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J. Geod. Sci. 2014; 4:136–149
Research Article Open Access
R. Lehmann*
Detection of a sinusoidal oscillation of unknown
frequency in a time series – a geodetic approach
Abstract: Geodetic and geophysical time series may con-
tain sinusoidal oscillations of unknown angular fre-
quency. Often it is required to decide if such sinusoidal
oscillations are truly present in a given time series. Here
we pose the decision problem as a statistical hypothesis
test, an approach very popular in geodesy and other scien-
tic disciplines. In the case of unknown angular frequen-
cies such a test has not yet been proposed. We restrict our-
selves to the detection of a single sinusoidal oscillation
in a one-dimensional time series, sampled at non-uniform
time intervals. We compare two solution methods: the like-
lihood ratio test for parameters in a Gauss-Markov model
and the analysis of the Lomb-Scargle periodogram. When-
ever needed, critical values of these tests are computed
using the Monte Carlo method. We analyze an exemplary
time series from an absolute gravimetric observation by
various tests. Finally, we compare their statistical power.
It is found that the results for the exemplary time series
are comparable. The LR test is more exible, but always
requires the Monte Carlo method for the computation of
critical values. The periodogram analysis is computation-
ally faster, because critical values can be approximately
deduced from the exponential distribution, at least if the
sampling is nearly uniform.
Keywords: data analysis; least squares spectral analysis;
likelihood ratio test; Lomb-Scargle periodogram; signi-
cance test; time series
DOI 10.2478/jogs-2014-0015
Received February 11, 2014; accepted September 28, 2014
1Introduction
Time series analysis is an important tool for modern geode-
sists and geophysicists. It is used to analyze observations
in almost all branches of geodesy: In structural health
*Corresponding Author: R. Lehmann: Dresden, E-mail:
r.lehmann@htw-dresden.de
monitoring (SHM) we take repeated geodetic observations
of engineering structures like bridges (e.g. Psimoulis et al.
2008). In geodynamics we monitor e.g. Earth rotation or
sea level variations (e.g. Erol 2011). In geodetic naviga-
tion we trace the trajectory of e.g. vehicles or ships (e.g.
Lehmann and Koop 2009).
All those geodetic and even more geophysical tasks
produce time series. They are often sampled at non-
uniform time intervals including data gaps, inuenced by
random observation errors (noise) including correlations,
aected by outliers and sometimes even non-stationary.
The observed phenomena often manifest themselves as
a superposition of sinusoidal oscillations of various fre-
quencies. It is the task of modern geodesists and geophysi-
cists to detect such oscillations in the time series. This in-
cludes a decision if the oscillations identied in the time
series are truly present in the observed phenomenon or if
they are purely the eect of noise.
If the frequencies of the potential oscillations are
known, then the procedure is well-known and elaborated.
Important geodesists have paved the way to this proce-
dure, rst of all Vaniček (1969, 1971), who created the least
squares spectral analysis (LSSA). See also (Wells et al.
1985). This approach allows us to treat time series analy-
sis in the framework of least squares theory, which is well-
elaborated and popular. LSSA can handle most practical
problems, e.g. non-uniform sampling and noise. The de-
tection of sinusoidal oscillations in this framework is ba-
sically a statistical hypothesis test, particularly a signi-
cance test of spectral peaks in the least squares spectrum
(Pagiatakis 1999).
In geodesy we most often test hypotheses in Gauss-
Markov or Gauss-Helmert models. A signicance test of
parameters in a Gauss-Markov model is often called v-
test (Teunissen 2000,2001). This statistical test is a special
application of the likelihood ratio (LR) test (e.g. Tanizaki
2004 p. 54 ). The rationale of the LR test is provided by the
famous Neyman–Pearson lemma (Neyman and Pearson
1933), which demonstrates that under various assump-
tions such a test has the highest statistical test power
among all competitors. It is often applied even if we cannot
exactly or only approximately make these assumptions in
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Detection of a sinusoidal oscillation of unknown frequency in a time series – a geodetic approach |137
practice (Teunissen 2000, Kargoll 2012). Most importantly,
this lemma applies only to simple hypotheses, while most
of the hypotheses used in geodesy are composite.
Since the frequencies of the sinusoidal oscillations to
be detected are xed quantities in the LSSA, it is necessary
to know them a priori. This holds true e.g. for geodynamic
phenomena with diurnal, seasonal or annual oscillations
(e.g. Erol 2011). But in other applications like SHM this is
often not the case. Extending the LSSA by unknown fre-
quency parameters makes the model nonlinear. Although
more dicult, it is possible to do statistical tests also in
non-linear models. The LR test can also be applied here.
One could argue that by rening the grid of xed frequency
parameters to be solved for, we get closer and closer to
the true frequencies. But such an approach invalidates an
important approximation made in the hypothesis test, see
eq. 30 below.
As computers become faster, geodesists see the oppor-
tunity to really solve for unknown frequencies. Mautz and
Petrovic (2005) consider a single as well as multiple sinu-
soidal oscillations in time series by least squares adjust-
ment with unknown frequency parameters. In the case of a
single sinusoidal oscillation, the Interval Newton Method
is applied to the computation of the oscillation param-
eters by global minimization. In the case of multiple si-
nusoidal oscillations, a stepwise detection procedure of
single oscillations is applied with global minimization in
each step, providing a reasonable initial guess for the -
nal joined computation of the oscillation parameters. The
signicance problem is not addressed in this study.
Kaschenz and Petrovic (2009) later extend the method
to two-dimensional time series. The proposed decision
problem is solved in a purely heuristic manner, not mak-
ing use of any statistical test. The methodology is illus-
trated by the analysis of an Earth orientation parameter
series of the International Earth Rotation and Reference
System Service (IERS). Also in this study the decision prob-
lem, whether the oscillations identied in the time series
are truly present in the observed phenomenon or whether
they are purely the eect of noise, is not posed.
A well-established approach, although not so popular
in geodesy, is periodogram analysis. Unlike Fourier anal-
ysis, periodogram analysis can be immediately applied
even if the time series is non-uniformly sampled. Peaks in
the periodogram indicate that the observed phenomenon
contains sinusoidal oscillations with the corresponding
angular frequencies (Priestley 1982). Scargle (1982) modi-
es the classical periodogram to what is now known as the
Lomb-Scargle periodogram. See also (Lomb 1976). If in this
paper we speak of “periodogram” then we always refer to
the “Lomb-Scargle periodogram”. Scargle (1982) demon-
strates that there is a strong relationship between the pe-
riodogram and LSSA. Both methods even yield identical
results (Emery and Thomson 2001). See subsection 3.3 be-
low.
It remains to be shown how to test statistically that a
periodogram peak is signicant, i.e. it is almost certainly
(with some decision error) produced by a sinusoidal oscil-
lation of that particular angular frequency, and not only
the product of random noise. Scargle (1982) demonstrates
that under standard assumptions the periodogram values
follow an exponential distribution. However, if we want to
use many periodogram values as test statistics in a mul-
tiple statistical test or equivalently use their maximum in
a single statistical test, then it is required that these pe-
riodogram values are statistically mutually independent.
