Suppose the data are a collection of points (x1, y1),
..., (xN, yN) lying in some square region of the plane. |
Subdivide the square into smaller squares of side length s1, then into still smaller
squares of side length s2, and so on, down to some fairly tiny size sT. |
|
Count the number of points that lie in each square of side length s1. Call these
n(1,s1), n(2,s1), n(3,s1), ... . |
Count the number of points that lie in each square of side length s2. Call these
n(1,s2), n(2,s2), n(3,s2), ... . |
Repeat the same counting for all the square sizes down to sT. |
  |
Now compute the moments for a range of q-values. That is, let q range from,
say, -20 to 20 in steps of 0.25. For each q value compute the
qth moment |
M(s1,q) = (n(1,s1)/N)q +
(n(2,s1)/N)q + ... |
where the sum omits any term with n(i,s1) = 0. (Think of a negative power of 0
and you'll see why.) |
Now for each q-value compute the moments M(s2,q),
M(s3,q), ... M(sT,q). |
  |
We hope to discover a power law relation between M(si,q) and
(1/si)τ(q), so for each q find the best-fitting line through the
points |
(log(1/s1), log(M(r1,q))),
(log(1/s2), log(M(r2,q))), ...
(log(1/sT), log(M(rT,q))) |
The slope of this line is τ(q). |
  |
Finally, to find the f(α) curve, start with α small, say around 0.1. Find the minimum value of |
τ(q) + α⋅q |
using all the q-values. (Why does this give f(α)?) |
Now we have a fairly delicate point. What if we
have started with an α-value below the minimum, or continued to an α-value above the maximum? For the moment
we adopt an imperfect solution: |
If this minimum of τ(q) + α⋅q is negative, ignore that α-value. |
Otherwise, the minumum of τ(q) + α⋅q is the f(α) for that α. |