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Some Theory of StatisticalInference for Nonlinear Science:Expanded VersionWilliam A. BrockEhung G. Baek
SOME THEORY OF STATISTICAL INFERENCE FOR NONLINEAR SCIENCE:
EXPANDED VERSION'
William A. Brock
University of Wisconsin-Madison
Ehung G. Baek
Iowa State University
January 1991
Abstract
This article shows how standard errors can be estimated for a measure of the number of excited degreesof freedom (the correlation dimension), and a measure of the rate of information creation (a proxy for theKolmogorov entropy), and a measure of instability. These measures are motivated by nonlinear science andchaos theory. The main analytical method is central limit theory of U-statistics for mixing processes. Thispaper takes a step toward formal hypothesis testing in nonlinear science and chaos theory.
Acknowledgments
The first author thanks the Guggenheim Foundation, the Wisconsin Graduate School, the Wisconsin AlunmiResearch Foundation, and the National Science Foundation (Grant # SEC-8420872) for essential financialsupport for this work. We thank two referees and the editor, C. Bean, for very helpful comments andTaeho Lee for the numerical integrations. Support for this paper was provided in part by the Santa FeInstitute Economic Research Program which is funded by grants from Citicorp/Citibank and the RussellSage Foundation and by grants to SFI from the John D. and Catherine T. MacArthur Foundation, theNational Science Foundation (PHY-8714918) and the U.S. Department of Energy (ER-FG05-88ER25054).
1
1. INTRODUCTION
Empirical nonlinear science has enjoyed a boom In economics.
Examples are Brock (1986), Barnett and Chen (1988), Brock and Sayers
(1988), Frank and Stengos (1988a,b,1989), Gennotte and Marsh (1987),
Hsieh (1989), Sayers (1986), Scheinkman and LeBaron (1989a,b) , Ramsey
and Yuan (1989,1990).
Much of the excitement has to do with the potentiality of
quantifying such vague notions as "level of complexity", "degree of
instability", and "number of active nonlinear degrees of freedom".
At a general level nonlinear science has a rich storehouse of ideas
to inspire the field of nonlinear time series analysis, and, vice
versa.
Most of the work to date has relied on diagnostics such as
correlation dimension, Kolmogorov entropy, and Lyapunov exponents.
Expository papers in this area are Brock (1986), Frank and Stengos
(1988b) for economics, and Eckmann and Ruelle (1985), Theiler (1990b)
for natural science. Eckman and Ruelle (1985) is an especially
detailed and comprehensive review of nonlinear science. Brock (1986)
contains some applications to economics and a discussion of some
pitfalls to avoid. Frank and Stengos (1988b) surveys some of the
useful literature and techniques and studies empirical chaos in
economics by using daily rates of return on gold.
Unfortunately no formal theory of statistical inference exists
for the dimension measures and the instability measures of nonlinear
science. Brock, Dechert, and Scheinkman, hereafter, BDS (1987)
developed some statistical theory (discussed below) for the
2
correlation integral of GrassbergerjProcacciajTakens (a measure of
spatial nonlinear correlation) and used this theory to formulate a
test of the null hypothesis of independently and identically
distributed (lID) for a univariate series against an unspecified
alternative. This work was extended to the vector case by Baek and
Brock (1988). Brock and Dechert (1988a) provided some ergodic
theorems for the correlation integral and some convergence theorems
for a Lyapunov exponent estimation algorithm.
The new contribution of this paper is to provide some
statistical inference theory for dimension measures and Kolmogorov
entropy. Central limit theorems for weakly dependent stochastic
processes and for U-statistics provide the tools needed for this
paper. They are presented in section two. Asymptotic standard
errors of dimension and Kolmogorov entropy estimates are derived as
applications of the theory. Nuisance parameter problems occuring in
these measures are discussed. In section three we calculate the
correlation dimension estimates, the Kolmogorov entropy estimates,
and their standard errors by using returns on weekly stock market
index studied by Scheinkman and LeBaron (1989a). Final remarks and
conclusions are in section four.