The task is to nd a maximum set of angular frequencies
at which the corresponding periodogram values enjoy this
statistical property. Scargle (1982) calls them “natural fre-
quencies”, but is only able to determine such a set for
the case of uniform sampling (see subsection 3.3 below).
The term “natural frequencies” may be perceived as a mis-
nomer, because there is nothing “natural” involved here.
It seems that the idea of natural frequencies can be
traced back to Hannan (1960, pp. 76-83). In a time se-
ries the author looks for sinusoidal components only at
these discrete frequencies and nds a loss of the statisti-
cal power of recovering them, if the true frequency falls
between them. In this publication this result will be con-
rmed (subsection 4.4).
Within the next decades little progress has been made
towards a more rigorous implementation of hypothesis
tests in periodogram analysis. Horne and Baliunas (1986)
make experiments with time series of non-uniform sam-
pling and derive an empirical formula for the number of
natural frequencies. The authors call them “independent
frequencies”, which is again a misnomer, because not the
frequencies themselves are statistically independent, but
the corresponding periodogram values.
Koen (1990) discusses the theory and assumptions
made by Scargle (1982) for seeking periodicities in astro-
nomical time series. It is shown that the concept of natural
frequencies severely underestimates the potential statisti-
cal errors, and that the most often used statistical hypoth-
esis is inappropriate for testing for the presence of specic
periodicities.
Hernandez (1999) undertakes a comprehensive study
of statistical signicance tests in periodogram analysis. A
critical test for the maximum number of signicant fre-
quencies is designed and the unnecessary diculties that
the manipulation of the data brings into the statistical
signicance interpretation is demonstrated. The method
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138 |R. Lehmann
is applied to measurements of wind with a medium fre-
quency radar.
As computers become faster, new numerical tools
come within reach, namely the Monte Carlo (MC) method,
which is re-used in our present paper (see subsections 4.2
and 4.4 below). Frescura et al. (2007) propose a practi-
cal method for estimating the signicance of periodogram
peaks, applicable also to irregularly sampled time series,
based on the MC method.
Another approach to time series analysis of non-
equispaced observations is known as CLEAN transforma-
tion. It was developed by Roberts et al. (1987) and performs
an iterative deconvolution of the spectral window in the
frequency domain. CLEAN does not require a stochastic
model of the observations. Therefore, the decision prob-
lem, whether the oscillations identied in the time series
are truly present in the observed phenomenon or whether
they are purely the eect of noise, cannot be posed.
CLEAN is used in many branches of geosciences, e.g.
in seismology by Baisch and Bokelmann (1999) and in
Paleoclimatology by Heslop and Dekkers (2002). The lat-
ter contribution is interesting because it employs the MC
method for the determination of errors. In geodesy no ap-
plications of CLEAN are known.
The outline of our present paper is as follows: We re-
strict ourselves to the detection of a single sinusoidal os-
cillation in a one-dimensional time series. The formula-
tion as a hypothesis test is given in section 2. The scope
of the paper is to compare the LR test for parameters in a
Gauss-Markov model (subsections 3.1 and 3.2) with the pe-
riodogram analysis (subsection 3.3), which can be viewed
as a special implementation of the LSSA. In subsection 3.4
we show how the MC method can be used to determine crit-
ical values of hypothesis tests. An exemplary time series
from an absolute gravimetric measurement is introduced
in subsection 4.1 and analyzed by the LR test (subsection
4.2) and by the periodogram (subsection 4.3). In subsec-
tion 4.4 we compare the statistical power of both tests.
2Formulation of the problem
Let
y=(y1· · · yn)T(1)
be a n-vector of observations of sample values of a uni-
variate function y(t), aected by random observation er-
rors (noise) and sampled at xed time instances t1,. . . ,tn.
These instances are not necessarily equispaced. We intend
to investigate if y(t)attains the form of a single sinusoidal
oscillation
y(t)=a0+a1·cos (ωt)+a2·sin (ω t)(2)
with unknown non-random coecients a0,a1,a2
and an unknown non-random angular frequency ω. This
is also known as the sinusoidal model, sometimes param-
eterized by amplitude and phase angle instead of two co-
ecients a1,a2.
Before the spectral analysis the time series yis often
“centered” in such a way that a mean of yas an estimate of
a0is subtracted. Sometimes even higher order trend func-
tions are pre-subtracted. This might often be a good ap-
proximation of the general solution. But we will not im-
mediately use this approach.
The solution of this problem takes the form of a statis-
tical hypothesis test. As a null hypothesis H0it is proposed
that the observations, eq. 1, uctuate purely due to the ef-
fect of random noise ε:
H0:yi=a0+εi,i= 1,. . . ,n(3)
The noise vector εis modelled by a Gaussian random
vector
ε=(ε1. . . εn)T∼N(0,σ2Q)(4)
with known positive denite matrix Qand a known
or unknown variance factor σ2, such that σ2Qis the co-
variance matrix of ε. In geodesy Qis often called “cofactor
matrix”. Ndenotes the Gaussian normal distribution. In
the following we will treat both the case of known and un-
known variance factor σ2. Both cases are practically im-
portant and will be called case 1 and case 2 in the fol-
lowing. In case 1, σ2is often assumed to be known from
long standing experiences with observations of this kind.
In case 2, this assumption is not made due to lack of such
experiences.
As an alternative hypothesis HA, it is proposed that the
observations contain, besides the noise ε, a signal sin the
form of a sinusoidal oscillation:
HA:yi=a0+a1·cos (ωti)+a2·sin (ωti)+εi
=: si(a0,a1,a2ω) + εi,i= 1,. . . ,n(5)
The angular frequency ωis assumed to be from an
interval [ωmin,ωmax]with properly specied bounds 0<
ωmin <ωmax. A parameter vector θis set up and this vec-
tor is sought in the parameter space Θ.
case 1 : θ=(a0,a1,a2,ω),Θ=R3×[ωmin,ωmax ](6a)
case 2 : θ=a0,a1,a2,ω,σ2,Θ=R3×[ωmin,ωmax ]×R+
(6b)
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Detection of a sinusoidal oscillation of unknown frequency in a time series – a geodetic approach |139
As usual, Rndenotes the cartesian power of the set of
real numbers R,R+denotes the set of positive real numbers
and ×is the cartesian product operator. A proper formula-
tion of the proposed hypotheses according to the formal-
ism of statistics (e.g. Teunissen 2000) reads
case 1 : H0:θ∈Θ0=R×02×[ωmin,ωmax ](7a)
HA:θ∈ΘA=R×R2∖02×[ωmin,ωmax ](7b)
case 2 : H0:θ∈Θ0=R×02×[ωmin,ωmax ]×R+(7c)
HA:θ∈ΘA=R×R2∖02×[ωmin,ωmax ]×R+(7d)
As usual, 0ndenotes the cartesian power of the set : This
means that the parameter space Θis subdivided into two
disjoint subspaces Θ0and ΘA. If H0holds true then θ∈Θ0
and if HAholds true then θ∈ΘA. Nonetheless, we formally
keep ωin the parameter vector θ.
