A Nearest Neighbor Bootstrap for Resampling Hydrologic Time Series

8/11/2019 A Nearest Neighbor Bootstrap for Resampling Hydrologic Time Series

1/15

WATERRESOURCES RESEARCH, VOL.32, NO. 3,PAGES 679-693,MARCH1996

A

nearest neighbor bootstrap for resampling hydrologic time

series

Upmanu Lall and Ashish Sharma

Utah Water Research Laboratory, Utah State University, Logan

Abstract A nonparametric method for resampling scalar or vector-valued time series is

introduced. Multivariate nearest neighbor probability density estimation provides the basis

fo r the resampling scheme developed. The motivation for this work comes

from

a desire

to preserve the dependence structure of the time series while bootstrapping (resampling it

withreplacement). The method is data driven and is preferred where the investigator is

uncomfortable

with prior assumptions as to the

form

(e.g., linear or nonlinear) of

dependence and the form of the probability densityfunction (e.g., Gaussian). Such prior

assumptionsareoften made in an ad hocmanner foranalyzing hydrologic data.

Connections

of the

nearest neighbor bootstrap

to

Markov processes

as

well

as its

utility

in

a general Monte Carlo setting are discussed. Applications to resampling monthly

streamflow

and some synthetic data are presented. The method is shown to be

effective

withtime series generated by linear and nonlinear autoregressive models. The utility of

the

method

for

resampling monthly streamflow sequences with asymmetric

and

bimodal

marginalprobability densities

is

also demonstrated.

Introduction

Autoregressive moving average (ARMA) models for time

series analysis are

often

used by hydrologists to generate syn-

thetic streamflow and weather sequences to aid in the analysis

of

reservoir

an d

drought management. Hydrologic time series

can exhibit the

following

behaviors, which can be a problem for

commonly

used linearARMAmodels: (1) asymmetric and/or

multimodal

conditional

a nd

marginal probability distributions;

(2) persistent large amplitude variations at irregular time in-

tervals; (3) amp litude-frequency dependence (e.g., the ampli-

tude of the oscillations increases as the oscillation period in-

creases); (4) apparent long memory (this could be related to

(2 )

and/or (3));

(5 )

nonlinear dependence between

x

t

versus

x

t

_

T

fo r

some

lagr; and (6)

t ime irreversibility (i.e.,

the

time

series plotted

in

reverse time

i s

"different" from

th e

time series

in forward

time).The physicsofmost geophysical processes is

time irreversible. Streamflow hydrographs often

rise

rapidly

and attenuate slowly, leading to time irreversibility.

Kendall

and

Dracup [1991] have argued for simple resam-

pling schemes, such as the index sequential method, for

streamflow simulation in place of ARMA models, suggesting

that

th e

ARM A streamflow sequences usually

do not

"look"

like real stream flow sequences. An alternative is presented in

the workofYakowitz[Yakowitz, 1973, 1979;

Schusterand Ya-

kowitz , 1979;Yakow itz , 1985, 1993;

Karlsson

and Yakowitz,

1987a,b ],Smith

etal [1992],

Smith

[1991],

and Tarboton

et

a l.

[1993] who consider the time series as the outcome of a

Markov process and estimate the requisite probability densities

using nonparametric methods. A resampling technique or

bootstrap for scalar or v ector-valued, stationary, ergodic time

series data that recognizes

th e

serial dependence structure

of

the time series is presented here. The technique is nonpara-

Copyright1996by the American Geophysical Union.

Paper number 95WR02966.

0043-1397/96/95WR-02966$05.00

metric; that

is ,

no prior assumptions

as to the

distributional

form of the

underlying stochastic process

ar e

made.

The

bootstrap

[Efron, 1979;

Efroh an d

Tibishirani,1993]is a

technique that prescribes a d ata resam pling strategy using the

random mechanism that generated the data. Its applications

for estimating confidence intervalsa nd parameter uncertainty

are well known

[see

Hardle

an d

Bowman,

1988;

Tasker,

1987;

Woo, 1989;

Zucchini andAdamson, 1989].

Usually the boot-

strap resamples with replacement from the empirical distribu-

tion function

of

inde penden t, identically distributed data.

The

contribution

o f

this paper

is the

development

of a

bootstrap

for

dependent data that preserves the dependence in a probabi-

listic sense. This method should beuseful for the Monte Carlo

analysis

of a

variety

of

hydrologic design

and

operation prob-

lems where time series data on one or more interacting vari-

ables are available.

The

underlying concept

of the

methodology

is

introduced

through Figure 1. Consider that the serial dependence islim-

ited to the two previous lags; that is,x

t

depends on the two

prior values

x

t

_

l

andx

t

_

2

.

Denote this ordered pair,

or

bituple, at a timet

t

byD,. Let the corresponding succeeding

value be denoted byS.Consider thek nearest neighbors of

D,

as thek

bituples

in the

time series that

are

closest

in

terms

of

Euclidean distance to D,. The first three nearest neighbors are

marked as

D

1?

D

2

an d

D

3

.

The

expected value

of the

forecast

S

can be estimated as an appropriate weighted averageof the

successors

x

t

(marked as

1,2,

and 3, respectively) to these th ree

nearest neighbors. The weights may depend inversely on the

distance between

D, and its

k nearest neighbors

D

1 ?

D

2

, ,

D^.

Aconditional probab ilitydensityf x\D

f

) may beevaluated

empirically using a nearest neighbor density estimator [see

Silverman,

1986,

p. 96]

with

th e

successors

x

l9

,

x

k

.

For

simulation thex

i

can be drawn randomly from one of the

k

successors to the D

1?

D

2

, , D^using this estimated condi-

tional density.Here,this operation

willbe

done

by

resampling

th e

original data with replacement.

Hence th e

procedure

de -

veloped is termed a nearest neighbor time series bootstrap. In

679


2/15

680

LALL AND SHARMA:

NEAREST NEIGHBOR BOOTSTRAP

I

I S S

f i l j

|

f i l

j|

Values o f

:

s ^ >

t i m e

Figure

1 Atime

series

fromthe

modelx

f

+

1

=

(1 - 4 x

t

-

0.5)

2

).

This is a deterministic, nonlinear model with a time

series

that looks random. A forecast of the

successor

to the

bituple D; , marked

as

S

in the

figure,

is of

interest.

The

"pat-

terns"

or bituples of

interest

are the

filled

circles,

near

th e

three

nearest neighborsD

1 }

D

2

andD

3

to the pattern

D

z

.

T he

successors to

these

bituples are marked as 1, 2, and 3, respec-

tively.

