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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).

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