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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 t0 as at t. Let { yi } denote the annual inflows, and sn yi y2 yn (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). 1 PILLAI 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 ( y n y N ) 2 . N n 1 In this case RN u N vN (17-4) (17-5) (17-6) 2 PILLAI 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) 3 PILLAI 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 sn n n sN N d B(t ) tB(1), 0 t 1. (17-13) N N N N Hence by the functional central limit theorem, using (17-3) and (17-4) we get 4 PILLAI uN vN d max{B(t ) tB(1)} min{B(t ) tB(1)} Q, 0 t 1 0 t 1 N (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 rescaled range statistic RN / DN is of the order of O( N 1/ 2 ). 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 5 PILLAI 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 6 PILLAI 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 yn 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 / ni 1 gi represents the running mean of the slow trend. 7 From (17-5) and (17-17), we obtain PILLAI 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 n1 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) 8 we get (17-20) converges to zero. Similarly, since PILLAI 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 n1 ( xn ) and the Caesaro means N1 n1 ( 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 9 (17-24) PILLAI 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 vN 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 10 PILLAI 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) 11 PILLAI Consequently for the sequence { yn } 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 { yn } if and only if the slow trend sequence {g n } satisfies GN lim c0 0, H N N H 1 / 2. (17-38) 12 PILLAI 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) 13 PILLAI To obtain its max and min, notice that 1 N M n M n1 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 14 PILLAI (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 15 PILLAI 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 b1 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 16 PILLAI 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. 17 observations. Hence using a limiting argument the Hurst exponent PILLAI 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) 18 PILLAI 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 19 PILLAI