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I've been playing around with real-world data such as GPS tracks to try to see whether, e.g., the function $z(x)$ is fractal, where $x$ is the cumulative horizontal distance traveled along a hiking trail and $z$ is the elevation. I was interested in testing the hypothesis that this function is characterizable as fractional Brownian motion with the power spectrum depending on the frequency as $f^{-\beta}$ for some fixed exponent $\beta$.

I did FFTs on some data and found that a log-log plot of power versus frequency usually did look roughly linear, and the slopes I got corresponded to $\beta$ in the range of about 2.2 to 3.9. These values seem to cluster around $\beta=3$, which is a fractal dimension of 1. So the simple, nonglamorous explanation is probably just that this function is not fractal, and that kind of makes sense because the people building the trails are intentionally trying to make sure that the energy expended in hiking that trail is minimized (and certainly not infinite!). It also seems pretty difficult to tell whether these values are "real." There are many, many spurious effects that could dominate these results, e.g., the data may be discretized or interpolated, GPS data have probably already been through a Kalman filter, digital elevation models (DEMs) may have already been filtered, and so on.

But anyway, fractional Brownian motion is supposed to have a Hurst exponent $H\in(0,1)$, which corresponds to $\beta\in(1,3)$. Some of my beta values are greater than 3. So if we have a function with $\beta>3$, how do we interpret it? It seems like it would have a lower fractal dimension $D=(5-\beta)/2$ than a line, which would be odd. Do we expect artifacts of the analysis (aliasing, filtering of the data before I obtain it, ...) to dominate the fitted value of $\beta$ when the true $\beta$ is the non-fractal value of 3?

related: test for fractional brownian motion

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I've done some more reading, and I think I understand this better now. A useful reference is:

Fay et al., Estimators of Long-Memory: Fourier versus Wavelets, https://arxiv.org/abs/0801.4329

These large values of $\beta$ indicate nonstationarity, which means that it could be, e.g., the integral of fBm. For parameter estimation, the basic idea seems to be to differentiate it, and then use a standard method of estimation that is known to work for fBm. In my own application, it is actually the derivative that is of practical interest anyway, so this makes a lot of sense.

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