On 20 Jun, 23:44, Tim Wescott <t...@seemywebsite.com> wrote:
> On Sat, 20 Jun 2009 11:37:27 -0700, Rune Allnor wrote:
> > On 20 Jun, 20:14, Tim Wescott <t...@seemywebsite.com> wrote:
> >> On Sat, 20 Jun 2009 11:04:38 -0700, Rune Allnor wrote:
> >> > On 20 Jun, 19:50, Tim Wescott <t...@seemywebsite.com> wrote:
>
> >> >> My choice of oddball integrals was intentional, as I want to go on
> >> >> to calculating various moments for the probability distributions of
> >> >> the surface of the hypersphere when 3D probability distributions are
> >> >> mapped onto it. *Clearly if I map a tight Gaussian distribution onto
> >> >> the hypersphere with a standard deviation that's much smaller than
> >> >> the hypersphere radius the resulting probability distribution will
> >> >> be easy; it's figuring out what happens as that probability
> >> >> distribution opens up that's making my brain cramp.
>
> >> > I'm a bit curious about what kind of problem leads you out in such
> >> > kinds of calculations?
>
> >> Unscented transformations for quaternion PDFs used to represent angles
> >> in a (hopefully soon-to-be) unscented Kalman filter.
>
> > The expected value is a 4-vector pointing in some desired direction and
> > the PDF represents the probability distribution of actual directions?
>
> Yup. *Given a PDF I'd _like_ to be able to solve for the expected value
> of the four vector elements, as well as their cross-correlation. *I'd
> like this to be in a form that's tractable enough that I can code it into
> an algorithm without either making people's heads explode when they read
> it and without making the processor bog down.
>
> But for now I'll just settle with being able to get the element means and
> variances out of the thing for a variety of variances of the Gaussians,
> or a clear indication that if I insist on using Gaussians the math is
> going to be hopelessly intractable.

I did have a go at this for a Gaussian on the unit circle .
Note that I did this at 4AM (no intoxication though), arithmetics
has never been a force of mine, my pencil broke and I generally
am not able to read my own handwriting.

In other words, don't trust the details in what follows:

================================================== ===========
Leaving out the scaling factors, the Gaussian PDF on the
circle becomes (view with fixed-width font)

inf
g(phi) = sum exp-((phi - nu)/sigma + 2 n pi)^2 [1]
n = -inf

since there is a possibility that the angle error in reality
includes any number of revolutions around the unit circle.

For simplicity, define

a = (phi - nu)/sigma [2a]
b = 2pi [2b]

and substitute into [1]:

inf
g(phi) = sum exp-(a + bn)^2 [3]
n = -inf

inf
= sum exp-(a^2+2abn+b^2n^2) [4]
n = -inf

inf
= exp(a^2) sum exp(-2abn)exp(-b^2 n^2) [5]
n = -inf

Next, convert from a double-sided infinite sum to a
one-sided infinite sum, using

inf inf
sum exp(-x) = 1 + 2 sum cosh(x). [6]
n = -inf n=1

Apply [6] to the first exponential inside the summation
in [5] to find

inf
g(phi) = exp(a^2){1 + 2 sum cosh(2abn)exp(-b^2 n^2)} [7]
n=1

To get the end result, substitute [2a-b] into [7] and insert
the missing scaling factors.
================================================== ===========

That's as far as I can get. Because of the n^2 fector in
the exponent inside the sum, the summation formula for
geometric series can't be used (at least that's how I
understand it).

Now, the form of [7] seems to be rather nice: It's a basic
Gaussian (the outer term) and a lot of correction terms.
Since the exponential correction term is dominated by n^2,
you probably don't need to many correction terms to get an
impression of the PDF. Note also that the cosh term contains
the (phi-nu)/sigma correction, so let sigma -> inf and
investigate what hapepns when the PDF 'opens up' towards
uniform. Maybe one is able to work this out further,
using power series etc.

Or you can mage a worst-case assumption about the series
being a sum of cosh(2abn)exp(-b^2 n) terms (substituted
n for n^2 in the exponent) and use the formula for a
geometric series to proceed.

Now, you do introduce an error, but you push the PDF
towards something 'worse' in the sense

g(phi)_n > g(phi)_n^2,

where the subscript indicates what n term you used in
the exponential.

Whatever blunders and errors I might have made in the
above, here is the maths challenge:

Now, this excercise (or the corrected / amended version
of it...) works well when working on a unit circle in
2D space. Even generalizing it to the spheric shell in
3D space is beyond me: The above works for the azimuth
angle in the spherical coordinate system. I have no idea
how to handle the elevation angle: It covers only half
the sphere (from 'pole' to 'pole') and once it extends
beyond +/- pi, it couples back in on the azimuth angle,
which flips by pi.

I don't see how to handle this for the problem at hand.

But, as somebody made a point of earlier this week:
That's what mathematicians are for - handle the
technicalities in the theory we mere mortal engineers
have to use.

Any takers among the mathematicians? How does one
extend the 'infintely many wrap-arounds' on the
circle in 2D space to a sphere in 3D space?

Rune