2. THEORY OF STATISTICAL INFERENCE
Let {at}, t=1,2, ... ,T be a sample from a strictly stationary and
ergodic stochastic process which we abuse notation by also denoting
3
by {at}' or a deterministic chaos with unique, ergodic invariant
measure as in Brock (1986). This assumption allows us to replace all
limiting time averages by corresponding phase averages. Also the
limiting value of all time averages will be independent of initial
conditions. The data, {at} can be "embedded" in m- space by
constructing "m-futures"
a~ = (at , ... ,at +m_1), t=1,2,oo.,T-m+1.
The correlation integral for embedding dimension m is defined by
(2.1 ) C(€,m,T) =
where Tm=T-m+1, I(x,Yi€)=1 if IIx-yll~€ and 0 otherwise, IIxll denotes
the maximum norm, Le. II xII = max Ix·1 on Rm. The correlationO~i~m-1 1
integral measures the fraction of total number of pairs (a~,a~) such
h d b m d m. hthat t e istance etween at an as IS no more t an €. In other
words, it is a measure of spatial correlation. Note that C(€,m,T) is
a double average of an indicator function. Hence one expects it to
converge as T~. Denker and Keller (1986, Theorem 1 and (3.9)) and
Brock and Dechert (1988a) show that
(2.2) dC(€,m,T) ~ C(€,m).
It is worthwhile to give some intuition into the measure C(€,m).
where Cx=EI(Xt,Xs;f), Cy=EI(Yt'Ys;f) , Kx=EI(Xt,XS;f)I(Xs,Xr;f) ,
Ky=EI(Yt'YS;f)I(Ys'Yr;f), and' denotes the sample estimate.
Note that in order to get the simple variance formula (A.16) we
replaced the null we were interested in by IIDI. However we
constructed the test of IIDI so that it has zero power against
dependence of Xt +1 on past X's, but has power against dependence of
Xt +1 on past Y's. Monte Carlo work is probably the most effective
way of evaluating whether we made a wise move in trading off a
complicated variance formula and an accurate null for a simple
variance formula and a less accurate null. This is all "research in
progress". lie mention it here only because it illustrates the
general theme of this paper: The measure of spatial correlation from
chaos theory, called the correlation integral, can be combined with
U-statistics theory to produce useful statistical tests for the
presence of nonlinear predictability.
43
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46
BDS Statistic I: Size", forthcoming in Brock, V., Hsieh, D. and
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(Department of Economics, University of Visconsin - Madison).
SCHEINKMAN, J and LEBARON, B. (1989a), "Nonlinear Dynamics and Stock
Returns" Journal of Business, 62, 311-337.
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Complexity: Chaos, Sunspots, Bubbles, and Nonlinearity (Cambridge
University Press, Cambridge), 213-227.
47
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48
EXTENDED TABLES AND HISTOGRAIS FOR READERS' REFERENCE OFSOlE THEORY OF STATISTICAL INFERENCE FOR NONLINEAR SCIENCE
Note: All Monte Carlo experiments are based on 2500 iterations in thefollowing tables and figures.
