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One of the most widely used spectroscopic quantitation methods is Inverse Least Squares, also known as Multiple Linear Regression and P-Matrix. Most methods based on Beer's Law assume that there is little or no interference in the spectrum between the individual sample constituents or that the concentrations of all the constituents in the samples are known ahead of time. In real world samples, it is very unusual, if not entirely impossible to know the entire composition of a mixture sample. Sometimes, only the quantities of a few constituents in very complex mixtures of multiple constituents are of interest. Obviously, these simpler calibration methods will fail for these samples since the full compositional chemistry and pure spectroscopic composition is not known. One solution to this problem is to take advantage of algebra to rearrange Beer’s Law and express it as:
where the same relationships between the spectral response at a single wavelength ( ), the constituent absorptivity constant ( ), the pathlength of light (b) and the constituent concentration (C) are maintained. By combining the absorptivity coefficient () and the pathlength (b) into a single constant, this can also be expressed as:
where E is a matrix of concentration prediction error.
Due to the powers of mathematics, this seemingly trivial variation has tremendous implications for the experiment. This expression of Beer’s Law implies that the concentration is a function of the absorbances at a series of given wavelengths. This is entirely different from Classical Least Squares, where absorbance at a single wavelength is calculated as an additive function of the constituent concentrations. Consider the following two equations:
Notice that even if the concentrations of all the other constituents in the mixture are not known, the matrix of coefficients (P) can still be calculated correctly. This model, known by many different names including Inverse Least Squares (ILS), Multiple Linear Regression (MLR), or P-Matrix seems to be the best approach for almost all quantitative analyses since no knowledge of the sample composition is needed beyond the concentrations of the constituents of interest.
The selected wavelengths must be in a region where there is a contribution of that constituent to the overall spectrum. In addition, measurements of the absorbances at different wavelengths are needed for each constituent. In fact, in order to accurately calibrate the model, measurements of at least one different wavelength is needed for each additional independent variation (constituent) in the spectrum.
Again for those interested in matrix algebra, the P matrix of coefficients can be solved by computing:
but as with CLS before, if the A matrix is not square, the pseudo-inverse must be used instead:
This model seems to give the best of all worlds. It can accurately build models for complex mixtures when only some of the constituent concentrations are known. The only requirement is selecting wavelengths that correspond to the absorbances of the desired constituents.
Unfortunately, the ILS calibration approach does have some drawbacks. Due to the dimensionality of the matrix equations, the number of selected wavelengths cannot exceed the number of training samples. In theory, it should be possible to just measure many more training samples to allow for additional wavelengths, but this causes a new problem. The absorbances in a spectrum tend to all increase and decrease together as the concentrations of the constituents in the mixture change. This effect, known as collinearity, causes the mathematical solution to become less stable with respect to each constituent.
Another problem with adding more wavelengths to the model, is an effect known as overfitting. Generally, starting from very few wavelengths, and adding more to the model (provided they are chosen to reflect the constituents of interest) will improve the prediction accuracy. However, at some point, the predictions will start to get worse. When the number of wavelengths increases in the calibration equations, the likelihood that "unknown" samples will vary in exactly the same manner decreases. When too much information in the spectrum is used to calibrate, the model starts to include the spectral noise which is unique to the training set and the prediction accuracy for unknown samples suffers.
In ILS, the averaging effect gained by selecting many wavelengths in the CLS method is effectively lost. Therefore wavelength selection is critically important to building an accurate ILS model. Ideally, there is a crossover point between selecting enough wavelengths to compute an accurate least squares line and selecting few enough so that the calibration is not overly affected by the collinearity of the spectral data.
Many of today’s software packages that perform ILS (MLR) calibrations use sophisticated algorithms to find the "best" set of wavelengths to use for each individual constituent of interest. They attempt to search through the wavelengths and try different combinations to locate that cross-over point. Luckily, the calculations involved in computing the P matrix are very fast. However, when you consider that a spectrum may have as many as 2000 wavelength data points, it’s obvious to see that calculating ILS models for all possible combinations of wavelengths can be an excruciating task.
Inverse Least Squares in an example of a multivariate method. In this type of model, the dependent variable (concentration) is solved by calculating a solution from multiple independent variables (in this case, the responses at the selected wavelengths). It is not possible to work backwards from the concentration value to the independent spectral response values as an infinite number of possible solutions exist. However, the main advantage of a multivariate method is the ability to calibrate for a constituent of interest without having to account for any interferences in the spectra.
Since it is not necessary to know the composition of the training mixtures beyond the constituents of interest, the ILS method is better suited to more complex types of analyses not handled by the CLS approach. It has been used for samples ranging from natural products (such as wheat, wool, cotton and gasoline) to manufactured products.
ILS Advantages
· Based on Beer's law.
· Calculations are relatively fast.
· Multivariate model allows calibration of very complex mixtures since only knowledge of constituents of interest is required.
ILS Disadvantages
· Wavelength selection can be difficult and time consuming. Must avoid collinear wavelengths
· Number of wavelengths used in the model limited by the number of calibration samples.
· Generally, a large number of samples are required for accurate calibration.
· Collecting calibration samples and measuring via a primary calibration can be difficult and tedious.
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