Note

how the

successor

(1) to the

closest

nearestneigh-

bo r D J L ) is closest to the successor S) of D,. A sense of the

marginal probability distributionof

x

f

is

obtained

b y

looking

at

the

values

of

x

t

shown

on the

right

side of the

figure.

As the

sample size

n

increases, the sample space of

x

gets filled in

between 0 and 1, such that the sample

values

are arbitrarily

close to each other, but no

value

is ever repeated exactly.

summary,o ne

finds

k

patterns

in the

datathat

ar e

"similar"

to

th e

current

pattern

and then

operates

on their respective

suc-

cessors to define a

local regression,

conditional

density,

or

resampling.

The nearest neighbor

probability density estimator

and its

use

withMarkov

processesis

reviewed

in the

next

section.The

resampling

algorithm is

described after that. Applications

to

synthetic an d

streamflow data

are then

presented.

Background

It is natural to pursue

nonparametric estimation

of

proba-

bilitydensitiesandregression functions through weighted local

averages of the target

function.

This is the foundation fo r

nearestneighbormethods.

T he

recognition

of the

nonlinearity

of

the underlying dynamics of

geophysical

processes,

gains

in

computational

ability,

and the

availability

of large

data sets

have spurred the

growth

of the nonparametric literature. The

reader isreferred to

work

b ySilverman [1986],Eubank[1988],

Hiirdle

[1989,

1990], and Scott [1992] for

accessible

mono-

graphs. G y o r f i eta l.[1989]provide

a

theoreticalaccount

that is

relevant for timeseriesanalysis.

Lall

[1995]surveys hydrologic

applications. For timeseries analysisamovingblockbootstrap

(MBB) waspresented by

Kunsch

[1989].Here a block ofm

observations is resam pled with replacement, as opposed to a

single

observation in the

bootstrap.

Serial dependance is

pre-

served

within,

but not across, a block. The block length

m

determines the order of the serial dependence that can be

preserved. Objective

proceduresfor the selection of the

block

lengthm

ar e

evolving.

Strategies for

conditioning

the MBB on

other processes

(e.g., runoff on

rainfall)

are not obvious. Our

investigations

indicated that the MBB may not be able to

reproduce the sample

statistics

as well as nearest neighbor

bootstrap presentedhere.

The k nearest neighbor (A>nn)

density

estimator is defined

as

[Silverman,

1986, p. 96]

/ NN( X) =

kin

kin

(1)

wherek

is thenumbero f

nearestneighbors

considered,

d

is the

dimension of the

space,

c

d

is the volume of a unit sphere ind

dimensions

c

l

= 2 , c

2

=

T T , c

3

=

47T/3,

, c

d

=

[7r

d/2

/Y d/2 + 1)]), r

k

x) is the Euclidean distance to the

A:th-nearest data point, and V

k

\) is the volume of a d-

dimensional sphere

of

radiusr

k

\).

This estimatoris readilyunderstoodby observing that for a

sample

of size

n,

we expect

approximately

{nf x)V

k

x)} ob -

servations

to lie in the

volume V

k

x).

Equating

this

to the

number

observed,

that

is ,k ,

completes

the definition.

A

generalized

nearest

neighbor

density

estimator

[Silver-

man,

1986,

p .

97],defined

in

(2),

c an

improve

thetailbehavior

of

the

nearestneighbor

densityestimator by

using

a monoton-

ically an d

possibly rapidly

decreasing smooth kernelfunction.

7GNN(

X

)

1

rt x)n

x

x/

r

k

x)

(2)

The "smoothing" parameter is the number of

neighbors

used,k,

and the

tail behavior

is determined by the

kernelK i).

The

kernel

has the

role

of a

weight function

(data

vectors X ,

closer

to the point of estimate x are

weighted more),

and can

be chosen to be any validprobability

density

function.Asymp-

totically,

under

optimal mean square

error (MSE)

arguments,

k

should be chosen

proportional

to

^

4/

(^

+4

)

for a probability

density

that

is twice

differentiable.

However, given a

single

sample from an unknown

density,

such a rule is of little prac-

tical

utility.The

sensitivity

to the choiceo f A :issomewhat lower

as akernel that ism onotonically decreasingwith

r

k

x)

isused.

A new

kernel function

that

weightsthej'thneighbor

ofx,

using

a

kernel

that

depends

on the

distance between x,

and itsy th

neighbor is developed in the resampling methodology section.

Yakowitz

(references cited

earlier) developed

a theoretical

basis for using nearest neighbor an dkernel methods fo r time

series forecasting and applied

them

in a hyd rologic context. In

those papers

Yakowitz considers

a finite order,

continuous

parameter

Markov

chain as an

appropriate

model fo r

hydro-

logic time series. He observes that discretization of the state

space

ca n

quickly lead

to

either

an

unmanageable number

of

parameters

(the curse of dimensionality) orpoor approxima-

tion of the transition functions,

while

theARMA approxima-

tions

to such a

process call

fo r

restrictive

distributional and

structural assumptions. Strategies for the

simulation

of daily

flowsequences,

one-step-ahead

prediction, and the conditional

probability of

flooding

(flow crossing a threshold) are exem-

plified with

river flows and shown to be

superior

to

A R M A

models.

Seasonality

is

accommodated

by including thecalen-

da r date as one of the

predictors. Yakowitz

claims that this

continuous

parameter Markov chain

approach is

capable

of

reproducing any possible

Hurst

coefficient.

Classical

ARMA

models

are optimalonly

under squared

error loss and onlyfor

linear operations on the

observables.

The loss/risk functions

associated

with

hydrologic decisions (e.g., wheth er

to

declare

a

flood

warning or not) are usually asymmetric. The nonpara-

metric

framework

allows

attention to be

focused directly

on


3/15

LALL AND SHARMA:

NEAREST

NEIGHBORBOOTSTRAP

681

calculating

these

loss functions and evaluating the conse-

quences.

The

example

of Figure 1 is now extended to showhow the

nearest neighbor

method

is

used

in the Markov framework.

One step Markov transition functions are considered. The

relationship betweenx

t l

andx

t

is shown in Figure2. The

correlation between

x

t

andx

t l

is 0,

even though

there is

clear-cut dependence between the two variables.

Consider an approximation of the

model

in Figure 1 by a

multistate, first-order

Markov

chain,

where

transitions

from,

say, state 1 forx

t

(0 to 0.25 in Figure 2) to states 1, 2, 3, or 4

forx

t l

are ofinterest.The state / to

state; transition

prob-

ability p

tj

is evaluated by

counting

the

relative fraction

of

transitions

from

state i to

state

j. The

estimated

transition

probabilities depe ndon thenumberofstates chosenas

well

a s

their

actual

demarcation

(e.g., one may

need

a nonuniform

grid that recognizes variations in data density). For the non-

linear model used in our

example,

a

fine discretization

would

be

needed.

