Understanding Random Vectors and Hypersphere Distributions

[rabbed]

Hi

There are so many different kinds of probability distributions regarding a uniform distribution
of points on the surface or inside a hypersphere in 2D and 3D and it's hard to see the big picture
or any pattern between them. I'm confused and exhausted :)

The overall fuzzy plan is to go from 2D polar coordinates (integrating by r*dr*dθ) to 3D suitable
coordinates (integrating by whatever the equivalent might be).
In both cases, I want to multiply the independent components PDF's, get the CDF and invert it
to get the cartesian coordinates of a point that will be uniformly distributed inside the
hypersphere. I guess normalization after this will give me a random vector?
Then I want to show that independent standard normal distributed coordinates will also accomplish
this after normalization, in a more unified way.

Does that plan make sense from a mathematician point of view and have I got something wrong?

Right now I'm stuck at the 2D case, I guess I want to multiply V_PDF(v) = 1/(2*pi) with some R_PDF(r).
How do I get such an R_PDF(r)?
I should end up with (x,y) = r*sqrt(u)*e^(i*2*pi*v) with u and v ~ U(0,1).

/rabbed

 

[mfb]

How do I get such an R_PDF(r)?

Your area between r and r+dr is proportional to r, so the right pdf should have the same property. That leads to r^2 plus normalization for the cumulative probability distribution. Compare this to the area where the radial coordinate is smaller than r...
For low dimensions, suitable distributions for cartesian coordinates are so much easier.

 

[rabbed]

First lame attempt:

R_PDF(r) = k
V_PDF(v) = 1/(2*pi)

RV_PDF(r,v) = R_PDF(r)*V_PDF(v) = k*r/(2*pi)

RV_CDF(r,v) =
P(R < r AND V < v) =
integral wrt v from 0 to 2*pi of integral wrt r from 0 to x of RV_PDF(r,v) * r*dr*dv =
integral wrt v from 0 to 2*pi of integral wrt r from 0 to x of k/(2*pi) * r*dr*dv =
integral wrt v from 0 to 2*pi of dv * [k*r^2/(2*pi)] wrt r from 0 to x =
integral wrt v from 0 to 2*pi of k*x^2/(2*pi) * dv =
[v*k*x^2/(2*pi)] wrt v from 0 to 2*pi =
k*x^2 = 1
so k = 1/(x^2) ?

Not sure what that means :)

Also, do I have to do this in two steps where the second step gives the cartesian coordinate Y using Y_PDF(y) calculated below, and also include R_PDF(r) to get the CDF and then invert it?

V_PDF(v) = 1/(2*pi)
Y = r*sin(V)

Y_CDF(y) =
integral wrt v from pi-arcsin(y/r) to pi+arcsin(y/r) of V_PDF(v)*dv =
integral wrt v from pi-arcsin(y/r) to pi+arcsin(y/r) of 1/(2*pi)*dv =
[v/(2*pi)] wrt v from pi-arcsin(y/r) to pi+arcsin(y/r) =
(pi+arcsin(y/r))/(2*pi) - (pi-arcsin(y/r))/(2*pi) =
( (pi+arcsin(y/r)) - (pi-arcsin(y/r)) )/(2*pi) =
( pi+arcsin(y/r) - pi + arcsin(y/r) )/(2*pi) =
2*arcsin(y/r)/(2*pi) =
arcsin(y/r)/pi

Y_PDF(y) =
d[arcsin(y/r)/pi]/dy =
1/(pi*sqrt(r^2-y^2))

 

[mfb]

so k = 1/(x^2) ?

Not sure what that means :)

If x is your radius, that prefactor will give the right normalization.

Y_CDF(y) = [...] = arcsin(y/r)/pi
Y_PDF(y) = [...] = 1/(pi*sqrt(r^2-y^2))

Looks like a reasonable result. I don't think it is helpful if you want to generate points, however, as X and Y are correlated then.

A fast way if you have access to random numbers with uniform distribution: scale/shift the distribution so it goes from -r to r. Calculate a point (x,y), compare x^2+y^2 to r^2: if the point is in the circle, you are done, if it is outside, repeat the process. While it is not guaranteed to succeed, you have a pi/4 > 75% chance each time, so on average you just need ~2.6 random numbers and a few multiplications and additions.

 

[rabbed]

Ok, thanks..
I spotted some errors and did some research and there is still something not adding up.
So if we have a disc with radius K on which we randomize points by using the random variables R (radius) and V (angle):

R_PDF(r) = c where (0 < r < K)
V_PDF(v) = 1/(2*pi) where (0 < v < 2*pi)

RV_PDF(r,v) = R_PDF(r)*V_PDF(v) = c/(2*pi)

RV_CDF(r,v) =
P(R < r AND V < v) =
integral wrt v from 0 to 2*pi of integral wrt r from 0 to K of RV_PDF(r,v) * r*dr*dv =
integral wrt v from 0 to 2*pi of integral wrt r from 0 to K of c/(2*pi) * r*dr*dv =
integral wrt v from 0 to 2*pi of dv * [c*r^2/(4*pi)] wrt r from 0 to K =
integral wrt v from 0 to 2*pi of c*K^2/(4*pi) * dv =
[v*c*K^2/(4*pi)] wrt v from 0 to 2*pi =
2*pi*c*K^2/(4*pi) =
c*K^2/2 = 1

so c = 2/(K^2) and we get:
R_PDF(r) = 2/(K^2)

but this should be proportional to r?

am I using the area element r*dr*dv wrong in the integration?

 

[mfb]

You forgot the factor r in R_PDF(r). It has to be proportional to r.

am I using the area element r*dr*dv wrong in the integration?

No, that is fine.

 

[rabbed]

But won't that give me the following?

R_PDF(r) = c*r where (0 < r < K)
V_PDF(v) = 1/(2*pi) where (0 < v < 2*pi)

RV_PDF(r,v) = R_PDF(r)*V_PDF(v) = c*r/(2*pi)

RV_CDF(r,v) =
P(R < r AND V < v) =
integral wrt v from 0 to 2*pi of integral wrt r from 0 to K of RV_PDF(r,v) * r*dr*dv =
integral wrt v from 0 to 2*pi of integral wrt r from 0 to K of c*r/(2*pi) * r*dr*dv =
integral wrt v from 0 to 2*pi of integral wrt r from 0 to K of c*r^2/(2*pi) * dr*dv =
integral wrt v from 0 to 2*pi of dv * [c*r^3/(6*pi)] wrt r from 0 to K =
integral wrt v from 0 to 2*pi of c*K^3/(6*pi) * dv =
[v*c*K^3/(6*pi)] wrt v from 0 to 2*pi =
2*pi*c*K^3/(6*pi) =
c*K^3/3 = 1
so c = 3/(K^3)

R_PDF(r) = r*3/(K^3)

 

[mfb]

Looks good.

 

[rabbed]

Something is not adding up.. I found this while doing some research:
http://stackoverflow.com/questions/5837572/generate-a-random-point-within-a-circle-uniformly
the answer containing "C*r" sounds pretty close to what we're getting.

After I get R_PDF(r), the next step should be to get it's CDF and invert it to get r = K*sqrt(u)

 

[rabbed]

Can anyone explain this?

I'm pretty sure R_PDF(r) = 2*r/(K^2)

But how can it be derived?