CONTENTSTable 1: Correlation dimension estimates of weekly returnsTable 2: Correlation dimension estimates of standard normal random
numbersTable 3: Kolmogorov entropy estimates of weekly stock returnsTable 4: Kolmogorov entropy estimates of standard normal random
numbersTable 5: Correlation dimension estimates and standard errors
sample size=1000j £+d£=0.9j £=0.92Table 6: Kolmogorov entropy estimates and standard errors
sample size=1000j £=0.9Table 7: Correlation dimension estimates and standard errors
sample size=500j £+d£=0.9j £=0.92Table 8: Kolmogorov entropy estimates and standard errors
sample size=500j £=0.9Table 9: Correlation dimension estimates and standard errors
sample size=250j £+d£=0.9j £=0.92Table 10: Kolmogorov entropy estimates and standard errors
sample size=250j £=0.9Table 11: Correlation dimension estimates and standard errors
sample size=1000j £+d£=0.92j £=0.93Table 12: Kolmogorov entropy estimates and standard errors
sample size=1000j £=0.92Table 13: Correlation dimension estimates and standard errors
sample size=500j £+d£=0.92j £=0.93Table 14: Kolmogorov entropy estimates and standard errors
sample size=500j £=0.92Table 15: Correlation dimension estimates and standard errors
sample size=250j £+d£=0.92j £=0.93
Table 16: Kolmogorov entropy estimates and standard errorssample size=250j £=0.92
Figure 1: Histogram of the Correlation dimension estimatessample size=1000j £+d£=0.9j £=0.92 j m=2Histogram of the Kolmogorov entropy estimatessample size=1000, £=0.9j m=2
Figure 2: Histogram of the Correlation dimension estimatessample size=1000j £+d£=0.9j £=0.92 j m=4Histogram of the Kolmogorov entropy estimatessample size=1000, £=0.9j m=4
Figure 3: Histogram of the Correlation dimension estimates
49
sample size=1000; £+d£=0.9; £=0.92; m=8Histogram of the KolmogoroY entropy estimatessample size=1000, £=0.9; m=8
Figure 4: Histogram of the Correlation dimension estimatessample size=500; £+d£=0.9; £=0.92; m=2Histogram of the Kolmogoroy entropy estimatessample size=500, £=0.9; m=2
Figure 5: Histogram of the Correlation dimension estimatessample size=500; £+d£=0.9; £=0.92; m=4Histogram of the KolmogoroY entropy estimatessample size=500, £=0.9; m=4
Figure 6: Histogram of the Correlation dimension estimatessample size=500j £+d£=0.9; £=0.92; m=8Histogram of the KolmogoroY entropy estimatessample size=500, £=0.9; m=8
Figure 7: Histogram of the Correlation dimension estimatessample size=250; £+d£=0.9; £=0.92; m=2Histogram of the KolmogoroY entropy estimatessample size=250, £=0.9; m=2
Figure 8: Histogram of the Correlation dimension estimatessample size=250; £+d£=0.9; £=0.92; m=4Histogram of the Kolmogoroy entropy estimatessample size=250, £=0.9; m=4
Figure 9: Histogram of the Correlation dimension estimatessample size=250; £+d£=0.9; £=0.92; m=8Histogram of the KolmogoroY entropy estimatessample size=250, £=0.9; m=8
Figure 10: Histogram of the difference of two dimension estimates
sample size=500; £+d£=0.9; £=0.92; m=6Figure 11: Histogram of the difference of original and bootstrap
samples
sample size=500; £+d£=0.9; £=0.92; m=6
50
Table 1
CORRELATION DIMENSION ESTIMATES OF WEEKLY STOCK RETURNS
j=l1 ' 2 ' 21 = [log(E+&E) - log(E)r , A = K(E+&E)jC(E+&E) , B = K(E)jC(E) ,
C = V(E+&E,E)j(C(E+&E)C(E)). Standard normal random numbers ofsize of 1000 were used ~o ~alcul~te K(E+&E), K(E), C(E+&E), C(E)and V(E+~E,E) at which A, B and C are evaluated. The standarderror of the mean ASE is reported in parenthesis.
• True ASE is ~bt~ined ~rom the same ASE formula which is givenabove. But A, B and C were evaluated at numerically calculatedvalues of K(E+&E)=0.2511, K(E)=0.2098, C(E+&E)=0.4755, C(E)=0.4332,
2and V(E+&E,E)=0.2295 for E+&E=0.9 and E=0.9 .• J! SD = J! x standard error of [D(.) - mJ of the 2500 replications.
55
Table 6: Monte Carlo Simulation
KOLMOGOROV ENTROPY ESTIMATES AND STANDARD ERRORS
Number of Replications=2500j Sample Size=1000j f=0.9
Notes:• IMSL subroutine DRNNOA was called to generate 1000 standard
normal random numbers, and RNSET was called to set an initial seed.• Avera~e of Entropy = mean of the Kolmogorov Entropy estimate
[Km(·)+log C(f,l,T)] of the 2500 replications. The standard errorof sample mean out of the 2500 replications is reported inparenthesis.
