Re: Parametric curve fitting

*To*: mathgroup at smc.vnet.net*Subject*: [mg18601] Re: [mg18568] Parametric curve fitting*From*: "Mark E. Harder" <harderm at ucs.orst.edu>*Date*: Tue, 13 Jul 1999 01:01:30 -0400*Sender*: owner-wri-mathgroup at wolfram.com

Virgil; Just a suggestion; I don't have time now to demonstrate this idea now, but it is similar to data analyses I do all the time, and I'll describe it in words. Instead of treating your sampled time series as functions with noise and searching them pairwise for correlations, treat them as a set of m samples from Real N-space (assuming that the entries are Real numbers). Place the N-vectors as columns in a matrix, call it A. Then the correlation between the i-th and j-th columns in A is proportional to the i,j-th element of the correlation matrix (Transpose[A].A) . To discover significant correlations, we need to filter noise out of this matrix, which I believe can be done using Mathematica's SingularValueDecomposition function. The SVD of matrix A returned as the product of 3 matrices, i.e. A=U.S.Transpose[V]. U and V are orthonormal matrices. S is a diagonal matrix with its largest elements listed first, and the columns of U and the rows of V are listed in order of decreasing significance in approximating A. See a good book on applied linear algebra for the algebraic and statistical details, eg. Noble & Daniel. Now, the correlation matrix, Transpose[A].A= V.[S^2] .Transpose[V]. (The columns of V are eigenvectors and the elements of S^2 the corresponding eigenvalues of the correlation matrix. ) To eliminate noise, examine elements of S, looking for a sharp drop-off where adding more vectors to the expansion only reconstructs noise, then drop all elements of S^2 and corresponding columns of V beyond that point, and reconstruct the correlation matrix only with the significant elements remaining. The correlations in the filtered correlation matrix should now be mostly noise-free, and therefore visible. NB. Mathematica returns U and V as their transposes! If you have any questions, e-mail me and I will try to answer. -mark -----Original Message----- From: Virgil Stokes <virgil.stokes at neuro.ki.se> To: mathgroup at smc.vnet.net Subject: [mg18601] [mg18568] Parametric curve fitting >Suppose that I have a set of experimental time series data, say > > x1(t) , t = 1,2,...N > x2(t), t = 1,2,...N > x3(t), t = 1,2,...N > . > . > . > xm(t), t = 1,2,...N > >where, each represents a sequence of sampled values (containing >measurement error). I would like to see what pairs (if any) >are linearly related. That is, if I plotted x1(t) vs x2(t) and this >appeared that it could be approximated by a straight line then >one might assume that these could be linearly related. However, this >is not a simple regression problem since both of the measured >time series which we are trying to fit contain measurement errors >and thus one of the basic assumptions of ordinary regression is violated. > >Are there any Mathematica functions available that could be used >for this type of time series fitting problem? > >-- Virgil > >