In statistics, the parameters in vector θare called nui-
sance parameters. Note that in the case θ∈Θ0the angu-
lar frequency ωis cancelled in eq. 2. This means that the
nuisance parameter ωis only present under the alterna-
tive hypothesis. Such a non-standard test is treated by An-
drews and Ploberger (1994) and Hansen (1996). It turns
out that "the classical asymptotic optimality properties of
Lagrange multiplier (LM), Wald, and likelihood ratio (LR)
tests do not hold in these nonstandard problems" and “the
LR test is not found to be an optimal test” (Andrews and
Ploberger 1994). Nonetheless, we use the LR test (see sub-
section 3.1) because there are no well-established alterna-
tives.
We desire to test statistically the hypotheses H0versus
HA, i.e. none versus one sinusoidal oscillation in y. For this
purpose a test statistic T(y)needs to be introduced. This is
a function of the observations y. Its actual value allows us
to decide if H0or HAholds true, with some small probabil-
ity of decision error. Extreme values of Tindicate that H0
must be rejected. Next we derive a probability distribution
of T(y)and choose a probability of type I decision error α
(signicance level). Therefrom we derive a critical value c,
beyond which we will nd values of Tonly with probabil-
ity α, provided that H0holds true. If the actual value of T
exceeds cthen we are inclined to reject H0, otherwise we
fail to reject it. The various tests dier with respect to the
proposed test statistics T(y).
3Methodology
3.1 Likelihood ratio test
We consider the likelihood function L(θ|y)of the param-
eter vector θ. The test statistic TLR of the LR test is the ratio
of the maximum of L(θ|y)within the subspace Θ0associ-
ated with H0and its total maximum:
TLR (y) = max{L(θ|y):θ∈Θ0}
max{L(θ|y):θ∈Θ}(8)
TLR being small means that if H0holds true then it is
much more unlikely to observe ythan if HAholds true. If
TLR is below some critical value cLR derived from a desired
signicance level α, then we are inclined to reject H0:
PTLR <cLR |H0=α(9)
If no prior information is available for θthen the likelihood
function Las a function of θequals the probability den-
sity function (PDF) as a function of y. This case is the most
common one and will be exclusively considered further.
The PDF of yequals that of ε, which can directly
be taken from eq. 4, shifted by the non-random vector
s(a0,a1,a2,ω)of signals in eq. 5:
L(θ|y)=σ−ndet (2πQ)−1/2exp −Ω(a0,a1,a2,ω|y)
2σ2
(10)
with
Ω(a0,a1,a2,ω|y):= ||y−s(a0,a1,a2,ω)||Q2(11)
where ||x||Q2:= xTQ−1xdenotes the L2-norm with weight
matrix Q−1(xis simply a wildcard here). Ωis often called
“least squares error function”. If H0holds true then this
simplies to
L(θ|y)=σ−ndet (2πQ)−1/2exp −Ω
′(a0|y)
2σ2,
Ω
′
(a0|y):=||y−a0·1n||Q2(12)
1ndenotes the n-vector of ones.
The computation of eq. 8 requires the solution of two
nonlinear optimization problems, in the denominator of
eq. 8 for the full set of parameters θin eq. 6a or eq. 6b and
in the numerator of eq. 8 only for the parameter a0, in case
2 supplemented by the parameter σ2.
Case 1: If σ2is known then maximizing Lin eq. 10 and
eq. 12 equals minimizing Ωand Ω′, and in turn coincides
with the common least squares solution of eq. 3 and eq. 5,
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140 |R. Lehmann
respectively:
TLR
1(y)
= exp ⎛
⎝−
min Ω
′
(a0|y)−min Ω(a0,a1,a2,ω|y)
2σ2⎞
⎠
= exp −Ω
′(^
a
′
0|y)−Ω(^
a0,^
a1,^
a2,^
ω|y)
2σ2(13)
where ^
a0,^
a1,^
a2,^
ωdenote the common least squares esti-
mates of a0,a1,a2,ωand ^
a
′
0denotes the least squares es-
timates of a0under H0, i.e. with the restriction a1=a2= 0.
Alternatively we may use the fully equivalent test
statistic
T1(y) = Ω
′(^
a
′
0|y)−Ω^
a0,^
a1,^
a2,^
ω|y
σ2=: Ω
′
min (y)−Ωmin(y)
σ2
(14)
Case 2: A necessary condition for a stationary point of
Lis ∂L(θ|y)/∂σ2= 0, which yields the well-known maxi-
mum likelihood estimate of σ2as
^
σ2=Ω(a0,a1,a2,ω|y)
nand ^
σ2=Ω
′(a0|y)
n(15)
in the denominator and in the numerator of eq. 8, respec-
tively. In both cases the argument of exp() in eq. 10 and
eq. 12 at the maximum equal −n/2, such that the exp-terms
cancel in eq. 8. Obviously, the det-terms cancel as well. The
remaining terms σ−nyield the test statistic
TLR
2(y)=
max (Ω
′(a0|y)/n)−n/2
max (Ω(a0,a1,a2,ω|y)/n)−n/2(16)
=⎛
⎝
min Ω
′(a0|y)
min Ω(a0,a1,a2,ω|y)⎞
⎠
−n/2
=Ω
′(^
a
′
0|y)
Ω(^
a0,^
a1,^
a2,^
ω|y)−n/2
=1 + Ω
′(^
a
′
0|y)−Ω(^
a0,^
a1,^
a2,^
ω|y)
Ω(^
a0,^
a1,^
a2,^
ω|y)−n/2
Alternatively we may use the fully equivalent test
statistic
T2(y) = Ω
′(^
a
′
0|y)−Ω(^
a0,^
a1,^
a2,^
ω|y)
Ω(^
a0,^
a1,^
a2,^
ω|y)(17)
=: Ω
′
min (y)−Ωmin(y)
Ωmin(y)
If T1or T2is above some properly chosen critical value
c1or c2, respectively, then we are inclined to reject H0:
P(Ti>ci|H0) = α i = 1,2(18)
For the same α-level this yields the same test decision
as using TiLR (y),i= 1,2in eq. 13 or eq. 16.
3.2 Computation of the LR test statistics
So we are left with the computation of Ω
′
min(y)and Ωmin(y)
in eq. 14 or eq. 17, which are both ordinary least squares
problems.
The computation of Ω
′
min (y)reduces to the special
least squares problem y=a0·1nwith weight matrix Q−1
related to H0. Its solution ^
a
′
0is known to be the weighted
mean of observations.
