From: Spencer Graves <spencer.graves_at_pdf.com>

Date: Fri 05 Aug 2005 - 04:20:08 EST

*>
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> [1] -0.1059246

*>
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*> #And for those paths already at zero, we are adding
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*>
*

*>
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> [1] 0.3482376

*>
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*> To see a simulation a bit closer to what you were expecting, replace the
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*> starting values with a random distribution with mean Qp0.
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*>
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*> i.e. replace
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*>
*

*>
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*> with
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*>
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*>
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*>
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> Robert

*>
*

*>
*

*> -----Original Message-----
*

*> From: Spencer Graves [mailto:spencer.graves@pdf.com]
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*> Sent: Thursday, August 04, 2005 12:16 PM
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*> To: r-help@stat.math.ethz.ch
*

*> Subject: [R] Counterintuitive Simulation Results
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*>
*

*>
*

*> I wonder if someone can help me understand some
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*> counterintuitive
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*> simulation results. Below please find 12 lines of R code that
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*> theoretically, to the best of my understanding, should produce
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*> essentially a flat line with no discernable pattern. Instead, I see an
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*> initial dramatic drop followed by a slow rise to an asymptote.
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*>
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*> The simulation computes the mean of 20,000 simulated
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*> trajectories of
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*> 400 observations each of a one-sided Cusum of independent normal
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*> increments with mean EZ[t] = (-0.1) and unit variance. Started with any
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*>
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*> initial value, the mean of the Cusum should approach an asymptote as the
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*>
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*> number of observations increases; when started at that asymptote, it
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*> should theoretically stay flat, unlike what we see here.
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*>
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*> I would think this could be an artifact of the simulation
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*> methodology, but I've gotten essentially this image with several
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*> independently programmed simulations in S-Plus 6.1, with all six
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*> different random number generators in R 1.9.1 and 2.1.1 and with MS
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*> Excel. For modest changes in EZ[t] < 0, I get a different asymptote but
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*>
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*> pretty much the same image.
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*>
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*> #################################################
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*> simCus5 <- function(mu=-0.1, Qp0=4.5, maxTime=400, nSims=20000){
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*> Qp.mean <- rep(NA, maxTime)
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*> Qp.t <- rep(Qp0, nSims)
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*> for(i in 1:maxTime){
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*> z.t <- (mu + rnorm(nSims))
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*> Qp.t <- pmax(0, Qp.t+z.t)
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*> Qp.mean[i] <- mean(Qp.t)
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*> }
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*> Qp.mean
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*> }
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*> set.seed(1)
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*> plot(simCus5(Qp0=4.5))
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*> #################################################
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*>
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*> 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
*

*>
*

*> ______________________________________________
*

*> R-help@stat.math.ethz.ch mailing list
*

*> https://stat.ethz.ch/mailman/listinfo/r-help
*

*> PLEASE do read the posting guide!
*

*> http://www.R-project.org/posting-guide.html
*

*>
*

Date: Fri 05 Aug 2005 - 04:20:08 EST

Hi, Robert:

Yes, I understand now. Thanks very much for your insight.

Best Wishes, Spencer Graves

McGehee, Robert wrote:

> Spencer,

*> On the first iteration of your simulation, all of the Qp.t + z.t < 0, so
**> you're adding a vector of rep(4.5, 20000) to a random distribution with
**> mean -0.1. So one would expect on iteration 2, your mean would have
**> dropped by about 0.1 (which it does). This process continues until about
**> the 20th iteration when we start seeing that a large number of our
**> initial starting points are floored at zero (because of the pmax). For
**> points greater than zero, we continue to subtract an average of 0.1
**> (actually less than this), but for those points already at zero, we're
**> actually adding a mean of 0.348 (since we can never subtract from a zero
**> number in this case), which starts the trajectory moving upward towards
**> its asymptote.
**>
**> #That is, for those paths far above 0.1, we are subtracting
**>
*

>>mean(rnorm(10000, mean = -0.1))

> [1] -0.1059246

>>mean(pmax(0, rnorm(10000, mean = -0.1)))

> [1] 0.3482376

>>Qp.t <- rep(Qp0, nSims)

>>Qp.t <- rnorm(nSims, Qp0, sd = 3.7)

> Robert

-- 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 ______________________________________________ R-help@stat.math.ethz.ch mailing list https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide! http://www.R-project.org/posting-guide.htmlReceived on Fri Aug 05 04:25:55 2005

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