Quantitative Spectroscopy Quantitative Spectroscopy 2 - Algorithms

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1 Quantitative Spectroscopy Quantitative Spectroscopy 2 - Algorithms Copyright 2006 Mettler-Toledo AutoChem, Inc. 1

2 Chapter 2 Outline Beer-Lambert Law or Beers Law Beer s Law, Classical Least Squares (K-Matrix) Beer s Law, Inverse Least Squares (P-Matrix) Partial Least squares (Beer s Law) 2

3 Beer s Law The Beer-Lambert Law or Beer s Law is the linear relationship between Absorbance and concentration of an absorbing species. A = ε λ x b x c Note: ε refers specifically to units of molarity. In the general case a is used instead. Where A is measured absorbance (dimensionless), ε is the wavelength dependent absorptivity coefficient (units of M -1 cm -1 ), b is the pathlength in cm, and c is the concentration in M. 3

4 Beer s Law Classical Least Squares (K-Matrix) (1) A = a λ x b x c (2) A = abc (3) A = KC and (4) C = A / K, in a one component system, this gives us a one point calibration A one point calibration is generally not a good idea (noise and other deviations from perfect linearity are best handled by multiple samples). As soon as at least two components are present we need as many equations as unknowns to figure out both concentrations: (5) A λ1 = K a,λ1 C a (6) A λ2 = K b,λ2 C b A λ1 is the absorbance associated with component A at a particular wavelength. A λ2 is the absorbance associated with component B at second wave- Length. 4

5 Beer s Law Classical Least Squares (K-Matrix) (1) A = a λ x b x c (2) A = abc (3) A = KC and (4) C = A / K, in a one component system, this gives us a one point calibration A one point calibration may or may not give us good accuracy as we discussed earlier. Noise and other deviations from perfect linearity are best handled by multiple samples. As soon as at least two components are present we need as many equations as unknowns to figure out both concentrations: (5) A λ1 = K a,λ1 C a (6) A λ2 = K b,λ2 C b A λ1 is the absorbance associated with component A at a particular wavelength. A λ2 is the absorbance associated with component B at second wave- Length. 5

6 Beer s Law Classical Least Squares (K-Matrix) Component A Component B Component A + Component B A λ1 A λ2 These are isolated peaks. What would we do if they were overlapped? 6

7 Beer s Law Classical Least Squares (K-Matrix) Component A Component B Component A + Component B A λ1 A λ2 How will we deal with overlap? 7

8 Beer s Law Classical Least Squares (K-Matrix) (7) A λ1 = K a,λ1 C a + K b, λ1 C b (8) A λ2 = K a,λ2 C a + K b, λ2 C b A λ1 K a,λ1 K b, λ1 C a = A λ2 K a,λ2 K b, λ2 C b This is how we deal with overlap. In general. If we Leave the two component, Two wavenumber case and go to the general case, we can solve for K which is our calibration matrix. Once we have K, we collect a non- Calibration spectrum (either validation or reaction) and use The K matrix along with the A matrix (the spectrum) to solve for C of all the components in The solution. 8

9 Beer s Law Classical Least Squares (K-Matrix) (9) K = AC -1 We have now switched to matrix notation. We can solve for the K matrix this way, except that because we are often going to use more calibration than there are components in the solution the C matrix will have more rows than columns. If it is not square then we can t take the inverse (C -1 ) so we will take the pseudo-inverse and the equation looks like this. (10) K = AC (CC ) -1 Now that the K matrix is solved we may collect a spectrum with an unknown concentration of chemical species of interest and predict the concentration(s). 9

10 Beer s Law Classical Least Squares (K-Matrix) What are some advantages and disadvantages of Classical Least Squares? Advantage: It is very simple, there is little risk that the model will be able to over fit data (model noise or other correlations). This will not be the case with Inverse Least Squares based models (p-matrix or PLS). Disadvantages: All of the concentrations of components that influence the spectrum must be known. This is usually not possible and even in cases where it is possible to know this, how do we provide the model with information about temperature and molecular interactions (two variables that often affect the spectrum non-negligibly? DISCUSS CLASSICAL LEAST SQUARES 10

11 Beer s Law, Inverse Least Squares, p-matrix What are some advantages and disadvantages of Inverse Least Squares? Disadvantage: The risk for basing the model on properties correlated to, but not representative of, the concentration we think we are predicting. Advantages: All of the concentrations of components that influence the spectrum need not be known. By appropriate selection of wavenumber values we are able to calibrate the model so that it works over temperature and molecular interaction variances. 11

12 Beer s Law, Inverse Least Squares, p-matrix (1) A = a λ x b x c (2) A = abc (11) c = A/ab (12) p = 1/ab We no longer consider a to be the absorptivity constant of a single molecule or functional group, but instead consider it to be the absorptivity constant of a the solution at a particular wavenumber value. (13) c = pa This would suffice for a non-overlapped peak corresponding to a material in which we have interest. However as soon as another material has a peak overlapped with it, we need at least one more variable to account for the overlap. However, we still only need to know the concentration of the material we are interested in. 12