• Mean ASE = mean of the 2500 empirical ASE's,ASE=[4f(K(f)/C(f)2)m+l- (K(f)/C(f)2)m+K(f)/C(f)2 _1}]1/2.
Standard normal random numbers of size of 1000 were used tocalculate K(f), C(f) at which ASE is evaluated. The standarderror of the mean ASE is reported in parenthesis.
• True ASE is obtained from the same ASE formula which is givenabove. But it was evaluated at numerically calculated values ofK(f)=0.2511, C(f)=0.4755 for f=0.9.
• JT SD = Jt x standard error of [Km(·) + log C(f,l,T)] of the 2500replications.
56
Table 7: Monte Carlo Simulation
CORRELATION DIMENSION ESTIMATES AND STANDARD ERRORSNumber of Replications=2500; Sample Size=500; €+d€=0.9, E=0.92
Notes:• IMSL subroutine DRNNOA was called to generate 500 standard
normal random numbers, and RNSET was called to set an initial seed.• Average of Dimension Estimates = mean of the correlation dimension
estimates of the 2500 replications. The standard error of samplemean out of the 2500 replications is reported in parenthesis forgiven dimension.
• Mean ASE = mean of the 2500 empirical ASE's,
ASE=[4,2{Am + Bm _ 2Cm + 2mi 1
(Am-j + Bm-j _ 2Cm-j)}]lj2, wherej=l
1 • 2 • 21 = [log(E+dE) - log(E)r , A = K(E+dE)jC(E+dE) , B = K(E)jC(E) ,C = W(€+dE,E)j(C(E+dE)C(E)). Standard normal random numbers ofsize of 500 were used t~ c~lcula~e K(E+dE), K(E), C(E+dE), C(E)and W(E+dE,E) at which A, B and C are evaluated. The standarderror of the mean ASE is reported in parenthesis.
• True ASE is ~bt~ined !rom the same ASE formula which is givenabove. But A, B and C were evaluated at numerically calculatedvalues of K(E+dE)=0.2511, K(€)=0.2098, C(E+dE)=0.4755, C(E)=0.4332,
2and W(E+dE,€)=0.2295 for E+dE=0.9 and E=0.9 .• vT SD = vT x standard error of [D(.) - m] of the 2500 replications.
57
Table 8: Monte Carlo Simulation
KOLMOGOROV ENTROPY ESTIMATES AND STANDARD ERRORS
Number of Replications=2500j Sample Size=500j E=0.9
Notes:• IMSL subroutine DRNNOA was called to generate 500 standard
normal random numbers, and RNSET was called to set an initial seed.• Avera~e of Entropy = mean of the Kolmogorov Entropy estimate
[Km(·)+log C(E,1,T)] of the 2500 replications. The standard errorof sample mean out of the 2500 replications is reported inparenthesis.
• Mean ASE = mean of the 2500 empirical ASE's,ASE=[4f(K(E)/C(E)2)m+1- (K(E)/C(E)2)m+K(E)/C(E)2 _1}]1/2.
Standard normal random numbers of size of 500 were used tocalculate K(E), C(E) at which ASE is evaluated. The standarderror of the mean ASE is reported in parenthesis.
• True ASE is obtained from the same ASE formula which is givenabove. But it was evaluated at numerically calculated values ofK(E)=0.2511, C(E)=0.4755 for E=0.9.
• vT SD = IT x standard error of [Km(·) + log C(E,1,T)] of the 2500replications.
58
Table 9: Monte Carlo Simulation
CORRELATION DIMENSION ESTIMATES AND STANDARD ERRORSNumber of Replications=2500j Sample Size=250j e+Ae=0.9, e=0.92
1 • 2 • 21 = [log(e+Ae) - log(e)r , A = K(HAe)/C(e+Ae) , B = K(e)/C(e) ,C = V(e+Ae,e)/(C(e+Ae)C(e)). Standard normal random numbers ofsize of 250 were used t~ c~lcula!e K(e+Ae), K(e), C(e+Ae), C(e)and V(e+Ae,e) at which A, B and C are evaluated. The standarderror of the mean ASE is reported in parenthesis.
