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Rev .Ac a d .C a nar.Cienc., VIII (Núm. 1), 1 9 - 29 ( 1 99 6)
·JiOOTSTRAP ESTIMATION OF POPULATION SPECTRUM WITH REPLICATED
TIME SERIES.
Abstract
C. N. Hemández Flores, J. Artiles Romero and P. Saavedra Santana
Department of Mathematics.
University of Las Palmas de Gran Canaria.
35017 Las Palmas de Gran Canaria, Canary Islands
e-mail: cflores@dma.ulpgc.es
In this work a set with r objets is considered and a stationary process X; (t) with spectral
distribution absolutely continuous is observed for each of them at same time interval. Each spectral
density function f; (ro) may be considered ita realization of a stationary process R(ro) .The spectrum
population f (ro) is defined by E[ R(ro)).We estímate f (ro) by means of a bootstrap method and we
proof the asymptotic validity when the number of objects r tend to infinity.
Keywords Spectrum Estimator, Bootstrap, Replicated Time Series, Population Spectrum,
Periodogram .
l. Introduction.
Although spectral analysis is a very highly déveloped methodology, almost ali of this
development has been in the context of a single, long time series. This perhaps the fact reflects that
the origins of the subject were signal processing and the physical sciences. However, the usefulness
of time series methodology is becoming more widely accepted in biomedical sciences, where
replicatetl experiments are the rule rather than the exception. Dig~le and Al-Wasel (1993) studied
replicated time series of measurements of the concentration of luteinizing hormone in blood
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samples. First, they considered each time serie as the realization of a stationary process with spectral
density function absolutely continuous f(ro). Nevertheless, the strong variability between subjects
lead to inconsistent data with this model and they proposed the altemative model I ;(w 1) = f ;(W 1) ·U ij
i=l, ... ,r; j=l, ... , [N/2] , being I¡(w j ) the periodogram at the frequency roj for the ith individual,
f ;(w1) the subject specific spectrum and Uij mutually independent, unit mean exponential variates
with ·common pdf e-u, u~O. Suppose now that r units are selected at random from a given
population. Then we regard the f ¡( w j ) as independent realizations of a random function R(ro), and
set f( ro)=E[R( ro)), where the expectation is defined with respect to the population of subjects and
f(ro) is called the population spectrum. They estimated f (ro) supposing certain parametrizations for
the processes involved in the model and they obtained the average periodogram as a maximum
likehood estimator of the population spectrum. Obviously, this estimator is unbiased and consistent
for the number of objects.
We have considered a more generalised model than the one developped by Diggle and AlWasel
without making any parametrizations and we haved used a bootstrap method for the
estimation of the population spectrum. Moreover, we have compared the confidence intervals
obtained by means of the bootstrap method with those obtained using the normal approximation
when r is large and the central limit theorem is taken in to account. Efron and Tibshirani ( 1986) used
bootstrap for estimation of a parametric time series model. Franke and Hlirdle (1992) worked with
bootstrap too for estimating the spectral density function when there is only one realization of a
stationary process. They proved the theoretical asymptotic validity of the bootstrap principie
according to Bickel and Freedman (1981), Freedman (1981) and Romano (1988). We have also
proved the validity of the bootstrap principie for a fixed number of observations by object and an
increased number of objects. The simulations we present illustrate that our procedure works for
moderate sample sizes by object and a large number of objects.
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2. The model.
Let {X¡r: t= l ,. .. ,N) be r time series, each one with spectral density functionf¡( ú>j). According
to Diggle and Al-Wasel we consider the abo ve mentioned model
I¡ (ro j) = ¡; (ro j )U u, i = 1, .. ., r; j = 1,. . ., [ N/2], being f; ( w 1 ) the periodogram at the frequency ú>j for the
ith subject, /;( w 1) the specific spectrum and Uij mutually independent, unit mean variates; we omit
the hypothesis of Diggle and Al.-Wasel tljat the U¡1 are exponential variates. We also assume that
/; ( w) = f ( w )Z¡( ro) where { Z¡( ú>)}: i= l ,. . .,r are independent copies of a stochastic process { Z( ú>j)}
with E[Z(ú>j)]=l, for ali ú>j, where f(w) is called the population spectrum. So, f¡(w1J=f(w1J%,
being % = Z¡( ro 1 )U;1, random variables i.i.d. for each fixed j with distribution function Fj, unit mean
and finite variance a ~ ~ a 2 .
We propose as an estimator of the population spectrum f ( ú>) the average periodogram
A 1 ~ -
f(w) = - L..J;(w) = /(w)
r i=t
The following procedure gives a bootstrap aproximation for, ](iíl~
Step 1 . The variables Wij are estimated as:
A l¡ (ú> ) 0 0 W. = ---'- , 1 = l,. . .,r; J = l,. . .,[N / 2]
,, l (ú>)
Obviously, for each frequency ú>j , L; W;j / r = 1.
