I'm currently learning chemometrics for my work and I have a simple question about Multiple Linear Regression (MLR).
Just to explain the context: I am simply using UV-Vis-NIR spectra (2500 wavelengths) to quantify a molecule in presence of interfering species. I have built a calibration set which describes my concentration intervals in a complete and balanced way (50 samples), and a validation set that is basically real samples taken from a process (50 samples, independent from the calibration set). After some try/retry exercices and some optimizations with a chemometrics add-on for MATLAB, I've come up with a parsimonious PLS model (SIMPLS algorithm) that accurately predicts concentrations of the validation set. For now, because my validation samples are significantly different in concentrations and interfering species than my calibration set, I consider a model to be good if it correctly predicts my validation solutions: I do not use statistical tests such as t-tests.
However, after I tried MLR, I realized that the MLR model was significantly more robust with respect to interfering species (the root-mean-square error of prediction is twice smaller and some validation samples where the PLS model gives a prediction relatively far from reality are correctly predicted by MLR).
Here comes my question:
In almost every textbook or publication that I have read it is said that MLR is not applicable if we have more variables than samples, because the inverse of the $X'X$ matrix, where $X$ is the predictor block, doesn't exist. Yet, my MLR model is actually working better than my PLS model when, if I understand what I read correctly, the MLR shouldn't even work because I have more variables than samples (and my variables are supposedly very collinear).
Does the fact of having more variables than samples absolutely not prevent the model from being calculated, and therefore from making good predictions, but just makes the regression coefficients unstable and difficult to interpret? Or am I messing around, and having a well-functioning MLR model under these conditions should worry me about the relevance of my approach?
Thank you very much.
PS: I've learn (well, maybe unaccurately) the basics of chemometrics mainly with Tormod Naes and Harald Martens books, along with some publications. Do you have any book suggestion to pursue my learning? Ty again!