Given a

finite data set, estimates

of the

multistate

transition

probabilities

may be

unreliable.

Clearly, this situa-

tion

is

exacerbated

if one

considershigher dimensions

for the

predictor space. Further, a

reviewer

has observed that a dis-

cretization of a

continuous space

Markovprocess is not nec-

essarily

Markov.

Now consider the

nearest

neighbor

approach. Consider

tw o

conditioning points x*

A

an d x*

B

. The k

nearest

neighbors of

thesepointsare in the

dashed

windows and

B,

respectively.

The neighborhoods are

seen

to

adapt

to variations in the

sam-

pling density

of*,.

Since such neighborhoodsrepresentmoving

windows (a s opposed to fixed

windows

for the multistate

Markov chain)

at

each

point of

estimate,

we can expect re-

duced bias in the recovery of the

target

transition

functions.

The one

step

transition probabilities at x* can be obtained

through an application of thenearest

neighbor

density estima-

to r

to thex

t+

values

that

fall in

windows likeA

and

B.

A

conditional bootstrap of the data can be obtained by resam-

pling from this set ofx

t l

values. Since

each transition prob-

ability estimate is

based

on k points, the problem faced in a

multistate Markov

chain

model of sometimes not

having

an

adequate

number

of

events

or state

transitions

to develop an

estimate is

circumvented.

The Nearest Neighbor

Resampling Algorithm

In thissectiona new

algorithm

fo rgen erating synthetic

time

series samples by

bootstrapping

(i.e., resampling the original

time series with replacement) ispresented.Denote the time

seriesbyx

t

,t=l,

,

n, an d

assume

aknown

dependence

structure, that is, which and how m any lags the

future

flowwill

depend

on. This

conditioning

set is

termed

a

"featurevector,"

an d

th e simulated or forecasted

value,

th e

"successor."

The

strategy is to find the historical

nearest

neighbors of the cur-

rent

feature

vector and resample

from

theirsuccessors. Rathe r

than

resampling uniformly from the k successors, a discrete

resampling kernel is introduced to weight th e

resamples

to

reflect th e similarityof the neighbor to the conditioning point.

This

kernel

decreases

mon otonically

with

distance and adapts

to the local sampling

density,

to the dimension of the feature

vector, and to

boundaries

of the sample space. An attractive

probabilistic

interpretation

of this

kernel consistent with

the

nearest

neighbor

density estimator

is also

offered.

The resam-

pling

strategy

ispresented through the following flowchart:

State

1 2 3

0.75.

0.5-

0.25.

State

4

0 0.25 0.5 0.75 1

Figure

2 Aplotof

x

t l

versusx

t

for the

time

series gener-

ated from themodel

x

t+

=(1 -

4 x

t

- 0.5)

2

).Thestate

space

for

x

isdiscretized

intofour states,

a s

shown.Also

shown

ar e

windows^

an d

B

with

"whiskers"located

over two

points

x*

A

an d

x*

B

.

These

windows

represent

a

k

nearest

neighbor-

hood of the corresponding

x

t

.

In

general,

these

windows

will

not be symmetric about the

x

t

of interest. One can

think

of

statetransition

probabilities

using thesewindows in

much

th e

same

w ay aswiththe

m ultistate Markov chain.

Avalueo f

x

t

+1

conditional

to point

A

or

B

can be

bootstrapped

by

appropri-

atelysamplingwith

replacementone of the

valueso fx

t

+

that

fall

in the

correspondingwindow.

1. Define the

composition

of the "feature vector"

t

of

dimensiond, e.g.,

Case 1 D,: (*,_ , J t ,_

2

);

d

= 2

Case2

=

Ml+M2

Case3 D, : (* l ,_

T l

,

l ,_

M l r l

;

x2

t

,*2,_

T

,

*2,_

M2T2

);

d = Ml

+ M2 + 1

where

rl (e.g., 1 month) and r2

(e.g.,

12 months) are lag

intervals, and

Ml,

M 2 > 0 are the number of

such

lags

considered in the model.

Case 1 represents dependence on two prior values. Case 2

permits direct

dependence

on

multiple

time scales, allowing

one to

incorporate monthly

and

interannual

dependence. For

case 3,xl an dx2 may

refer

to rainfall an d runoff or to two

different

streamflow

stations.

2.

Denote

the current feature vector as

D,

and determine

its

k

nearest neighbors among the D,, using the weighted

Euclidean

distance

1/ 2

(3)

wherev

tj

is

the;

th componento f

D

r

,

an d

W j

are scaling

weights

(e.g., 1 or 1 / S j , where s

}

- issome measure of scale such as the

standard deviation or range of V j ) .

The weights W j may be specified apriori,as indicated above,

or m ay be

chosen

toprovide th e

best

forecast

for a particular

successor

in a least

squares

sense [see Yakowitz an d Karlsson,

1987].

Denote theordered set ofnearest neighbor indices byJ

t

^ .

An

element

y ( /) of this se t

records

the

time t

associated with

t h e j t h closest D

r

to D,. Denote jc

y( / )

as the successor to D

y ( / )

.


4/15

68 2

LALL

AND SHARMA: NEAREST NEIGHBOR BOOTSTRAP

0.4

0.0

0.3

-

0.2-

It

r

. .*

Figure

3 . Illustration of

resam pling weights,

K j i ) ) , at

selected

conditioning

points*,,

using

k =

10 with

a

sample

of

size

10 0

from

an

exponential distributionwithparameter

of 1. The

original sampled values

are

shown at the top of the

figure. Note

how the bandwidth and kernel shape

vary with

sampling density.

If

the

data

are

highlyquantized,

it ispossible

that

a number of

observations

may be the

same distance from

the conditioning

point.

The resampling kerneldefined instep3 (below) is

based

on theorder ofelements in

J^

k

.

Wherea number of

observa-

tions are the same

distance away,

th e

original ordering

of the

data can

impact

the ordering inJ^

k

. To

avoid such

artifacts,

the time

indices

t ar e copied into a

temporary

array that is

randomlyperm uted prior to distance calculations and creation

of thelist]

ik

.

3. Define a

discrete

kernel K j i ) ) fo r resampling one of

the j c

y

-

/ )

as

follows:

(4)

exactly of volume

F(r

/J(/)

).

Assuming that the observations

are independent (which theymay not be), the likelihood with

whichthej(0

tn

observationshouldberesampledasrepresen-

tative of D; is

proportional

to

HV r

ij i}

).

Now,consider that in a small locale of D,, the local density

can be approximatedas aPoisson process,withconstant

rate

A .

