|
The Principal Component Regression method combines the Principal Component Analysis (PCA)spectral decomposition with an Inverse Least Squares (ILS) regression method to create a quantitative model for complex samples. Unlike quantitation methods based directly on Beer's Law which attempt to calculate the absorbtivity coefficients for the constituents of interest from a direct regression of the constituent concentrations onto the spectroscopic responses, the PCR method regresses the concentrations on the PCA scores.
The eigenvectors of a PCA decomposition represent the spectral variations that are common to all of the spectroscopic calibration data. Therefore, using that information to calculate a regression equation (in place of the straight spectral responses) will produces a robust model for predicting concentrations of the desired constituents in very complex samples. It is interesting to note that the PCA factors matrix, F, performs a similar task to the K matrix in the Classical Least Squares (CLS) model; it stores the "constituent" spectral data. This does not mean that the rows of the F matrix are the spectra of the pure constituents; because they are not. However, they cannot be used alone without the scores matrix S to represent the original data (as in CLS, it needs the C matrix to perform the same function).
On the other hand, the scores in the S matrix are unique to each calibration spectrum, and just as a spectrum is represented by a collection of absorbances at a series of wavelengths, it can also be series of scores for a given set of factors. Much like the classical models performed a regression of the concentration C matrix directly on the spectral absorbances in the A matrix, it is also possible to regress C against the scores S matrix.
In this case, the regression technique from the ILS model is obviously the best choice. This gives the model the best qualities of the ILS method, such as, no a priori knowledge of the complete sample composition and some robustness in predictions with respect to contaminant constituents not present in the original calibration mixtures. The model equation is therefore:
where C is the m by n matrix of constituent concentrations, B is an m by f matrix of the regression coefficients and the S matrix is the scores from the PCA model. The dimensions of the matrices are n for the number of samples (spectra), m for the number of constituents used for calibration, and f for the number PCA eigenvectors. As with ILS, the B coefficients matrix can be solved by the regression:
Thus the name for this type of model is Principal Components Regression; it combines Principal Component Analysis and Inverse Least Squares Regression to solve the calibration equation for the model. All that remains is to come up with a single unified equation that represents the PCR model. Therefore, rearranging the matrix model equation from before to represent the scores as a function of the spectral absorbances and the eigenvectors produces:
It is not necessary to use the inverse (or pseudo inverse) of F to solve this equation. This is due to the fact that when PCA is used to solve the spectral model, the resulting F matrix of eigenvectors is a special type of matrix called an orthonormal matrix. This type of matrix has a very interesting quality: when the matrix is multiplied by its own transpose, the identity matrix is the result. Multiplying any matrix by the identity matrix is the same as multiplying a single number by one; the result is always the number again. So, to get the equation for the scores, both sides of the earlier PCA model equation for the spectral data were simply multiplied by the transpose of the F matrix.
Finally, by combining the concentration equation with the scores equation, the final PCR model equation emerges:
where C is the m by n matrix of constituent concentrations, B is an m byf matrix of the regression coefficients, A is an n by p matrix of spectral absorbances, and F is an f by p matrix of eigenvectors. The dimensions of the matrices are n for the number of samples (spectra), m for the number of constituents used for calibration, p for the number of data points (wavelengths) used for calibration, and f for the number PCA eigenvectors.
The PCR calibration model is not completely free of problems, however. It is important to note that PCR is a two-step process; the PCA eigenvectors and scores are calculated and then the scores are regressed against the constituent concentrations using a regression method similar to ILS. Remember that the ILS method can build accurate calibrations, provided that the selected variables (in the earlier discussion, the variables were the responses at selected wavelengths) are physically related to the properties (constituent concentrations) they are regressed against. However, the PCA factors/scores are calculated independently of any knowledge of these concentrations. They merely represent the largest common variations among all the spectra in the training set. Presumably, these variations will be mostly related to changes in the constituent concentrations, but there is no guarantee this will be true.
In fact, many PCR models include more factors than are actually necessary as some of the eigenvectors are not related to any of the constituents of interest. Ideally, a PCR model should be built by performing a selection on the scores (much like a selection of wavelengths for an ILS model) to determine which factors should be used to build a model for each constituent. In practice, this is a difficult process, both in terms of developing the selection rules and making it simple to perform. Most commercial chemometric software packages do not support this type of PCR model.
In addition, like the ILS method, the predictive ability of the PCR model will suffer if the constituent concentrations are collinear. Again, this means that a relatively large number of the training set samples are required, and they must be tested by the primary calibration method to determine the "randomness" of the constituent concentrations.
PCR Advantages
· Does not require wavelength selection. Any number can be used; usually the whole spectrum, or large regions.
· Larger number of wavelengths gives averaging effect, making model less susceptible to spectral noise.
· PCA data compression allows using inverse regression to calculate model coefficients; can calibrate only for constituents of interest.
· Can be used for very complex mixtures since only knowledge of constituents of interest is required.
· Can sometimes be used to predict samples with constituents (contaminants) not present in the original calibration mixtures.
PCR Disadvantages
· Calculations are slower than most Classical methods.
· Optimization requires some knowledge of PCA; models are more complex to understand and interpret.
· No guarantee PCA vectors directly correspond to constituents of interest.
· Generally, a large number of samples are required for accurate calibration.
· Collecting calibration samples can be difficult; must avoid collinear constituent concentrations.
| 
0 Items