The computation of Ωmin(y)coincides with the so-
lution of the weighted least squares problem y=
s(a0,a1,a2,ω)with weight matrix Q−1. Here we can take
advantage of the fact that sis non-linear only as a function
of ω, and linear otherwise. Therefore, for xed ω, the least
squares solution can be readily written down as
⎛
⎜
⎝
^
a0(ω)
^
a1(ω)
^
a2(ω)⎞
⎟
⎠=A(ω)TQ−1A(ω)−1A(ω)TQ−1y(19)
with
A(ω) := ⎛
⎜
⎜
⎝
1 cos(ωt1) sin(ωt1)
.
.
..
.
..
.
.
1 cos(ωtn) sin(ωtn)
⎞
⎟
⎟
⎠
In this way, a0,a1,a2can be eliminated from Ω. As a result
we obtain
Ωω(ω|y) := ||y−s(^
a0(ω),^
a1(ω),^
a2(ω),ω)||2
Q(20)
The minimum of this univariate function is sought in
the interval [ωmin,ωmax]by
Ωmin(y) = min[ωmin ,ωmax]Ωω(ω|y)(21)
The minimum is obtained at an argument called ^
ω.
We can try to compute it by numerical functions like MAT-
LAB’s function fminbnd (MathWorks 2013), which is based
on golden section search and parabolic interpolation.
However, often Ωωhas multiple local minima, which re-
quires a global optimizer like the Interval Newton Method
as proposed in (Mautz and Petrovic 2005).
3.3 Periodogram analysis
Scargle (1982) modied the classical periodogram to what
is now known as the Lomb-Scargle periodogram
P(ω|y)
:= 1
2[iyicos ω(ti−τ(ω))]2
icos2ω(ti−τ(ω)) +[iyisin ω(ti−τ(ω))]2
isin2ω(ti−τ(ω))
(22)
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where τ(depending on ω)is dened as
tan 2ωτ(ω) = isin 2 ωti
icos 2ωti
τ(ω)is a time shift, which makes the Fourier base vec-
tors orthogonal even if the time instances t1,. . . ,tnare not
equispaced.
It is found by Scargle (1982) that for the least squares
solution of the simplied problem
1. Q=I(i.e. statistical independence of noise, ho-
moscedasticity)
2. a0= 0 (i.e. a weighted mean is pre-subtracted)
the corresponding least squares minimum value Ωω(ω|y)
in eq. 20 relates to P(ω|y)by
Ωω(ω|y) + P(ω|y) =
i
yi2=: ||y||2= constant (23)
This is to say: Computing Ωmin(y)in eq. 21 is equiv-
alent here to maximizing P(ω|y)in eq. 22. On the other
hand, Ωmin
′
(y)in absentia of the constant term a0and
Q=Iequals ||y||2, see eq. 12. The test statistic eq. 14 of
case 1 assumes the special form
TP(y) = max(P(ω|y))
σ2(24)
In periodogram analysis something similar to case 2 is
not considered. If TPis above some critical value cPthen
we are inclined to reject H0:
P(TP>cP|H0) = α(25)
For the same α-level this may yield a dierent test deci-
sion as compared to T1in eq. 14, due to Ωmin
′
(y)not being
constant.
The argument for dropping the constant term a0in
eq. 2 is often that a mean value is pre-subtracted from the
observations. However, this more or less invalidates Q=I
(if valid at all) because such an operation necessarily cor-
relates the reduced observation. Hopefully, this neglect is
tolerable at least if nis large.
Moreover, if ωwas known (consider that the inter-
val [ωmin,ωmax]reduces to a point) then the test statistic
would simplify to
TP(y,ω) = P(ω|y)
σ2(26)
which under H0is well known to follow an exponential
distribution (Scargle 1982, Frescura 2007):
P(TP<cP|H0) = 1 −exp −cP= 1 −α(27)
Therefore, the critical value cPis easily derived from
the inverse exponential cumulative distribution function.
In order to use this convenient property even in the case
that ωis not known, it is proposed to set up a multiple tests
(Scargle 1982) for a set of frequencies like
ωmin <ω1<· · · <ωN<ωmax (28)
If one of the test statistics TP(y,ωj)in eq. 26 or equiv-
alently their maximum
TP
max := maxj=1,...,NTPy,ωj=σ−2maxj=1,...,NPωj|y
(29)
exceeds some critical value cmaxPthen we are inclined to
reject H0.
PTP
max <cmaxP|H0=PTP(y,ω1)<cmax P|H0∧
(30)
· · · ∧PTP(y,ωN)<cmax P|H0≈1−exp(−cmaxP)N
The latter relationship would hold as an equality if
and only if all TPy,ωjare statistically independent. In
the case of equispaced sampling this is the case for the fre-
quencies
ωj=2πj
T,j= 0,. . . ,⌊n/2⌋(31)
where Tis the total time span and ⌊·⌋denotes the largest
previous integer. This has been reported by Scargle (1982),
who calls ωjthe “natural frequencies”. Otherwise the
identication of such frequencies has not yet been fully
successful (Scargle 1982, Horne and Baliunas 1986, Fres-
cura 2007). In the case of non-equispaced time instances
the test statistics TPy,ωjin eq. 26 with eq. 31 are in gen-
eral not fully independent which makes a rigorous mul-
tiple test very dicult. It is generally supposed that the
dependencies introduced by the uneven sampling are in
some sense negligible.
Note that in the multiple test HAreads: “The obser-
vations contain a signal in the form of a sinusoidal oscil-
lation with a frequency ω1or . . . or ωN.” This is remark-
ably dierent to HAused in eq. 7b or eq. 7d, where the fre-
quency could attain any value in the interval [ωmin,ωmax ].
It seems as if this is no practical drawback as long as the set
in eq. 28 is chosen suciently dense. But since P(ω|y)is a
smooth function, the test statistics TPy,ωjTPy,ωj+1
of neighboring frequencies ωjωj+1 are then highly corre-
lated, which fully invalidates eq. 30.
3.4 Determination of critical values
If in eq. 14 or eq. 17 we obtain Ti(y)>ci, where ciis a
proper critical value derived from a desired α-level, then
we are inclined to reject H0.
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In statistics it is known that if sdened in eq. 5 would
be a linear function then
T1|H0∼χ2(2) (32a)
n−4
2·T2|H0∼F(2,n−4) (32b)
where χ2(k1)denotes the χ2-distribution with k1de-
grees of freedom and F(k1,k2)denotes the F-distribution
with k1and k2degrees of freedom (cf. Koch 1999, Teunis-
sen 2000).
In the following we will verify if we can use these dis-
tributions for the computation of the critical values at least
approximately. Here we restrict ourselves to case 2 (σ2un-
known). The critical value c2of the test of the hypotheses
H0in eq. 7c versus HAin eq. 7d would equal the quantile
of F(2,n−4), such that
c2=FF−1(α|2,n−4) (33)
where FF−1(α|2,n−4) denotes the inverse cumulative dis-
tribution function of F(2,n−4).
