Transcript Document

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 t  t 0
as at t. Let { yi } denote the annual inflows, and
s n  yi  y 2    y n
(17-1)
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).
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their cumulative inflow up to time n so that
sN
1 N
y N   yi 
N i 1
N
(17-2)
represents the overall average over a period N. Note that { yi } may
as well represent the internet traffic at some specific local area
network and y N the average system load in some suitable time frame.
To study the long term behavior in such systems, define the
“extermal” parameters
u N  max{sn  ny N },
(17-3)
1 n  N
vN  min {sn  ny N },
1 n  N
as well as the sample variance
1 N
DN   ( yn  y N ) 2 .
N n 1
In this case
RN  u N  v N
(17-4)
(17-5)
(17-6)
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defines the adjusted range statistic over the period N, and the dimensionless quantity
RN
u  vN
 N
DN
DN
(17-7)
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 RN / DN where
yi , i  1, 2,  N are independent identically distributed random
variables with common mean  and variance  2 , note that for large N
by the strong law of large numbers
d
sn 

N (n , n 2 ),
d
yN 

N ( , 2 / N )  
(17-8)
(17-9)
and
d
DN 

2
(17-10)
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with probability 1. Further with n  Nt , where 0 < t < 1, we have
s Nt   Nt  d
sn  n
lim
 lim

 B(t )
N 
N 
N
N
(17-11)
where B(t ) is the standard Brownian process with auto-correlation
function given by min (t1 , t2 ).To make further progress note that
sn  ny N  sn  n  n( y N   )
n
 ( sn  n )  ( s N  N )
N
(17-12)
so that
sn  ny N
N

sn  n 
n sN  N  d


 B(t )  tB(1), 0  t  1. (17-13)
N
N
N
Hence by the functional central limit theorem, using (17-3) and
(17-4) we get
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uN  v N
N
d

 max{B(t )  tB(1)}  min{B(t )  tB(1)}  Q,
0 t 1
0 t 1
(17-14)
where Q is a strictly positive random variable with finite variance.
Together with (17-10) this gives
RN
u  vN d
 N

 N Q,
DN

(17-15)
a result due to Feller. Thus in the case of i.i.d. random variables the
1/ 2
rescaled range statistic RN / DN is of the order of O( N ). It
follows that the plot of log( RN / DN ) versus log N should be linear
with slope H = 0.5 for independent and identically distributed
observations.
log( R N / DN
Slope=0.5
)




  


    
log N
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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 log( RN / DN ) versus log N are linear with slope H  0.75.
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 H  1 / 2 for N in (17-15).
The first positive result in this context was obtained by Mandelbrot
and Van Ness (1968) who obtained H  1 / 2 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
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RN
d


Q,
H
DN N
N  ,
(17-16)
where Q is a nonzero real valued random variable.
IID with slow Trend
Let { X n } be a sequence of i.i.d. random variables with common mean
 and variance  2 , and g n be an arbitrary real valued function
on the set of positive integers setting a deterministic trend, so that
y n  xn  g n
(17-17)
represents the actual observations. Then the partial sum in (17-1)
n
becomes
sn  y1  y2    yn  x1  x2    xn   g i
i 1
 n ( xn  g n )
(17-18)
where g n  1 / ni 1 g i represents the running mean of the slow trend.
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From (17-5) and (17-17), we obtain
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n
1 N
DN   ( y n  y N ) 2
N n 1
1 N
1 N
2 N
2
2
  ( xn  x N )   ( g n  g N )   ( xn  x N )(g n  g N )
N n 1
N n 1
N n 1
1 N
2 N
(17-19)
2
2
 ˆ X   ( g n  g N )   (xn  x N )( g n  g N ).
N n 1
N n 1
Since { xn } are i.i.d. random variables, from (17-10) we get
d
ˆ X2 

 2 . Further suppose that the deterministic sequence { g n }
N
converges to a finite limit c. Then their Caesaro means N1 n1 g n  g N
also converges to c. Since
1 N
1 N
2
2
2
(
g

g
)

(
g

c
)

(
g

c
)
,


n
N
n
N
N n 1
N n 1
(17-20)
2
2
applying the above argument to the sequence ( g n  c) and ( g N  c)
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we get (17-20) converges to zero. Similarly, since
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1 N
1 N
(xn  xN )( g n  g N )   ( xn   )( g n  c)  ( xN   )( g N  c),

N n1
N n1
(17-21)
by Schwarz inequality, the first term becomes
2
N
N
1
1 N
2 1
2
(
x


)(
g

c
)

(
x


)
(
g

c
)
.



n
n
n
n
N n 1
N n 1
N n 1
(17-22)
2
2
2
But n1 ( xn   )   and the Caesaro means N1 n1 ( g n  c)  0.
Hence the first term (17-21) tends to zero as N  , and so does the
second term there. Using these results in (17-19), we get
1
N
N
gn  c
N

