App ears in Neural Networks in the Capital Markets

Pro ceedings of the Third International Conference London Octob er

A Refenes Y AbuMostafa J Mo o dy and A Weigend eds World Scientic London

IMPROVED ESTIMATES FOR

THE RESCALED RANGE AND HURST EXPONENTS

John Mo o dy and Lizhong Wu

Computer Science Dept Oregon Graduate Institute

Portland OR

Email mo o dycseogiedu lwucseogiedu

ABSTRACT

Rescaled Range RS analysis and Hurst Exp onents are widely used as measures

of longterm memory structures in sto chastic pro cesses Our empirical studies

show however that these statistics can incorrectly indicate departures from

random walk b ehavior on short and intermediate time scales when very short

term correlations are present A mo dication of rescaled range estimation RS

analysis intended to correct bias due to shortterm dep endencies was prop osed

by Lo Weshow however that Los RS statistic is itself biased and

intro duces other problems including distortion of the Hurst exp onents We

prop ose a new statistic RS that corrects for mean bias in the range Rbut

do es not suer from the short term biases of RS or Los RS We supp ort our

conclusions with exp eriments on simulated random walk and AR pro cesses

and exp eriments using high frequency interbank DEM USD exchange rate

quotes We conclude that the DEM USD series is mildly trending on time

scales of to ticks and that the mean reversion suggested on these time

RS analysis is spurious scales by RS or

Intro duction and Overview

There are three widely used metho ds for longterm dep endence analysis auto

correlation analysis fractional dierence mo dels Granger Joyeux Hosking

and scaling law analysis including rescaled range RS analysis Hurst

Hurst exp onents Hurst Mandelbrot Van Ness and drift exp onents

Muller Dacorogna Olsen Pictet Schwarz Morgenegg This pap er stud

ies RS analysis and Hurst exp onents whichhave b ecome recently p opular in the

nance community largely due to the empirical work of Peters Compared

to auto correlation analysis the advantages of RS analysis include detection of

longrange dep endence in highly nongaussian time series with large skewness and

kurtosis almost sure convergence for sto chastic pro cesses with innite variance

and detection of nonp erio dic cycles However there are also two deciencies

asso ciated with rescaled range analysis and the estimation of Hurst exp onents

estimation errors exist when the time scale is very small or very large relative to the

numb er of observations in the time series Mandelbrot Wallis Wallis Mata

las Feder Ambrose Ancel Griths Mo o dy WuaMuller

Dacorogna Pictet and the rescaled range is sensitive to shortterm de

p endence McLeo d Hip el Hip el McLeo d Lo The second

shortcoming will sometimes lead to completely incorrect results

Lo analyzed the mean bias in the range statistic R due to shortterm

dep endencies in the time series and prop osed a mo died rescaling factor S that is

intended to remove or reduce these eects Wehave found however that Los statistic

is itself biased and causes some new problems on short time scales while attempting

to correct the mean bias of the range R including distortion of the Hurst exp onents

S N statistic for a given While Los approach fo cuses on the actual value of the R

time scale of interest N Hurst and Mandelbrot test for long term dep endency by

comparing the slop e of RS N curveto Our empirical results show however

that Hurst exp onents standard rescaled range analysis and Los mo died rescaled

range can yield incompatible results with the conventional interpretations of these

statistics due to the biases contained in the R S andS statistics

We prop ose a new unbiased rescaling factor S that is able to correct for the mean

biases in R at all time scales without inducing new distortions of the rescaled range

and Hurst exp onents at short time scales

The outline of this pap er is as follows In Section we will briey intro duce

the rescaled range analysis and the Hurst exp onent The analysis and estimation

pro cedures are then demonstrated on tickbytickinterbank foreign exchange data

Through empirical comparisons we showhow seriously shortterm dep endencies in a

time series can aect the rescaled range analysis In Section we explain why there

is a mean bias in the range estimation and intro duce Los mo died approach Some

simulation results with the mo died algorithm are shown and compared to results

using the original algorithm In Section weevaluate Los mo died rescaled range

analysis list and analyze the problems asso ciating with it and showhow it distorts

the Hurst exp onent In Section we present our new unbiased rescaling factor S

along with empirical results that demonstrate the improvements that it yields relative

