Characterising lognormal fractional-Brownian-motion density fields with
  a Convolutional Neural Network
  In attempting to quantify statistically the density structure of the
interstellar medium, astronomers have considered a variety of fractal models.
Here we argue that, to properly characterise a fractal model, one needs to
define precisely the algorithm used to generate the density field, and to
specify -- at least -- three parameters: one parameter constrains the spatial
structure of the field; one parameter constrains the density contrast between
structures on different scales; and one parameter constrains the dynamic range
of spatial scales over which self-similarity is expected (either due to
physical considerations, or due to the limitations of the observational or
numerical technique generating the input data). A realistic fractal field must
also be noisy and non-periodic. We illustrate this with the exponentiated
fractional Brownian motion (xfBm) algorithm, which is popular because it
delivers an approximately lognormal density field, and for which the three
parameters are, respectively, the power spectrum exponent, $\beta$, the
exponentiating factor, ${\cal S}$, and the dynamic range, ${\cal R}$. We then
explore and compare two approaches that might be used to estimate these
parameters: Machine Learning and the established $\Delta$-Variance procedure.
We show that for $2\leq\beta \leq 4$ and $0\leq{\cal S}\leq 3$, a suitably
trained Convolutional Neural Network is able to estimate objectively both
$\beta$ (with root-mean-square error $\epsilon_{_\beta}\sim 0.12$) and ${\cal
S}$ (with $\epsilon_{_{\cal S}}\sim 0.29$). $\;\Delta$-variance is also able to
estimate $\beta$, albeit with a somewhat larger error ($\epsilon_{_\beta}\sim
0.17$) and with some human intervention, but is not able to estimate ${\cal
S}$.
