Need a candidates for a random-walk, but limited, perturbation.

Question

I have some time-dependent data that I would like to perturb. Just pulling numbers out of a uniform or normal distribution won't suffice. The jump in perturbation from one time to the next should have a certain smoothness to it. So, this sounds like a random walk. However, the perturbations cannot grow beyond a certain physically realistic size. So pure random walk won't work either. I guess I could put in a random walk perturbation and just chop it off if it gets too big. But this seems artificial. Is there any sort of random-walk-type perturbation I could use to achieve both of my goals here? Thanks.

Answer 1

If anyone's interested, after some thought, it looks like Brownian motion fits that bill.

If $w_i$ is a set of random numbers like in a random walk, e.g. 1,1,-1,1,-1,-1, etc. a random walk sequence can be constructed like so:

$x_i = \sum_{j=0}^i w_j$

Brownian motion just puts an exponential damping on the sum so that terms back beyond some specified time are not important:

$x_i = \sum_{j=0}^i \exp((j-i)/N)w_j$

Less physically, but maybe more effectively, you can just truncate the sum:

$x_i = \sum_{j=i-i_0}^iw_j$

These are two ways of killing the importance of data in the "distant past" and they both kill the low frequency parts of the random walk that cause unphysically (in my case) large perturbations.