Multifractals

The Method of Moments applied to a Time Series example

Given a time series x1, x2, ..., xN,

Denote by m the minimum value of all the xi, and by M the maximum value of all the xi.

For q = -20 to 20 in steps of 0.25,

(1) Select a bin size r and partition [m,M] into intervals of size r:

[m, m+r), [m+r, m+2r), ..., [m+kr, M], where m+kr < M <= m+(k+1)r.

Call these bins B0, ..., Bk.

(2) For each j, 0 <= j <= k, count the number of xi that lie in bin Bj. Call this number nj. Ignore all nj that equal 0 after all the xi have been counted..

(3) Now compute the qth moment, Mrq = (n0/N)q + ... + (nk/N)q, where the sum is over all nonzero ni.

Repeat (1) - (3) for a range of r values getting closer to 0.

We hypothesize the power law relation, Mrq scales as (1/r)τ(q).

So for all r values, find the slope of the best-fitting line through the points (Log(1/r), Log(Mrq)). The goodness of fit measures the success of the power law hypothesis.

Call the slope of the best-fitting line τ(q).

We have computed τ(-20), τ(-19.75), ..., τ(20).

Start with α = 0.1. Compute min{τ(q) + α⋅q}, where the minimum is taken over all q values, q = -20, -19.9, ..., 20. (Why do we want to do this?) Call this minimum f(α).

Ignore negative values of f(α). Increase α in steps of 0.1 and plot the points (α), f(α)) for positive f(α). Continue until f(α) becomes negative.

Here is the f(α) curve for the time series above.

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