ttyThe BDS statistic and residual test
H.S.Kim,D.S.Kang,J.H.Kim
Abstract.The conventional nonparametric tests have been widely ud in
many fields for the residual analysis of a fitted model on obrvations.Also,in recent,a new technique called the BDS (Brock–Dechert–Scheinkman)statistic has been shown that it can be ud as a powerful tool for the residual analysis,especially,of a nonlinear system.The purpo of this study is to compare the powers of the nonparametric tests and BDS statistic by residual analysis of the fitted models.This study evaluates stochastic models for four monthly rainfalls in Korea through the residual analysis by using the conventional nonpara-metric and BDS statistics.We u SARIMA and AR Error models for fitting each rainfall and perform the residual analysis by using the test techniques.As a result,we find that the BDS statistic is more reasonable than the conventional nonparametric tests for the residual analysis and AR Error model may be more appropriate than SARIMA model for modeling of monthly rainfalls.
Keywords:Nonparametric tests,Correlation integral,BDS statistic,SARIMA model,AR Error model
1
Introduction
After Yule (1927)or Box and Jenkins (1970),the stochastic models have been ud in scientific,economic and engineering applications for the analysis of time ries.Hydrologists have also widely ud stochastic analog for the analyzing and modeling of hydrologic time ries.However,when the stochastic model is fitted to the hydrologic data ts,the conventional nonparametric techniques may be lacking for a clear residual analysis.This was proved by Brock et al.(1991),who Stochastic Environmental Rearch and Risk Asssment 17(2003)104–115ÓSpringer-Verlag 2003DOI 10.1007/s00477-002-0118-0104
H.S.Kim Department of Civil Engineering,Inha University,Incheon,Korea
D.S.Kang Department of Water Resources,Kun-il Engineering Co.,Ltd.,Seoul,Korea
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J.H.Kim (&)Department of Civil and Environmental Engineering,Korea University,Seoul,Korea,E-mail:jaykim@korea.ac.kr
This work was supported by grant No.R01-2001-000-00474-0from the Basic Rearch Program of the Korea Science &Engineering Foundation.
ud a new test technique,called the Brock –Dechert –Scheinkman (BDS)statistic,for residual analys.bombay
The BDS statistic can be applied to the estimated residuals of any time ries model and ud as a model lection tool.The BDS statistic has its origins in the recent work on deterministic nonlinear dynamics and chaos theory.It is not only uful in detecting deterministic chaos,but also rves as a residual di-agnostic test.If the model does capture all the properties of hydrologic data,the residuals should be independently and identically distributed errors (IIDs),which have no patterns.Under the null hypothesis of independent and identical distribution (IID),the BDS statistic has shown its ability in distinguishing random time ries from the time ries generated by low dimensional chaotic or nonlinear stochastic process.Hsieh (1989)tested the nonlinearities in daily foreign exchange rates using the BDS statistic and detected a strong nonlinear dependence.Takens (1993)surveyed whether a given stationary time ries is adequately described by a linear stochastic model or contains nonlinearities using the BDS statistic.Also,the BDS statistic rves as a residual diagnostic that can be ud for testing the ‘‘goodness of fit ’’of an estimated model.Brock et al.(1996)showed that the BDS test is not just a test of nonlinearity,but might also be a good test of model speci fication.The speci fic objective of this study is to compare nonparametric statistics with the BDS stati天然气ng
stic and to detect an appropriate model for monthly rainfalls using the BDS statistic.Monthly rainfall is still a uful time scale to summarize water availability for many applications in hydrology,agriculture and ecology.Seasonal autoregressive in-tegrated moving average (SARIMA)(Box et al.,1970)and autocorrelated error (AR Error)(White,1984)models are ud for detecting an appropriate model for monthly rainfalls in Korea.
2
Test statistics
2.1
Nonparametric statistics
When a quence of obrvations is uncorrelated,the autocorrelation function for all lags should be theoretically equal to zero.However,the estimated autocorre-lation function of the sampled uncorrelated ries is not exactly equal to zero,but has a sampling distribution which depends on the sample size.This sampling distribution may be ud to test the hypothesis that the sampled time ries has independently and identically distributed (IID)errors (Salas et al.,1993).In this ction we prent four nonparametric statistics which are widely ud for time ries analysis.
