality of a software system, it is desirable to test all possible combinations of the input parameters under various configurations. But the exhaustive testing of software systems with a large number of input parameters is practically impossible due to the combinatorial explosion problem. In order to address and mitigate this problem, combinatorial t-way strategy can be used to generate an array of test cases covering all combinations of only t input parameters. Since the minimum covering array generation (MCAG) is an NP-hard optimization problem, recently some strategies based on meta-heuristics have been used to solve this problem. Despite the usefulness of these strategies, they cannot completely solve the MCAG problem in systems with a large number of input parameters. This paper solves the mentioned MCAG problem by adapting the Asexual Reproduction Optimization (ARO) algorithm. Additionally, the ARO algorithm is improved (named ImpARO) by changing the population size, the mutation and crossover functions. The statistical analysis of the experimental results on several different configurations shows that ImpARO outperforms ARO, GALP, GS, BAPSO, DPSO, WOA, and GSTG as meta-heuristics, TConfig as a mathematical strategy, and PICT and IPOG as greedy strategies by 89%, 46%, 68%, 65%, 69%, 62%, 65%, 64%, 65%, and 59%, respectively. Moreover, various experiments show that ImpARO has faster convergence speed compared to ARO.

A fast belief rule base generation and reduction method for classification problems

The belief rule-based (BRB) system has been one of the most significant rule-based systems for its ability to deal with various kinds of information under uncertainty, and it has shown great potential for classification problems. However, the combinatorial explosion problem hugely limits the application of the BRB system as excessive rules could not only increase the computation cost, but also impact the performance of the BRB system. Therefore, motivated by this problem, this paper proposed a fast and accurate belief rule base generation and reduction method. Firstly, a fast belief rule base generation method is introduced, where similar rules are grouped and combined to ensure all the possible situations are covered without generating excessive rules. Then, a redundancy-based belief rule base reduction method is proposed, where the redundancy degree of the belief rule that represents the degree to which a belief rule is affected by other rules is introduced, and it is calculated to identify redundant rules. Furthermore, the conventional evidential reasoning (ER)-based inference process is retained from conventional BRB systems. Thirty classification benchmarks from the well-known UCI machine learning repository are tested to validate the effectiveness of the proposed method, and the results are compared with other rule-based systems, improved BRB systems, and other machine learning methods. Comparison results show that the proposed method could effectively reduce the size of the BRB without costing its accuracy. Furthermore, sensitivity analysis and robustness analysis are conducted, which further show the effectiveness and robustness of the proposed method.

Data association for maneuvering targets through a combined siamese network and XGBoost model

Data association plays an important role in forming target tracks when false alarms exist. Its accuracy is key to reducing the computational burden of the combinatorial explosion problem inherent to target tracking in environments where false alarms are densely distributed. Traditional methods are usually developed according to some assumed motion models and suffer performance degradation when there is mismatch between assumed models and the actual motion, which may often be the case for maneuvering targets. To cope with this problem, we propose a data-driven method that learns the association criteria directly from data. By use of the domain knowledge and a convolutional Siamese network, features of different attributes present in the observations are extracted. Then the features are fused through the use of XGBoost. Simulation results show that the proposed method performs better than the traditional model-based methods in correlating observations of maneuvering targets and avoiding correlations of false alarms. By visualizing the decision process of the feature fusion model, it is found that the proposed method learns from the data to make comprehensive use of multiple features and does not simply set hard thresholds for the features. The computational complexity is also analyzed both theoretically and experimentally.

A deep reinforcement learning method for structural dominant failure modes searching based on self-play strategy

In the research area of structural reliability analysis (SRA), the dominant failure modes (DFMs) of a structural system make significant contributions to life-span failure prediction and safety assessment. However, the high computational cost caused by the combinatorial explosion is the main problem in DFMs searching that hinders its application and further development. Recently, many successful applications have proved that the self-play deep reinforcement learning (DRL) has a strong ability to obtain action policy in the face of combinatorial explosion problems. Inspired by this, a self-play strategy is designed to optimize the DRL-based DFMs searching process and reduce the computational effort. A scoring function is designed and used as the refereeing standard of the self-play games and helps improve the efficiency of Monte Carlo tree search (MCTS) in an asynchronous training process. In comparison with the β-unzipping method and exploration-based DFMs searching method, the proposed method significantly improved training efficiency with an accuracy of over 95% and a lower requirement of the number of finite element analysis (FEA), both of which contribute to the policy learning of failure component selection. In summary, the method shows potential applications for actual structures and makes valuable contributions to the problem with high computing costs.