Under

this

assumption

the expected value ofllV r

ij i}

) is

E l l V r

i j l }

} } =

A/;

(6)

The kernel K j i ) ) isobtainedb y

norm alizingtheseweights

over

th e

k nearest

neighborhood.

c lj

1/7

(7)

where K j i ) ) is the

probabilitywith which

x

j i}

is

resampled.

This resampling kernel

is the

same

for any

/

and can be

computed

an d

stored prior

to the

start

of the

simulation.

4. Using the discrete probability mass function (p.m.f.)

K j i ) ) , resample anj c

y( O

, update th e current

feature

vector,

an d

return

tostep2 if additional simulated values areneeded.

A similar

strategy

fo r

time

series forecasting ispossible.A n

m-step-ahead forecast is obtained byusing the corresponding

generalizednearest neighbor regression estimator:

(5 )

where*,

m

andz

y(/)m

denotethe rath successortoiand;(0

respectively.

Parameters of the Nearest Neighbor Resampling

Method

Choosing the Weight Function K v)

The goals fo r designing a resampling

kernel

are to (1) re-

duce

the

sensitivity

of the

procedure

to the actual

choice

of

k,

(2 ) keep

the estimator

local,

(3) have k

sufficiently

large to

avoid

simulating

nearly identical

traces,

and (4)

develop

a

weight function

that adapts

automatically to

boundaries

of the

domain and to the

dimensiond

of the

featurevectorD ,.These

criteria suggesta resampling kernel thatdecreases monotoni-

cally

as

r

tj

increases.

Consider

a

d-dimensional ball

of

volume V r)

centered at

D,.. The observation D

;(0

falls in this ball

when

the ball is

where c is a constant of proportionality.

These weights do not

explicitly

depend on the dimension

d

of the

feature vector

D,. The

dependence

of the

resampling

scheme

on

d

is

implicit

through the behavior of the distance

calculations

usedto

find

nea rest neighborsa sd

varies.

Initially,

we avoided

m aking the

assumption

of alocal

Poissondistribu-

t ion and def ined K j i ) ) t h r o u g h a no rma l iza t ion o f

l/K(r,

7(0

). Thisapproach

gave satisfactory

results

as

well

bu t

was computationally more

demanding. The

results obtained

using(7)were comparablefor a

given

k.

The behavior of this kernel in the boundary region, the

interior,

and the tails isseen inFigures3 and 4.

From

Figure

3,

observe that

th e

nearest

neighbor

method

allows

consider-

ablevariation in the"bandwidth"( intermsof arangeof values

of

jc)

as a function of position and underlying

density.

The

bandwidth becomes automatically

larger

as the density be-

comes

sparser an d

flatter.

In

regions

of

high data

density

(left

tail or

interior)

the

kernel

is nearly symmetric(the

slightasym-

metry

follows

th e

asymmetry

in the

underlying distribution).

Along

t he sparse

right tail

the kernels are qu ite asymmetric, as

expected. Some

attributes

of these kernels relative to a uni-

form

kernel (with the same k), used by the ordinary nearest

neighbor method, are

shown

inFigure4.

For bounded data (e.g.,

streamflow

that is constrained to be

greater

than

0), simulation of

values

across the boundary is

often a

concern. This problem

is avoided in the

method

pre-

sented

since

the resampling weights are defined only for the

sample

points.

A second

problem

with bounded

data

is

that

bias inestimating the target

function using

local

averages

in -

creases near

the boundaries. This bias can be recognized by


5/15

LALL AND SHARMA:NEAREST NEIGHBOR BOOTSTRAP

683

0.4

0.3-

0.2-

0.1

-

0.0

Centroido fK (j(i))

Uniform K erne]

yCentroid

0.00

0.02

0.04 0.06 0.08

0

.22 0.26 0.30 0.34

(a)

X

(b )

K ()

0.2-

2.0

6.0

(c)

Figure 4.

Illustration

of weights

K j i ) )

versus

weights

from

the

uniform

kernel applied

at

three

points

selected

from

Figure 3. The uniform kernel

weights

are l/k fo r

each

jsJ i, k). The effective centroid

corresponding to

eachkernel

fo r

eachconditioning

point / (in

each case

with the

highest value

of K j i ) ) ) is

shown.

For

/

in the interior of the data

(Figure

4b) the

centroids

of both th e

uniform

kernel an d K j i ) )

coincide with i.

Toward

the edges of the

data (Figures

4a and 4c) the centroid corresponding to K j i ) ) is

closer to / than that for the uniform kernel. The K j i ) ) ar e thus less biased than th e uniform kernel for a

givenk . The kernel K j i ) ) also has a lower varianc e than the uniform

kernel

for a givenvalue ofk.

observingthat

the

centroid

of the

data

in thewindowdoesn ot

lie

at the

point

ofestimate. From

Figure

4note that while the

kernel

is

biased toward

th e

edges

of the

data, that

is, the

centroid of the kernel does not match the

conditioning

point,

th e

bias

is much

smaller than

for the

uniform kernel.

If one

insists

on

kernels that

ar e

strictly positivewith

monotonically

decreasing weights with distance, it may not be possible to

devise

kernels

that are substantially

better

in terms of bias in

th e boundary region.

The

primary advantages

of the

kernel

introduced here ar e

that (1) it adapts automatically to the

dimension

of the data,

(2) the

resampling weights

need to be computed

just

once fo r

a

given

k , (there is no need to

recompute

th e

weights

or

their

normalization

to 1), and (3) bad effectsof data

quantization

or

clustering (this

could lead to

near

zero values ofr-

J(/)

at some

points) on the resampling strategy, that arise if one were to

resample

using

a kernel

that depends

directlyon

distance

(e.g.,

proportional

to\IV r

i

j^)), areavoided.Thesefactors trans-

late

into

considerable

savings

in computational

time

that ca n

be im portant for large datasetsan d high dimensional settings,

an d

into

improved

stability

of the

resampling

algorithm.

Choosing

the Number of Neighborsk and Model Orderd

The

order

of

ARMA

models is

often picked

[Loucks

etal.,

1981]using theAkaike information criteria(AIC). Suchcrite-

ria

estimate

th e

variance

of the residual to

time

series

forecasts

from amodel, ap propriately penalizedfor the

effective degrees

of

freedom in the model. A similarperspective based on cross

validation isadvocated

here.

Crossvalidation

involves

"fitting"

the model byleavingout onevaluea t a

time

from the data and

forecasting it using the remainder. The model

that yields

th e

least-predictive sum of squares of errors across all such fore-

castsi spicked. One can

approximate

the averageeffect of such

an

exercise

on the sum of

squares

of

errors without

going

through th eprocess.

