Long term trend and hurst phenomenonlectr17.

1. 1
17. Long Term Trends and Hurst Phenomena
From ancient times the Nile river region has been known for
its peculiar long-term behavior: long periods of dryness followed by
long periods of yearly floods. It seems historical records that go back
as far as 622 AD also seem to support this trend. There were long
periods where the high levels tended to stay high and other periods
where low levels remained low1.
An interesting question for hydrologists in this context is how
to devise methods to regularize the flow of a river through reservoir
so that the outflow is uniform, there is no overflow at any time, and
in particular the capacity of the reservoir is ideally as full at time
as at t. Let denote the annual inflows, and
1A reference in the Bible says “seven years of great abundance are coming throughout the land of
Egypt, but seven years of famine will follow them” (Genesis).
0
t
t 
}
{ i
y
n
i
n y
y
y
s 


 
2 (17-1)
PILLAI

2. 2
their cumulative inflow up to time n so that
represents the overall average over a period N. Note that may
as well represent the internet traffic at some specific local area
network and the average system load in some suitable time frame.
To study the long term behavior in such systems, define the
“extermal” parameters
as well as the sample variance
In this case
}
{ i
y
N
y
},
{
max
1
N
n
N
n
N y
n
s
u 



.
)
(
1 2
1
N
N
n
n
N y
y
N
D 
 

},
{
min
1
N
n
N
n
N y
n
s
v 



(17-3)
(17-4)
(17-5)
N
N
N v
u
R 
 (17-6)
N
s
y
N
y N
N
i
i
N 
 
1
1
(17-2)
PILLAI

3. 3
defines the adjusted range statistic over the period N, and the dimen-
sionless quantity
that represents the readjusted range statistic has been used extensively
by hydrologists to investigate a variety of natural phenomena.
To understand the long term behavior of where
are independent identically distributed random
variables with common mean and variance note that for large N
by the strong law of large numbers
and
N
N
N
N
N
D
v
u
D
R 
 (17-7)
N
N D
R /
N
i
yi 
,
2
,
1
, 
 ,
2

),
,
( 2

 n
n
N
s d
n 



 

 )
/
,
( 2
N
N
y d
N
2


d
N
D
(17-8)
(17-9)
(17-10)
PILLAI

4. 4
with probability 1. Further with where 0 < t < 1, we have
where is the standard Brownian process with auto-correlation
function given by min To make further progress note that
so that
Hence by the functional central limit theorem, using (17-3) and
(17-4) we get
,
Nt
n 
    )
(
lim
lim t
B
N
Nt
s
N
n
s d
Nt
N
n
N













(17-11)
)
(t
B
).
,
( 2
1 t
t
)
(
)
(
)
(




N
s
N
n
n
s
y
n
n
s
y
n
s
N
n
N
n
N
n









(17-12)
( ) (1), 0 t 1.
d
n N n N
s ny s n s N
n
B t tB
N
N N N
 
  
  
  
    (17-13)
PILLAI

5. 5
where Q is a strictly positive random variable with finite variance.
Together with (17-10) this gives
a result due to Feller. Thus in the case of i.i.d. random variables the
rescaled range statistic is of the order of It
follows that the plot of versus log N should be linear
with slope H = 0.5 for independent and identically distributed
observations.
0 1
0 1
max{ ( ) (1)} min{ ( ) (1)} ,
d
N N
t
t
u v
B t tB B t tB Q
N  
 


    
,
Q
N
v
u
D
R d
N
N
N
N





N
N D
R / ).
( 2
/
1
N
O
)
/
log( N
N D
R
(17-14)
(17-15)
)
/
log( N
N
D
R
N
log
 






  

 
 

 
 




Slope=0.5
PILLAI

6. 6
The hydrologist Harold Erwin Hurst (1951) generated
tremendous interest when he published results based on water level
data that he analyzed for regions of the Nile river which showed that
Plots of versus log N are linear with slope
According to Feller’s analysis this must be an anomaly if the flows are
i.i.d. with finite second moment.
The basic problem raised by Hurst was to identify circumstances
under which one may obtain an exponent for N in (17-15).
The first positive result in this context was obtained by Mandelbrot
and Van Ness (1968) who obtained under a strongly
dependent stationary Gaussian model. The Hurst effect appears for
independent and non-stationary flows with finite second moment also.
In particular, when an appropriate slow-trend is superimposed on
a sequence of i.i.d. random variables the Hurst phenomenon reappears.
To see this, we define the Hurst exponent fora data set to be H if
)
/
log( N
N D
R .
75
.
0

H
2
/
1

H
2
/
1

H
PILLAI

7. 7
where Q is a nonzero real valued random variable.
IID with slow Trend
Let be a sequence of i.i.d. random variables with common mean
and variance and be an arbitrary real valued function
on the set of positive integers setting a deterministic trend, so that
represents the actual observations. Then the partial sum in (17-1)
becomes
where represents the running mean of the slow trend.
From (17-5) and (17-17), we obtain
,
, 