Data-Based Estimation of the Dynamic Reliability and Performance Indicator of an Industrial Manufacturing System

The aim is to develop a more simple and effective method's performance and dynamic reliability assessment for complex industrial systems. By using the operating data of the industrial system characterized by a strong desynchronization and applying to it prediction algorithms of artificial intelligence applied to the time series, the model will have learned from the behavior of the complex manufacturing system allowing the operator or decision-maker to better orientate the maintenance, production, and quality policies. Furthermore, we propose this approach to avoid tedious mathematical methods related to dynamic reliability calculations and performance evaluation to make forecasts of the company's operation over a long period by identifying future bottlenecks in the system's behavior. The low-performance indicators and irrelevant reliability presented by many third-generation industries are due to the lack of efficient and simple tools for reliability assessment taking into account the dynamic aspect of the different elements of the production chain, maintenance department, production department, and quality department. We propose to develop a model that will abstract from conventional, complex, and inefficient mathematical methods for systems subject to combinatorial explosion problems in the manufacturing industry.

A new method for disease diagnosis based on hierarchical BRB with power set

Disease diagnosis occupies an important position in the medical field. The diagnosis of the disease is the basis for choosing the right treatment plan. Doctors must first diagnose what the patient has based on the clinical characteristics of various diseases, and then they can administer the right medicine. When building models for disease diagnosis, models are required to be able to handle various uncertainty information. The belief rule base (BRB) can effectively handle various information under uncertainty by introducing belief distributions. However, in current research, BRB-based disease diagnosis models still have problems of combinatorial rule explosion and inability to deal with local ignorance effectively. Therefore, a hierarchical BRB with power set (H-BRBp)-based disease diagnosis model is proposed in this paper. First, the physiological indexes and data of the patients were analyzed, and the data were preprocessed using the principal component regression (PCR) algorithm. Second, the H-BRBp disease diagnosis model was constructed to solve the deficiencies in the above BRB disease diagnosis model. Finally, the validity and advantages of the model were verified by experiments on lumbar spine disease diagnosis and a large number of comparison experiments.

Belief rule-base expert system with multilayer tree structure for complex problems modeling

Belief rule-base (BRB) expert system is one of recognized and fast-growing approaches in the areas of complex problems modeling. However, the conventional BRB has to suffer from the combinatorial explosion problem since the number of rules in BRB expands exponentially with the number of attributes in complex problems, although many alternative techniques have been looked at with the purpose of downsizing BRB. Motivated by this challenge, in this paper, multilayer tree structure (MTS) is introduced for the first time to define hierarchical BRB, also known as MTS-BRB. MTS-BRB is able to overcome the combinatorial explosion problem of the conventional BRB. Thereafter, the additional modeling, inferencing, and learning procedures are proposed to create a self-organized MTS-BRB expert system. To demonstrate the development process and benefits of the MTS-BRB expert system, case studies including benchmark classification datasets and research and development (R&D) project risk assessment have been done. The comparative results showed that, in terms of modelling effectiveness and/or prediction accuracy, MTS-BRB expert system surpasses various existing, as well as traditional fuzzy system-related and machine learning-related methodologies.

A novel optimization method for belief rule base expert system with activation rate

Although the belief rule base (BRB) expert system has many advantages, such as the effective use of semi-quantitative information, objective description of uncertainty, and efficient nonlinear modeling capability, it is always limited by the problem of combinatorial explosion. The main reason is that the optimization of a BRB with many rules will consume many computing resources, which makes it unable to meet the real-time requirements in some complex systems. Another reason is that the optimization process will destroy the interpretability of those parameters that belong to the inadequately activated rules given by experts. To solve these problems, a novel optimization method for BRB is proposed in this paper. Through the activation rate, the rules that have never been activated or inadequately activated are pruned during the optimization process. Furthermore, even if there is a complete data set and all rules are activated, the activation rate can also be used in the parallel optimization process of the BRB expert system, where the training data set is divided into some subprocesses. The proposed method effectively solves the combinatorial explosion problem of BRB and can make full use of quantitative data without destroying the original interpretability provided by experts. Case studies prove the advantages and effectiveness of the proposed method, which greatly expands the application fields of the BRB expert system.