Here, the

forecast

is formed

(equation

(5)) as a

weighted

average of the successors. When using the full sample, define

th e

weight

used

with

the successor to the current point asw

7;

.

Thisweight

recognizes

the influenceof that point on the esti-

mate

at the same location.Hence the influence of the rest of

th e pointson the fit at that point is (1 -

n >

7 7

) .

This suggests

thatifestimated full sampleforecast error < ?

y

is

divided

by(1 -


6/15

684 LALL AND SHARMA:

NEAREST

NEIGHBOR

BOOTSTRAP

Table 1. Statistical

Comparison

of

A ; - n n

and AR1

Model

SimulationsApplied to an A R1 Sample

Simulations

5% Quantile Median

95%

Quantile

AR1

Sample

k-nn AR1

k-nn

AR1

k-nn

AR1

Mean

0.04

Standard

111

deviation

Skew

-0.17

Lag

1 0.63

correlation

-0.14 -0.12 0.02

0.04

1.02 1.03 110 111

0.24 0.20

1.18

1.20

-0.32

-0.25 -0.18 0.00 -0.03 0.21

0.56

0.57 0.62 0.63 0.68 0.69

W j j) , ameasureofwhat the

error

may be if the datapoint (D

7

,

X j )

was not used in developing the

estimate

is

provided.

Note

that the degrees of

freedom

(e.g., 0 for ak = 1 and 4/3 for a

k = 3

using

a

uniform

kernel)

of

estimate

are implicit in this

idea.

Craven and Wahba

[1979]

present

a

generalized

cross

validation (GCV) score function that considers

the average

influence

of

excludedobservations

fo r

estimation

at

eachsam-

ple

point

an d approximates the

predictive

squared error of

estimate. The GCV score is

given

as

I

e]ln

(8)

The GCV

scorefunction

can be used to

choose

both

k

and

d.

For the kernelsuggested in thispaper, w

;7

is aconstant fo r

a givenk, and the GCV can bewritten as

G CV

=

(9 )

A prescriptive

choice

of

A :

=n

l/2

from experience isalso

suggested.

This is agood choicefor 1


7/15

LALL AND SHARMA: NEAREST NEIGHBOR BOOTSTRAP

685

Xt-1

Figure

6. (a) An ASH estimate of thebivariate

probability

densityf x

t

,

*,_i)

for the

SETARsample,

th e

thick curve

denoting

aLOWESS

smooth,

and (b) an average of the ASH estimates of the bivariate

probability

density

f x

t

,

x

t

_-

across 100

/c-nn

resamples

from

the

SETAR

sample.

ings are reproduced. One can try various combinations of k

and d and

visually

compare

resamp ling attributes with

histor-

ical sample attributes

(sample

mom ents and

marginal

or

joint

probability densities).

Applications

Two synthetic

examples,

on e

from

a

linear

and one

from

a

nonlinear model,

a re

presented

first.

These

are

followed

by an

application

to

monthly

flows

from

Weber River near Oakley,

Utah.

In all cases a lag 1 model

with

k

chosen

a s

n

1/2

was

used.

Comparative

performance

of the

simulations

is

judged

using

sample

moments and samplep.d.f.'s

estimatedusing adaptive

shifted

histograms

(ASH;Scott

[1992,

chap.

5]).

In all applica-

tions usingtheunivariate

ASH,

a binwidthof (*

max

- *

min

)/

9.1,

wherex

max

and*

min

are the

respective

maximum and

minimumvalues of the data, and five shifted histogramswere

used.For bivariate densitiesa binwidthof x

max

-

*

min

)/3.6

an d

five

shifted histograms in each

coordinate

direction were

used.These are the default settings for the computer code dis-

tributed by D.Scott.Conditional expectations,E x

t

\x

t

_

1

), are es-

timated

using

LOWESS

[Cleveland,

1979].LOWESSis a popular

robust, locallyweighted linear regression technique that allowsa

flexible

curve

to be fitbetween two variables. We used default

parameter choices (threeiterations for computing the robust es-

timates

basedon two thirds of the da ta)

with

the "lowess"func-

tion

availableon

S-Plus [Statistical Sciences,

Inc.,

1991].


8/15

686


200

400 600

Time (months)

2 3 4 5 6 7 8 9 10 11 12

Time

(monthsf or

year

2 )

3 4 5 6 7 8 9 10 11 12

Time

(months

for

year

29)

Figure 7. TheWeber River

monthly

streamflow t imeseries.Flowsfor 2 years (1906

(year

2) and1933(year

29)) are shown in the lower panels.Note the

asymmetry

about thepeak

monthly

flow in 1906

clearly shows

that the t ime

series

is irreversible; that is, its

propert ies

will be quite different in

reverse

t ime.

ExampleEl

The

first

data se t considered is a

sample

of size 500

from

a

linear autoregressive model of order 1,AR1, defined as

x

t

= (10)

wheree

t

is a meanzero,variance 1, Gaussian randomvariable.

One hundred real izat ions,

each

of length 500, were

then

generated

from

an AR1 model

fitted

to the

sample

an d

from

th e

nearest neighbor (/c-nn) bootstrap. Selected statisticsfrom

these simulations are compared inTable1. The sample statis-

tics

considered

ar e

reproduced

by the /c-nn

method ,

whileonly

th e

sa mple statistics

used in fitting the parametr ic AR1 model

are reproduced.The AR1

simulations

insteadreproduce pop-

ulation

or model

statistics

(e.g.,skew = 0) for parameters that

are not explicitly fitted to the

sample. With

repeated applica-

tions to anumber of samples from th e same distribution, the

A:-nn procedure reproduces

the

population

statistics as

well.

On theother

hand,

a parame tric model only

reproduces

fitted

statistics. The ASH estimated median

p.d.f.

ofx

t

, from th e

/c-nn resamples, matches th e sample

p.d.f.,

and the scatter of

the estimated

p.d.f.'s

across

resamples

is comparable to the

scatter

from the AR1 samples. In order to save

space,

these

results

are not reproducedhere.

Example E2

A sample of size 200 was generated from a

self-exciting

threshold autoregressive

(SETAR)

model described by Tong

[1990, pp.

99-101].

The general structure of such models is

similar to

that

of a

linear autoregressive model,

with the

dif-

ference being that the parameters of the model change

upon

crossing one ormore thresholds.