 N
Q
N
D
R d
H
N
N
(17-16)
}
{ n
X
 ,
2
 n
g
n
n
n g
x
y 
 (17-17)
)
(
1
2
1
2
1
n
n
n
i
i
n
n
n
g
x
n
g
x
x
x
y
y
y
s










 



(17-18)


n
i i
n g
n
g 1
/
1
PILLAI

8. 8
Since are i.i.d. random variables, from (17-10) we get
Further suppose that the deterministic sequence
converges to a finite limit c. Then their Caesaro means
also converges to c. Since
applying the above argument to the sequence and
we get (17-20) converges to zero. Similarly, since
).
)(
(
2
)
(
1
ˆ
)
)(
(
2
)
(
1
)
(
1
)
(
1
1
2
1
2
1
1 1
2
2
2
1
N
n
N
n
N
n
N
N
n
n
X
N
n
N
n
N
n
N
n
N
n
N
n
N
n
N
N
n
n
N
g
g
x
x
N
g
g
N
g
g
x
x
N
g
g
N
x
x
N
y
y
N
D


















 




 

 (17-19)
}
{ n
x
.
ˆ 2
2

 
d
X
}
{ n
g
N
N
n n
N g
g 
 1
1
,
)
(
)
(
1
)
(
1 2
1
2
2
1
c
g
c
g
N
g
g
N
N
N
n
n
N
n
N
n




 



(17-20)
2
)
( c
gn  2
)
( c
gN 
PILLAI

9. 9
by Schwarz inequality, the first term becomes
But and the Caesaro means
Hence the first term (17-21) tends to zero as and so does the
second term there. Using these results in (17-19), we get
To make further progress, observe that












N
n
N
N
n
n
N
n
N
n
N
n
c
g
x
c
g
x
N
g
g
x
x
N 1
1
),
)(
(
)
)(
(
1
)
)(
(
1


.
)
(
1
)
(
1
)
)(
(
1
1
2
2
1
2
1











N
n
n
N
n
n
N
n
n
n c
g
N
x
N
c
g
x
N


(17-21)
(17-22)
2
1
2
1
)
( 
 

 
N
n n
N x .
0
)
(
1
2
1


 
N
n n
N c
g
,


N
.
2




 d
N
n D
c
g (17-23)
(17-24)
)}
(
{
max
)}
(
{
max
)}
(
)
(
{
max
}
{
max
0
0 N
n
N
n
N
n
N
n
N
n
N
n
N
n
N
g
g
n
x
x
n
g
g
n
x
x
n
g
n
s
u













 PILLAI

10. 10
and
Consequently, if we let
for the i.i.d. random variables, then from (17-6),(17-24) and (17-25)
(17-26), we obtain
where
From (17-24) – (17-25), we also obtain
)}.
(
{
min
)}
(
{
min
)}
(
)
(
{
min
}
{
min
0
0 N
n
N
n
N
n
N
n
N
n
N
n
N
n
N
g
g
n
x
x
n
g
g
n
x
x
n
g
n
s
v














)}
(
{
min
)}
(
{
max 0
0 N
n
N
n
N
n
N
n
N
x
x
n
x
x
n
r 


 



(17-25)
(17-26)
N
N
N
N
N G
r
v
u
R 


 (17-27)
)}
(
{
min
)}
(
{
max 0
0 N
n
N
n
N
n
N
n
N
g
g
n
g
g
n
G 


 



(17-28)
PILLAI

11. 11
From (17-27) and (17-31) we get the useful estimates
and
Since are i.i.d. random variables, using (17-15) in (17-26) we get
a positive random variable, so that
)},
(
{
min
)}
(
{
max 0
0 N
n
N
n
N
n
N
n
N
g
g
n
x
x
n
v 


 



)},
(
{
max
)}
(
{
min 0
0 N
n
N
n
N
n
N
n
N
g
g
n
x
x
n
u 


 



.
N
N
N r
G
R 

(17-29)
(17-30)
(17-31)
,
N
N
N r
G
R 

.
N
N
N
G
r
R 

(17-32)
(17-33)
}
{ n
x
y
probabilit
in
Q
N
r
N
r N
X
N
,
ˆ 2




(17-34)
y.
probabilit
in
Q
N
rN


(17-35)
hence
and
)]
(
max
)
(
min
}
)
min
{(
max
)}
{(
max
[use i
i
i
i
i
i
i
i
i
i
i
y
x
y
x
y
x 




PILLAI

12. 12
Consequently for the sequence in (17-17) using (17-23)
in (17-32)-(17-34) we get
if H > 1/2. To summarize, if the slow trend converges to a finite
limit, then for the observed sequence for every H > 1/2
in probability as
In particular it follows from (17-16) and (17-36)-(17-37) that
the Hurst exponent H > 1/2 holds for a sequence if and only
if the slow trend sequence satisfies
}
{ n
y
0
/
2
/
1





H
H
N
H
N
N
N
N
Q
N
r
N
D
G
R 

(17-36)
}
{ n
g
},
{ n
y
0



 H
N
H
N
N
H
N
N
H
N
N
N
G
N
D
R
N
D
G
N
D
R

(17-37)
.