• True ASE is ~bt~ined ~rom the same ASE formula which is givenabove. But A, B and C were evaluated at numerically calculatedvalues of K(e+Ae)=0.2511, K(e)=0.2098, C(e+Ae)=0.4755, C(e)=0.4332,
2and V(e+Ae,e)=0.2295 for e+Ae=0.9 and e=0.9 .• yT SD = yT • standard error of [D(.) - m] of the 2500 replications.
59
Table 10: Monte Carlo Simulation
KOLMOGOROV ENTROPY ESTIMATES AND STANDARD ERRORS
Number of Replications=2500j Sample Size=250j f=0.9
Notes:• IMSL subroutine DRNNOA was called to generate 250 standard
normal random numbers, and RNSET was called to set an initial seed.• Avera~e of Entropy = mean of the Kolmogorov Entropy estimate
[Km(·)+log C(f,l,T)] of the 2500 replications. The standard errorof sample mean out of the 2500 replications is reported inparenthesis.
• Mean ASE = mean of the 2500 empirical ASE's,ASE=[4f(K(f)/C(f)2)m+l- (K(f)/C(f)2)m+K(f)/C(f)2 _1}]1/2.
Standard normal random numbers of size of 250 were used tocalculate K(f), C(f) at which ASE is evaluated. The standarderror of the mean ASE is reported in parenthesis.
• True ASE is obtained from the same ASE formula which is givenabove. But it was evaluated at numerically calculated values ofK(f)=0.2511, C(f)=0.4755 for f=0.9.
• VI SD = VI x standard error of [Km(·) + log C(f,l,T)] of the 2500replications.
.,60
Table 11: Monte Carlo Simulation
CORRELATION DIMENSION ESTIMATES AND STANDARD ERRORSNumber of Replications=2500; Sample Size=1000; €+&f=0.92, €=0.93
1 - 2 - 21 = [log(f+&f) - log(f)r , A = K(f+&f)/C(f+&€) , B = K(f)/C(f) ,C = V(€+&f,f)/(C(f+&f)C(f)). Standard normal random numbers ofsize of 1000 were used ~o ~alcul~te K(f+&f), K(f), C(f+&f), C(f)and V(f+&f,f) at which A, B and C are evaluated. The standarderror of the mean ASE is reported in parenthesis.
• True ASE is ~bt~ined ~rom the same ASE formula which is givenabove. But A, B and C were evaluated at numerically calculatedvalues of K(f+&f)=0.2098, K(f)=0.1743, C(f+&f)=0.4332, C(f)=0.3938,
2 3and V(f+&f,f)=0.1902 for f+&f=0.9 and f=0.9 .• VI SD = VI x standard error of [D(.) - m] of the 2500 replications.
."61
Table 12: Monte Carlo Simulation
KOLMOGOROV ENTROPY ESTIMATES AND STANDARD ERRORS
Number of Replications=2500; Sample Size=1000; £=0.92
Notes:• IMSL subroutine DRNNOA was called to generate 1000 standard
normal random numbers, and RNSET was called to set an initial seed.• Avera~e of Entropy = mean of the Kolmogorov Entropy estimate
[Km(·)+log C(£,l,T)] of the 2500 replications. The standard errorof sample mean out of the 2500 replications is reported inparenthesis.
• Mean ASE = mean of the 2500 empirical ASE's,ASE= [4{ (K( £) /C( £)2)m+l_ (K( £) /C( E) 2)m+K(£) /C( £) 2 -1}] 1/2.
Standard normal random numbers of size of 1000 were used tocalculate K(£), C(£) at which ASE is evaluated. The standarderror of the mean ASE is reported in parenthesis.
• True ASE is obtained from the same ASE formula which is givenabove. But it was evaluated at numerically calculated values of
2K(£)=0.2098, C(£)=0.4332 for £=0.9 .• JT SD = JT x standard error of [Km(·) + log C(£,l,T)] of the 2500
replications.
