核主元分析KPCA的降维特征提取以及故障检测应用-KPCA_v2.zip

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核主元分析KPCA的降维特征提取以及故障检测应用-KPCA_v2.zip
本帖最后由 iqiukp 于 2018-11-9 15:02 编辑    核主元分析（Kernel principal component analysis ，KPCA）在降维、特征提取以及故障检测中的应用。主要功能有：（1）训练数据和测试数据的非线性主元提取（降维、特征提取）
（2）SPE和T2统计量及其控制限的计算
（3）故障检测

参考文献：
Lee J M, Yoo C K, Choi S W, et al. Nonlinear process monitoring using kernel principal component analysis[J]. Chemical engineering science, 2004, 59: 223-234.

1. KPCA的建模过程（故障检测）：
（1）获取训练数据（工业过程数据需要进行标准化处理）
（2）计算核矩阵
（3）核矩阵中心化
（4）特征值分解
（5）特征向量的标准化处理
（6）主元个数的选取
（7）计算非线性主成分（即降维结果或者特征提取结果）
（8）SPE和T2统计量的控制限计算
function model = kpca_train
% DESCRIPTION
% Kernel principal component analysis
%
%    mappedX = kpca_train
%
% INPUT
% X          Training samples
%             N: number of samples
%             d: number of features
% options    Parameters setting
%
% OUTPUT
% model       KPCA model
%
%
% Created on 9th November, 2018, by Kepeng Qiu.

% number of training samples
L = size;

% Compute the kernel matrix
K = computeKM;

% Centralize the kernel matrix
unit = ones/L;
K_c = K-unit*K-K*unit unit*K*unit;

% Solve the eigenvalue problem
[V,D] = eigs;
lambda = diag;

% Normalize the eigenvalue
V_s = V ./ sqrt';

% Compute the numbers of principal component

% Extract the nonlinear component
if options.type == 1 % fault detection
dims = find) >= 0.85,1, 'first');
else
dims = options.dims;
end
mappedX  = K_c* V_s ;

% Store the results
model.mappedX =  mappedX ;
model.V_s = V_s;
model.lambda = lambda;
model.K_c = K_c;
model.L = L;
model.dims = dims;
model.X = X;
model.K = K;
model.unit = unit;
model.sigma = options.sigma;

% Compute the threshold
model.beta = options.beta;% corresponding probabilities
[SPE_limit,T2_limit] = comtupeLimit;
model.SPE_limit = SPE_limit;
model.T2_limit = T2_limit;

end复制代码2. KPCA的测试过程：
（1）获取测试数据（工业过程数据需要利用训练数据的均值和标准差进行标准化处理）
（2）计算核矩阵
（3）核矩阵中心化
（4）计算非线性主成分（即降维结果或者特征提取结果）
（5）SPE和T2统计量的计算
function [SPE,T2,mappedY] = kpca_test
% DESCRIPTION
% Compute the T2 statistic, SPE statistic,

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