to the standard RS and Los RS statistics In Section we conclude our pap er

with a discussion

RS Analysis for High Frequency FX Data

Among the various approaches for quantifying correlations and deviations from

gaussian b ehavior for sto chastic pro cesses several approaches have b een suggested

that are based on scaling laws Unlike traditional correlation analysis these scaling

law metho ds are intended to quantify structure that p ersists on a sp ectrum of time

scales

The Rescaled Range RS analysis and Hurst Exp onents were rst develop ed

by Hurst and rened and p opularized by Mandelbrot etal in the late s

and early s These b ecame p opular in nance due to the clear exp osition of the

methodsinFeder and the empirical work of Peters A related approach

based on the drift exp onentwas indep endently pioneered byMuller et al

and scaling laws for directional change frequency have b een suggested by Guillaume

In this pap er we restrict our attention to RS analysis and Hurst exp onents

RS Analysis and Hurst Exponents

The RS statistic is the range of partial sums of deviations of a time series from its

mean rate of change rescaled by its standard deviation Denoting a series of returns

one p erio d changes by r the average m and biased standard deviation S of the

t

a

returns from t t to t t N are

 

t N

X

mN t r N

 t

tt 



t N

X



r mN t S N t

t  

N

tt 

The partial sum of deviations of r from its mean and the range of partial sums

t

are then dened as

t 

X

X N t r mN t for N

 t 

tt 

R N t max X N t min X N t

  

erage values The RS statistic for time scale N is simply the ratio b etween the av

of R N t and S N t

 

P

R N t



t

RS N

P

S N t



t

a

The quantity S N t conventionally used in RS analysis is an estimate of the standard deviation

p

that is biased downward by a factor N N The unbiased estimate of the true standard

deviation N t is



t N

X



r mN t b N t

t

N

tt 

In section we present improved results using the unbiased estimate b N t

Assuming that a scaling law exists for RS N we can write

H

RS N aN

where a is a constantandH is referred to as the Hurst exp onent By estimating H

we can characterize the b ehavior of time series as follows

H random walk

H meanreverting

H meanaverting

For a more detailed but readable discussion of RS analysis and Hurst exp onents

see Feder

RS Analysis for High Frequency FX Data

High frequency interbank FX data consists of a sequence of BidAsk prices quoted

byvarious rms that function as market makers While BidAsk price quotes from

many market makers are displayed simultaneously by wire services such as Reuters

and Telerate a single price series can b e constructed from the sequence of newly

up dated quotes

We are analyzing a full year of such tickbytickInterbank FX price quotes for

three exchange rates the Deutschmark US Dollar rate DEMUSD the Japanese

Yen US Dollar rate JPYUSD and the Deutschmark Yen DEMJPY cross

rate The data were obtained from Olsen Asso ciates of Zuric h The data sample

includes every tick from Octob er through Septemb er For the DEMUSD

the year has ticks

To study the b ehavior of returns on a sp ectrum of time scales we p erform rescaled

range analysis and compute Hurst exp onents Figure shows the rescaled range anal

ysis for Octob er DEMUSD Bid returns Both the original data and scrambled

data were analyzed The upward shift in the curve for the scrambled data is evidence

for mean reversion of the original series on all time scales measured

Further results are presented in Mo o dy Wu To summarize them

the b ehavior of the Hurst exp onents in the returns of DEMUSD exchange rates

is qualitatively dierent from the Hurst exp onents of gaussian series and scrambled

series of the returns The b ehaviors of RS N and H are more similar to those of

an AR pro cess with negative co ecient

To understand the nature and meaning of the apparent mean reverting b ehavior in

the high frequency FX data wehave p erformed a series of investigations as describ ed

in Mo o dy Wu b The question is whether the observed mean reversion over

a range of intermediate times scales is due to shortterm price oscillations on time

scales of a few ticks or is evidence of intrinsic dep endencies in the price movements

on those intermediate time scales DEM_USD Bid_Returns Oct.92 2.2

2

1.8

1.6 Scrambled

1.4

1.2

Log10(R/S) 1 Original

0.8

0.6

0.4

0.2 0.5 1 1.5 2 2.5 3 3.5 4 4.5

Log10(# of Ticks)