13 Beer s Law, Inverse Least Squares, p-matrix Component A (component of Interest). Component B Component A + Component B A λ1 A λ2 13

14 Beer s Law, Inverse Least Squares, p-matrix (14) C A = p λ1,a A λ1 Let us imagine that our component A has a p value at wavenumber value λ1 of 10. (15) 10 = (10)(1) In the absence of component B a 10% wt solution gives a peak of 1.0 Abs. Let us (16) 10 = (10)(1.2) further imagine that component B has an absorbance at the same wavelength that adds 0.2 Abs. to the value of A λ1 causing the model described in equation 14 to fail. But B also has an absorbance at at A λ2 which we will exploit to fix the model. 14

15 Beer s Law, Inverse Least Squares, p-matrix (17) C A = p λ1,a A λ1 (18) C A = p λ1,a A λ1 + p λ2,a A λ2 (19) C A = p λ1 A λ1 + p λ2.a A λ2 (20) 10 = (10)(1.2) + (-2)(1) We have shown that this model doesn t work when another component B overlaps with the A peak at (λ1). However if we include another wavenumber value that gives the model information about B we can exploit this to correct for the presence B and make the model work. So now that we need two variables to make the model work we need two equations. In order to generate another equation we simply collect another spectrum. In practice we will want to use at least twice as many spectra as wavenumber values as a rule of thumb. So now using p-matrix we have a model that does not require us to know the concentrations of every component (or property (such as temperature)) in the solution. It simply requires us to collect calibration standards that show the model The effect of the component or property. 15

16 Beer s Law, Inverse Least Squares, p-matrix A concern with p-matrix is that we could model something correlated with, but not representative of component A. Some examples could be the solvent that was used to carry A into the reaction or a temperature ramp that approximates the disappearance or dosing of the component. Here are some facts about ILS. 1. We will need at least as many sample spectra as wavenumber values used. 2. As we will see in the next section, ILS does not have the noiserejection capability of a factor based method such as PLS. DISCUSS INVERSE LEAST SQUARES 16

17 Partial Least Squares (PLS) (Beer s Law) Absorbance Wavenumber This diagram shows two noise free spectra, from a two component solution. Each spectrum has three resolved data points. In a real calibration sample set we will use between 7-90 spectra (10-40 is typical) and we will use many discreet wavenumber values. 17

18 Partial Least Squares (PLS) (Beer s Law) We re-plot the spectra as points in three dimensions where each measured variable is an axis. In a noise-free, twocomponent system the points would all lie in a plane (R. Kramer). We will next reorient the axes and create new axes that describe the data more efficiently 1000 cm cm cm -1 18

19 Partial Least Squares (PLS) (Beer s Law) We rotate the axes so that the plane containing the data lies in the plane of the paper. 19

20 Partial Least Squares (PLS) (Beer s Law) Next we draw a line that minimizes the squares of the distance of the data from the line. This line is referred to as. 1. The first factor 2. The first principal component 3. The first eigenvector 4. The first spectral loading Or 5. The first loading vector For our discussion we will refer to this as a factor. 20

21 Partial Least Squares (PLS) (Beer s Law) A second orthogonal factor is drawn to meet the origin of the first factor. Notice that with the addition of the second factor the dimensionality of the data set has been reduced from three to two. In a real model the dimensionality might be reduced from 60 wavenumber values to five factors. Next we will add noise to the system to make it more realistic. The value that each spectrum has with regard to a particular factor is called a score. So in this system we could take any spectrum and represent it as: Spectrum y = (score * factor 1) + (score * factor 2) DISCUSS FACTORS AND SCORES 21

22 Partial Least Squares (PLS) (Beer s Law) Ideal. In the plane Real. The real points have been displaced so that they are above and below the plane 22

23 Partial Least Squares (PLS) (Beer s Law) The real points have been displaced so that they are above and below the plane, so two axes are no longer sufficient to describe the data. Therefore we add a third factor. However should we really include this factor in our model? No! We will explore this question in more detail later. 23

24 Partial Least Squares (PLS) (Beer s Law) Spectra: Rows are spectrum 1,2, n and columns are wavenumber values = Spectral scores Spectra X factors Spectral Factors: factors x wavenumber values Concentrations: Spectra X Components Concentration scores Spectra X factors Conc factors:factors X components PLS takes advantage of correlations in the spectral data and the concentration data by swapping scores during the calculation. PLS has two advantages over ILS (p-matrix) 1. It reduces the dimensionality of the data 2. It rejects some of the noise present in spectral data block and the concentration data block. 24

25 Partial Least Squares (PLS) (Beer s Law) PLS Care must be taken to validate the PLS model. The risk of modeling correlations that are not representative of the concentrations being predicted is higher in PLS than in the previous algorithms we have discussed. 25