Andrew M. Ross wrote: > Nischal Piratla wrote: >> Given the random process X = [X1,X2,.Xt,Xt+1..XN]'. I have got >> scatter plots with Xt, Xt+i for all i = 1,2,3... If these plots do >> not have any pattern and I see a bunch of patches, can I conclude >> that the random variables X1, X2, X3, .. are independent? If not, >> can you please provide a counter argument. >> Being a computer engineer, I would really appreciate a reference. >> Nischal Piratla http://lamar.colostate.edu/~nischal > > It is a good idea to do the plots you suggested. If one was more > paranoid, you could also do plots at lag 2, and at lag 3, etc. > If you compute the AutoCorrelation Function (ACF), this will give you > the correlation at each lag value. Many programs (such as S/Splus/R) > will also draw dashed lines on each side of zero that are significance > bands; if the ACF is outside those lines, then it is statistically > significantly different from zero. So if the ACF falls within the > lines, you can be fairly confident that the random variables are > independent (at least linearly, since the usual correlation is a > measurement of linear relationship). > > Depending on the distribution of the X values, you might also want to > do your plots after a transformation. In particular, if your > variables have an Exponential distribution, they tend to cluster > toward the axes, and it's hard to spot any patterns in the mess. > However, if you plot exp(-Xt) versus exp(-(Xt+1)), the plot should > have a U(0,1) distribution on both axes. > > Sorry that I don't have any references for you. I look forward to > hearing other people's ideas too. > > Andrew
Some additional ideas ... (i) if the ACF's do not show correlations, then possibly try ACF's for the series constructed by subracting an overall mean and squaring (or taking the absolute value)... but initially try the cross-correlation function of the constructed series with the original. (ii) as a simple and more general alternative to the possibility suggested for the "Exponential case", try transforming using the ranks of the data in the original series (convert to (0,1) for convenience). For this constructed series try both ACF suggestion and scatter plots. David Jones . . ================================================================= Instructions for joining and leaving this list, remarks about the problem of INAPPROPRIATE MESSAGES, and archives are available at: . http://jse.stat.ncsu.edu/ . =================================================================
