[SciPy-User] [OT] Transform (i.e., Fourier, Laplace, etc.) methods in Prob. & Stats.

David Goldsmith d.l.goldsmith@gmail....
Wed Nov 25 18:31:00 CST 2009

On Wed, Nov 25, 2009 at 3:45 PM, <josef.pktd@gmail.com> wrote:

> On Wed, Nov 25, 2009 at 6:24 PM, David Goldsmith
> <d.l.goldsmith@gmail.com> wrote:
> > Good info, thanks; I'll look up "your" thread, Josef, on the archive and
> run
> > down what look like relevant references.  (FWIW, my interest is that I'm
> > helping out (nominally, "tutoring," but this level, it's more akin to
> being
> > a sounding board, checking his derivations, and "reminding" him of
> various
> > subtleties that are emphasized in math, but not necessarily in EE, etc.)
> > this guy working on his dissertation on air traffic control automation
> using
> > wireless communication protocols, very probability heavy stuff, and for
> the
> > second time yesterday, he presented me with a transform application - in
> > this instance, the "Z" transform - in this probability-heavy stuff, and
> this
> > is outside of my training in probability, so I want to "bone-up.")
> Thanks
> > again,
> I always have to look for your reply because you don't follow our
> bottom-posting
> policy.

Sorry, I tend to "follow" when I'm saying something in direct response to
something I'm replying to and/or when I think that I'm likely _not_
terminating the thread, but when I'm responding generally and/or think that
I am likely terminating the thread, then I tend to just reply at the top.
I'll try to remember that we have a policy. :-)

I have seen the z-transform only in the context of time series analysis
> http://en.wikipedia.org/wiki/Z-transform
> especially this
> http://en.wikipedia.org/wiki/Z-transform#Linear_constant-coefficient_difference_equation
> covered to some extend in scipy.signal, lfilter and lti

Part of the problem was that it wasn't clear to either of us - myself or my
"student" - why the authors of this particular paper were using the
z-transform at all where they were - it seemed their result was easily
derivable w/out it, so we were both baffled.

so the other literature to Laplace transforms and characteristic functions
> might not be very closely related.

Perhaps not directly (in any event, presently, I'm interested in
theoretical/"analytical," i.e., not numerical, applications anyway), but my
philosophy has always been, if I can be directed to something that is closer
to on target than what I've been able to find on my own, then even if it's
not a bulls-eye, I can often find a bulls-eye in the reference's
references.  For example, "Chung (or any other book on graduate
probability)" sounds like a good starting point.  So thanks for reminding me
about the thread.  (I knew it sounded familiar: I contributed to it!  And on
that note, I "let it lie" at the time, but now feel I should say, admittedly
a little defensively, that of course Anne's comments were on the mark; the
only reasons I felt it necessary to add what I did about complex integration
over a closed path were: A) you had indicated that you were a bit of a
novice in the field, and the result I was giving is, perhaps arguably, the
subject's most fundamental result, and B) I felt that it was important that
you were aware of it because, if any of your functions _were_ analytic and
your paths closed, then you shouldn't be doing any (explicit) numerical (or
symbolic, for that matter) integration at all - you should just be
"hard-wiring" those integrals to zero!  And for what it's worth: every time
you integrate with respect to one (continuous) real variable, you're doing a
path integration - one so comparatively trivial that we don't call it that,
but a path integration nevertheless.) :-)


> Josef
> >
> > DG
> >
> > On Wed, Nov 25, 2009 at 2:41 PM, nicky van foreest <vanforeest@gmail.com
> >
> > wrote:
> >>
> >> Hi,
> >>
> >> 2009/11/25  <josef.pktd@gmail.com>:
> >> > On Wed, Nov 25, 2009 at 2:53 PM, David Goldsmith
> >> > <d.l.goldsmith@gmail.com> wrote:
> >> >> Are there enoug applications of transform methods (by which I mean,
> >> >> Fourier, Laplace, Z, etc.) in probability & statistics for this to be
> >> >> considered its own specialty therein?  Any text recommendations on it
> >> >> (even
> >> >> if it's only a chapter dedicated to it)?  Thanks,
> >> >>
> >> >
> >> > Some information is in the thread on my recent question
> >> > "characteristic functions of probability distributions"
> >> >
> >> > There is a large literature in econometrics and statistics about using
> >> > the characteristic function for estimation and testing.
> >> > The reference of Nicky for queuing theory uses mostly the Laplace
> >> > transform (for discrete distributions),
> >>
> >> It has been some time ago (more than 5 years...), but I recall that
> >> Whitt, in his articles on the numerical inversion of Laplace
> >> transforms, discretized Laplace transforms to facilitate the
> >> inversion, The distributions themselves are not necessarily discrete.
> >> One example would be the waiting time distribution of customers in a
> >> queue, which is continuous for most service and arrival processes.
> >>
> >> There is certainly potential for dedicated numerical inversion algo's
> >> for the Laplace transforms of density and distribution functions. The
> >> latter form a somewhat specialized sort of function. Distribution
> >> functions are 0 at -\infty, and 1 at \infty, and are non decreasing.
> >> They may also have discontinuities, but not too many. These properties
> >> may affect the inversion.  Besides these properties, the transforms
> >> are used to obtain insight into the behavior of the sum of independent
> >> random variables. Such sums can be rewritten as the product of the
> >> transforms of distribution. This product in turn requires inversion
> >> to, as some people call it, take away the Laplacian curtain.
> >>
> >> Nicky
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