Khaled Rifai wrote: > Hi there, > > I want to generate correlated random numbers in the course of a monte > carlo simulation. > > I always read that Cholesky Factorization requires positive definite > correlation matrices. I can�t see the reason why it shouldn�t work for > positive semidefinite matices as well! Any insights? >
A lot depends on what you (and everyone else) mean here. If you are thinking about a particular computer implementation then such things may or may not work, depending on how much care is taken in implementation. There is the question of "uniqueness" of the decomposition, but this usually doesn't matter for random number generation. Some things may depend on whether you are using the X=UV formulation, or X=UDV, where in the latter it may be easier to classify/identify cases where zero-variances are found. I would suggest finding a suitable text to refer to. One possibility is: Matrix Algebra from a Statistician's Perspective. D.A Harville , Springer 1997 > I heard about eigenvalue and singular value decomposition as > alternative approaches. As I understand it SVD is only the extension > of an eigenvalue decomposition for non squared matrices. Is there any > other use for SVD within a monte carlo simulation of correlated random > numbers? > I think that the approaches give equivalent answers, but there can be large differences in computer-time involved. I think Cholesky Decomposition is prefered for small, medium and large problems, with SVD prefered for very large problems. > If the Eigenvalue Decomposition yields negative eigenvalues caused by > rounding errors or other inconsistencies what is the best method to > proceed in an monte carlo simulation? (any literature references?) > > Since Cholesky, Eigenvalue and SVD only work for normal random > variables because their concept is based on the fact that linear > combinations of normal random variables are themselves normally > distributed I wonder if there are any other methods to induce > correlation (either Pearson, Spearman or Kendall) into a monte carlo > simulation? Yes, lots of methods. If simple transformations of each marginal variable does not produce acceptable multivariate-normality (implemented in the obvious way), then you might refer to: Continuous Multivariate Distributions Vol 1 . Kotz, S, Balakrishnan N and Johnson NL, Wiley , 2000 Families of frequency distributions. Ord JK, Griffin, 1972 You might be able to find an established family of multivariate distributions suitable for your applications, or else there are some general-purpose families that might be usable if you can specify the marginal distributions you want. Such things are in the above references. > Has maybe anyone heard about Stein�s Algorithm that works > with latin hypercube sampling and is able to explain it to somebody > like me with a rather non mathematical background? > > Thanks in advance > > Khaled . . ================================================================= Instructions for joining and leaving this list, remarks about the problem of INAPPROPRIATE MESSAGES, and archives are available at: . http://jse.stat.ncsu.edu/ . =================================================================
