one simple idea is to generate correlated normals (vector multivariate normal), and then use the cumulative distribution function F_i of component i such: F_i(X_i), which is uniform.
Kjetil (this will not preserve tha value of the correlation coefficient, so you must experiment) On Mon, Feb 21, 2011 at 7:30 PM, Mike Marchywka <marchy...@hotmail.com> wrote: > > > > > > > ---------------------------------------- >> Date: Mon, 21 Feb 2011 15:53:26 +0100 >> From: erich.neuwi...@univie.ac.at >> To: marchy...@hotmail.com >> CC: soren.fau...@biology.au.dk; r-help@r-project.org >> Subject: Re: [R] Generating uniformly distributed correlated data. >> >> We want to generate a distribution on the unit square with the following >> properties >> * It is concentrated on a "reasonable" subset of the square, >> and the restricted distribution is uniform on this subset. >> * Both marginal distributions are uniform on the unit interval. >> * All horizontal and all vertical cross sections are sets of lines >> segments with the same total length >> >> If we find a geometric figure with these properties, we have solved the >> problem. >> >> So we define the distribution to be uniform on the following area: >> (it is distorted but should give the idea) >> >> x***/-----------------/***x >> |**/-----------------/****| >> |*/-----------------/*****| >> |/-----------------/******| >> |-----------------/******/| >> |----------------/******/-| >> |---------------/******/--| >> |--------------/******/---| >> |-------------/******/----| >> |------------/******/-----| >> |-----------/******/------| >> |----------/******/-------| >> |---------/******/--------| >> |--------/******/---------| >> |-------/******/----------| >> |------/******/-----------| >> |-----/******/------------| >> |----/******/-------------| >> |---/******/--------------| >> |--/******/---------------| >> |-/******/----------------| >> |/******/-----------------| >> |******/-----------------/| >> |*****/-----------------/*| >> |****/-----------------/**| >> x***/-----------------/***x >> >> There is the same number of stars in each horizontal row and each >> vertical column. >> >> >> So we define >> g(x1,x2)= 1 abs(x1-x2) <= a or >> abs(x1-x2+1) <= a or >> abs(x1-x2-1) <= a >> 0 elsewhere >> >> The total area of the shape is 2*a. >> The admissible range for a is <0,1/2> >> therefore >> f(x1,x2)=g(x1,x2)/(2*a) >> is a density functions. >> This is where simple algebra comes in. >> This distribution has >> expected value 1/2 and variance 1/12 for both margins >> (uniform distribution), and it has >> covariance = (1-3*a+2*a2)/12 >> and correlation = 1 - 3*a + 2*a2 >> >> The inverse function of 1 - 3*2 + 2*a2 is >> (3-sqrt(1+8*r))/4 >> >> Therefore we can compute that our distribution with >> a=(3-sqrt(1+8*r))/4 >> will produce a given r. >> >> >> Ho do we create random numbers from this distribution? >> By using conditional densities. >> x1 is sampled from the uniform distribution, and for a give x1 >> we produce x2 by a uniform distribution on the along the vertical cross >> cut of the geometrical shape (which is either 1 or 2 intervals). >> And which is most easily implemented by using the modulo operator %%. >> >> This mechanism is NOT a convolution. Applying module after the addition >> makes it a nonconvolution. Adding independent random variables >> without doing anything further is a convolution, by applying a trimming >> operation, the convolution property gets lost. >> >> > The thing inside the mod allows convolution, as I mentioned the effect of > the mod is to move back the pieces that fall outside the desired range > and they happen to restore the uniform distribution. I thought my > explanation was simple and easy after the fact but not sure > it would have motivated the original design too well. > > > >> >> >> >> >> > > ______________________________________________ > R-help@r-project.org mailing list > https://stat.ethz.ch/mailman/listinfo/r-help > PLEASE do read the posting guide http://www.R-project.org/posting-guide.html > and provide commented, minimal, self-contained, reproducible code. > ______________________________________________ R-help@r-project.org mailing list https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide http://www.R-project.org/posting-guide.html and provide commented, minimal, self-contained, reproducible code.