A New Model for Network Security Situation Assessment of the Industrial Internet

To address the problem of network security situation assessment in the Industrial Internet, this paper adopts the evidential reasoning (ER)algorithm and belief rule base (BRB) method to establish an assessment model. First, this paper analyzes the influencing factors of the Industrial Internet and selects evaluation indicators that contain not only quantitative data but also qualitative knowledge. Second, the evaluation indicators are fused with expert knowledge and the ER algorithm. According to the fusion results, a network security situation assessment model of the Industrial Internet based on the ER and BRB method is established, and the projection covariance matrix adaptive evolution strategy (P-CMA-ES) is used to optimize the model parameters. This method can not only utilize semiquantitative information effectively but also use more uncertain information and prevent the problem of combinatorial explosion. Moreover, it solves the problem of the uncertainty of expert knowledge and overcomes the problem of low modeling accuracy caused by insufficient data. Finally, a network security situation assessment case of the Industrial Internet is analyzed to verify the effectiveness and superiority of the method. The research results show that this method has strong applicability to the network security situation assessment of complex Industrial Internet systems. It can accurately reflect the actual network security situation of Industrial Internet systems and provide safe and reliable suggestions for network administrators to take timely countermeasures, thereby improving the risk monitoring and emergency response capabilities of the Industrial Internet.

Extended Belief Rule-Based System With Accurate Rule Weights and Efficient Rule Activation for Diagnosis of Thyroid Nodules

Extended belief-rule-based (EBRB) system is a representative rule-based system and has attracted much attention due to its capability of solving the problems of combinatorial explosion and time-consuming optimization incurred by belief-rule-based system. Despite their advantages, the development of EBRB suffers from some shortcomings, such as the unreasonable calculation of similarity between input and antecedent belief distributions (BDs), inaccurate calculation of individual matching degrees and rule weights, and inefficient determination of activation rules. To address these shortcomings, this article proposes a new EBRB system, in which an existing similarity measure between two BDs is adopted to accurately calculate the individual matching degrees and rule weights. Furthermore, the activation weight of a rule is efficiently calculated within an activation group generated using the affinity propagation algorithm, which is utilized to determine whether the rule is activated. The activation rules are then integrated using their weights and the evidential reasoning algorithm to generate the inference result. To demonstrate its accuracy and efficiency, the proposed EBRB system is employed to help diagnose thyroid nodules based on the historical examination reports collected from a tertiary hospital located in Hefei, Anhui, China. The findings are highlighted by the comparison of the proposed EBRB system with existing EBRB systems based on the historical reports and datasets derived from the University of California at Irvine.

A novel motor fault diagnosis method based on principal component analysis (PCA) with a discrete belief rule base (DBRB) system

Motor vibration signal data sets are characteristically random and nonlinear, and its features are difficult to extract for fault identification. To reduce the uncertainty of fault diagnosis, a method based on principal component analysis (PCA) and discrete belief rule base (DBRB) was developed for the first time. Initially, the vibration signal was first denoised using a wavelet threshold algorithm to eliminate interference. Second, overlapping signals were segmented into 15 time windows and a total of 13 typical time domain features and mathematical statistical features were extracted. Third, the dimensions of the features were reduced to three principal components by PCA and were taken as the antecedent attributes of the DBRB. However, the amount of information in each principal component is different, so the variance contribution rate was taken as an antecedent attribute weight to restore the original data characteristics. Fourth, a PCA-DBRB model was established, which effectively avoided the combinatorial explosion problem of rule base in the DBRB model. In addition, to obtain appropriate reference values, the k-means algorithm was introduced to take the cluster centers as reference values. The method was then validated by collecting typical fault data from motor bench experiments. The results demonstrated that compared with other traditional classifiers, this approach is more effective and superior in classification performance and more accurate in diagnosing faults from motor vibration data.

Perspectives of Machine Learning Development on Kerogen Molecular Model Reconstruction and Shale Oil/Gas Exploitation