62
Table 13: Monte Carlo Simulation
CORRELATION DIMENSION ESTIMATES AND STANDARD ERRORS2 3Number of Replications=2500; Sample Size=500; E+~E=0.9 , E=0.9
1 ' 2 ' 21 = [log(E+~E) - log(E)r , A = K(E+~E)jC(E+~E) , B = K(E)jC(E) ,C = W(E+~E,E)j(C(E+~E)C(E)). Standard normal random numbers ofsize of 500 were used t? c~lcula~e K(E+~E), K(€), C(€+~€), C(€)and W(E+~€,€) at which A, B and C are evaluated. The standarderror of the mean ASE is reported in parenthesis.
• True ASE is ?bt~ined ~rom the same ASE formula which is givenabove. But A, B and C were evaluated at numerically calculatedvalues of K(€+~€)=0.2098, K(E)=0.1743, C(€+~€)=0.4332, C(€)=0.3938,
2 3and W(E+~€,€)=0.1902 for €+~E=0.9 and €=0.9 .• vT SD = vT x standard error of [D(.) - m] of the 2500 replications.
63
Table 14: Monte Carlo Simulation
KOLMOGOROV ENTROPY ESTIMATES AND STANDARD ERRORS
Number of Replications=2500; Sample Size=500; €=0.92
Notes:• IMSL subroutine DRNNOA was called to generate 500 standard
normal random numbers, and RNSET was called to set an initial seed.• Avera~e of Entropy = mean of the Kolmogorov Entropy estimate
[Km(·)+log C(€,l,T)] of the 2500 replications. The standard errorof sample mean out of the 2500 replications is reported inparenthesis.
• Mean ASE = mean of the 2500 empirical ASE's,ASE=[4{(K(€)/C(£)2)m+l_(K(€)/C(£)2)m+K(£)/C(€)2 _1}]1/2.
Standard normal random numbers of size of 500 were used tocalculate K(£), C(£) at which ASE is evaluated. The standarderror of the mean ASE is reported in parenthesis.
• True ASE is obtained from the same ASE formula which is givenabove. But it was evaluated at numerically calculated values of
2K(€)=0.2098, C(£)=0.4332 for £=0.9 .• VT SD = yT x standard error of [Km(·) + log C(€,l,T)] of the 2500
replications.
64
Table 15: Monte Carlo Simulation
CORRELATION DIMENSION ESTIMATES AND STANDARD ERRORS2 3Number of Replications=2500; Sample Size=250; f+~f=0.9 , f=0.9
Embeddlllg Average of Mean True 'If snDimension Dim Est ASE ASE
1 = [log(f+~f) - log(f)r , A = K(f+~f)jC(f+~f) , B = K(f)jC(f) ,C = W(f+~f,f)j(C(f+~f)C(f)). Standard normal random numbers ofsize of 250 were used t? c~lcula!e K(f+~f), K(f), C(f+~f), C(f)and W(f+~f,f) at which A, B and C are evaluated. The standarderror of the mean ASE is reported in parenthesis.
• True ASE is ?bt~ined ~rom the same ASE formula which is givenabove. But A, B and C were evaluated at numerically calculatedvalues of K(f+~f)=0.2098, K(f)=0.1743, C(f+~f)=0.4332, C(f)=0.3938,
2 3and W(f+~f,f)=0.1902 for f+~f=0.9 and f=0.9 .• JT SD = JT x standard error of [D(.) - m] of the 2500 replications.
65
Table 16: Monte Carlo Simulation
KOLMOGOROV ENTROPY ESTIMATES AND STANDARD ERRORS
Number of Replications=2500; Sample Size=250j f=0.92
Embeddmg Average of Mean True :;r SDDimension Entropy ASE ASE
Notes:• IMSL subroutine DRNNOA was called to generate 250 standard
normal random numbers, and RNSET was called to set an initial seed.• Average of Entropy = mean of the Kolmogorov Entropy estimate
[Km(·)+log C(f,l,T)] of the 2500 replications. The standard errorof sample mean out of the 2500 replications is reported inparenthesis.