Such

a m odel may be appro-

priate

for daily

streamflow, since crossing

a

flow threshold

(defined on a single

past

flow or collectively on a set of past

flows) with

flow

increasing

may signal

runoff

response to

rain-

fall

or snow

melt ,

and crossing the thresho ld

withflow decreas-

ingm ay

signal

return

to base flow or recessionbehavior.Here

a lag 1

SETAR model

was

used:

x

t

= 0.4 + 0.

x

t

= 1 . 5

*,-

+ e

t

.5*,- +e,

if x

t

P O

O3P

@3 S -

^ ^ C/3 O

a p o

Q

Probability Density

0.0 0.0004

0.0008 0.0012

& S 8 -

^

S

-^

^ 8

v ?

3

* ^ o

I"

cr

O^

x H -.

8 E C

Probability

Density

0.0 0.0004 0.0008

0.0012

O

I o

HOSHOIHN

1SHHV3N

^

QNV1TV1


11/15


689

1 15

O c t o b e r

1 15

O c t o b e r

Figure

10.

(a) An ASH estimateof the

b ivariate probability

densityf x

t

,

x

t

_ )

for the

October/November

historicalflows,

the thickcurvedenotingaLOWESSsmooth; (b) anaverageof

theASH estimates

of/(;t,,

x

t

_ across100

simulations

from

an

ARl

model;

and (c) an

average

of the ASH

estimates

of

f x

f

, J c

f

_

1

)

across 100k-rm resamples. Values are given in

cubic

feet pe r second (1 cfsequals 0.028

m

3

/s).

snow

m elt. January

is

typically

the month

with

the

lowest

flow.

The

1905-1988

monthly

t ime

series

and

flows

for two specific

years

ar e presented in Figure7.

A monthly ARl model was fitted to logarithmically trans-

formed

monthly flows,

with

monthly

varyingparameters

esti-

mated

asdescribedbyLoucks eta l.

[1981,

p.285] topreserve

moments in real space. This entails sequentially simulating

monthly

flow

moving

through

th e calendar

year,

using (for

example)only the 83

monthly

values for

J anuary

an d February

over the 83-year record to simulate February flows

givenJan-

uary flows. O ne hundred

simulations

of 83 yearswere gener-

ated in

each case.

The A:-nn

bootstrap

is

applied

in a similar

mannerusing(forexample)

J anuary

flows to findneighborsfo r

a given

January

flow and the corresponding February succes-

sor.

The

flow

data

are not

logarithmically transformed

for the

k-rmbootstrap.

Selected results

ar e

presented.

A

comparison

of

means,

standard deviations,skew,and lag 1 correlations are presen ted

for

the 12

months

in

Figure

8.

Both

thek-rmand the ARl

model

seem to be comparable in reproducing th e mean

monthlyflow

and its variation

acrosssimulations.

The standard

deviations

of

monthly flows

are

somewhat more

variable in

k-rm simulations than those

from the ARl

model . Some

months (low flows) have

a

slight

downward

bias

in the

simu-

lated standard deviation,

while

others (March and

July,

th e

months following a minimum and maximum in monthly flow,

respectively)

show

a

larger spread

in standard

deviation

than in

the ARl case. While both models seem to do well in repro-

ducing

the historical lag 1 correlation, the

k-rm

statistics ap -

pear to be

more variable across

realizations. A difference be -

tween the two simulators is apparent in Figure 8c ,where the

ARl

model fails

to

reproduce

th e monthly and the

annual

skews

aswell as the

k-rm

mo del does. Recall that the ad hoc

prescriptive choiceof A : =n

l/2

wasused here,withnoat tempt

at fine tuning th ek-rmsimulator.

The

marginal probability density

functions were estimated

by

ASH for flow in

each month. Selected

results for

simula-

tions

from

th ek-rmand for the AR l model applied to loga-

rithmically transformed flows for 3

months

ar e

illustrated

in

Figure

9. We see

from

Figure

9 that the

k-rm

samples are

indeed

aboo tstrap; thatis, the

simulated marginal probability

densities behave much like the empirical sample probability

density.

The usual shortcoming of thebootstrapin reproducing

only historical sample values

is

also apparent.

We see

that

while the lognormal

density

usedby the ARl modelis

plausi-

ble in a number of months

(e.g., October),

the lognormal

modelseemsto be inappropriateinotherm onths, for example,

March,

where the

skew

is too extreme for the

ARl,

andJ une ,

whichexhibits a distinct bimodalitythatm ay be related to the

timing

or

amount

of snow

melt.

T he

latter

is

interesting, since

the 10 0 simulationsfrom the ARl model

fail

tobracketthe two

prominent modes of the ASH density estimate,

lending

sup-

port to the

idea

of

bimodality

under apseudohypothesis

test

obtained from thisM onte Carlo experiment.

The

bivariate probability

density

functions

fo r flows ineach

consecutive pair of months (e.g., May and J une) were also

computed

by

ASH.

Results forselected months arepresented

in Figures 10 through 12. Other month pairswere

found

to

exhibit

features similar

to those in Figures 10 through 12. In

each

casew epresent ascatterplot of the flows (in

cfs)

in the 2

months,with

a

LOWESS

[C leveland,

1979] smooth

of thecon-

ditionalexpectationE x

t

\x

t

_

1

).

Anexaminationof the

Octo-

ber/Novem ber density inFigure10 reveals that the ARl model

may

be quite

appropriate

for this

pair

of

months.

The ASH

estimated

density

from

the

sample

and the

averages

of the

ASH

estimated densitiesfrom

the ARl and the

k-rm

samples

are all

very similar.

The

LOWESS estimate

of the

conditional

expectation of the November flow, given the October flow, is

very nearly a straight

line.

Figures 11 and 12 refer to the months of

April/May

and

May/June , where runoff

from

snow

melt becomes important.

The timing of the start and of thepeak rate of snow meltvary

over

this

period. Consequently, one can expect some hetero-

geneity in the sampling distributions of flows in thesemonths.

From Figure

lla

we see tha t the LOWESS estimate of

E x

t

\x

t

_

1

), exhibits some degree of nonlinearity fo r April/

May.The

slope

o f E x

t

\ x

t

_

l

) f o r * , _ < 150 cfs

(4.2

m

3

/s) is

quite

different

from

th e

slope

forx

t

_

l

>

150

cfs.

This

is

reminiscent of the

SETAR

model examinedearlier.We could

belabor this

point through formal tests

of

significance

for

dif-

ference

in slope. Our purpose here is to show that the

k-rm

approach

ca n

adapt

to such

sample

features, while the ARl

model

may

not.

The

averagebivariate densities

of the

Simula-


12/15

690

LALL

AND

SHARMA: NEAREST NEIGHBOR BOOTSTRAP

1 0 0

200 300

April

300

April

400

500

200 300

April

500

Figure

11 (a) An ASHestimateof the

bivariate

probability densityf x

t

, x

t

_ ) for the April/May

historical

flows, the thick

curve denoting

aLOWESSsmooth, (b) an average of the ASH estimatesof/(*,,x

t

_

x

) across

100 simulationsfrom an AR1

model,

and (c) an average of the ASH estimates

o f f x

t

, x

t

_

x

) across 100k-nn

resamples.

