Hi Petr,
Thank you very much . I will let you know my progress!
Do you think that sampling from the traingular distribution will also be
good enough? Will it provides similar results?
Val
On Wed, Feb 1, 2012 at 12:48 PM, Petr Savicky <savi...@cs.cas.cz> wrote:
> On Wed, Feb 01, 2012 at 11:01:21AM -0500, Val wrote:
> > Hi Petr,
> >
> > Thank you very much. It looks we are almost there.
> >
> >
> > #an example, use a function approximating your graphs
> > * f <- function(x) { 0.7 - 1.25*((x - 1)^2 - 0.4)^2 }
> > f <- function(x) { 0.2 - 1.5*((x - 1)^2 - 0.1)^2 }
> >
> > #plot function f(x)
> > # x <- seq(0, 2, length=51)
> > x <- seq(0.01, 1.75, length=51)
> >
> > y <- f(x)
> > plot(x, y, type="l")
> >
> > The above plot may give the desired result.
> >
> > Here are the the conditions.
> >
> > 1. Y- value is always positive.
>
> I used a polynomial as f(x), since this is simple.
> However, for a better tuning of the shape, a spline
> may be better.
>
> > 2. Can I set the mean and SD of the value of X?
> > Example mean of x= 24 and SD =5
>
> For these conditions, it may be better to shift the
> function by 24 to the left, so the mean will be 0
> and the second moment will be 25. Then try to find
> a spline with a shape, which you require, and with
> first moment 0 and second moment 25 computed as follows.
>
> # example spline
> library(splines)
> xOrig <- 2*(-5:5)
> yOrig <- c(0, 3, 3.9, 4, 3.9, 3.9, 3.9, 4, 3.9, 3, 0)
> ispl <- interpSpline(xOrig, yOrig)
>
> # see the shape
> x <- seq(-10, 10, length=10001)
> y <- predict(ispl, x)$y
> stopifnot(y >= 0)
> plot(x, y, type="l")
>
> # compute the moments
> moment1 <- sum(x*y)/sum(y)
> moment2 <- sum(x^2*y)/sum(y)
>
> Here, we get
>
> moment1
> [1] -2.484611e-17
>
> moment2
> [1] 25.88646
>
> I think that a trial and error method may be sufficient
> for adjusting the original points to get [0, 25]
> within a small error.
>
> Another option is to create several candidate splines
> and consider their mixture. The moments are linear
> functions of the mixture parameters, so the mixture
> parameters may be obtained as a solution of a system
> of linear equations. If we have three splines such
> that the triangle between the corresponding points
> [moment1, moment2] contains the point [0, 25], then
> the mixture parameters will be nonnegative and the mixture
> well defined.
>
> Petr.
>
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