Introduction
When we at Sentronic started investigating alternatives to empirical MVDA approaches, like partial least squares (PLS), I was initially skeptical that pure component MVDA approaches, like iterative optimization technology (IOT), would be useful given the difficulty I’ve had deploying another common pure component MVDA approach: classical least squares (CLS) in NIR PAT.
To my great surprise, IOT seems to provide accurate results in many NIR PAT applications where CLS generates biased results. Why is this? Let’s dig into the math of each approach.
A Deep Dive Into the Math of CLS
CLS is a well-known method of estimation in chemometrics. Under the assumptions of no or limited collinearity and a linear relationship between measured and predictor variables, CLS assumes that a measured spectrum can described as:
Amix = C ST + ε
Where
Amix = A measured mixture spectrum
C = A vector of spectral weights, often assumed to be either mole fractions or mass fractions
S = A matrix of the spectra of pure components that make a mixture
ε = The error-of-fit, or any spectral intensity not well described by linear combinations of the spectra in S.
T indicates the transpose of a matrix or vector
To estimate the concentrations of the mixture components using the equation above, the CLS approach uses the following equation:
Ĉ = Amix S (ST S) -1
Where is a vector of estimated concentrations for each component in S and -1 indicates the matrix in parentheses is inverted.
A Deep Dive Into the Math of IOT
IOT uses similar assumptions, but formulates its problem slightly differently. It defines a simulated spectrum, Asim, as:
Asim = C ST + ε
Assuming an existing set of pure component spectra, measured under similar conditions to the mixture measurement, IOT redefined ε as:
ε = Amix - Asim = Amix - C ST
ε under a set of constraints, typically ensuring that the sum of the values of C is unity (i.e., you can’t have more or less than 100% mass fraction or mole fraction) and each of the values of C can’t be less than 0 or greater than 1 (i.e., you can’t have negative concentration or concentration greater than 100%).
This may seem like a small difference in approaches; however, in practice these small differences can have big impacts on estimated concentrations. In the next part of this blog post (stay tuned!), we’ll look at a case study.