If you want to solve the inverse problem, you have make yourself free from thinking of chemometrics. Instead, you have to think of dispersion theory and wave optics to understand optical spectra. E.g. in the simplest case you have to use the Fresnel equations for the forward calculation of the reflectance of a strongly absorbing cubic crystal. To do that you have to know the index of refraction function or the dielectric function. The latter can be described by the damped harmonic oscillator model. Here is where the inverse problem comes into play, because how else would you know the oscillator parameters (except through quantum mechanical calculations, but this is another story). The solution to this inverse problem is known to be delivered by a method called dispersion analysis. It has been applied as early as 1930 (I do not know which kind of computer was used... ;-) and it is the origin of band fitting (even if only few know that), which is certainly the much less sophisticated.

Unfortunately dispersion analysis requires that there is first a fully analytical way for the forward calculation. Therefore it will not work with diffuse reflectance if this is what you are going for...

We have used PLSR of Raman spectra for example for determination of glucuse levels in human blood serum.

In spiked samples, a good linear regression is observed, and the regression co-efficient is pretty much exactly the spectrum of glucose.

Therefore, notwithstanding the previous comments, if the regression is from Yi > Yf, I would estimate the spectrum for Yn as Xi x (Regression co-efficient *(Yn-Yi))

Thank you all for your answers! Indeed, I understand why it is not possible to have one analytical solution. I also understand that we potentially have an infinite solutions. But maybe this problem has a stochastic answer to infer what would be the mean X and prediction interval for a particular y given the observed values and the PLSR model?

As Hugh J Byrne mentioned (if I understand well..), we can simulate random spectra using regressions between the X and Y and adding the correlated error terms from each wavelength. So I thought something similar could be used but this time taking advantage of the outputs of the PLSR model. In Wold et al. 2001: PLS-regression: a basic tool of chemometrics, it is written in paragraph 3.3 " The X-scores are a predictors of Y and also model X i.e., both, Y and X are assumed to be, at least partly, modelled by the same LV’s ". This is the sentence that made me think that maybe some info from the PLSR could help infer X.

Cheers,

Hi Julien, what Hugh stated may be working in Raman spectroscopy, but there is a big difference to optical spectroscopy, which is that in optical spectroscopy both intensity (i.e. reflectance or transmittance) and absorbance are additive.

I know this sounds odd, but what is usually overlooked is that in spectrophotometric concentration determination you have solutions which are perfectly homogeneous. This meands the additivity of transmittance (actually it is an averaging over transmittances of volume elements of the sample with the thickness of the sample and edge lengths of about one tenth of the wavelength) does not come into play. Furthermore, the solutions are usually diluted, which means that Beer's law is just an approximation of the Lorentz-Lorenz equation does not play a role.

However, if you have heterogeneous samples, the optical averaging leads to a spectrum that can be extremely complicated because you can have imperfect demixing (e.g. in a glass ceramics). See e.g. Article Beyond Beer's Law: Spectral Mixing Rules

I really do not see how you could solve inverse problems in this case, in particular if you assume chemometric methods that are based on the applicability of Beer's law...

Just to follow up, Julien showed me his results and given the fact that a leaf is a micro-heterogeneous sample in the wavelength range of interest (NIR), I think that indeed a linear dependence, very much like the one suggested by Hugh J Byrne for Raman spectroscopy, is present. It is just important to keep in mind that you really have to stick with reflectance spectra and not to try to convert them somehow to absorbance, because for micro-heterogeneous samples it is reflectance which is linearly dependent on concentration etc. and not absorbance.

Thanks Thomas Mayerhöfer . So basically, it is possible to predict X based on y using multiple linear regressions of y and X, or, by using the results of the PLSR, modelling the multiple linear relations between the scores of X and y. It is however necessary to model the autocorrelation of the error to produce random spectra. The prediction of y by the PLSR model using those random spectra correspond to y so this is a good approximation of the inverse problem.

Returning to my original suggestion -

If the regression is from Yi to Yf, I would estimate the spectrum for an intermediate Yn as

the initial spectrum, Xi + (Regression co-efficient *(Yn-Yi))

In this case the Regression (beta) Co-efficient is a spectrum.

Hello Hugh J Byrne, in your message the Regression co-efficient are the coefficients obtained by the PLSR, right? In that case, this solution doesn't work for reflectance spectra.

Julien Lamour - why not?

I tried on LMA (leaf mass per area) as the y variable, and reflectance as the X predictors (400 : 2400 nm). So I did the PLSR, obtained the Bi coefficients (i corresponds to a wavelenght), calculated a mean leaf spectra for a low LMA (LMA of 25 g. m-2). If I try to simulate the leaf spectra of a LMA of 100, I do as you suggest LMA 100 = Reflectance_LMA25 + Bi*75 but that does not produce a spectra. I don't see neither how that would work in my case. I can send you the data by mail if you need.

I have a meeting now...

The Regression co-efficients..... or Components (see the link below) should be spectra

https://uk.mathworks.com/help/stats/partial-least-squares-regression-and-principal-components-regression.html;jsessionid=6be9925bb119234705652e1765a7