Regression fitting using reference data

Question

This might sound like an experimental physics question but I am more interested in a mathematical algorithm solution to the problem.

I have measured spectra of reference compounds which I would like to fit to the spectra of unknown compounds to determine the proportion of each reference in the unknown. The spectra cannot be described by a simple analytical function but rather are fingerprints of endmembers. The unknown compound is likely a mixture of one or more reference compounds. I am interested in the mathematical formalism of an algorithm which might achieve this. Psuedo mathematically the problem might be described as:

$$Experimental(x) = A(Reference_{1}(x)) + B(Reference_{2}(x)) + C$$

where $Experimental(x)$ is a vector containing the experimental data. $Reference_1(x)$ & $Reference_2(x)$ are the reference vectors suitably normalised. $A$ & $B$ are the coefficients to be determined which will be constrained such that $A$ & $B$ are positive and finite. $C$ is a d.c offset.

Can someone point me in the direction of the correct mathematical terminology which might help me find a this? All discussions on least-squares fitting I've come across only deal with the fitting of analytical functions to data not data to data.

Thanks

Answer 1

My interpretation: You have 3 data sets, and you are wanting to know how closely a combination of $Ref_{1x}$ and $Ref_{2x}$ you stated is to your $Experimental$ data.

Without seeing your data, and the quantity for each of the 3 variables (which is important), I will give general advice.

From what you have stated thus far this is Regression and Hypothesis testing.

I suggest you do / read up on the following:

1. Graph each variable using box plots, and exclude any outliers.

After removal of outliers:

2. Look at the spread and skewness of each. This is important for assumptions for models later. Example: a positively skewed small sample of data would not indicate that variable was normally distributed.
3. Mathematical software such as SAS and SPSS - that allows you to model the data. You could also use MS Excel.
4. Look at Errors and plots of errors for different regression models applied to your data sets - they indicate what models you should use.
5. Consider the amount of data you have for each variable. Eg Some models assume a normal distribution for data, but if the data isn't in sufficient quantity, a normal model shouldn't be used. Similar with all the assumptions the model you are using presumes - research these & see if they are satisfied.
6. Value of $R^2$ for each model (how accurate the model is - the higher to 1 the better, with 1 perfect correlation).
7. P-values and confidence intervals. - needed for model and results interpretation.