63

Fractals

Conversely, if BH (t) decreased in the past, it will most probably

continue to fall. Thus, persistent processes maintain trend. In the

opposite case (0 < H < 1„2, C < 0), the process is named anti-persist-

ent. It is said also that anti-persistent processes are mean reverting; for

example, if the current process innovation is positive, then the next

one will most likely be negative, and vice versa. There is a simple

relationship between the box-counting fractal dimension and the

Hurst exponent

D¼2ÀH (6:1:8)

The fractal dimension of a time series can be estimated using the

Hurst™s rescaled range (R/S) analysis [1, 3]. Consider the data set

xi (i ¼ 1, . . . N) with mean mN and the standard deviation sN . To

define the rescaled range, the partial sums Sk must be calculated

X

k

Sk ¼ (xi À mN ), 1 k N (6:1:9)

i¼1

The rescaled range equals

R=S ¼ [ max (Sk ) À min (Sk )]=sN , 1 k N (6:1:10)

The value of R/S is always greater than zero since max (Sk ) > 0 and

min (Sk ) < 0. For given R/S, the Hurst exponent can be estimated

using the relation

R=S ¼ (aN)H (6:1:11)

where a is a constant. The R/S analysis is superior to many other

methods of determining long-range dependencies. But this approach

has a noted shortcoming, namely, high sensitivity to the short-range

memory [4].

6.2 MULTIFRACTALS

Let us turn to the generic notion of multifractals (see, e.g., [5]).

Consider the map filled with a set of boxes that are used in the box-

counting fractal dimension. What matters for the fractal concept is

whether the given box belongs to fractal. The basic idea behind the

notion of multifractals is that every box is assigned a measure m

that characterizes some probability density (e.g., intensity of color

64 Fractals

between the white and black limits). The so-called multiplicative

process (or cascade) defines the rule according to which measure is

fragmented when the object is partitioned into smaller components.

The fragmentation ratios that are used in this process are named

¨

multipliers. The multifractal measure is characterized with the Holder

exponent a

a ¼ lim [ ln m(h)= ln (h)] (6:2:1)

h!0

where h is the box size. Let us denote the number of boxes with given

¨

h and a via Nh (a). The distribution of the Holder exponents in the

limit h ! 0 is sometimes called the multifractal spectrum

f(a) ¼ À lim [ ln Nh (a)= ln (h)] (6:2:2)

h!0

The distribution f(a) can be treated as a generalization of the fractal

dimension for the multifractal processes.

Let us describe the simplest multifractal, namely the binomial

measure m on the interval [0, 1] (see [5] for details). In the binomial

cascade, two positive multipliers, m0 and m1 , are chosen so that

m0 þ m1 ¼ 1. At the step k ¼ 0, the uniform probability measure

for mass distribution, m0 ¼ 1, is used. At the next step (k ¼ 1), the

measure m1 uniformly spreads mass in proportion m0 =m1 on the

intervals [0, 1„2 ] and [1„2 , 1], respectively. Thus, m1 [0, 1„2 ] ¼ m0 and

m1 [ 1„2 , 1] ¼ m1 . In the next steps, every interval is again divided into

two subintervals and the mass of the interval is distributed between

subintervals in proportion m0 =m1 . For example, at k ¼ 2: m2 [0, 1„4 ]

¼ m0 m0 , m2 [ 1„4 , 1„2 ] ¼ m2 [1„2 , 3„4 ] ¼ m0 m1 , m2 [3„4 , 1] ¼ m1 m1 and so on.

At the kth iteration, mass is partitioned into 2k intervals of length 2Àk .

Let us introduce the notion of the binary expansion 0:b1 b2 . . . bk for

the point x ¼ b1 2À1 þ b2 2À2 þ bk 2Àk where 0 x 1 and

0 < bk < 1. Then the measure for every dyadic interval I0b1b2 : : : bk of

length 2Àk equals

Y

k

mbi ¼ m0 n m1 kÀn

m0b1b2 : : : bk ¼ (6:2:3)

i¼1

_

where n is the number of digits 0 in the address 0b1 b2 . . . bk of the

interval™s left end, and (k À n) is the number of digits 1. Since the

subinterval mass is preserved at every step, the cascade is called

65

Fractals

3 3

(a) (b)

2.5 2.5

2 2

1.5 1.5

1 1

0.5 0.5

0 0

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31

3

3

(c) (d)

2.5 2.5

2 2

1.5 1.5

1 1

0.5 0.5

0 0

14 7 10 13 16 19 22 25 28 31 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31

3

3

(f)

(e)

2.5

2.5

2 2

1.5 1.5

1 1

0.5 0.5

0 0

1 4 7 10 13 16 19 22 25 28 31 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31

Figure 6.3 Binomial cascade with m0 ¼ 0.6: a) k ¼ 0, b) k ¼ 1, c) k ¼ 2, d)

k ¼ 3, e) k ¼ 4, f) k ¼ 5.

conservative or microcanonical. The first five steps of the binomial

cascade with m0 ¼ 0:6 are depicted in Figure 6.3.

The multifractal spectrum of the binomial cascade equals

amax À a amax À a a À amin a À amin

f(a) ¼ À À

log2 log2

amax À amin amax À amin amax À amin amax À amin

(6:2:4)

The distribution (6.2.4) is confined with the interval [amin , amax ]. If

m0 ! 0:5, then amin ¼ À log2 (m0 ) and amax ¼ À log2 (1 À m0 ). The

binomial cascade can be generalized in two directions. First, one

can introduce a multinomial cascade by increasing the number of

subintervals to N > 2. Note that the condition

X

NÀ1

mi ¼ 1 (6:2:5)

0

66 Fractals