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Multivariate data analysis (view source)
Revision as of 14:35, 9 January 2014
, 14:35, 9 January 2014→Filtering
The relationship (matrix) defining the principal components ''y''<sub>''j''</sub> can be inverted, yielding the following inverse linear relationship:
The relationship (matrix) defining the principal components ''y''<sub>''j''</sub> can be inverted, yielding the following inverse linear relationship:
:<math>\mathbf{Equation}</math>
:<math>X_i = b_{0i} + \sum_{j=1}^P b_{ij}Y_j</math> for all ''i'' = 1,...,''P''
This inverse relationship can be used in estimating (interpolating) a variable ''x''<sub>''i''</sub> from prior estimates of the principal components ''y''<sub>''j''</sub>. The first principal components ''y''<sub>''j''</sub>, j ≤ ''P''<sub>0</sub>, can be estimated by some type of regression procedure (such as kriging), while the higher components ''y''<sub>''j''</sub>, j > ''P''<sub>0</sub>, corresponding to random noise, can be estimated by their respective means.
This inverse relationship can be used in estimating (interpolating) a variable ''x''<sub>''i''</sub> from prior estimates of the principal components ''y''<sub>''j''</sub>. The first principal components ''y''<sub>''j''</sub>, j ≤ ''P''<sub>0</sub>, can be estimated by some type of regression procedure (such as kriging), while the higher components ''y''<sub>''j''</sub>, j > ''P''<sub>0</sub>, corresponding to random noise, can be estimated by their respective means.