2.1.1
Anderson’s correlogram test
The sample autocorrelation coef ficient q k for the obrvations f x i g ;i ¼1;...;N ;in which N is the sample size,may show the normal distribution with mean zero and variance 1=N when the sample size N is large.The two-sided interval for the test of the null hypothesis q k ¼0versus the alternative hypothesis q
k ¼0is given as follows Àu 1Àa =2ffiffiffiffiN p ;u 1Àa =2ffiffiffiffiN p
!
ð1Þ105
where u 1Àa =2is the 1Àa =2quantile of the standard normal distribution.Therefore,if q k is located within the interval,the null hypothesis may not be rejected at an a signi ficance level.
2.1.2Run test A run is de fined as a succession of ones and zeros for the quence of obrva-tions.A quence of ones and zeros for the obrvations f x i g ;i ¼1;...;N can be formed as follows u i ¼1for x i > x u i ¼0for x i x
&'ð2Þwhere x
is the mean for the sample size of N .If ‘m ’is the number of runs of ones and ‘n ’is the number of runs of zeros,the total number of runs is ‘R ¼m þn ’.The run test is bad on the as-sumption that if a ries is IID,the total number of runs ‘R ’is approximately normal.Then,the statistic R c is de fined as follows
R c ¼R À1À2mn =ðm þn Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið2mn ð2mn Àðm þn ÞÞÞ=ððm þn Þ2Þðm þn À1ÞÞ
q ð3Þ
where R 1Àa =2is the 1Àa =2quantile of the standard normal distribution.2.1.3
Spearman ’s rank correlation coefficient test
The obrvations f x i g ;i ¼1;...;N ,in which N is the sample size,are ordered in increasing manner,with u i the rank of the ordered quence.The rank correlation R between the pairs (i ;u i )is ud for the Spearman ’s test.This coef ficient is obtained by
R ¼1À6P N i ¼1ði Àu i Þ2N ðN À1Þ
ð4ÞThe Spearman ’s rank correlation coef ficient R follows a standard normal distri-bution if the obrvations are random and 1ÀR 2has a Chi-square distribution with N À2degrees of freedom.Then,the student ’s t -distribution is applied to the ratio
T c ¼R ffiffiffiffiffiffiffiffiffiffiffiffiN À2p ffiffiffiffiffiffiffiffiffiffiffiffiffi1ÀR 2p ð5Þ
The null hypothesis of randomness may be rejected at the signi ficance level a if j T c j !T 1Àa =2ðN À2Þ;where T 1Àa =2ðN À2Þis the 1Àa =2quantile of the student ’s t -distribution with N À2degrees of freedom.
2.1.4
Turning point test
For a given ries of obrvations f x i g ;i ¼1;...;N ,in which N is the sample size,a peak is de fined as the occurrence of a value f x i g such that
106
x iÀ1<x i>x iþ1
and a trough by
x iÀ1>x i<x iþ1
If the sample ries is random,the total number of peaks and troughs M is approximately normally distributed.The test statistic u c can be computed bybha
u c¼MÀEðMÞ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
VarðMÞ
pð6Þ
someonThen,the hypothesis of randomness cannot be rejected at the a significance level if u c j j u1Àa=2where u1Àa=2is the1Àa=2quantile of the standard normal dis-tribution.
