| A simple way to construct multifractals is to use an IFS with transformations
{T1, ..., TN}, contraction ratios r1, ..., rN,
and probabilities p1, ..., pN. |
| We will show how to construct the f(α) curve from this information, through an
auxiliary function β(q) defined by the generalized Moran equation |
| (p1q)(r1β(q)) + ...
+ (pNq)(rNβ(q)) = 1 |
| A reason for this approach is that for large positive q the larger pi dominate;
for large negative q the the smaller pi dominate. |
| To simplify the algebra, we focus on the special case where all the ri
take a common value, r. Then the generalized Moran equation can be solved explicitly
for β(q) |
| β(q) = -Log(p1q + p2q +
... + pNq) / Log(r) |
| The Holder exponent α is the negative of the slope of the tangent line of β. That is, |
| α = -dβ/dq |
| When all ri = r, this can be solved explicitly |
| α(q) = (p1qLog(p1) + ...
+ pNqLog(pN)) / (Log(r) (p1q + ... + pNq)) |
| Then compute f(α) by |
| f(α(q)) = q⋅α(q) + β(q) |
| The f(α) curve can be approximated by letting q range from some negative value, say q = -20, to some positive
value, say q = 20, in fairly small steps. This avoids the substantial headaches involved in trying to write f as an
explicit, even if only approximate, function of α. |
| |
| For N = 4 and r = 0.5, we have |
| α(q) | = (p1qLog(p1) + ...
+ p4qLog(p4))/(Log(0.5)⋅(p1q + ... + p4q)) |
| f(α(q)) | = q⋅(p1qLog(p1) + ...
+ p4qLog(p4))/(Log(0.5)⋅(p1q + ... + p4q))
- Log(p1q + ... + p4q)/Log(0.5) |
|
| |
| Finally, there is a simple interpretation of the highest point of the f(α) curve. |