Values

are given in

cubic

feet per second (1 cfs

equals

0.028

m

3

/s).

tions based on ASH

once

again reinforce this difference be -

tween the A ; - n n and the AR1

models

and their

attributes.

Note

also

that the AR1

simulations

do not

reproduce

the sample

skews

for this

period

either.

The May/June analysis,

shown

inFigure 12, is

marked

by

considerably

increased variability in stream

flow

as the snow

melt

runoff develops.Once again, LOWESS shows some

de -

gree ofnonlinearity inE x

t

\x

t

_

1

), with theslope of the rela-

tionship smaller fo r high flows than for low flows. A

compar-

ison of the ASH bivariatedensitycontours inFigures 12a and

12 c reveals that the AR1 density is oriented quite differently

from

the ASH estimate for the raw sample and is unable to

reproduce

th e degree of

heterogeneity

in the sample density.

Recall that

the marginal density of June

flows

was bimodal,

with an antimode around 1000 cfs (28 m

3

/s). The

antimode

suggests that the

data

is

clustered

into two

classes

of events:

thosewith

flow below 1000

cfs (mode at 700 cfs

(19.6

m

3

/s))

and those with

flow

above 1000 cfs (mode at 1300 cfs (36.4

m

3

/s)). The LOWESS fi t

suggests

that the June flows

have

an

expectationcloseto1000cfs for M ayflowsgreater

than

about

70 0cfs.It appears that the conditional expectation averages

across the two modes fo r J u n e flows an d that the conditional

density

(Figure 12b) of June flows,

given

May flow, may be

bimodal, as seen in the marginal densityplot fo r J u n e flows.

The

significance

of the

findings

reported

above

is

that

the

nearest

neighborbootstrapprovides a

rather

flexible an d

adap-

tive

method for reproducing the historical

frequency

distribu-

tion

of streamflow. The possibly tenuous issue of choosing

between

a

variety

of candidate

parametric models month

by

month is

avoided. Matching

the historical

frequency

distribu-

tion

of

flows properly

is

important

fo r

properly estimating

storage requirements for a reservoir and analyzing reservoir

release

options. For snowmelt-driven streams in arid

regions

the

timing

an d

amount

of melt is

important

in

determining

reservoir operation.

The bimodality in the

probability density

of monthly streamflow during

the

melt months

may be con-

nected

to

structured

low-frequency

( interannual

and interdec-

adal)

climatic

fluctuations

in this

area [see Lall and Mann,

1995]. This would be a significant factor for reservoir opera-

tion,sincethe timing and amoun t of snowmelt maycorrespond

to a circulation pattern that corresponds to

specific

flow pat-

terns

in

subsequentmonths

as

well.

The

nearestneighborboot-

strap

would be an appropriate technique for simulating se-

quences conditioned on such factors. Work in this

direction

is

inprogress.

Summary and Discussion

A

nearest neighbor method

for a conditional

bootstrap

of

time

series waspresented an d exemplified. A corresponding

forecasting

strategy was

indicated.

O ur

contributions here

lie

primarily in the development of a newkernel, suggestion of a

parameter selectionstrategy,application to a

conditional

boot-

strap, an d demonstration of the methodology. It was shown

that sample attributes

ar e

reproduced

quite

well

by this

non-

parametric method

f or

both synthetic

and

real da ta sets. Given

the

flexibility

of these techniques, we consider them to have

tremendous

practical

potential.

The

parametricversus

nonparametric

statistical

method de-

bate

oftenveers toward

sample sizerequirements

an d

statisti-


13/15


691

\

500 1000

15

June

flows

(cfs)

2

2 4 6 8 1 12

May

2

8

2

2

8 12

May

May

Figure 1 2. (a) An ASH estimate of the bivariate probability

densityf x

t

, *

r

_

x

)

for the May/Jun e historical

flows,

the thick

curve denoting

a

LOWESS smooth;

(b) ASH

estimate

of the probability density of

J u n e

flows

conditionalto a Mayflowof 867 cfs(24.3

m

3

/s);

(c) an averageof the ASH estimateso f f x

t

, x

t

_ ) across1 00

simulations

from an AR1

model;

and (d) an

average

of the ASH

estimates off x

f

, x

f

_ across

10 0

k-rm

resamples. Values are given in cubic feet pe r second (1 cfsequals 0.028 m

3

/s).


14/15

692


cal efficiency arguments.

In the

context

of a

resampling strat-

egy,

as

espoused here,

these arguments

take

a somewhat

different

complexion. For some

processes,

such as daily

streamflow, identification

or

even definition

of anappropriate

parametric model is

problematic.

In

these

cases, data are

rel-

atively plentiful. For such cases, methods such as

those

pre-

sented here

ar e enticing.F or

monthly

an d annual

flows,

there

is

progressively

less structure, and sample sizes are smaller. In

these situations, parametric methods m ay indeed be

statisti-

cally

more

efficient provided the correct model is identifiable

an d

parsimonious.

In our

view, particularly

for the snow-fed

rivers

of the

westernUnited

States,

this

may not alwaysbe the

case. Indeed,for theapplication presented hereat the

monthly

timescaleit ishard tojustify choosingtheparam etric approach

over the

nearest

neighbor method. The consideration of pa-

rameter uncertainty is

justifiably

considered a good idea in

parametric

time series resampling of streamflow [Grygier and

Stedinger, 1990]. Likewise

it may be

useful

to think

about

model

uncertainty

when developing parametric models. The

latter consideration

is

implicit

in the

nonparametric approach,

since a

rather broad class

of

models

is

approximated.

The

impact ofvarying the"parameters"k and the modelorder on

specific attributes

of the resamples bears

further

investigation.

Our preliminary analyses suggest that the sensitivity of the

scheme

is

limited

over a

range

ofk values

near

the "optimal"

with the kernel used

here. Formal

investigations of this issue

are being pursued.

One can devise a

strategy

that allows

nearest neighbor

re -

sampling withperturbation

of the historical

data

in the

spirit

o f

traditional

autoregressive

m odels, that is, conditional expecta-

tion

withan addedrandominnovation. First, one evaluates the

conditional

expectationusing

the generalizednearestneighbor

regression estim atorf or

each

vectorD,in the

historicalrecord.

A residuale

t

can be computed as the

difference between

th e

successor*;o fD,and the

nearest

neighbor regression forecast.