2.2
The BDS statistic
The BDS statistic is derived from the correlation integral and has its origins in the recent work on deterministic nonlinear dynamics and chaos theory.According to Packard et al.(1980)and Takens(1981),the method of delays can be ud to embed a scalar time ries f x i g;i¼1;2;...;N into an m-dimensional space as follows
~x i¼ðx i;x iþt;...;x iþðmÀ1ÞtÞ;~x i2R mð7Þwhere t is the index lag.Grassberger and Procacccia(1983)introduced the correlation integral as a method of measuring the fractal dimension of deter-ministic data.It is a measure of the frequency with which temporal patterns are repeated in the data.The correlation integral at the embedding dimension m is given by
Cðm;N;rÞ¼
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MðMÀ1Þ
spring festival英语作文X
1i<j M
H rÀ~x iÀ~x j
ÀÁ
;r>0ð8Þ
HðaÞ¼0;if a0
HðaÞ¼1;if a!0
where N is the size of the data ts,M¼NÀðmÀ1Þt is the number of embedded points in m-dimensional space,andÁk k denotes the sup-norm.Cðm;N;rÞmea-sures the fraction of the pairs of points~x i,i¼1;2;...;M,who sup-norm paration is not greater than r.If the limit of Cðm;N;rÞas N!1exists for each r,we write the fraction of all state vector points that are within r of each other as
Cðm;rÞ¼lim
ssat网络课程
N!1Cðm;N;rÞ.
If the data is generated by a strictly stationary stochastic process that is absolutely regular,then this limit exists.In this ca the limit is as follows
Cðm;rÞ¼
Z Z
X HðrÀ~xÀ~y
k kÞd Fð~xÞd Fð~yÞ;r>0:ð9Þ
107
When the process is IID,and since HðrÀ~xÀ~y
k kÞ¼
Q m
k¼1HðrÀx kÀy k
j jÞ,Eq.(9)
implies that Cðm;rÞ¼C mð1;rÞ.Also Cðm;rÞÀC mð1;rÞhas asymptotic normal distribution,with zero mean and variance as follows
r2ðm;M;rÞ=4¼mðmÀ1ÞC2ðmÀ1ÞðKÀC2ÞþK mÀC2m
þ2
X mÀ1
i¼1
C2iðK mÀiÀC2ðmÀiÞÞÀmC2ðmÀiÞðKÀC2Þ
h i
ð10Þ
We can consistently estimate the constants C by Cð1;rÞand K by
Kðm;M;rÞ¼
6
MðMÀ1ÞðMÀ2Þ
X
1i<j M
HðrÀ~x iÀ~x j
ÞHðrÀ~x iÀ~x j
Þ
ÂÃ
ð11Þ
Under the IID hypothesis,the BDS statistic for m>1is defined as
BDSðm;M;rÞ¼
ffiffiffiffiffi
M
p
r
½Cðm;rÞÀC mð1;rÞ ð12Þ
It has a limiting standard normal distribution under the null hypothesis of
IID as M!1and obtains its critical values using the standard normal
distribution.
Even though the BDS statistic cannot be ud to distinguish between a non-linear deterministic system and a nonlinear stochastic system,it is a powerful tool
for distinguishing random time ries from the time ries generated by low
dimensional chaotic or nonlinear stochastic process.Not only is the resulting
test uful in detecting deterministic chaos,but it also rves as a residual di-
agnostic.If the model is correct,then the estimated residuals will pass the test for
IID.A failure to pass the test is an indication that the lected model is mis-
specified.
1660什么意思Before applying the BDS statistic,thefirst issue is what region of‘r’yields the BDS statistic that is well approximated by an asymptotic distribution.As the
sample size is incread,the distribution of the BDS statistic becomes more
normal,so the minimal number of data should be provided.Next,the region of
the embedding dimension‘m’should be suggested.If the sample size isfixed,
we expect thefinite sample property to worn as‘m’increas.This study
follows the recommendation of Brock et al.(1991)for lecting the ranges of m,
r,N.Therefore,500or more obrvations are prepared and the embedding
dimension m is ud in the range of2m 5.Then,the value of‘r’is lected
as half the standard deviations of the data ts.
3
Effectiveness of statistics
The effectiveness of nonparametric and BDS statistics are simply tested by using
the ries of Gaussian noi and Logistic map which is a chaotic process.Typical
time ries plots of Gaussian and chaotic process are shown in Fig.1.Gaussian
process f x t g has zero mean and variance r2and is written as
f x t g$Nð0;r2Þð13Þ108