Since sdened in eq. 5 is a nonlinear function, this is
not exactly true. But it may be valid at least approximately,
if the curvature of sis small in the subset of Θ, where pa-
rameters are somehow “relevant” for the solution of the
nonlinear least squares problem. This is the standard as-
sumption in geodesy when applying hypotheses tests to
nonlinear models. However, this can be problematic if the
relevant interval [ωmin,ωmax]for ωis so large that Ωωis
not at least approximately parabolic. Therefore, it is safer
not to rely routinely on the approximate validity of eq. 33.
In general, the distribution of T2may also depend on
the time instances t1,. . . ,tnand on the cofactor matrix
Q. Although H0is not a simple but composite hypothesis,
i.e. it contains nuisance parameters a0σ2, the distribution
of T2does not depend on those parameters. The reason is
that they cancel in T2. This is seen as follows:
1. A a0-shift of ydoes aect neither Ωmin nor Ω
′
min in
eq. 17.
2. A change of σ2scales Ωmin and Ω
′
min equally, such
that it cancels in the ratio of eq. 17.
Therefore, the desired distribution of T2can be computed
for arbitrary parameters a0,σ2. In the following we set
a0:= 0,σ2:= 1 for the sake of simplicity. Note that the
same applies to T1in eq. 14 and TP
max in eq. 29. Both test
statistics do not dependent on a0and σ2.
A numerical method to derive the distribution of some
test statistic Tand therefrom the corresponding critical
value c, even if his nonlinear, is the Monte Carlo (MC)
method. This method has been successfully used e.g. by
Lehmann (2012) for the computation of critical values of
normalized and studentized residuals employed in geode-
tic outlier detection. In principle, it replaces
– random variates by computer generated pseudo ran-
dom numbers (PRN),
– probability distributions by histograms and
– statistical expectations by arithmetic means
computed from a large number of MC experiments, i.e.
computations with PRN instead of noisy observations. It
works as follows:
1. Specify Q,t1,. . . ,tn,ωmin ,ωmax.
2. Dene a suciently large number Mof MC experi-
ments to be carried out (say M= 105,106or107, de-
pending on the available computer power).
3. Generate Mnoise vectors ε∼N(0,Q)making use
of σ2:= 1.
4. Set y=εfor each of the Mvectors εmaking use of
a0:= 0.
5. Solve the two nonlinear least squares problems for
each of the Mvectors y.
6. Compute the Mvalues of T. Their histogram approx-
imates the PDF of T.
7. Let the sorted list of Tbe (T1,. . . ,TM). The critical
value cis then approximated by the α-quantile of
this histogram: c= (T⌊αM⌋+T⌊αM⌋+1 )/2.
If we doubt that Mhas been chosen large enough then we
can simply re-compute cwith dierent PRN and compare
it to the previous value. Both values should be reasonably
close. (More advanced adaptive MC methods will be nei-
ther discussed nor used here.)
If the signicance level αis not yet determined then
we can generate a statistical lookup table of critical values
cbelonging to various αby repeating step 7. Thus, the time
consuming operation in step 5 must not be repeated.
4Practical applications
4.1 An absolute gravimetric time series
Absolute gravity values giwere determined in the geode-
tic laboratory of the University of Applied Sciences Dres-
den by means of the absolute gravity meter FG5 (Micro-
g LaCoste Inc., U.S.A.) (Reinhold 2013). Over a period of
26 hours we obtained a time series with 25 gravity values
given in Fig. 1. The observation values are conveniently
transformed to yi= (gi−9.8111936 m/s2)·108. The units
of yiare then µGal and will be omitted below. The obser-
vation time instances tiare not equispaced. We suppose
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the observations to be uncorrelated and of uniform accu-
racy, i.e. Q=I. We desire to test if the time series contains a
single sinusoidal oscillation. (Since tides and other eects
are corrected, such an oscillation could only be produced
by an instrumental eect not accounted for, which did not
show up before. Consequently, we expect that H0is true.
If H0must later be rejected then we know that this is likely
a decision error.)
Figure 1: Blue = absolute gravity values (*) (g- 9.8111936m/s2)*108,
Red = least squares solution with ^
ω= 2π·7.29d−1
Under H0we nd ^
a
′
0to be the simple mean of all
25 gravity values and
Ω
′^
a
′
0= 29.418
By visual inspection of the blue curve in Fig. 1 we guess
that there is an oscillation with 8 periods, i.e. a frequency
between 7d−1and 8d−1. The sampling interval is roughly
0.042d with two data gaps. The sampling rate is about
24d−1. According to the sampling theorem (e.g. Priestley
1982) we can recover frequencies in the signal up to ωmax ≈
2π·12d−1. Due to non-equispaced time instances, we can-
not exactly apply this theorem. Considering also the length
of the time series we choose [ωmin,ωmax]= 2π·[6,9]d−1.
The manufacturer of the instrument provides us with
a noise estimate of σ= 2 ·10−8m/s2= 2µGal (Reinhold
2013).
A similar study, purely based on the LSSA, has been
undertaken for the Canadian Superconducting Gravimeter
Installation by Yin Hui and Pagiatakis (2004), with gaps,
osets, unequal sampling decimation of the data and un-
equally weighted data points. The authors show that for
such data the LSSA is more suitable than Fourier analysis.
We want to stress that this exemplary time series is in-
troduced here for the sake of simplicity. It is short and with-
out gross errors or data gaps. This makes the computations
simple and comprehensible.
4.2 Likelihood ratio test of the exemplary
time series
In Fig. 2 the least squares objective function Ωω(ω|y)
in eq. 20 is plotted as a function of ωon the interval
[ωmin,ωmax ]. It is observed that there are possible multiple
local minima of Ωω(ω|y). For minimization of this func-
tion we use a kind of brute force method here: The function
is evaluated on a grid of equispaced frequencies with spac-
ing 0.1d−1. This is dense enough to come close to the global
minimum eq. 21 at least with one grid point. The minimum
grid point is then used as an initial guess of the Broyden-
Fletcher-Goldfarb-Shanno Quasi-Newton method (an op-
timization algorithm) with a cubic line search procedure,
as implemented in MATLAB’s function fminbnd (Math-
Works 2013). In Fig. 1 we have plotted the observation val-
ues versus the least squares solution. The frequency es-
timate being the argument of the minimum in eq. 21 is
^
ω= 2π·7.29d−1The minimum of the least squares ob-
jective function is
Ωmin (y)=Ω^
a0,^
a1,^
a2,^
ω|y= 21.139
and is clearly below Ω
′^
a
′
0. But can we conclude there-
from that H0is to be rejected? This must be decided by a
statistical hypothesis test. The corresponding test statis-
tics, eqs. 14, 17, amount to
T1(y)=29.418 −21.139
4= 2.070 (34)
T2(y)=29.418 −21.139
21.139 = 0.392 (35)
In Fig. 3 the histogram of the test statistic, eq. 17, de-
termined by the MC method with M= 105MC experiments
together with the PDF of the related F-distribution F(2,21)
from eq. 32b is plotted. It is found that there is a remarkable
dierence between them. The test statistic has much larger
values than expected by the F-distribution. This means
that eq. 32b cannot even approximately be applied. The
reason is that the linear model is not valid, not even ap-
proximately. This could have been expected by the mul-
timodal structure of the least squares problem. If the lin-
ear model would be approximately valid then Ωω(ω|y)as a
function of ωwould be close to a parabola. Figure 2 proves
the invalidity of the linear model. As a consequence, a crit-
ical value for the desired test cannot be derived from any
F-distribution eq. 32b.