The shale revolution has provided abundant shale oil/gas resources for the world, but the efficient, sustainable, and environmentally friendly exploitation of shale oil/gas is still challenging. Kerogen is the primary hydrocarbon source of shale oil/gas. The research on the kerogen chemo-mechanical properties significantly influences the development of shale oil/gas extraction technology. Rapid reconstruction of the kerogen molecular models is the most effective way to study the generation mechanism of shale oil/gas from the bottom-up molecular level. However, due to the combinatorial explosion problem, the reconstruction complexity of kerogen increases sharply because of the kerogen’s characteristics of complex origin, large molecular weight, and diverse functional groups. The traditional kerogen molecular reconstruction methods require professionals to comprehensively analyze various experimental information to approximate the actual kerogen molecular models through trial-and-error. So, the traditional methods are time and material-consuming and extremely inefficient. These shortcomings make researchers spend too much strength on the reconstruction of kerogen molecular models and cannot focus on the study of kerogen chemo-mechanical properties. For the past few years, state-of-the-art machine learning (ML) methods have been applied to intelligently reconstruct the kerogen molecular models through high-throughput and predict shale oil/gas production mechanisms. Although the current work is still in the infancy stage, ML methods are believed to be the most promising way to solve the drawbacks of traditional methods and reconstruct kerogen in reliable and large molecular weight. Hence, mechano-energetics is proposed to study the efficient development and utilization of energy based on mechanics and ML. This paper briefly reviews the development history of kerogen molecular model reconstruction methods and the research of ML in the fields of kerogen reconstruction and shale oil/gas exploitation. Some recommendations for further ML-based work are also suggested. We are convinced that the ML methods will accelerate the research of kerogen and promote the significant development of unconventional oil/gas exploitation technologies.

SCIPOG: Seeding and constraint support in IPOG strategy for combinatorial t-way testing to generate optimum test cases

Combinatorial t-way technique is efficient in generating test data and addressing the problem of combinatorial explosion. When constructing a test case, numerous literatures classified t-way strategies into two basic approaches as either One-test-at-a-time approach (OTAT) and One-parameter-at-a-time approach (OPAT). At least three major challenge groups can be encountered when creating test cases. The first one is provision of parameters seeding support that will improve the software quality. The second involves automatically obtaining data regarding parameter constraints and identifying interactions between systemcomponents. The last one is the execution speed and the test suite size. However, in all the existing OPAT t-way strategies, given that the system is loaded with this information, testing present-day software systems is made difficult or impossible. This study presents an effective combinatorial t-way test case generation strategy named Seeding and Constraint support in In-Parameter-Order Generalized (SCIPOG), to develop an improved paired testing approach. However, the study examines the present state-of-the-art and compares several OPAT strategies found in the literature. Moreover, experiments are discussed as part of this process to demonstrate the correctness of the implementation. When statistically analyzing the findings, two non-parametric tests—the Wilcoxon Rank and Friedman tests–were run. SCIPOG, however, produced competitive results. Finally, SCIPOG showed the efficiency of the two proposed methods, which are seeding and constraint support in IPOG strategy.

An efficient point-set registration algorithm with dual terms based on total least squares

Point set registration (PSR) is competitive with related techniques because it purposefully captures the overall structure between two point-set patterns. Typically, the point set registration problem can be divided into two sub-problems: (1) search point set correspondence (PSC); (2) estimate spatial transformation matrix (STM). Searching for the best PSC is a classical combinatorial explosion problem, and estimating the STM is a continuous space optimization problem. Also, two feature point sets detected by two low-quality images include point-position errors and often involve bilateral outliers composed of such feature points that cannot form a correspondence relationship. To address the above problems, we propose an efficient PSR algorithm with dual (symmetrical) terms based on the total least squares (DT-TLS), which can correct errors-in-variables and suppress multiple outliers. Meanwhile, the framework of soft decision-making is presented, and a TLS-based criterion is constructed to efficiently exploit the probability and global structures of two-point sets. Such TLS-based criterion with single row-orthonormal STM includes two interesting dual (symmetrical) terms that can be conveniently exploited to suppress bilateral outliers. The experimental results show that DT-TLS achieves better performance than the state-of-the-art algorithms in some multi-view computer vision tasks, indicating that our proposed algorithm is suitable for solving PSR problems.

BRN: A belief rule network model for the health evaluation of complex systems

A belief rule base (BRB) expert system provides a generic inference framework for approximating the complicated nonlinear relationships between inputs and outputs. Such systems have been widely applied in the system health management community. However, limited by the method of construction rules, existing methods encounter the combinatorial explosion problem in the modeling of complex systems. Thus, determining how to effectively use the collected signals, system mechanism, and expert knowledge to realize the health evaluation of complex systems has become a key issue limiting the development of BRB systems. To solve this problem, a modified micro belief rule structure is proposed, and a new belief rule network (BRN) model is developed. Then, the cautious conjunctive rule is introduced to realize the fusion and reasoning of the nonindependent nodes in the BRN. In addition, the structure and parameter learning strategies of the proposed BRN are given, thereby providing a systemic mechanism to enhance the capability of the BRN when both expert knowledge and observed data are available. The effectiveness of the BRN model is verified in an aerospace relay experiment.