[R] Counterintuitive Simulation Results

From: Spencer Graves <spencer.graves_at_pdf.com>
Date: Fri 05 Aug 2005 - 02:16:06 EST

          I wonder if someone can help me understand some counterintuitive simulation results. Below please find 12 lines of R code that theoretically, to the best of my understanding, should produce essentially a flat line with no discernable pattern. Instead, I see an initial dramatic drop followed by a slow rise to an asymptote.

          The simulation computes the mean of 20,000 simulated trajectories of 400 observations each of a one-sided Cusum of independent normal increments with mean EZ[t] = (-0.1) and unit variance. Started with any initial value, the mean of the Cusum should approach an asymptote as the number of observations increases; when started at that asymptote, it should theoretically stay flat, unlike what we see here.

          I would think this could be an artifact of the simulation methodology, but I've gotten essentially this image with several independently programmed simulations in S-Plus 6.1, with all six different random number generators in R 1.9.1 and 2.1.1 and with MS Excel. For modest changes in EZ[t] < 0, I get a different asymptote but pretty much the same image.

#################################################
simCus5 <- function(mu=-0.1, Qp0=4.5, maxTime=400, nSims=20000){

   Qp.mean <- rep(NA, maxTime)
   Qp.t <- rep(Qp0, nSims)
   for(i in 1:maxTime){

     z.t <- (mu + rnorm(nSims))
     Qp.t <- pmax(0, Qp.t+z.t)
     Qp.mean[i] <- mean(Qp.t)

   }
   Qp.mean
}
set.seed(1)
plot(simCus5(Qp0=4.5))
#################################################
	  Thanks for your time in reading this.
	  Best Wishes,
	  Spencer Graves

Spencer Graves, PhD
Senior Development Engineer
PDF Solutions, Inc.
333 West San Carlos Street Suite 700
San Jose, CA 95110, USA

spencer.graves@pdf.com
www.pdf.com <http://www.pdf.com>
Tel: 408-938-4420
Fax: 408-280-7915



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