[SciPy-User] characteristic functions of probability distributions
Thu Nov 5 23:02:47 CST 2009
On Thu, Nov 5, 2009 at 11:19 PM, Anne Archibald
> 2009/11/5 David Goldsmith <firstname.lastname@example.org>:
>> On Mon, Nov 2, 2009 at 12:51 PM, nicky van foreest <email@example.com>
>>> Hi Josef,
>>> > Second related question, since I'm not good with complex numbers.
>>> > scipy.integrate.quad of a complex function returns the absolute value.
>>> > Is there a numerical integration function in scipy that returns the
>>> > complex integral or do I have to integrate the real and imaginary
>>> > parts separately?
>>> You want to compute \int_w^z f(t) dt? When f is analytic (i.e.,
>>> satisfies the Cauchy Riemann equations) this integral is path
>>> independent. Otherwise the path from w to z is of importance. You
>>> might like the book Visual Complex Analysis by Needham for intuition.
>> Furthermore, if f is analytic in an (open) region R homotopic to an (open)
>> disc, then the integral (an integer number of times) around *any* _closed_
>> path wholly in R is identically equal to zero; there's a similar statement
>> (though the end value is a multiple of 2ipi) if f has only poles of finite
>> order in R. (Indeed, these properties should be used to unit test any
>> numerical complex path integration routine.) Are any of your paths closed?
> This may well be a red herring. It happens fairly often (to me at
> least) that I want to integrate or otherwise manipulate a function
> whose values are complex but whose independent variable is real.
> Such a function can arise by substituting a path into an analytic
> function, but there are potentially many other ways to get such a
> thing - for example you might choose to represent some random function
> R -> R2 as R -> C instead. Even if it's obtained by feeding a path
> into some function from C -> C, it happens very often that that
> function isn't analytic - say it involves an absolute value, or
> involves the complex conjugate.
> There are definitely situations in which all the clever machinery of
> analytic functions can be applied to integration problems (or for that
> matter, contour integration may be the best way available to evaluate
> some complex function), but there are also plenty of situations where
> what you want is just a real function whose values happen to be
> complex numbers. (Or vectors of length n for that matter.) But I don't
> think that any of the adaptive quadrature gizmos can handle such a
> case, so you might be stuck integrating the real and imaginary parts
> If you *are* in a situation where you're dealing with an analytic
> function, then as long as you're well away from its poles and your
> path is nice enough, you may find that it's very well approximated by
> a polynomial of high degree, which will let you use Gaussian
> quadrature, which can very easily work with complex-valued functions.
> The Romberg integration might even work unmodified.
Sorry for not coming back to this earlier,
Thanks Nicky, I looked at some papers by Ward Whitt and they look
interesting but much more than what I want to chew on right now. There
is more background, that I would have to read, than I have time right
now for this. I finally added "matlab" to my google searches, and I
think I found some references that use discretization and fft more
The integration problem should be pretty "nice", just a continuous
fourier transform and the inverse
For many distributions there is an explicit formula for both the
density and the characteristic function, e.g. normal
For some distributions only the characteristic functions has a closed
form expression, and the pdf or cdf has to be recovered numerically,
and I would have liked to have a generic method to go between the two.
I don't think I ever needed a path integral in my life, and I'm pretty
much a newbie to complex numbers, so parts of your explanations are
still quite a bit over my head. I think, I will come back to this
after I looked more at the examples where the estimation of a
statistical model or of a distribution is done in terms of the
characteristic function instead of the density.
The immediate example that I had tried, was (integration from -large
number to +large number)
integral exp(i t x)dF(x) = integrate.quad(exp(itx)*f(x))
or do I have to do
integral exp(i t x)dF(x) = integrate.quad(real(exp(itx)*f(x))) + j *
or is there another way?
The solution/integral might be either real or complex.
>>> > Thanks,
>>> > Josef
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