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Figure 2: Least squares objective function Ωω(ω|y)in eq. 20 plot-
ted over the abszissa (horizontal coordinate) ω/2π.
The critical values c2derived from the histogram in
Fig. 3 and the critical values c1derived from the analogous
histogram computed by the same MC method for T1are
α0.10 0.05 0.01
c18.92 10.5 14.2
c20.53 0.64 0.95
These values are veried by a second independent run
with M= 105MC experiments. Comparing c1and c2with
T1and T2in eqs. 34, 35 we nd in both cases that H0is
accepted, even with α= 0.10. Result: By the LR-test, we
do not detect any signicant sinusoidal oscillation in the
gravimetric time series.
4.3 Periodogram analysis of the exemplary
time series
In Fig. 4 the Lomb-Scargle periodogram P(ω|y)in eq. 22 is
plotted as a function of ωon the interval [ωmin,ωmax ]. It
seems to be the mirror image of the least squares objective
function as suggested by eq. 23. But this is not exactly true
because the constant term a0has not been dropped here.
Therefore, eq. 23 does not hold exactly.
The Lomb-Scargle periodogram indicates that a sinu-
soidal oscillation of frequency ^
ω= 2π·7.29d−1may be
contained in the observations. This must be conrmed by
a statistical hypothesis test. Computing the periodogram
eq. 22 at the natural frequencies eq. 31 (neglecting ω0= 0)
ωj= 2πj/T,j= 1,. . . ,12
Figure 3: Histogram of the test statistic, eq. 17, determined by MC
method together with the quantiles of 90%,95%,99% and the PDF
of the related F-distribution (vertical axis appropriately scaled).
Figure 4: Lomb-Scargle periodogram, eq. 22, plotted over the ab-
szissa (horizontal coordinate) ω/2π.
with total time span T= 315.44d−314.36d= 1.08dwe
nd a total maximum value
maxjPωj|y= 5.14
attained at ω10 = 2π·9.26 and inserting into eq. 29 yields
TP
max =5.14
4.0= 1.28 (36)
The critical value cmaxPderived from eq. 30 by
1−exp(−cP
max)N= 1 −α(37)
exp(−cP
max) = 1 −(1−α)1
N≈α/N
cP
max ≈ − log(α/N)
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and N= 12 assumes the following values:
α0.10 0.05 0.01
cP
max 2.08 2.38 3.08
Comparing this with eq. 36 we nd that H0is accepted,
even with α= 0.10. Again we do not detect any signicant
sinusoidal oscillation in this time series.
However, the frequency ω10 is so close to the approx-
imate Nyquist frequency that the periodogram may suf-
fer from the aliasing eect. (Of course, aliasing is an ef-
fect of an equispaced time series only. But the exemplary
gravimetric time series is almost equispaced, which re-
sults in a small aliasing eect.) Only three of the 12 nat-
ural frequencies eq. 31 are within the considered inter-
val [ωmin,ωmax]. These are the frequencies ω7= 2π·
6.46,ω8= 2π·7.38,ω9= 2π·8.31. The three correspond-
ing periodogram values are 1.62, 3.96, 0.37. If, as before, we
only consider them as admissible frequencies detectable
in the time series then this would lead to
TP
max =3.96
4.0= 0.99
With N= 3 we obtain from eq. 37:
α0.10 0.05 0.01
cP
max 3.40 4.09 5.70
The result is unchanged: H0is accepted, even with α=
0.10. By periodogram analysis we do not detect any sig-
nicant sinusoidal oscillation in this time series.
It remains to be checked if eq. 37 is a sucient ap-
proximation. For this purpose we compute the exact val-
ues cmaxPby the MC method. We apply the same proce-
dure as in the previous subsection to the case N= 3 of
only three admissible frequencies and obtain
α0.10 0.05 0.01
cP
max 3.35 4.05 5.70
The dierences to the approximate critical values from
eq. 37 are unimportant. But remember that the natural fre-
quencies remain nearly independent as long as the time
instances are nearly equispaced, as is the case here. There-
fore, we can expect much larger dierences e.g. in case of
large data gaps.
4.4 Test power of the likelihood ratio test
versus periodogram analysis
The key question in practical applications is whether the
LR test statistics, eqs. 14, 17, or the periodogram analysis,
eq. 29, has larger test power. Note that here we do not re-
fer to the power of the signal, but to the power Π= 1 −β
of the statistical test, which equals the probability that a
false H0is rejected by the test, i.e. an existing sinusoidal
oscillation in a time series is detected as such. βis known
as the probability of a type II decision error (false negative
rate) of the test.
A more powerful test is to be practically preferred be-
cause it is more often able to reject a false H0. From the
Neyman–Pearson lemma mentioned in subsection 3.1 we
cannot rigorously deduce that the LR test is more power-
ful because this lemma is not exactly applicable here. At
the moment it is unknown if the results of Andrews and
Ploberger (1994) and Hansen (1996) could lead to more sig-
nicantly powerful tests.
All test statistics were shown to be independent of
the parameters a0,σ2, but HAhas additional nuisance
parameters a1,a2,ω, which unfortunately do not can-
cel. Therefore, Πgenerally depends on those parameters.
In statistics, this relationship is known as the statistical
power function Π(a1,a2,ω).
We investigate the power of testing a sinusoidal os-
cillation in the exemplary gravimetric time series by both
the LR tests (case 1 and 2) and by the periodogram analy-
sis. For all tests we compute the function Π(a1,a2,ω)by
the MC method and compare their values. This will answer
the question which test can detect a sinusoidal oscillation
truly present in the observed time series, and which test
produces type II (false negative) errors. Higher values of Π
indicate a more powerful test. Loosely speaking, we com-
pute in a MC simulation how often a sinusoidal oscillation
is correctly detected by the tests. This ratio approximates
Π.
Unfortunately, we cannot compute Πfor all arguments
a1,a2,ω. We guess that Πmainly depends on
1. the ratio of amplitude of the disturbing signal and
magnitude of the noise (signal-noise-ratio) and
2. in the case of periodogram analysis only: how the
angular frequency ωrelates to the natural frequen-
cies ωjin eq. 31.
The phase of the signal is unimportant. Therefore, we re-
strict ourselves to a2= 0 (pure cosine oscillation).
