Faq how are T contributions calculated

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Issue:

How are T-contributions calculated?

Possible Solutions:

In PCA, T-contributions represent how the original variables contribute to give each sample its T2 value in a given model. They are calculated as if you are reconstructing the data (relative to the mean of the calibration data) except that each factor is first normalized by the variance it captured in the original data. This gives the reconstruction of the data as if all principal components captured equal amounts of variance in the original data. In other words: this is how the original variables project into the normalized multivariate space of the model.

To calculate the T-contributions for a given sample in a PLS_Toolbox PCA model, use the tconcalc function. Given the sample's data in variable data and the model in variable model, the following will calculate T-contributions.

  T_con = tconcalc(data,model);

Note that if data is a matrix of all your data and you want only a single sample's T contributions, pass only that sample's row:

data(row_number,:)

Numerical Calculation Details

To calculate the T-contributions for the "i"th sample (T_con_i) in Matlab notation:

T_con_i = t_i*L*U 

where t_i is a row vector of the scores (size: 1 x k) for a given sample, U is the transposed matrix of loadings (size: n x k), and L is a diagonal matrix containing the inverse of the square root of the eigenvalues for the k components. For example, with a three PC model, L would be:

  1/λ11/2  0        0
  0       1/λ21/2   0
  0       0        1/λ31/2


Where λj is the eigenvalue for component j. Note the similarity of the T-contributions equation to data reconstruction:

T_con_i = t_i*L*U      %calculate T-contributions
x_hat_i = t_i*  U      %calculate data approx.

x_hat_i is data approximation for sample i (relative to the mean of the calibration data).

Putting all this together, you can calculate T_con_i by hand for a given sample in the calibration data used for a PLS_Toolbox PCA model using:

T     = model.loads{1};  %Grab scores (note: for all samples!)
t_i   = T(i,:);          %grab scores for one sample
U     = model.loads{2}';  %Grab loadings
ncomp = size(U,1);
L     = diag(1./sqrt(model.detail.ssq(1:ncomp,2)));
T_con_i = t_i*L*U;       %calculate T-contributions


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