The

simulation

proceeds by

estimating

th e nearest neighbor

regression

forecast relative to a conditioning vector D, an d

then adding to this one of the < ?

y

correspondingt o adata point

j

that

lie in the

k nearest neighborhoodJ

tk

. The innovation e ^

is chosen using the resampling kernel K j i ) ) .

This

scheme

will per turb

th e historical

data

points in the

series,

with inno-

vations

that

ar e representative of the neighborhood, and will

thus

"fill in" between the

historical

data

values

as well as

extrapolating

beyond the

sample.

The computational

burden

is

increased and

there

is a

possibility that

the bounds on the

variables

will be violated during

simulation. However,

there

may be

situations

where the

investigator

m ay

wish

to adoptthis

strategy.

Further exploration

of

this

strategy is

planned.

Issues

such as

disaggregation

of streamflows

bear

further

investigation. O ne

strategy

is

trivial: resample

the

flow vector

that aggregates

to the

aggregate flow simulated.

A question

that

arises

is

whether there

is even any

need

to

work with

models that disaggregate(especially

in

time)usingthesemeth-

ods. One may

wish

to work directlywith,

say,

the daily flows,

conditioned on a

sequence

of

past daily flows

an d

weekly

or

monthly

flows.

The real

utility

of the method presented

here

may lie in

exploiting a

dependence

structure (e.g., in

daily

flows)

that is

difficult

to treat by traditional methods, as well as

complex

relationships betwe en variables, and in estimating confidence

limitsor riskin problems that

have

a

time

series structure. The

traditional

time

series

analysis

framework

directs

the

research-

er's attention toward an efficient estimation of model param-

etersund er som e m etric (e.g., least squares or maximum like-

lihood).

T he

performance metric

of interest to the

hydrologist

may not be the one optimal for the estimationof a certain set

of

parameters an d

selected model form. There

is

reason

to

directly explore other aspects of the problem that may be of

directinterest fo r reservoir

operation

and flood control, using

flexible,

adaptive, data exploratory methods.

Such investiga-

tionsusing

th e

/c-nnbootstrap

are in

progress.

Acknowledgments. The work reported herewas supported in part

by

t he

USGS through g rant nu mb er 1434-92-G-2265. Discussions

with

an d

review

of

this work

by David

Tarboton

are

acknowledged.

The

comments

of the

anonymous reviewers resulted

in a significant im -

provement of the

manuscript.

References

Cleveland,

W. S.,

Robust

locally weighted

regression

and smoothing

scatterplots,/.

Am. Stat.

Assoc.,

7 4,

829-836,

1979.

Craven,

P., and G.

Wahba ,

Smoothingnoisy

data

withsplinefunctions,

Numer. Math.,

31 ,

377-403, 1979.

Efron, B.,Bootstrap methods:Another look at the Jacknknife,Ann.

Stat.,

7,

1-26, 1979.

Efron, B., and R. Tibishirani,A n Introduction to theBootstrap,

Chap-

man and

Hall,

Ne w

York,1993.

Eubank, R. L.,

Spline Smoothingand Nonparametric

Regression,

Statis-

tics: Textbooks

and Monographs, Marcel Dekker,

New

York,1988.

Grygier,

J. C, and J. R.

Stedinger, Spigot,

a synthetic streamflow

generation package,technicaldescription,version2.5,School

o f

Civ.

an d

Environ.Eng.,Cornell

Univ.,

Ithaca, N. Y.,1990.

Gyorfi,

L., W .

Hardle,

P.

Sarda,

and P. Vieu,

Nonparametric Curve

Est imat ion f rom Time Series,

vol. 60 ,

Lecture Notes in Statistics,

Springer-Verlag,

New York,1989.

H a rd le ,

W ., A pplied Nonp arametric Reg ression, Econometr ic

Soc.

Monogr., Cambridge Univ. Press,

New

York, 1989.

Hardle ,

W .,

Smoothing

Techniques

With

Implementation

in S,

Springer-

Verlag,

New

York ,1990.

Hardle, W .,

and A. W. Bowman,

Bootstrapping

in

nonparametric

regression:

Local adaptive smoothing

and confidence bands,/.Am.

Stat.

Assoc.,

83,

102-110,

1988.

Karlsson, M., and S.

Yakowitz, Nearest-Neighbor methods

fo r

non-

parametric rainfall-runoff forecasting, Water Resour.Res.,

23 ,

1300-

1308, 1987a.

Karlsson,

M .,

and S. Yakowitz, Rainfall-runoff forecasting

methods,

old and

new,Stochastic Hydrol. Hydraul . , J ,303-318, 1987b.

Kendal l ,

D. R., and J . A. Dracup, A comparison of

index-sequential

an dAR(1)generated

hydrologicsequences,/

Hydrol.,122,335-352,

1991.

Kunsch , H.R.,The Jackknife and the Bootstrap for General

Stat ion-

ary Observations,

,4wL

Stat.,

1 7,

1217-1241, 1989.

Lall, U., Nonparamet r ic

function

est imation:

Recent

hydrologic appli-

cations,

U.S.

Natl.

Rep. Int.

Union Ge o d . Geophys. 1991-1994,

Rev.

Geophys., 33,1093, 1995.

Lall, U.,

and M.

Mann ,

The

Grea t Salt Lake:

A

barometer

of

low-

frequency climate

variability,

Water

Resour.Res.,31 W) , 2503-2515,

1995.

Loucks, D. P., J. R.

Stedinger,

and D. A.Hai th ,WaterResourceSystems

Planning and Analysis,

55 9

pp.,

Prentice-Hall ,

Englewood

Cliffs,

N. J . ,

1981.

Schuster, E., and S.

Yakowitz,C ont r ibut ions

to the

theory

of nonpara-

metric regression,

with

application

to

system identif ication, Ann.

Stat., 7,139-149,1979.

Scott,

D. W ., Multivariate Density Estimation: Theory, Practice an d

Visualization, J o hn Wiley, New York ,1992.

Silverman,

B.

W .,

Density

Estimation for Statistics and Data Analysis,

Chapman and

Hall,

New York,

1986.

Slack,

J . R., J . M.

Landwehr,

and A.

Lumb,

A

U.S.

Geological

Survey

streamflow data set for the United States for the

study

of climate

variations,1874-1988,Rep.92-129,U.S.Geol.Surv.,

Oakley,

Utah ,

1992.

Smith,

J . A.,

Long-range streamflow

forecasting

using

nonparametric

regression,

Water

Resour.

Bull. ,

27,39-46,1991.

Smith,J.,

G. N.

Day,

and M . D.

K ane , Nonparametr ic framework

fo r


15/15

Documents

A Nearest Neighbor Bootstrap for Resampling Hydrologic Time Series