Here, the MC method works very similar to subsection
3.4: After generating M= 105observations by
yi=a1·cos (ωti)+εi,i= 1,. . . ,25
with Gaussian pseudo random noise εi∼N(0,σ2),σ=
2µGal, we apply the LR tests and the periodogram analy-
sis. Note that in this investigation the gravity values gifrom
subsection 4.1 do not enter, but only the time instances ti.
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The results here do therefore not dependent on those ob-
servations.
The statistical power functions Πσ/2,0,ωand
Π(σ,0,ω)are displayed in Fig. 5 and 6. Remember that
values close to 1 mean that a truly exisiting sinusoidal os-
cillation in a time series is detected as such almost without
fail. Note that Πis a continuous function of ω, i.e. it can
be evaluated at any angular frequency ω, not only at the
natural frequencies ωjin eq. 31 in the case of periodogram
analysis.
Figure 5: Statistical power function Π(σ/2,0,ω)for α= 0.10 (red),
α= 0.05 (green) and α= 0.01 (blue) for LR-test case 1 (dash-
dotted), LR-test case 2 (solid) and periodogram analysis (dashed).
Figure 6: Statistical power function Π(σ,0,ω)with same line style
as in Fig. 5.
The following results are deduced from Fig. 5 and 6:
1. Πσ/2,0,ω<Π(σ,0,ω), which is trivial because
a stronger oscillation is more easily detected.
2. The power is larger if αis also large, which is again
trivial because for larger α,H0is more often rejected,
also if it is false.
3. The power of the periodogram analysis strongly de-
pends on the angular frequency ω. It is high if ω
coincides with a natural frequency ωj. This is also
not surprising because here the periodogram has
been evaluated in eq. 29. The power of the LR-tests
is nearly independent of ω.
4. The power of the LR-test case 1 considerably exceeds
case 2. This is the eect of σ2known.
5. It is most instructive to compare LR-test case 1 and
periodogram analysis. Averaging over ω, the power
is about the same, except for the margins of the in-
terval [ωmin,ωmax ], where the periodogram analysis
performs badly.
4.5 Final example with a data gap
Finally, we will test both methods at an irregularly sam-
pled time series with a considerable data gap. To check
if the correct test decision has been made, it is important
here to use synthetic data.
Assume a diurnal oscillation ωtrue = 2π·d−1has been
sampled almost hourly over a time span of 126 h (T=
5.25d). The time instances tare uniformly distributed over
the rst quarter of each hour and there is a simulated data
gap at the rst half of the third day. We assume that ε∼
N(0,I)is correctly known. Furthermore, the true unknown
coecients in eq. 1 are a0=a2= 0 and a1is varied in the
range of 0. . . 0.7. This is also the signal-to-noise-ratio. Fig-
ure 7 displays this time series for a1= 0.7.
The frequency taken as unknown is looked for be-
tween ωmin =π·d−1and ωmax = 4π·d−1. According to
eq. 31, the eight natural frequencies are
ω3= 1.14π·d−1,ω4= 1.52π·d−1,. . . ,
ω10 = 3.81π·d−1
This means, the true frequency ωtrue is between ω5and
ω6.
By the MC method in the same way as in subsection
4.2 we nd the critical values for T1of the LR test. With
N = 8 we obtain from eq. 37 the critical values for TP
max of
the periodogram. The results are
α0.10 0.05 0.01
c110.5 12.2 15.8
cmaxP4.38 5.08 6.68
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Figure 8 displays the results in terms of the test statistics.
For α= 0.10 the LR test is able to detect an oscillation
with amplitude a1>0.47, while the periodogram method
needs a1>0.61 to come to the decision that there is truly a
sinusoidal oscillation in the time series. With higher signif-
icance levels, the limits are increased according to Fig. 8.
This example proves that there is in principle no dras-
tic change of the situation when dealing with
– longer time series,
– more irregular time instances and
– larger data gaps
than considered in subsections 4.1-4.4.
Figure 7: Time series (blue) of section 4.5 with a data gap and true
oscillation (red) with amplitude a1= 0.7
5Conclusions
Our paper does not introduce a new method, but opposes
the analysis of the Lomb-Scargle periodogram for detect-
ing a single sinusoidal oscillation in a time series to the
LR test, which is the standard statistical test performed
in geodesy. The second method has never been applied to
the problem of detecting sinusoidal oscillations with un-
known frequencies.
It is demonstrated that both testing methods are
closely related, but are in general not identical. The dier-
ences are as follows:
Advantages of the LR test:
1. The LR test is more exible because it immediately
allows for heteroscedastic observations, even for
Figure 8: Test statistics T1of eq. 14 and TP
max of eq. 29 for the ex-
ample given in section 4.5 with corresponding critical values (dash-
dotted) and amplitude limits, where they are exceeded
correlated observations with a non-diagonal cofac-
tor matrix Q.
2. The LR test is more exible because it is easily ap-
plied to the case of an unknown variance factor σ2
(called case 2 here). The periodogram analysis could
also be extended to this case, but this has not yet
been done.
3. The LR test is more exible because it allows for
an unknown constant shift parameter (here a0) and
even higher order (e.g. linear) trend models.
4. Periodogram analysis has serious limitations, rst
of all the problems with the natural frequencies. If
the time series is highly unequispaced then the nat-
ural frequencies of the periodogram get statistically
more and more dependent. In contrast to this, by the
LR test we do not encounter any problems here.
Advantages of periodogramm analysis:
1. Periodogramm analysis is conceptually clearer. It
can be explained even to persons without statistical
background.
2. Periodogramm analysis is computationally less ex-
pensive because it avoids any nonlinear optimiza-
tion as well as the application of the MC method.
3. Periodogramm analysis is simple to implement into
a computer program.
However, according to our experiences, modern comput-
ers have enough computational power to compute critical
values by the MC method even for large time series. There-
fore, for most users the LR test is recommended.
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The exemplary gravimetric time series used here does
not require too much exibility: The time series is almost
equispaced, the noise is statistically independent and ho-
moscedastic. Therefore, the periodogram analysis is an
option and performs moderately well.
Using a standard signicance level we obtain the same
test decision by any test. From the geophysical background
of the example we have reasons to believe that it is the cor-
rect test decision. Note that the example only illustrates
the operating mode and proves the workability of the test-
ing methods. It cannot decide which method is generally
superior.
A further comparison shows that the statistical test
power of the periodogram analysis is about the same as
for the LR test with known variance factor σ2(called case 1
here), but unlike in the LR test, it depends on the frequency
(good near the natural frequencies and poor otherwise).
The experiment with a longer time series having a con-
siderable data gap and more irregular sampling time in-
stances shows that the results are in principle also valid in
this case. The LR test was able to detect an oscillation with
a smaller amplitude, but this might not always be the case
in other applications.