Figure Rescaled range analysis for DEMUSD Bid returns during Octob er

Between Original and Scrambled curves is a straight line with a slop e The lower

slop e for the original series for time scales less than ticks suggests that mean

reversion is present on short time scales

One of our studies is to observe the b ehavior of the blo ckaverages of prices The

blo ckaverage price is dened as

ik 

X

logBidt i logAsk t i p t

k

k

i

where k is the length of the sequence of ticks over which the mean price is calculated

The sequences are then downsampled by a factor of k sothattheblocks do not

overlap each other

Rescaled range analysis of the blo ckaverage prices is presented in Figure Here

the mean prices are calculated for blo cks of and ticks resp ectively As

explained in the gure the price b ehavior changes completely for blo cks of tick

versus blo cks of ticks the b ehavior in the tickbytick DEMUSD series shifts from

meanreverting to meanaverting

In summary the meanreverting price b ehavior app ears to b e due to the high

frequency oscillations When shortrun oscillations p ossibly caused byinventory ef

fects are smo othed the price movements shift from meanreverting to meanaverting (a) (b) 2 2

1.5 1.5

1 1 Log10(R/S) Log10(R/S) 0.5 0.5

0 0 1 1.5 2 2.5 3 1 1.5 2 2.5 3 Log10(# of Blocks) Log10(# of Blocks) (c) (d) 2 2

1.5 1.5

1 1 Log10(R/S) Log10(R/S) 0.5 0.5

0 0 1 1.5 2 2.5 3 1 1.5 2 2.5 3

Log10(# of Blocks) Log10(# of Blocks)

Figure Rescaled range analysis for DEMUSD during Octob er The price is

taken as the average of bids and asks over a short sequence of ticks From Figure

a to d the blo ck consists of and ticks resp ectively The dotted curves

are formed by the scrambled price sequences and their slop es equal to The lower

slop es of solid curves in Figure a and b suggest meanreverting price b ehavior

The higher slop es of solid curves in Figure c and d suggest meanaverting price

behavior

behavior

Los Mo died RS Analysis

Mean Bias in Range Statistic

While studying longterm memory structures in sto ck prices using RS analysis

Lo found that rejections of the null hyp othesis that the time series is a random

walk on long time scales can b e erroneous and can b e due instead to bias induced by

shortterm dep endencies He compared the asymptotic distributions of the rescaled

ranges b etween an iid random series and an AR shortterm dep endent series

When the time scale N increases without b ound the normalized rescaled range of an

iid series converges to the range of a Brownian bridge on the unit interval

R N t



p

B iid Series

N

However for a shortterm dep endent AR series with a regression co ecient the

normalized rescaled range converges to

R N t



p

B AR Series

N

q

The mean is biased by a factor of which for this sp ecial case is

The bias can b e signicant For example if the normalized rescaled range

is biased downward by a factor Therefore the shortterm dep endence will

bias the estimation of the longterm rescaled range

Modied RS Analysis

To remove the eect of mean bias due to shortterm dep endencies Lo

prop osed a mo died RS statistic His motivation was that if r is sub ject to short

t

term dep endence the auto covariances of r will not b e equal to zero and the range

t

R cannot simply b e normalized by the standard deviation alone The covariances

should b e considered also Andrews The rescaling term suggested byLo

b

includes weighted covariances up to lag q and has the form



q

t N t N

X X X



r m w q r mr m S N t q

t j t tj 

N N

tt 

j  tt j

b

Note that the S is biased downward The rst term in is biased by a factor N N while

the second term has negative bias This issue will b e addressed empirically in sections and

Section presents an unbiased replacement S

where w q is dened as

j

j

w q

j

q

This weighting function always yields a p ositive S The determination of q is rather

complicated Not b eing able to determine q by a simple closedform expression Lo

discussed the eect of varying q When q b ecomes large relativeto N the nite

sample distribution of the estimator can b e radically dierent from its asymptotic

limit However q cannot b e chosen to o small since the auto covariances b eyond

lag q may b e substantial and should b e included in the weighted auto covariances