d
DN 

 2.
(17-23)
To make further progress, observe that
u N  max{sn  ng N }
 max{n ( xn  xN )  n ( g n  g N )}
 max
{n ( xn  xN )}  max
{n ( g n  g N )}
0 n N
0 n N
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(17-24)
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and
vN  min{sn  ng N }
 min{n ( xn  xN )  n ( g n  g N )}
 min
{n ( xn  xN )}  min
{n ( g n  g N )}.
0 n N
0 n N
(17-25)
Consequently, if we let
rN  max
{n ( xn  xN )}  min
{n ( xn  xN )}
0 n  N
0 n N
(17-26)
for the i.i.d. random variables, then from (17-6),(17-24) and (17-25)
(17-26), we obtain
RN  u N  v N  rN  GN
(17-27)
GN  max
{n ( g n  g N )}  min
{n ( g n  g N )}
0 n N
0 n N
(17-28)
where
From (17-24) – (17-25), we also obtain
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u N  min
{n ( xn  xN )}  max
{n ( g n  g N )},
0 n N
0 n  N
(17-29)
(17-30)
vN  max
{n ( xn  xN )}  min
{n ( g n  g N )},
0 n  N
0 n  N
[use max{( xi  yi )}  max{( min xi )  yi }  min ( xi )  max ( yi )] and hence
i
i
i
i
RN  GN  rN .
From (17-27) and (17-31) we get the useful estimates
RN  GN  rN ,
and
i
(17-31)
(17-32)
RN  rN  GN .
(17-33)
Since { xn } are i.i.d. random variables, using (17-15) in (17-26) we get
rN
rN
(17-34)

 Q, in probability
2
ˆ X N  N
a positive random variable, so that
rN
 N Q in probability.

(17-35)
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Consequently for the sequence { y n } in (17-17) using (17-23)
in (17-32)-(17-34) we get
RN  G N
rN
Q /
(17-36)

 H 1/ 2  0
H
H
DN N
N
N
if H > 1/2. To summarize, if the slow trend { g n }converges to a finite
limit, then for the observed sequence { yn }, for every H > 1/2
RN
GN


H
H
DN N
DN N
RN
GN

0
H
H
DN N
N
(17-37)
in probability as N  .
In particular it follows from (17-16) and (17-36)-(17-37) that
the Hurst exponent H > 1/2 holds for a sequence { y n } if and only
if the slow trend sequence { g n } satisfies
lim
N 
GN
 c0  0,
H
N
H  1 / 2.
(17-38)
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In that case from (17-37), for that H > 1/2 we obtain
RN
 c0 / 
H
DN N
in probability as
N  ,
(17-39)
where c0 is a positive number.
Thus if the slow trend { g n } satisfies (17-38) for some H > 1/2,
then from (17-39)
RN
log
 H log N  c,
DN
as
N  .
(17-40)
Example: Consider the observations
yn  xn  a  bn ,
n 1
(17-41)

where xn are i.i.d. random variables. Here g n  a  bn , and the
sequence converges to a for   0, so that the above result applies. Let
 n  n N 
M n  n ( g n  g N )  b   k   k .
N k 1 
 k 1
(17-42)
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To obtain its max and min, notice that
  1 N 
M n  M n1  b  n   k   0
N k 1 

if n  (
at
1
N
k 1 k
N

)1/ , and negative otherwise. Thus max M N is achieved
1/ 
1

n0    k  
 N k 1 
N
(17-43)
and the minimum of M N  0 is attained at N=0. Hence from (17-28)
and (17-42)-(17-43)
 n0  n0 N  
GN  b   k   k .
N k 1 
 k 1
(17-44)
Now using the Reimann sum approximation, we may write
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
(1   ) 1 N  ,   1

N
1
1 N 
 logN

k  0 x dx  
,
  1

N k 1
N
N


 k 1 k 
,
  1


N
so that
(17-45)

(1   ) 1 /  N ,   1

N

n0  
,
  1
 log N
  k 
k 1
,
  1

1/
 N
(17-46)
and using (17-45)-(17-46) repeatedly in (17-44) we obtain
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 1 n0  1 N  
Gn  bn0   k   k 
N k 1 
 n0 k 1
 bn0 

1
(
n

N
)

c
N
,
  1
1
1   0


 1
1
 bn0  log n0  log N   c2 log N ,   1
N

  n0
  n    
  1
b1  0   k   c3 ,
  N  k 1 
(17-47)
where c1 , c2 , c3 are positive constants independent of N. From (17-47),
notice that if  1 / 2    0, then
Gn ~ c1 N H ,
where 1 / 2  H  1 and hence (17-38) is satisfied. In that case
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RN
 c1
(1 )
DN N
in probability as
N  .
(17-48)
and the Hurst exponent H  1    1 / 2.
Next consider   1 / 2. In that case from the entries in
(17-47) we get GN  o( N 1/ 2 ), and diving both sides of (17-33)
with DN N 1/ 2 ,
RN  rN
o( N 1 / 2 )
~
0
1/ 2
1/ 2
DN N
N
so that
RN
rN
~
Q
1/ 2
1/ 2
DN N
N
in probability
(17-49)
where the last step follows from (17-15) that is valid for i.i.d.
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observations. Hence using a limiting argument the Hurst exponent
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H = 1/2 if   1 / 2. Notice that   0 gives rise to i.i.d.
observations, and the Hurst exponent in that case is 1/2. Finally for
  0, the slow trend sequence { g n } does not converge and
(17-36)-(17-40) does not apply. However direct calculation shows
that DN in (17-19) is dominated by the second term which for
N
large N can be approximated as N1  x 2  N 2 so that
0
DN  c4 N 
as
N 
(17-50)
From (17-32)
RN  G N
rN
N Q


0
1
DN N
DN N
c4 N
where the last step follows from (17-34)-(17-35). Hence for   0
from (17-47) and (17-50)
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RN
GN
c1 N 1
c1



DN N
DN N c4 N 1
c4
(17-51)
as N  . Hence the Hurst exponent is 1 if   0. In summary,
1
 1/ 2

H ( )  
1  
 1 / 2
 0
 0
0    -1/2
  -1/2
(17-52)
H ( )
Hurst
phenomenon
1
1/2

-1/2
0
Fig.1 Hurst exponent for a process with superimposed slow trend
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