Parameter Learning Of Bayesian Networks From Small Data Sets And Its Applications In Paper-Making Machine Bearing Fault Diagnosis

The paper-making industry is one of the important industries supporting our national economy,and the bearing is often called "the joint of industry"because of its widely using in paper machine.With the increase of the speed and width of the paper machine,because of the large number of dryers and the large one-time investment,the bearing fault in the paper machine will bring the serious risk of loss to the paper enterprise.The monitoring of the running state of the paper machine bearing is the key point of the paper machine fault diagnosis.Due to the high cost of resources or time,it is difficult to obtain sufficient fault data from some practical complicated and uncertain bearing systems.It is a challenge for current data-based fault diagnosis methods to carry out effective fault diagnosis under the condition of small sample data,that is,small data sets.Bayesian network(BN)is a probabilistic graph model based on graph theory and probability theory.It is a powerful tool to deal with uncertainty in artificial intelligence field,and is widely used in disease diagnosis,fault monitoring and other practical problems.In order to improve the accuracy of network reasoning,it is very important to learn the Bayesian network parameters as accurately as possible by artificial intelligence.However,the lack of data can seriously affect the learning accuracy of the Bayesian network,leading to unsatisfactory results in network reasoning.In this paper,the problem of Bayesian network parameter learning is studied under the small data set.Based on the sample data and the expert qualitative constraints,respectively,the issue of BN parameter learning in small data sets is addressed.In addition,the problem of learning the target BN parameters based on the transfer learning mechanism is studied in this paper.This paper designs the corresponding bearing fault diagnosis algorithms based on BN parameter learning,and verifies the validity of the proposed algorithms.The main contributions of this paper about paper machine bearing fault diagnosis are summarized as follows:(1)Aiming at the problem of parameter learning in small data sets,a new Bayesian network parameter learning algorithm,Parameter Estimation under Constraints(PEUC)based on qualitative constraints and parameter extension is proposed.By integrating the sample data and the qualitative constraints of the experts,the parameter knowledge is transformed into the inequality constraints,and the BN parameter candidate set satisfying the constraint knowledge is obtained by using Bootstrap technique.Then the BN parameter is estimated by fusing the parameters from the sample data set and the candidate parameters in the constraint space.The variable weight balance coefficient is designed to balance the knowledge of qualitative constraints and the BN parameter information contained in the available sample set.When the sample data set is very limited,the parameter estimation tends to the candidate parameters in the constraint space.Once the sample data set is sufficient,the parameter estimation tends to the BN parameter information contained in the sample data set.Based on the standard BN model repositories,this paper discusses the influence of different sample size on BN parameter learning and the influence of different constraint on parameter search space.The experimental results show that the PEUC method can solve the problem of small or scarce modeling sample in BN parameter learning,which lays a foundation for the further application of PEUC in bearing fault diagnosis.(2)In order to diagnose early bearing faults in small data sets,a bearing fault diagnosis algorithm based on Bayesian network and PEUC algorithm is proposed.Firstly,the vibration signals of the bearing are processed by the Wavelet packet decomposition.Secondly,the feature vectors of the bearing fault are obtained by using the feature extraction function.The structure modeling of BN model is completed by combining the feature vector of bearing fault and the node to be diagnosed.Then,the parameters of fault diagnosis BN are modeled by PEUC method,and the BN model for fault diagnosis reasoning is obtained.Finally,using BN reasoning algorithm to complete the fault diagnosis of bearings.In the experimental analysis,using the early rolling bearing fault data set provided by Case Western Reserve University’s bearing data center,we verify that under the condition of small data set,in view of the normal rolling bearing,inner ring fault,rolling fault and outer ring fault,the validity of the bearing fault diagnosis algorithm based on PEUC is presented in this paper.In addition,the fault data set with sufficient data is also investigated,and the PEUC bearing fault diagnosis method proposed in this paper is used to carry out the diagnosis experiment,and the correctness of the bearing fault diagnosis approach is verified.The fault diagnosis experimental results show that the PEUC based bearing fault diagnosis algorithm can be used in the condition of sufficient data set or small data set.The presented method can solve the problem of effective diagnosis for early bearing fault.(3)A model of aggregation of target BN parameters based on transfer learning mechanism is designed.Then,a Variable Coefficient Transfer Learning(VCTL)algorithm based on aggregation and Transfer Learning is proposed.The sample size factor and the balance weight function are used to determine whether the auxiliary BN parameter learning task is activated or not.This algorithm takes into account not only the knowledge of resource domain,but also the contribution of resource correlation.Based on the parameter information of resource domain,a correlation weight factor is proposed to measure the correlation between the parameters of resource network and Target Network.Finally,by using the aggregation model of target BN parameters based on transfer learning mechanism,the initial target BN parameters and the parameter knowledge from the resource domain are aggregated to get the final target BN parameters.The experimental results show that the learning accuracy of the variable coefficient transfer learning algorithm based on aggregation and transfer learning is close to that of the classical MLE if the data set is sufficient.When the data set is small,the learning accuracy of VCTL algorithm is better than MLE algorithm,MAP algorithm or the most advanced parameter transfer method-local linear pool transfer learning(LoLP)algorithm.(4)Based on the target BN parameter aggregation model,a bearing fault diagnosis algorithm based on VCTL BN parameter estimation algorithm is advanced and verified via transfer learning mechanism.The experimental results show that VCTL can effectively accomplish the fault diagnosis task of bearing when the target domain data set is small.Using the knowledge of resource domain and the real fault diagnosis data,a transfer learning method based on multi-source domain knowledge is proposed for the fault diagnosis of paper machine bearings.To sum up,this paper is based on the background of the paper machine bearing fault diagnosis under the condition of small data set,using the method of Bayesian network theory research and experimental verification.This paper studies the effects of Bayesian network,sample data and expert qualitative constraints on Bayesian network parameter learning and transfer learning.On this basis,the bearing fault diagnosis algorithms based on Bayesian network and PEUC algorithm,and the bearing fault diagnosis algorithm based on VCTL Bayesian network parameter learning algorithm are designed respectively.The former can use the variable weight balance coefficient to balance the knowledge of qualitative constraints and the BN parameter information contained in the available sample set,and the latter,we can use the prior knowledge of source domain to dissolve the lack of BN learning parameter information in target domain,to improve the learning precision of BN model,thus improve the accuracy of bearing defect diagnosis.All these study results provide theoretical and technical reference for intelligent fault diagnosis in paper machine bearings,especially when the fault diagnosis modeling data set is small or limited.