Therefore the truncation lag mustbechosen with some consideration of the data at

hand

The mo died RS analysis rescales the range R using S instead of the standard

deviation S sowe refer to it as RS analysis As we shall demonstrate empirically

in sections and Los rescaling factor S has signicantdownward bias for small

N and this distorts b oth RS and the Hurst exp onent H

RS Analysis for Foreign Exchange Rates

In Section we found that the interbank tickbytick foreign exchange price

changes show meanreverting b ehavior on time scales from several ticks to hundreds

of ticks However when we consider the blo ckaverage prices averaged over a few

ticks the meanreverting b ehavior on longer time scales is not signicant suggesting

that the apparent meanreverting b ehavior is not due to the fundamental nature of

the price movements but rather is just an artifact induced by the high frequency

oscillations In the following weuseRS analysis and try to get an answer

Wenow consider Los mo died RS analysis to see whether it can help conrm

our conclusion ab ove that the observed mean reverting b ehavior on intermediate time

scales is actually due to the high frequency oscillations Unfortunatelyhowever we

nd that the RS statistic intro duces new problems and is not helpful in resolving

this issue

The RS analysis for the prices of DEMUSD exchange rates in Octob er is

plotted in Figure The data used consists of ticks Wetake the price as

the a verage of Bid and Ask quotes The eect of the lag q is also depicted in the

gure Since the high frequency oscillations in the foreign exchange data b elieved

by some to b e an inventory eect are on very short time scales in tick time weonly

use small q q f g in our analysis The case with q corresp onds to

the standard RS analysis From the gure we see that the RS curves shift

upwards compared to the RS curve and the upward shift when the time scale N

is small is signicantly larger than that when N is large This is due to the downward

biases in S DEM_USD Oct.92 1.6

1.4

1.2

1 Log10(R/S^) q=8 q=4 0.8 q=2 q=0

0.6

0.4 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3

Log10(N)

Figure RS analysis for DEMUSD data in Octob er Four solid curves

corresp ond to the analysis for the same data but using dierentlagq for computing

the auto covariances When q theRS analysis reduces to the RS analysis The

dashed straight line is with a slop e of and corresp onds to a random walk Note

that when the time scale is large N all RS curves havevery similar

slop es and the slop es are all smaller than that for the random walk

Table Estimated Hurst exp onents from the RS analysis for DEMUSD data in

Octob er

Lags Time Scales

q

RS Analysis and Hurst Exp onents

Table lists the estimated Hurst exp onents from the RS analysis over the time

scales and From the table we can see that for the smaller

time scales N the Hurst exp onents decrease signicantly as the lag q

increases for the larger time scales N the Hurst exp onents decrease

only slightly with increasing lag q and all Hurst exp onents are less than

By observing a series of empirical results in our studies wehave found the fol

lowing There indeed exists an estimation bias in the range statistic R due to

shortterm dep endencies in the series that shifts the standard rescaled range statistic

RS When the time scale N is large the mo died RS statistic can correct this

bias When the time scale N is small the rescaling factor S intro duces some new

errors in estimating b oth the rescaled range statistic RS and the Hurst exp onents

When the time scale N is large the slop es of RS curves are indep endent of the

lag q even though they shift vertically This means that the RS analysis when

q and the RS analysis when q yield the same Hurst exp onen ts

Toinvestigate the ab ove issues further wehave conducted the RS analysis for

simulated gaussian random walk and AR series

To demonstrate our rst observation we compare the RS analysis for a simulated

gaussian iid returns pro cess and an AR returns pro cess with regression co ecient

For suciently large N weknow that the estimated auto covariances for

the iid pro cess will b e very small and fall inside the signicance band for

j  

nonzero lags Similarly the auto covariances for the AR pro cess

decay exp onentially with lag j and their estimates for nite N fall within the

signicance band for large enough j Therefore we exp ect that for the iid pro cess

the RS curves for dierent q should b e the same for large time scales N and that

the RS curves for the AR pro cess for large N will approach those of the iid

pro cess with increasing q These eects are illustrated by Figures and where for Gaussian AR(1), a=−0.5 1.6 1.6

1.4 1.4

1.2 1.2

1 1 q=8 Log10(R/S^) q=8 Log10(R/S^) 0.8 0.8

0.6 0.6 q=0 q=0

0.4 0.4

1 1.5 2 2.5 3 1 1.5 2 2.5 3

Log10(N) Log10(N)

