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) ¼lim
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
¼log
2
(m
0
) and a
max
¼log
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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