N
}
{ n
y
}
{ n
g
.
2
/
1
,
0
lim 0





H
c
N
G
H
N
N
(17-38)
PILLAI

13. 13
In that case from (17-37), for that H > 1/2 we obtain
where is a positive number.
Thus if the slow trend satisfies (17-38) for some H > 1/2,
then from (17-39)
Example: Consider the observations
where are i.i.d. random variables. Here and the
sequence converges to a for so that the above result applies. Let
,
/
0 

 N
as
y
probabilit
in
c
N
D
R
H
N
N
 (17-39)
0
c
}
{ n
g
.
as
,
log
log 


 N
c
N
H
D
R
N
N
(17-40)
1
, 


 n
bn
a
x
y n
n

(17-41)
n
x ,

bn
a
gn 

,
0


.
)
(
1
1









 



N
k
n
k
N
n
n k
N
n
k
b
g
g
n
M 

(17-42)
PILLAI

14. 14
To obtain its max and min, notice that
if and negative otherwise. Thus max is achieved
at
and the minimum of is attained at N=0. Hence from (17-28)
and (17-42)-(17-43)
Now using the Reimann sum approximation, we may write
N
M
,
)
( /
1
1
1 

 

N
k
N k
n


/
1
1
0
1






 

N
k
k
N
n
(17-44)
0
1
1
1 








 


N
k
n
n k
N
n
b
M
M 

(17-43)
0

N
M
.
1
0
1
0







 



N
k
n
k
N k
N
n
k
b
G 

PILLAI

15. 15
so that
and using (17-45)-(17-46) repeatedly in (17-44) we obtain





















1
,
1
,
log
1
,
)
1
(
/
1
1
/
1
0







N
k
N
N
N
n
k
(17-46)

























1
,
1
,
log
1
,
)
1
(
1
1
1
1
0
1








N
k
N
N
N
dx
x
N
k
N
k
N
N
k
(17-45)
PILLAI

16. 16
where are positive constants independent of N. From (17-47),
notice that if then
where and hence (17-38) is satisfied. In that case
3
2
1 ,
, c
c
c
,
0
2
/
1 

 





























































1
,
1
1
,
log
log
1
log
1
1
,
)
(
1
1
1
3
1
0
2
0
0
0
1
1
0
0
1
1
0
0
0










c
k
N
n
b
N
c
N
N
n
n
bn
N
c
N
n
bn
k
N
k
n
bn
G
k
N
k
n
k
n
(17-47)
,
~ 1
H
n N
c
G
1
2
/
1 
 H
PILLAI

17. 17
and the Hurst exponent
Next consider In that case from the entries in
(17-47) we get and diving both sides of (17-33)
with
so that
where the last step follows from (17-15) that is valid for i.i.d.
observations. Hence using a limiting argument the Hurst exponent
.
1
)
1
(




N
as
y
probabilit
in
c
N
D
R
N
N
 (17-48)
),
( 2
/
1
N
o
GN 
.
2
/
1



,
2
/
1
N
DN
y
probabilit
in
N
N
o
N
D
r
R
N
N
N
0
)
(
~ 2
/
1
2
/
1
2
/
1



Q
N
r
N
D
R N
N
N

2
/
1
2
/
1
~
 (17-49)
.
2
/
1
1 

 
H
PILLAI

18. 18
H = 1/2 if Notice that gives rise to i.i.d.
observations, and the Hurst exponent in that case is 1/2. Finally for
the slow trend sequence does not converge and
(17-36)-(17-40) does not apply. However direct calculation shows
that in (17-19) is dominated by the second term which for
large N can be approximated as so that
From (17-32)
where the last step follows from (17-34)-(17-35). Hence for
from (17-47) and (17-50)
.
2
/
1


 0


}
{ n
g
N
D

 2
0
2
1
N
x
N
N 



 N
as
N
c
DN 4

(17-50)
0
1
4






N
c
Q
N
N
D
r
N
D
G
R
N
N
N
N
N
0


,
0


PILLAI

19. 19
as Hence the Hurst exponent is 1 if In summary,
4
1
1
4
1
1
c
c
N
c
N
c
N
D
G
N
D
R
N
N
N
N


 



.


N .
0
















-1/2
2
/
1
-1/2
0
1
0
2
/
1
0
1
)
(






H
(17-51)
(17-52)
Fig.1 Hurst exponent for a process with superimposed slow trend
1/2
1
-1/2
0

)
(
H
Hurst
phenomenon
PILLAI