Laplace and other transforms

ianm's picture

14. Laplace and other transforms.

Another topic for discussion. In the chemometric course I and an associate gave this summer, I was struck by the thought that Principal Components Regression is a mathematical transformation of the spectral information into regression space. Also, Fourier Analysis is a mathematical transformation from wavelength space to frequency space. So why aren't other mathematical transformations used, such as LaPlace transformation? I don't know if it would be of help, but does anyone know of any attempts to use LaPlace transformations or any other transformations?

From Jim Reeves

Factor analysis has been used by some, I know this is somewhat like PCA. Also there was an article in AI magazine several years ago on the use of Wavelets and spectral data. I suspect a lot of others have been tried, but in the end nothing really ever does a whole lot better than what's already used. Fred McClure showed that you could use the Fourier coeffients directly un regression analysis. This resulted in data compression among other things, but no one has ever really taken it up to my knowledge. I suspect unless some new method was shown to be the only way to go and the software companies followed up, nothing much will change.

Quite a valid observation. Other examples of transforms which are used are the various families of wavelets, the orthogonal spaces formed by Grahm-Schmidt orthogonalization, PLS factor spaces, the various symmetry-based factor spaces, to name a few. The eigenvector space and related PLS factor spaces which are commonly employed are valuable because of the property they have of maximizing the variance captured by each successive factor. This allows for rank-reduction and some noise removal without the loss of predictive information.


Richard Kramer