kernel PCA for process monitoring

Has any body got a working MATLAB code for kernel PCA for process monitoring which includes how to obtain the monitoring statistics(T2 an SPE), please?

Answer 1

[Aditya]

Hi Raphael,

Kernel PCA (KPCA) can be a powerful tool for process monitoring, especially when dealing with nonlinear data. To implement KPCA for process monitoring in MATLAB, you'll need to follow these steps, including calculating the monitoring statistics ( T^2 ) and SPE (Squared Prediction Error).

Here’s a basic outline of how you can implement KPCA and compute the monitoring statistics:

1. Prepare your data.
2. Define the Kernel Function
3. Perform KPCA on Training Data
4. Project Data and Calculate Monitoring Statistics
5. Monitor Test Data

Following is the example code for the same:

function K = rbf_kernel(X, Y, sigma)
    % X and Y are data matrices where each column is an observation
    % sigma is the bandwidth of the Gaussian kernel
    sqDist = pdist2(X', Y').^2;
    K = exp(-sqDist / (2 * sigma^2));
end
% Assume trainData is your training data matrix (size n x m)
sigma = 1.0; % Example value for the Gaussian kernel

% Compute the kernel matrix
K_train = rbf_kernel(trainData, trainData, sigma);

% Center the kernel matrix
N = size(K_train, 1);
oneN = ones(N, N) / N;
K_train_centered = K_train - oneN * K_train - K_train * oneN + oneN * K_train * oneN;

% Solve the eigenvalue problem
[eigenvectors, eigenvalues] = eig(K_train_centered);
eigenvalues = diag(eigenvalues);

% Sort eigenvalues and eigenvectors
[~, idx] = sort(eigenvalues, 'descend');
eigenvectors = eigenvectors(:, idx);
eigenvalues = eigenvalues(idx);
% Number of principal components to retain
numComponents = 5;

% Projection matrix
alpha = eigenvectors(:, 1:numComponents);
lambda = eigenvalues(1:numComponents);

% Project training data
projectedTrainData = K_train_centered * alpha;

% Calculate T^2 and SPE for training data
T2_train = sum((projectedTrainData ./ sqrt(lambda')).^2, 2);
reconstructedTrainData = projectedTrainData * alpha';
SPE_train = sum((K_train_centered - reconstructedTrainData).^2, 2);
% Compute the kernel matrix between test and training data
K_test_train = rbf_kernel(testData, trainData, sigma);

% Center the test kernel matrix
K_test_centered = K_test_train - oneN * K_train - K_test_train * oneN + oneN * K_train * oneN;

% Project test data
projectedTestData = K_test_centered * alpha;

% Calculate T^2 and SPE for test data
T2_test = sum((projectedTestData ./ sqrt(lambda')).^2, 2);
reconstructedTestData = projectedTestData * alpha';
SPE_test = sum((K_test_centered - reconstructedTestData).^2, 2);