Step 2 . B bootstrap samples {W \ ,. .... w•r J } are drawn from { ~j,. .. ., W,j } . For each frequency ~,,
we consider the bootstrap periodogram computed as follows:
Step 3 . Finally, the bootstrap estimation of the spectrum population is computed as follows:
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The w,; obtained through a boostrap resampling from { W;j, .... , W,j } ha ve unit mean with regard to
the empirical distribution.
Obviously the average periodogram is unbiased and a consistent estimator for f(w j) when r
increase to infinity. Within the bootstrap context, we obtain E'[J'(wJl=_!. Í,E[I,'(wj)]=J(wj) ;
r i=I
therefore, the bootstrap estimator is unbiased for J( m j). According to Hardle and Bowman ( 1988)
it is not necessary to correct the pivota) quantity.
3.- The bootstrap principie holds.
The basic idea of bootstrapping, as applied to the population spectrum estimation context, is
to infer properties of the distribution ot the estimator f(w) from the conditional distribution of its
bootstrap approximations j' (w), given the original data. To prove the theorical validity of this
bootstrap principie, we follow Bickel and Freedman (1981) and consider the Mallows distance
between the pivota) quantity .Jr(J(w; )- f(w;)) and its bootstrap approximation .Jr(J'{w; )- J(w; )) .
Here, the Mallows distance between distributions F and G is defined as
d2 (F,G) = inf{E[IX-YJ 2 )1'2}
where the infimum is taken over ali pairs of random variables X and Y having marginal
distributions F and G, respectively. We adopt the convention that where random variables appear as
arguments of d2, they represent the corresponding distributions. In particular, bootstrap quantities
represent their conditional distribution given the original data.
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Theorem.
Under the preceding conditions, the bootstrap principie holds in the following form:
Proof:
By definition having distribution and
.fr(J"(w1)- J(w1))= j~J)~(W.; -1) having distribution l/t(fr1,) . Therefore, replacing in (1), we
finally get
df ( l/f( F1 ), l/t( F1,)) ~O in probability (2)
Since d2 is a metric, and we obtain
We will prove that each term of the second member in the inequality converges to zero in
probability.
Firstly, d; ( 1/f( F¡ ), 1/f( F¡,)) ~ O in probability
Let Fj denote the distribution function of W;j and Fjr denote the empirical distribution function of
{ W¡1, .... , W,1 } • Let w;; be random variables with a distribution function Fjr .
An application of Bickel and Freedmans's (1981) Iemma 8.6 lead to
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Vl,k dJ(tt-;1 -1, tt-;; -1 )=dJ(Wic1 -1, ~1 -1 ), using it in (3)
f 2
-( (-¡) j) fL . d 2 ( ' ) 2 ( ) 2 ( ' ) 2 "'u - 1, W¡1 - 1 = f ill 1 d2 W;1 - !, W¡1 - 1
r i=I
by lemma 8.3 of Bickel and Freedman's (1981) dJ(W¡1 ,W'iJ )~o
Secondly, we ha ve to pro ve that di ( 11'( F1, ), ir( F1,)) ~O in probability for r ~ oo .
2 ( ( ) " ( )) 2 [ f ( (¡) j ) f ( ' ) J( (¡) j ) f ( ' )J dz 11' Fj, '11' FJ, = dz J-; {:i \.V;¡ - 1 ' J-; {:i W¡J - 1
By definition of the Mallows metric
J(wj) is a consistent estimator for f(ill 1) and converges in mean square for r~ oo
E[W;~ -1]2 :s;a~. applying this in (4), we obtain:
IJ( ill ¡ )- J( ill ¡ )i 2 E[ W¡~ - 1 r ~O
Finally, we ha ve to pro ve that di (ir( F1, ). ir( F1,)) ~ O in probability for r ~ 00 •
=!"2 (( ¡)¡)~ 2 (" 'u' ·""'u" ) =!"2 ( ill¡)~ 2 (~ .~" )
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By definition of the Mallows metric di ( F,, fr,), we may consider the joint distribution of { Wu} and
{ wij }' which assigns probability y, at each ( w u • w u ) for each j fixed to establish that
Since we know J( ro j) is a consistent estimator and J( ro j) ~ f( ro j) converges in mean square.
On the other hand, it can be proved that
Therefore, we can write:
Collecting together the three terms of the Mallows metric, we now have that this converges in
probability to O.
4. Simulation
In this section, a simulation study illustrates the performance of our bootstrap approach. We
have considered r subjects and for each one, a moving average process (MA(2)) has been simulated
at same time intervals, where the coefficients (</>¡,</>2 ) were chosen at. random from a bivariate
normal distribution with mean vector (0.2, -4), Var( </>¡.)= Var( </>2 )=0.01 and Cov( </>¡. </>2 )=-0.007.
We have represented simultaneously for each frequency the population spectrum and the
confidence band obtained by means of the bootstrap estimation proposed. For large values of r, we
have represented the confidence intervals obtained after approximating the pivot Fr(J(ro j )- f( ro j ))
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by a normal·· distribution N( O.J( w 1 )an, having replaced ª2 J by its estimator
\
, ... --
---
',
1 -
o .,,_~~t-~~1--~----4>--~-+~~-+~~_._~~-+-~~+-~~t-~___.
