Probability of Surface Realization Above Black Line in Gaussian Process Puzzle

[maka89]

I am looking at a surface where the height is described by a zero-mean gaussian process with $cov(h_1,h_2)=\sigma^2\exp[-0.5\frac{(x_1-x_2)}{a^2}]$.

Given that h(x=0) = 0, what is the probability that a surface realization will go above the black line going from x=0 to infinity in the figure? Thats all.

This is for a scattering problem I'm working on in physics :)

 

[andrewkirk]

What are $h_1$ and $h_2$ and how, if at all, do they relate to $x_1$ and $x_2$?

Which of the symbols represent random variables and which represent constants?

 

[maka89]

The height is modeled as a multivariate gaussian. $h_i$ refers to the height at the point $x=x_i$. The covariance function determince the elements of the covariance matrix. I.e. how to heights at different positions are correlated. The equation should be $cov(h_i,h_j) = \sigma^2\exp(-0.5\frac{(x_i-x_j)^2}{a^2})$. (Its more common to use i,j instead of 1,2 )

 

[andrewkirk]

Sorry to be obtuse, but I still don't understand the problem description.

How is the process multivariate?

As described, we have a scalar real random process that is the height, whose index variable is position.

If position is scalar then there are no vector variables in the specification, so the term multivariate does not appear to apply, and neither does the statement that it is a surface.

Alternatively, if position $x$ is vector, how are we to interpret the expression $\exp[-0.5\frac{(x_i-x_j)^2}{a^2}]$, since multiplication of vectors is not defined?

 

[maka89]

No problem. Its good that you are interested at least :D

So, you are right its a scalar Gaussian process with x as an index variable.

E[h(x)] = 0
E[h(x)h(x')]=$\sigma^2\exp(-0.5\frac{(x-x')^2}{a^2})$

Never mind the indices I wrote for h and x if you find them confusing. I conceptualize it like x is discretized(But with infinitely small spacing) which I find useful sometimes when using GP's for regression. Therefore the indices.

 

[andrewkirk]

I'm afraid I don't have an answer, but I can at least give a formal statement of the problem. The height process is a Brownian Motion (aka Wiener Process), that is typically denoted by $B_t$ or $W_t$. Let's use $B_t$.

The question is as follows:

For a Brownian Motion $B_t$, what is an expression for
$$Prob\left(\sup_{t\geq 0}\left(B_t-t\right)>0\right)$$

If we could get an expression for
$$Prob\left(\sup_{0\leq s\leq t}\left(B_s-s\right)>0\right)$$
then the answer would be the limit as $t\to\infty$ of that expression.

I suspect that the route to an answer will involve the Reflection Principle, but I don't currently have an idea about how to get there.

If nothing comes up soon, I'd suggest posting the problem, stated as in this post, on Maths Stack Exchange. There are lots of posters there with strong Brownian Motion skills. eg see this page, which is about a problem that has some similarity to yours, but is not the same.

 

[maka89]

Thanks for the answer :) Hadn't considered that angle. I will work on it some more and then write here if I post on stack exchange or solve it. I attach a new figure that is more realistic. The old one allowed all slopes on the point where the scattering occurs. This one is constraned to having a slope at the same point that obeys the law of reflection(local incident and scattering angle on the surface is equal). And includes the scattered beam.

For a specific choice of surface parameters and incident and scattering angle, it is quite easy to montecarlo sample a lot of surface realizations calculate the probablility of a beam being blocked. But I'm hoping to be able to formulate it a bit more, well formal.