Two books that I have found useful for these types of questions are: Beebe, Pell, and Seasholtz, Chemometrics: A Practical Guide, Wiley, 1998. (better at getting from raw data through validation and prediction)
Martens and Naes, Multivariate Calibration, Wiley, 1989. (better theoretical explanations of the algorithms) I won't try to answer your questions, since I can't answer all of them with great confidence. See these sites for mor info, tutorials, etc. http://www.acc.umu.se/~tnkjtg/chemometrics/ http://www.chemometrics.net/ http://www.spectroscopynow.com/Spy/basehtml/SpyH/1,,2-0-0-0-0-home-0-0,00.html ...and if you'd like to go somewhere interesting, and learn a lot about this stuff (for a very reasonable price), consider this: http://rcs.chph.ras.ru/wsc.htm Be sure and check out the report. I went this year and am planning to go again next year. mathstudent1001 wrote: > > I have some simple questions regarding PLS (Partial Least Squares) > Linear Prediction. > > Say that X is an n by k matrix where k is the number of dimensions of > the data and n is the number of trials. (Have I transposed the > standard coordinates?) > > Say that Y is an n dimensional column vector that encodes a value for > each trial. > We will only be predicting one value in the below. > > We want to produce a predictor P a k dimensional row vector. > > Say PLS(X,Y) produces a predictor P using the PLS algorithm. > > Say A is an n by n orthogonal matrix and say B is an k by k orthogonal > matrix. > > Is it true that PLS(AX, AY)=PLS(X,Y)? > > Is it true that PLS(XB,Y) = B PLS(X,Y) (using matrix multiplication)? > > (Or are these group invariance properties not exactly true because of > the approximations > and iterative estimation of the algorithm?) > > If not please explain. > > Do we have to mean center the data first? > > What is the correct understanding from the highest abstract point of > view of the relation of PLS to Principle Component Analysis? > > Is there an easy formula for PLS(X,Y) if X is diagonal? (This would > be the case in somebody PCA-ed X first and the left and right > orthogonal properties above were true. > > How do I write PLS(X,Y) iteratively using the PLS algorithm in a > simple way that related it to PLS( other simpler X's, other simpler > Y's)? > > Is there an n dimensional vector Y' such that > PLS(X,Y)=ExactPredictor(X,Y')? > Is there an easy formula for Y' if X is diagonal? > > Why isn't the average of the true and predicted stuff the same? Is > there some sort of L2 like average relating them? What is the formula > relating them? > > In what sense is PLS the optimal predictor? > > Is PLS stable with respect to noise in X and Y? > > If Y1 and Y2 are Y-like vectors is PLS(X,Y1+Y2) = PLS(X,Y1)+PLS(X,Y2)? > If not what is the correct relation between them? Do you have to > center Y1 and Y2 both first? > > If you apply PCA to the k+1 by n matrix made up of X and Y is there > any relation between these eigenvalues and the new coordinates and > PLS(X,Y) i.e. can the predictor be thought of as a new coordinated of > this PCA decomposition? Is not what is the formula? > > A similar question is in order for the Exact Predictor instead of PLS? > If you apply PCA to the k+1 matrix made up both X and Y is there any > relation between these eigenvalues and the new coordinates of Exact > Predictor(X,Y) . . . > > If PLS(X,Y) = 0 what does this mean conceptually? > > Is PLS(X,0) = 0? If not why? > > If W is a k to k2 matrix (possibly with some restrictions like sub > orthogonal?) > Is PLS(XW,YW) = PLS(X,Y) W? If not what is the relation? > Is there any special relation if W is a projection matrix i.e. some of > the data is > zeroed? Here k2 may be bigger or smaller than k! > > What is the best book explaining PCA and PLS from the point of view of > the kind of abstract questions and conceptual thought processes in the > spirit of these questions? > > Who are the world's leading authorities both applied and practice for > this stuff? . . ================================================================= Instructions for joining and leaving this list, remarks about the problem of INAPPROPRIATE MESSAGES, and archives are available at: . http://jse.stat.ncsu.edu/ . =================================================================
