Question about Gaussian distribution

[pamparana]

Hello everyone,

I am trying to understand situations under which Gaussian distribution would apply. For example, I read somewhere that if you have some ink drop on a porous paper, then the distribution of the displacement of ink particles is approximately gaussian.

I am trying to figure out why this should be? Is it because the diffusion in the 2 directions are independent and due to the central limit theoram they should follow Gaussian distribution.? Also, what happens if the paper is very narrow in one dimension and the ink can move freely in one direction but not the other. Would the distribution be still Gaussian?

Also, what does one mean when one says that a gaussian function has only a single directional maximum? I read this in a paper about diffusion MRI and the failure of the diffusion tensor model to capture multidirectional diffusion.

Would really appreciate your thoughts on this.

Thanks,

Luca

 

[bpet]

Hello everyone,

I am trying to understand situations under which Gaussian distribution would apply. For example, I read somewhere that if you have some ink drop on a porous paper, then the distribution of the displacement of ink particles is approximately gaussian.

I am trying to figure out why this should be? Is it because the diffusion in the 2 directions are independent and due to the central limit theoram they should follow Gaussian distribution.? Also, what happens if the paper is very narrow in one dimension and the ink can move freely in one direction but not the other. Would the distribution be still Gaussian?

Also, what does one mean when one says that a gaussian function has only a single directional maximum? I read this in a paper about diffusion MRI and the failure of the diffusion tensor model to capture multidirectional diffusion.

Would really appreciate your thoughts on this.

Thanks,

Luca

Sounds like your questions are on the right track - there's a lot of assumptions behind the Gaussian approximation, e.g. summing lots of small independent movements (for CLT), particles moving independently (for LLN), homogeneity (for gaussian) and directional independence (for 2D gaussian) to name a few. Which mathematical description are you working with? (e.g. PDE or SDE)

 

[pamparana]

Hello,

Thanks for replying to me. Actually I was just looking at the diffusion equation and I now sort of understand why this should be Gaussian due to the underlying process being modeled as a Brownian motion. I do not know much about PDEs and I am afraid your question is much deeper than my limited mathematical knowledge :(

I was thinking that if diffusion is unrestricted, then you have bivariate normal distribution with variance increasing with time. However, if diffusion is preferred in one direction and restricted in another, then what would be the nature of this distribution? Surely, it cannot be described as a Gaussian anymore.

Also, any thoughts on what it means by " a gaussioan function has single directional maximum"?

Thanks,
Luca

 

[bpet]

I was thinking that if diffusion is unrestricted, then you have bivariate normal distribution with variance increasing with time. However, if diffusion is preferred in one direction and restricted in another, then what would be the nature of this distribution? Surely, it cannot be described as a Gaussian anymore.

Also, any thoughts on what it means by " a gaussioan function has single directional maximum"?

Having two diffusion rates is ok if the directions are orthogonal - the level curves of the Gaussian will just be elliptical instead of circular. Limiting case will be effectively 1D diffusion.

Not sure what happens for the non-orthogonal case, or what "single directional maximum" means, sorry.

 

[pamparana]

Hi again,

Thanks for that. That makes sense. You raise one interesting point about the directions havng to be orthogonal. Why that restriction?

So going back to the ink blot diffusion through porous paper. One imagines free diffusion being in all directions. So, the gaussian model is not very adequate in that case?

Again, many thanks for taking the time to reply. The more I start reading, the more I realize what an idiot I am! :(

Cheers,
Luca

 

[bpet]

There's a theorem in linear algebra about symmetric matrices having orthogonal eigenvectors - this explains why ellipses have orthogonal axes.

With the ink blot example it's "equidirectional" because the axes are equal, so the ellipse is a circle and there is no preferred direction (also consider general eigenvectors of the identity matrix).

 

[pamparana]

Ahhhhh...thanks for that!

Much appreciated.

/Luca

 

[pamparana]

Hello,

One final question. Going back to the inkblot example... Say instead of one ink blot there are two ink blots that start at the same point and with different physical parameters that influence its distribution. Would it be reasonable to model the distribution as two separate gaussian functions and combine them through some sort of a weighted approach?

What about the fact that they might interact with each other and diffusion of one might impede the disffusion of the other? Then, I am guessing it would not be fair to still model the whole thing as a gaussian distribution...or would it?

Looking forward to hearing your views on this.

Thanks,
Luca

 

[bpet]

What about the fact that they might interact with each other and diffusion of one might impede the disffusion of the other?

Then you get nonlinear effects - I don't know much about that, sorry.

 

[SW VandeCarr]

First you need to understand the unbiased and uncorrelated model of random walks (in the discrete case) which is a diffusion model at the limit. The distribution of the position of a given particle is Gaussian in the continuous case for finite time t(i) in any given direction from the origin. The link below provides a good introduction. The Fokker-Planck model (2.4) is described for dimensions greater than one.

http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2504494/ [/URL]