• Mean ASE = mean of the 2500 empirical ASE's,ASE=[4{(K(f)/C(f)2)m+l_(K(f)/C(f)2)m+K(f)/C(f)2 _1}]1/2.
Standard normal random numbers of size of 250 were used tocalculate K(f), C(f) at which ASE is evaluated. The standarderror of the mean ASE is reported in parenthesis.
• True ASE is obtained from the same ASE formula which is givenabove. But it was evaluated at numerically calculated values of
2K(f)=0.2098, C(f)=0.4332 for f=0.9 .• VI SD = vT x standard error of [Km(·) + log C(f,l,T)] of the 2500
replications.
Figure 1
HISTOGRAM OF CORR. DIMENSIONib.------------------------,(T)
66
oN(T)
oeoN
0_eo
o '--_~_.....ILJ.-2.0 -1.8 -1.6
Midpoint
I.-1.4 -1.2
-
-
2T=lOOO, e=O,9, 10+ AE=O.9 , m=2
HISTOGRAM OF KOL. ENTROPYg.------------------------..(T)
oN~(T)
oeo~N
o
>-~()
Com 0 ,.:=IND'"moLCDI.J..~
oeo
o'<t
o-1.2
--I-0,8 -0.4 0,0
Midpoint0.4
-'-- -0,8
-
-
-
T=lOOO, E=O. 9, m=2
Figure 2
HISTOGRAM OF CORR. DIMENSIONgr-------------------------,M
oNM
oCX)N
67
o>~I(.)
Coale:>::1N0"aloLeoLL~
oCX)
01
"""o --tI
-2.2 -1.8 -1.4Midpoint
1I---1.0
-
-
2T=lOOO,G:=O.9, €+,oE=O.9 , m=4
HISTOGRAM OF KOL. ENTROPYo0r------------------------,"""oeoM
oCX)e
N
Goc"""al N::100"0Q)C\lLLLOeo:I:t:~
oC\l~
e:>CX)
o ·IJ-1.5 -1.0 -0.5 0.0 0.5
Midpoint
I-to 1.5 2.0
-
T=lOOO, €=O. 9, m=4
Figure 3
HISTOGRAM OF CORR. DIMENSION~r----------------------,rr>
aNrr>
a
>.~()
Co<Do:::IN
lifoLCDI..L.~
oCD
o
: '--_~_.J. ...I!-3 -2 -1
Midpoint2
T= 1000, E =0.9, E +4~ =0 • 9 , m=8
L....._~__---l
o 1
HISTOGRAM OF KOL. ENTROPY~r-------------------------,rr>
aNrr>
aCD~N
a
>.~()
Co<Do:::INcr<DoLCDI..L.~
aCO
o
:,--_~-.1-3 -2
T=lOOO, €=0.9, m=8
-1 . 0 1Midpoint
LI-2 3
Figure 4
HISTOGRAM OF CORR. DIMENSIONg.-------------------------,....oCD
'"oC\J
'"oOlC\J
Goc""<DC\J:::Jo0-0(DC\JLI.L..0
CD=I:l:~
oC\J~
1.-0.8 -0.6-1.8 -1.4 -12 -1.0
Midpoint2T=500, e: =0.9, €+4e=0.9 , m=2
ol.---~'-"'"-1.6
01Ol
o....
HISTOGRAM OF KOL. ENTROPYg.-------------------------,....oCD
'"~
'"oOlC\!
>'00 ....CC\J(D:::Jo0-0(DC\!LI.L..0
CD=I:l:~
~I-~
-
-
-
oOl
o....o L- • .I.. L...
-1.2 -0.8L... L... L...
