Conversely, if B

H

(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

D ¼ 2 H(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

x

i

(i ¼ 1, ...N) with mean m

N

and the standard deviation s

N

.To

define the rescaled range, the partial sums S

k

must be calculated

S

k

¼

X

k

i¼1

(x

i

m

N

), 1 k N(6:1:9)

The rescaled range equals

R=S ¼ [ max (S

k

) min (S

k

)]=s

N

,1 k N(6:1:10)

The value of R/S is always greater than zero since max (S

k

) > 0 and

min (S

k

) < 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

Fractals 63

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 Ho

¨

lder

exponent a

a ¼ lim

h!0

[lnm(h)= ln (h)] (6:2:1)

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

h and a via N

h

(a). The distribution of the Ho

¨

lder exponents in the

limit h ! 0 is sometimes called the multifractal spectrum

f(a) ¼lim

h!0

[lnN

h

(a)= ln (h)] (6:2:2)

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, m

0

and m

1

, are chosen so that

m

0

þ m

1

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

for mass distribution, m

0

¼ 1, is used. At the next step (k ¼ 1), the

measure m

1

uniformly spreads mass in proportion m

0

=m

1

on the

intervals [0,

1

⁄

2

] and [

1

⁄

2

, 1], respectively. Thus, m

1

[0,

1

⁄

2

] ¼ m

0

and

m

1

[

1

⁄

2

,1]¼ m

1

. In the next steps, every interval is again divided into

two subintervals and the mass of the interval is distributed between

subintervals in proportion m

0

=m

1

. For example, at k ¼ 2: m

2

[0,

1

⁄

4

]

¼ m

0

m

0

, m

2

[

1

⁄

4

,

1

⁄

2

] ¼ m

2

[

1

⁄

2

,

3

⁄

4

] ¼ m

0

m

1

, m

2

[

3

⁄

4

,1]¼ m

1

m

1

and so on.

At the k

th

iteration, mass is partitioned into 2

k

intervals of length 2

k

.

Let us introduce the notion of the binary expansion 0:b

1

b

2

...b

k

for

the point x ¼ b

1

2

1

þ b

2

2

2

þ b

k

2

k

where 0 x 1 and

0 < b

k

< 1. Then the measure for every dyadic interval I

0b1b2 :::bk

of

length 2

k

equals

m

0b1b2 :::bk

¼

Y

k

i¼1

m

b

i

¼ m

0

n

m

1

kn

(6:2:3)

where n is the number of digits 0 in the address 0

_

bb

1

b

2

...b

k

of the

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

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

64 Fractals

conservative or microcanonical. The first five steps of the binomial

cascade with m

0

¼ 0:6 are depicted in Figure 6.3.

The multifractal spectrum of the binomial cascade equals

f(a) ¼

a

max

a

a

max

a

min

log

2

a

max

a

a

max

a

min

a a

min

a

max

a

min

log

2

a a

min

a

max

a

min

(6:2:4)

The distribution (6.2.4) is confined with the interval [a

min

, a

max

]. If

m

0

0:5, then a

min

¼log

2

(m

0

) and a

max

¼log

2

(1 m

0

). 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

N1

0

m

i

¼ 1(6:2:5)

(a)

0

0.5

1

1.5

2

2.5

3

135791113151719212325272931 357911131517192123252729131

(b)

0

0.5

1

1.5

2

2.5

3

(c)

0

0.5

1

1.5

2

2.5

3

14 7 101316192225 2831

1 4 7 1013161922252831

(d)

0

0.5

1

1.5

2

2.5

3

357911131517192123252729131

(e)

0

0.5

1

1.5

2

2.5

3

(f)

0

0.5

1

1.5

2

2.5

3

135791113151719212325272931

Figure 6.3 Binomial cascade with m

0

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.

Fractals 65

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