Finally, we mention that the results can be extended
to the case of multiple sinusoidal oscillations. However,
the minimization problem eq. 21 is then a multivariate op-
timization problem, which makes the MC method compu-
tationally even more costly. If the computer power is avail-
able, we see no insurmountable obstacle.
In the future it is worth trying to replace the LR-test
by even more powerful tests following the line of Andrews
and Ploberger (1994) and Hansen (1996).
References
Andrews D.W.K. and Ploberger W., 1994, Optimal tests when a nui-
sance parameter is present only under the alternative, Economet-
rica, 62, 6, 1383-1414.
Baisch S. and Bokelmann G. H. R., 1999,Spectral analysis with incom-
plete time series: an example from seismology, Computers and
Geosciences, 25, 739-750.
Emery W.E. and Thomson R.E., 2001, Data Analysis Methods in Phys-
ical Oceanography, Elsevier. ISBN: 0-444-50756-6.
Erol S., 2011, Time-Frequency Analyses of Tide-Gauge Sensor Data.
Sensors 11:3939-3961. DOI: 10.3390/s110403939
Frescura F.A.M., Engelbrecht C.A. and Frank B.S., 2007, Signicance
Tests for Periodogram Peaks, arXiv:0706.2225 [astro-ph]. DOI:
10.1016/j.icarus.2012.06.015 http://arxiv.org/pdf/0706.2225v1.
pdf
Hannan E.J., 1960, Time Series Analysis, Methuen, London.
Hansen B.E., 1996, Inference when a nuisance parameter is not iden-
tied under the null hypothesis. Econometrica, 64, 2, 413-430.
Hernandez G., 1999, Time series, periodograms, and signicance,
Journal of Geophysical Research, Space Physics 104, A5, 10355–
10368. DOI: 10.1029/1999JA900026 http://onlinelibrary.wiley.
com/doi/10.1029/1999JA900026/abstract
Heslop D. and Dekkers M.J., 2002, Spectral analysis of unevenly
spaced climatic time series using CLEAN: signal recovery and
derivation of signicance levels using a Monte Carlo simulation,
Physics of the Earth and Planetary Interiors, 130, 103-116.
Horne J.H. and Baliunas S.L., 1986, A prescription for period analy-
sis of unevenly sampled time series, Astrophysical Journal, Part 1,
302, 757-763. ISSN: 0004- 637X
Kargoll B., 2012, On the Theory and Application of Model Misspeci-
cation Tests in Geodesy, German Geodetic Commission Series C,
No. 674, Munich.
Kaschenz J. and Petrovic S., 2009, A Methodology for the Identica-
tion of Periodicities in Two-dimensional Time Series, Zeitschrift für
Vermessungswesen, 134, 2, 105-112.
Koch K.R., 1999, Parameter Estimationand Hypothesis Testing in Lin-
ear Models, Springer, Berlin Heidelberg New York.
Koen Ch., 1990, Signicance testing of periodogram ordinates, ApJ,
348, 700-702. DOI: 10.1086/168277
Lehmann R. and Koop B., 2009, Conveyance of large cruise liners –
Geodetic investigation of rolling and track prediction, J Appl Geod,
3, 3, 131–141. DOI: 10.1515/JAG.2009.014
Lehmann R., 2012, Improved critical values for extreme normalized
and studentized residuals in Gauss-Markov models, J Geod, 86,
1137–1146. DOI: 10.1007/s00190-012-0569-0
Lomb N.R., 1976, Least-squares frequency analysis of unequally
spaced data. Astrophysics and Space Science, 39, 2, 447-
462. DOI: 10.1007/BF00648343 http://link.springer.com/article/
10.1007%2FBF00648343
MathWorks, 2013, MATLAB R2013a Documentation Center. http://
www.mathworks.de/de/help/documentation-center.html
Mautz R. and Petrovic S., 2005, Erkennung von physikalisch
vorhandenen Periodizitäten in Zeitreihen, Zeitschrift für Vermes-
sungswesen, 3/2005, 156-165.
Neyman J. and Pearson E.S., 1933, On the Problem of the Most E-
cient Tests of Statistical Hypotheses, Philosophical Transactions
of the Royal Society A: Mathematical, Physical and Engineering
Sciences, 231, 694–706, 289–337. DOI: 10.1098/rsta.1933.0009
Pagiatakis S.D., 1999, Stochastic signicance of peaks in
the least-squares spectrum, J Geod, 73, 67-78. DOI:
10.1007/s001900050220
Priestley M.B., 1982, Spectral Analysis and Time series, Academic
Press Inc., London. ISBN: 9780125649223
Psimoulis P., Pytharouli St., Karambalis D. and Stiros S., 2008, Poten-
tial of Global Positioning System (GPS) to measure frequencies of
oscillations of engineering structures, Journal of Sound and Vibra-
tion, 318, 606–623. DOI: 10.1016/j.jsv.2008.04.036
Reinhold A., 2013, Absolute und relative Schweremessungen in
der Hochschule für Technik und Wirtschaft Dresden vom 09.-
12.11.2012, Technischer Bericht G4-2012-3, Bundesamt für Kar-
tographie und Geodäsie (BKG), Frankfurt am Main.
Roberts D.H., Lehár J. and Dreher J.W., 1987, Time Series Analysis with
CLEAN: I. Derivation of a Spectrum, Astron. J., 93, 4, 968–989.
Scargle J.D., 1982, Studies in astronomical time series analysis. II -
Statistical aspects of spectral analysis of unevenly spaced data,
Astrophysical Journal, Part 1, 263, 835-853. DOI: 10.1086/160554
Tanizaki H., 2004, Computational Methods in Statistics and Econo-
metrics, Marcel Dekker, New York. ISBN-13: 978-0824748043
Unauthenticated
Download Date | 11/24/15 7:50 PM
Detection of a sinusoidal oscillation of unknown frequency in a time series – a geodetic approach |149
Teunissen P.J.G., 2000, Testing theory; an introduction, 2nd edition,
Series on Mathematical Geodesy and Positioning, Delft University
of Technology, The Netherlands. ISBN: 90-407-1975-6
Teunissen P.J.G., 2001, Adjustment theory; an introduction, 2nd edi-
tion, Series on Mathematical Geodesy and Positioning, Delft Uni-
versity of Technology, The Netherlands. ISBN: 90-407-1974-8
Wells D., Vaniček P. and Pagiatakis S., 1985, Least-squares spectral
analysis revisited, Issue 84 of Technical Report, University of New
Brunswick, Dept. of Surveying Engineering.
Vaniček P., 1969, Approximate spectral analysis by least-squares, As-
trophy Space Sci, 4, 387-391.
Vaniček P., 1971, Further development and properties of the spec-
tral analysis by least-squares, Astrophy Space Sci, 12, 10-33. DOI:
10.1007/BF00656134
Yin Hui and Pagiatakis S., 2004, Least squares spectral anal-
ysis and its application to superconducting gravimeter data
analysis, Geo-spatial Information Science, 7, 4, 279-283. DOI:
10.1007/BF02828552
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