,00000 '62832 1,25664 1, 88496 2, 51327 3, 14159
FREQUENCY
Fig. l. N=20 r= 100
Lower band
Theorical density
spectral function
Upper band
In Fig. 1, due to the fact that the interpolations are made with a low number of frequencies
used to estimate the population spectrum, one can think of the existence of a possible bias of the
estimator in the maximum of the population spectrum.
,00000 ,62832
, 31416
'
,
' ' ' , • ,
,, .. .. ~
'
'
' ' ,
, "
,,
: ' ' '
-'
' .... - - ..
---
-,
1,25664 1, 88496 2, 51327
' 94248 1, 57080 2, 19911
FREQUENCY
Fig. 2. N=60 r=IOO
26
._
2, 82743
3, 14159
Lower band
Theorical density
spectral function
Upper band
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In Fig 2. the increase of frequencies produces as a consequence a rise in the amount of peaks
of the confidence bands that could lead to overvalue the contribution of certain frequencies to the
spectral power.
4,5--------------------------.
4,0
J,5
J,O
2,5
2,0
1,5
1,0
,5._ _ _... _ ___.,___-+---+--+----------<o----+---+----o
,00000 '62832 1, 25664 1, 88496 2, 51327 3,14159
,31416 '94248 1, 57080 2, 19911 2,82743
FREQUENCY
Fig 3 N=60 r=lOOO
Lower bootstrap band
Theorical density
spectral function
Upper bootstrap band
Lower normal band
Upper normal band
Fig 3 shows that for large values of r the bootstrap approximation coincides practically with
the normal approximation.
5. Discussion.
The graphics obtained suggest that for large values of N, it is more suitable to estímate the
population spectrum by means of average periodogram smoothing. So, a more adequate perception
is obtained since the peaks in the estimator suggest that sorne frequencies have negligible
contributions to the spectral power. In the same way, a smaller variance for the estimator of
population spectrum is obtained although a bias is introduced. Hemández-Flores (1996) propases
sorne estimators for population spectrum based in smoothing of the average periodogram by means
of kernel estimators, for a large N.
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However, in real life problems, specially in medica! sciences, only a small number of
observations by individual is available, although a large number of individuals can be analysed.
Obviously, it is not adequate to make estimations through smoothings in this case.
In comparison to Diggle and Al-Wasel's model(I993), ours allows the Var(Z(ú>j)) to change
with the frequency Wj . The parametrization of the model they introduced is not justified completely
and it produces an estimator J( cv 1 ) whose variance does not take in to account the number N of
observations by individual as one could expect from a parametric procedure. Despite the fact of
using too the mean periodogram as the spectral density estimator and not using a parametric model,
the results we have obtained show that the variance does not increase. As mentioned in the above
section, we have used a bootstrap procedure that not only satisfies the asymptotic validity conditions
following Mallows metric, but it also achieves confidence regions similar to the ones obtained using
a normal approximation when we are dealing with a large number of individuals.
6. References
[1] Bickel, P. and Freedman, D. (1981 ). "Sorne Asymptotic Theory for the Bootstrap". The Annals
of Statistics. 9: 1196-1217.
[2] Diggle, P. J. and Al-Wasel, l. ( 1993). "On Periodogram-Based Spectral Estimation for
Replicated Time Series". Developments in Time Series Analysis. Chapman & Hall.
[3] Franke, J. and Hardle, W. (1992). "On Bootstraping Kernel Spectral Estimates". The Annals
of Statistical. 20: 121-145.
[4] Freedman, D. (1981) "Bootstrapping Regression Models". The Annals of Statistics. Vol9, nº 6,
1218-1228.
[5] Hardle, W. and Bowman, A. (1988). "Bootstrapping in Nonparametric Regression: Local
Adaptive Smoothing and Confidence Bands". Journal of the American Statistical Association.
83:102-110.
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© Del documento, de los autores. Digitalización realizada por ULPGC. Biblioteca Universitaria, 2017
[6] Hernández-Flores, C. N. (1996) In preparation. Ph D Thesis, Las Palmas de Gran Canaria
University.
[7] Priestley, M.B. (1 981). "Spectral Analysis and Time Series". Wiley.
[8] Romano, J. (1988). Bootstrapping the Mode. Inst. Statist. Math. 40: 565-586.
[9] Saavedra, P. y Hernández, C.N. (1994). "Un problema de estimación espectral con series
replicadas". XXI Congreso Nacional de Estadística e Investigación Operativa. Calella, 1994.
[10] Saavedra, P., Artiles, J. y Hernández, C.N. (1995). "Un problema de estimación espectral
utilizando técnicas bootstrap". XXII Congreso Nacional de Estadística e Investigación Operativa.
Sevilla, 1995.
Rec ibido : 1 7 Mayo 1 99 6
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