Figure RS analysis for a gaussian iid returns pro cess left panel and an AR

returns pro cess with negative co ecient right panel using dierent lag parameters

q Note that the RS statistic intro duces a new estimation error when the

observed time scale N is small

large time scales RS curves for the AR pro cess shift upward for increasing q and

overlap the RS curves for the iid pro cess only for q This demonstrates that

the bias of the range exists in the RS statistic and that the RS statistic can correct

this bias for large N with a prop er choice of the lag parameter q

The left panel of Figure illustrates our second observation Since there is no

dep endence in the time series we exp ect that all the RS curves should b e equivalent

for dierent lag parameters q for all time scales N However as shown by Figure

the curves for dierent q are not the same when the time scale N is small This eect

is due to the negative bias in the second term of Eq

To explain our third observation we rst rewrite the RS analysis

R

HlogN b log

S

as

C R

log log HlogN b



S S



Where b is constantandS and C are the rst and the second terms of Eq q=0 q=2

1.5 1.5

1 1 Log10(R/S^) Log10(R/S^)

0.5 0.5

1 1.5 2 2.5 3 1 1.5 2 2.5 3 Log10(N) Log10(N) q=4 q=8

1.5 1.5

1 1 Log10(R/S^) Log10(R/S^)

0.5 0.5

1 1.5 2 2.5 3 1 1.5 2 2.5 3

Log10(N) Log10(N)

Figure Comparison of the RS analysis for an AR series solid curve to that

for a gaussian series dashed curve for q The bias of the range due to

the shortterm dep endence in AR series can clearly b e seen in the curves with

q Whenq the bias is corrected and the RS curve of the AR series

overlaps that of gaussian series for long time scales N

resp ectivelyWehave

P P

q

t N

r mr m w q

C

t tj j

tt j j 

P

t N





S

r m

t

tt 

C

If there is no shortterm dep endence and Eq reduces to that for the RS



S

C R

analysis If N is small against log N will change with N The slop e of log



S S

C

in Eq will b e mo died If N is large enough will not dep end on the value of



S

 C

in Eq is the same as that of b Its existence will N The eect of log



 S

shift the RS curvevertically but will not change the slop e of the curve The ab ove

explanation is completely consistent with the empirical results shown in Figures

andTable We therefore conrm that the RS analysis and the RS analysis

have the same Hurst exp onents when the time scale N is large

In Los own simulation results in Lo for example Table Vb for an AR

pro cess and Table VIb for a gaussian fractional dierenced pro cess we also nd similar

evidence For the AR pro cess when N the RS curves with q have

very close slop es For the gaussian fractional dierenced pro cess all RS curves

have similar slop es on all time scales with q and Figure depicts the

RS curves of Los simulation results On the other hand the upp er panel of Figure

shows that Los RS N mayhave a negativeslopewhich is not theoretically

justiable using the standard denition of Hurst exp onents and is incompatible with

the scaling laws for sto chastic pro cesses Feder

In summary Lo tests the random walk hyp othesis directly based on the value of

the RS or R S statistic while Hurst and Mandelbrot do so by comparing the slop e

of RS N curve to Unfortunately biases in the denitions of R S and S can

lead to errors in the estimates of RS N RS N and H for short time scales

N or when short term dep endencies are present in the series under study Under

standard interpretations of RS N RS N and H these errors can lead to

misleading and sometimes inconsistent results

Rescaled Range Analysis with Unbiased S

To address the problems with the statistics RS N and RS N describ ed

ab ove we prop ose an unbiased rescaling factor S that corrects for mean biases in

the range R due to shortterm dep endencies without inducing the distortions on short

time scales that S and Los S do Denoting the standard unbiased estimate of the

variance as

t N

X

 

b

r m N t

t 

N

tt  Gaussian Fractionally Difference, d=−1/3

0.4

0.3 q=50

0.2 q=25

0.1 q=10

0 q=5

Log10(R/S^) −0.1

−0.2 q=0 −0.3

−0.4

2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3 Log10(N) Gaussian Fractionally Difference, d=1/3

0.6

q=0

0.5

0.4

Log10(R/S^) q=5 0.3

0.2 q=10

0.1 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3

Log10(N)

Figure Los RS curves for gaussian fractional dierenced pro cesses The curves

are constructed based on Los own simulation results see Table VIa and Table VIb in

