The scaling function t(q) is related to the multifractal spectrum f(a)

via the Legendre transformation

t(q) ¼ min

a

[qa f(a)] (6:2:11)

which is equivalent to

f(a) ¼ arg min

q

[qa t(q)] (6:2:12)

Note that f(a) ¼ q(a H) þ 1 for the self-affine processes.

In practice, the scaling function of a multifractal process X(t) can

be calculated using so-called partition function

S

q

(T, Dt) ¼

X

N1

i ¼0

X(t þ Dt) X(t)

jj

q

(6:2:13)

where the sample X(t) has N points within the interval [0, T] with the

mesh size Dt. It follows from (6.2.9) that

log {E[S

q

(T, Dt)]} ¼ t(q) log (Dt) þ c(q) log T (6:2:14)

Thus, plotting log {E[S

q

(T, Dt)]} against log (Dt) for different values

of q reveals the character of the scaling function t(q). Multifractal

models have become very popular in analysis of the financial time

series. We shall return to this topic in Section 8.2

6.3 REFERENCES FOR FURTHER READING

The Mandelbrot’s work on scaling in the financial time series is

compiled in the collection [1]. Among many excellent books on frac-

tals, we choose [2] for its comprehensive material that includes a

description of relations between chaos and fractals and an important

chapter on multifractals [5].

6.4 EXERCISES

*1. Implement an algorithm that draws the Sierpinski triangle with

r ¼ 0:5 (see Figure 6.2).

Hint: Choose the following fixed points: (0, 0), (0, 100), (100,

0). Use the following method for the randomized choice of the

Fractals 67

fixed point: i ¼ [10 rand()] %3 where rand() is the uniform

distribution within [0, 1] and % is modulus (explain the ration-

ale behind this method). Note that at least 10000 iterations are

required for a good-quality picture.

*2. Reproduce the first five steps of the binomial cascade with

m

0

¼ 0:6 (see Figure 6.3). How will this cascade change if

m

0

¼ 0:8?

68 Fractals

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