-0.4 0.0Midpoint
•-0.4 0.8
T=500, €=0.9, m=2
Figure 5
HISTOGRAM OF CORR. DIMENSION7°
8.-----------------------,"""otol")
oNl")
oCDN
>'0U"""C NQ)::::Jo0"0Q)NLl.l..0
to*l:~
oN~
oCD
-
lI--0.6 -0.2-1.4••-1.8 -1.0
Midpoint2
T=500,€=O.9,E+~€=O.9, m=4
o
HISTOGRAM OF KOL. ENTROPY~r----------------------,"""oo
"""olI)l")
o>.guCo(DlI)::::IN0"(DOLOl.l..N
*l:olI)~
olI)
o ••-2 -1 oMidpoint
1I_----'
2
T=500, €"O.9, m=4
Figure 67/
HISTOGRAM OF CORR. DIMENSION~,----------'"o(\J
'"oCI)(\J
oCI)
-I. --1 2-2---3
o L
-1 0Midpoint
2T=500, ,"=0.9, 6+.116=0.9 , m=8
HISTOGRAM OF KOL. ENTROPY~,---------------------------,'"o(\J
'"oCI)(\J
oCI)
o L-- ..-.~-4 -2 0
.Midpoint
lb• .-2 4
T=500, E=0.9, m=8
Figure 7
HISTOGRAM OF CORR. DIMENSION~r------------------------''"oN
'"o(l)N
o(l)
7.2...
o....
o '------'....... ~-1.6 -1.2 -0.82 Midpoint
T=250,€=O.9,€+~e=O.9 ,m=2
1.---0.4 0.0
HISTOGRAM OF KOL. ENTROPY~r-------------------------''"oN
'"o(l)N
o
>.r!i()Co<Do:IN0<Do'-CDl..I..~
o(l)
o _....
o --.-1.2 -0.8 -0.4 0.0 0.4Midpoint
1I.0.8 1.2
T=250, E=O.9, m=2
Figure 8
HISTOGRAM OF CORR. DIMENSION~~--------(T)
73
OL.....-.----·.-1.8 -1.4
oN(T)
o>.~(,)
COCDc::>::IN0CDoLCDl.L~
oCD
-1.0 -0.6 -0.2Midpoint
2T=250, E=O.9, E+4E=O.9 , m=4
-
.
lh- ----J
0.2 0.6
HISTOGRAM OF KOL. ENTROPYg~----------------------,'<t
oto(T)
oN(T)
oCD
o'<t
o --I-2 -1 o
Midpoint
lI.-1 2
-
T=250, €=O. 9, m=4
HISTOGRAM OF CORR. DIMENSION·'.
Figure 9
oOr---------:::-- ---,-q-
oCD
'"
7-¥-
oC\!
'"o00C\!
>'0U-q-eC\!a>::Jo0"0(l)C\!LLL.O
CD=l:i:~
oC\!~
o00
oMidpoint
2T=250,E"=O.9, €+4€=O.9 , m=8
lb---2 4
.
HISTOGRAM OF KOL. ENTROPYgr--------------------~CD
oolD
o>.~ fUC(D
::Jo0"0 f(I)'"LLL.
=l:i: g cC\!
oo~
o-2
T=250, E:o.g, m=8
2lI.
B
Midpoint10 14
-
Figure 10HISTOGRAM OF THE DIFFERENCE OF TIlO DIMENSION ESTlKATES
Notes:• The histogram was generated under the null hypothesis that (alt )
and (a2t ) are lID with common distribution function.
• Number of Replications-2S00; Size of each sample-SaO; m-6;2e+c,e-O. 9; e-O. 9
• IMSL subroutine DRNNOA was called to generate the first and thesecond half of each sample separately.
• The test statistic was standardized.
g,.---------------------------,'t
oto
'"
70
1.6
oC\J
'"
0_CD
~
o~-d-1.6 -1.2 -0.8
'- '-
-0.4 0.0 0.4
Midpoint
-
-
-
-
.
'- Ll...J.IK-' ~--'0.8 1.2
"
Figure 11HISTOGRAM OF THE DIFFERENCE OF ORIGINAL AND BOOTSTRAP SAMPLES
Notes:• The histogram was generated under the null hypothesis that (at) is
IID2• Number of Replications-2S00; Sample Size-SOO; m-6; <+6<-0.9; <-0.9
• IMSL subroutine DRNNOA was called to generate all the originalsamples.
• The bootstrap samples were constructed with replacement