Lo All the RS curves have similar slop es in all time scales with dierent

time lags For the pro cess with d all the RS slop es are negative This is

incompatible with the allowed scaling laws for sto chastic pro cesses

the prop osed unbiased rescaling factor with weighted covariances up to lag q is



q q

t N

X X X

 N  j



b N t  w q  S N t q  w q  r  mr  m 

t

j j tj



N N

j  j  tt j

j

This where w q is the weighting function as dened byLow q

j j

q 



weighting function yields a p ositive S provided that qN It is trivial to show

that the estimates of the auto covariances in have zero mean bias When q

b

S reduces to the unbiased standard deviation

Gaussian AR(1), a=−0.5 1.6 1.6

1.4 1.4

1.2 1.2

1 1 Log(R/S*) Log(R/S*) 0.8 0.8

q=8 q=8 0.6 0.6 q=0 q=0 0.4 0.4

1 1.5 2 2.5 3 1 1.5 2 2.5 3

Log(N) Log(N)

Figure RS N for a gaussian iid returns pro cess left panel and an AR

returns pro cess with negative co ecient right panel using dierent lag parameters

q f g Note that unlike the results for RS N in Figure the mean bias

in R for the AR pro cess is corrected with q without inducing distortions for

short time scales N Note also that unlike the results for RS N the results for

the gaussian iid pro cess are indep endentofq

Figures and present empirical results for the prop osed RS N statistic

The results for simulated gaussian iid and AR returns pro cesses shown in Figure

conrm the ecacy of our prop osed RS N analysis The results for the high

frequency DEMUSD FX series in Figure supp ort our conclusions obtained by blo ck

averaging in Figure that the DEMUSD series is actually trending rather than

meanreverting on short time scales to ticks and the spuriouslyobserved

meanreversion on these time scales in the RS N curves is actually due to high

frequency oscillations on time scales of a few ticks

DEM_USD Oct.92 1.6

1.4

1.2

q=5 1

0.8 q=2 Log(R/S*)

0.6 q=0

0.4

0.2 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3

Log(N)

Figure RS N for the DEMUSD bid returns for Octob er using lag pa

rameters q f g Comparing the results for q to the RS N curves for

the unscrambled FX data in Figures and a we see slightly less apparentmean

reversion at short time scales N This reduction in apparent mean reversion is due

to the unbiasedness of S relativeto S The results for q f g are qualitatively

similar to the blo ckaveraged results for blo cks of and ticks shown in Figures

c and d This supp orts our conclusion that the apparent mean reversion on short

time scales in the DEMUSD series is actually due to the high frequency oscillations

and that when these are removed the series is actually slightly trending

Concluding Remarks and Discussions

The RS RS and RS Statistics and Hurst Exponents

Due to the use of a biased estimate of the standard deviation the RS and RS

statistics are biased upward on short time scales resulting in downward errors in

estimated Hurst exp onents When shortterm dep endencies are present the estimated

RS analysis may b e biased The RS statistic adds auto covariance to the range in

standard deviation to correct this bias when the observed time scale is large enough

When the time scale is small however the RS statistic intro duces new estimation

errors due to negative biases in S This eect also results in downward errors in

estimated Hurst exp onents Our prop osed RS statistic overcomes deciencies in

both RS and RS correcting for short term dep endencies in the time series without

intro ducing additional biases on short time scales N

The TickbyTick DEMUSD Series

As demonstrated in Mo o dy Wu Mo o dy Wu a and Mo o dy Wu

b there exist very signicant one or twotickanticorrelations in the returns

of the DEMUSD series When analyzing the price b ehavior and forecasting price

changes on longer time scales such shortterm anticorrelations should b e removed

Wehave demonstrated in this pap er that not considering these eects results in

completely dierent conclusions ab out the b ehavior of the series as measured by the

Hurst exp onents on time scales of to ticks

As shown in Mo o dy Wu b simply downsampling the price series cannot

removed this shortterm anticorrelation However our prop osed RS N analy

sis conrms our previous results obtained by short term blo ckaveraging that the

DEMUSD series is actually mildly trending on time scales of to ticks and

that the suggested meanreversion in the RS and RS analyses on these time scales

is spurious

Acknowledgements

We thank Steve Rehfuss for valuable comments and discussions We gratefully

acknowledge supp ort for this work from ARPA and ONR under grant NJ

and NSF under grantCDA

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