1 ACCEPTED SOLUTION
Accepted Solutions
Richard Jackson wrote:
I cheerfully confess that I'd never even heard of the word factorization in connection with spectra until this thread arose.I have, and in fact I do it all the time. Think of a spectrum with N points as a vector that represents a point in N dimensional space. If you have several spectra, then they form a group of points in that space. Each of the N axes represents the intensity or amplitude at a given frequency. To solve any given problem, however, those may not be the optimal choice of axes. We can generate a new set of axes by creating a new set of "spectra", or factors, that are linear combinations of the current spectra, which corresponds to rotating our original axes. The simplest scheme is to make the new axes point in the directions of maximum variance in the data, which is principal component analysis. It's useful because if the number of underlying sources of variation in the data is less than N (often the case in spectra) then after rotation all the values on some axes are zero, or very close to zero, and we can throw them away. in other words, we can reduce the dimensionality of the data with minimal, and controllable, loss of information. If you look in the DAEP under "Principal Component Analysis" and "Principal Component Regression of NIR Spectra for Alcohol Mixtures" there are a couple of examples. There are lots of other factoring schemes though, that use different criteria for the axes rotation, including some that don't even guarantee that the new axes are orthogonal (which PCA does). Which is best depends on what the end goal is (for example, in the analysis of the alcohol mixtures, PCA is actually not optimal. A better method is partial least squares, which takes into account the known concentrations. That would be another (long) worksheet though, and I've never bothered to to write it in Mathcad).
I don't know if that's what Sabah actually wants to do though. Especially given that there are only three spectra, in which case I can't see much benefit to factoring as I understand it.
Thanks, Richard. A very good explanation - got it! 
I'll re-read the DAEP.
Cheers,
Stuart
Richard Jackson wrote:
Could you be more specific about what you want to do. Do you mean you want to do a principle component analysis of the spectra?
Hi Richard,
I cheerfully confess that I'd never even heard of the word factorization in connection with spectra until this thread arose. However, a quick, but not particularly fruitful (*), webcrawl suggests that it may occur in the context of spectrograms. In which case, the underlying Mathcad technique would seem to be reshaping the signal vectors to form an array of contiguous subsignals and then apply an FFT to each of these subsignals? Is that the general idea? If so, do you think the attached worksheet might be of some interest (it was a bit of a pain writing for a non-zero ORIGIN!)? ...
Stuart
(*) Most of the sites I looked at tended to assume you knew what factorization was in the first place!
Richard Jackson wrote:
I cheerfully confess that I'd never even heard of the word factorization in connection with spectra until this thread arose.I have, and in fact I do it all the time. Think of a spectrum with N points as a vector that represents a point in N dimensional space. If you have several spectra, then they form a group of points in that space. Each of the N axes represents the intensity or amplitude at a given frequency. To solve any given problem, however, those may not be the optimal choice of axes. We can generate a new set of axes by creating a new set of "spectra", or factors, that are linear combinations of the current spectra, which corresponds to rotating our original axes. The simplest scheme is to make the new axes point in the directions of maximum variance in the data, which is principal component analysis. It's useful because if the number of underlying sources of variation in the data is less than N (often the case in spectra) then after rotation all the values on some axes are zero, or very close to zero, and we can throw them away. in other words, we can reduce the dimensionality of the data with minimal, and controllable, loss of information. If you look in the DAEP under "Principal Component Analysis" and "Principal Component Regression of NIR Spectra for Alcohol Mixtures" there are a couple of examples. There are lots of other factoring schemes though, that use different criteria for the axes rotation, including some that don't even guarantee that the new axes are orthogonal (which PCA does). Which is best depends on what the end goal is (for example, in the analysis of the alcohol mixtures, PCA is actually not optimal. A better method is partial least squares, which takes into account the known concentrations. That would be another (long) worksheet though, and I've never bothered to to write it in Mathcad).
I don't know if that's what Sabah actually wants to do though. Especially given that there are only three spectra, in which case I can't see much benefit to factoring as I understand it.
Thanks, Richard. A very good explanation - got it! I'll re-read the DAEP.
Cheers,
Stuart
sabah al-fartosy wrote:
.... Honestly, I am not professional with Mathcad ...
You're not the only one. I didn't check that my modified vec,seq, index, vec2mat and reshape functions worked for all values of ORIGIN. 
(I normally use ORIGIN=0 and write my functions accordingly; however, your worksheet seemed like a good excuse to modify them for any ORIGIN ... ).
I *think* the revised sheets contains corrected versions of these functions.
Stuart
