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

josef.pktd@gmai... josef.pktd@gmai...
Wed Nov 25 17:45:38 CST 2009

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,

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

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

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 enough 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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