[fault diagnosis] bearing fault diagnosis based on Bayesian optimization support vector machine with matlab code

1 Content introduction

Bayesian network (Bayesian Network or BN) It is an effective tool for modeling and uncertainty reasoning in the field of artificial intelligence . The basic task of Bayesian network reasoning is ： Given a set of evidence variable observations , The posterior probability distribution of a set of query variables is calculated by searching the conditional probability table . In practical applications , The observed evidence value can be any value , That is, the evidence value may not be included in the conditional probability table . therefore , It is necessary to propose a method that can calculate the posterior probability distribution when any evidence value is given . In response to this question , This paper mainly discusses the construction and reasoning of Bayesian network with learning function ： When constructing Bayesian network structure from samples , Also learn the maximum likelihood parameter from the sample ( Maximum likelihood hypothesis ), To replace the corresponding conditional probability table , That is, the maximum likelihood parameter is regarded as a part of Bayesian Network .　　 After constructing Bayesian network structure from samples using traditional methods , This paper mainly focuses on how to learn maximum likelihood parameters from samples . Bayesian network contains two reasoning methods ： Forward reasoning and reverse reasoning . For forward reasoning , We propose a method based on support vector machine and Sigmoid Function to learn maximum likelihood parameters . For reverse reasoning , First, based on Bayesian formula , Turn the reverse reasoning problem into the forward reasoning problem ; Then the maximum likelihood parameters are linearly interpolated .　　 However , For the constructed Bayesian Network , They probably don't have the original samples . In this case , This paper presents a method of mapping the existing conditional probability table into samples , Then learn the maximum likelihood hypothesis from the obtained samples .　　 further , In order to apply Bayesian network with maximum likelihood assumption to approximate reasoning , This paper gives the corresponding Gibbs Sampling algorithm .　　 Last , We give an application example , The accuracy and verification of the test learning maximum likelihood hypothesis algorithm are given Gibbs Experiments on the convergence of sampling algorithm . Preliminary experimental results show that our method is feasible .　

　 The main contributions of this paper are as follows ：　　 ● This paper presents a method of learning Bayesian networks with learning function , That is, when using existing methods to construct Bayesian network structure from samples , Based on support vector machine and Sigmoid function , Also learn the maximum likelihood hypothesis from the sample , To replace the corresponding conditional probability table . Then based on Bayesian network with maximum likelihood assumption , This paper further proposes the corresponding forward and reverse reasoning methods . This solves the problem of reasoning given any evidence value .　　 ● This paper presents a method of mapping the existing conditional probability table into samples , The maximum likelihood hypothesis can also be learned from the existing conditional probability table , It solves the problem of Bayesian network that has been constructed ( There may be no original sample ), The problem that reasoning can be carried out even given any evidence value .　　 ● further , In order to apply Bayesian network with maximum likelihood assumption to approximate reasoning , This paper gives the corresponding Gibbs Sampling algorithm . To a certain extent, it solves the inefficient problem of Bayesian network accurate reasoning .​

2 Simulation code

clcclear allclose alladdpath(genpath(pwd))​%  Generate 3 Class sample （ Two dimensional Gaussian distribution ）?sigma = [0.6 0; 0 0.6];numData = 100;​mu = [6 5];X_1 = mvnrnd(mu, sigma, numData);label_1 = ones(numData, 1);​mu = [3 9];X_2 = mvnrnd(mu, sigma, numData);label_2 = 2*ones(numData, 1);​mu = [-2 7];X_3 = mvnrnd(mu, sigma, numData);label_3 = 3*ones(numData, 1);​data = [X_1; X_2; X_3];label = [label_1; label_2; label_3];​%%  Bayesian optimization parameters ​%  Upper and lower limits of variables and type settings c =  optimizableVariable('c',  [1e-2 1e2], 'Type', 'real');g =  optimizableVariable('g',  [2^-7 2^7], 'Type', 'real');parameter = [c, g];​%  Cross validation parameter settings （ Set to... When cross validation is turned off []）?kfolds = 5;% kfolds = [];​%   Objective function objFun = @(parameter) getObjValue(parameter, data, label, kfolds);​%  Bayesian optimization iter = 30;points = 10;results = bayesopt(objFun, parameter, 'Verbose', 1, ...                   'MaxObjectiveEvaluations', iter,...                   'NumSeedPoints', points);%  Optimization results [bestParam, ~, ~] = bestPoint(results, 'Criterion', 'min-observed');​%%  Retrain with optimal parameters SVM Model c = bestParam.c;  g = bestParam.g; ​%  Training and testing cmd = ['-s 0 -t 2 ', '-c ', num2str(c), ' -g ', num2str(g), ' -q'];model = libsvmtrain(label, data, cmd);[~, acc, ~] = libsvmpredict(label, data, model); ​%% SVM Boundary visualization ?d = 0.02;[X1, X2] = meshgrid(min(data(:, 1)):d:max(data(:, 1)), min(data(:, 2)):d:max(data(:, 2)));X_grid = [X1(:), X2(:)];grid_label = ones(size(X_grid, 1), 1);[pre_label, ~, ~] = libsvmpredict(grid_label, X_grid, model);​%  Keep the number ｇ Point diagram ?figurecolor_p = [150, 138, 191;12, 112, 104; 220, 94, 75]/255; %  The data goes back  Plutonium ?color_b = [218, 216, 232; 179, 226, 219; 244, 195, 171]/255; %  In the branch area, you can see the smoke in advance hold onax(1:3) = gscatter(X_grid (:,1), X_grid (:,2), pre_label, color_b);​%  Continue to control  Data ?ax(4:6) = gscatter(data(:,1), data(:,2), label);set(ax(4), 'Marker','o', 'MarkerSize', 7, 'MarkerEdgeColor','k', 'MarkerFaceColor', color_p(1,:));set(ax(5), 'Marker','o', 'MarkerSize', 7, 'MarkerEdgeColor','k', 'MarkerFaceColor', color_p(2,:));set(ax(6), 'Marker','o', 'MarkerSize', 7, 'MarkerEdgeColor','k', 'MarkerFaceColor', color_p(3,:));set(gca, 'linewidth', 1.1)title('Decision boundary (gaussian kernel function)')axis tightlegend('off')box onset(gca, 'linewidth', 1.1)

3 Running results

4 reference

[1] Wang Junyu . Research on fault diagnosis of rolling bearing based on wavelet packet and optimized support vector machine . 

[2] SuXiaoJie . Research on Bearing Fault Diagnosis Based on support vector machine under large-scale data .

[3] Yang Zhengyou , Peng Tao , Li Jianbao , etc. . Based on Bayesian inference LSSVM Rolling bearing fault diagnosis [J]. Journal of electronic measurement and instrumentation , 2010, 24(5):5. 

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