We develop a general framework for the frequency analysis of irregularly
sampled time series. It is based on the Lomb–Scargle periodogram, but
extended to algebraic operators accounting for the presence of a polynomial
trend in the model for the data, in addition to a periodic component and a
background noise. Special care is devoted to the correlation between the
trend and the periodic component. This new periodogram is then cast into the
Welch overlapping segment averaging (WOSA) method in order to reduce its
variance. We also design a test of significance for the WOSA periodogram,
against the background noise. The model for the background noise is a
stationary Gaussian continuous autoregressive-moving-average (CARMA) process,
more general than the classical Gaussian white or red noise processes. CARMA
parameters are estimated following a Bayesian framework. We provide
algorithms that compute the confidence levels for the WOSA periodogram and
fully take into account the uncertainty in the CARMA noise parameters.
Alternatively, a theory using point estimates of CARMA parameters provides
analytical confidence levels for the WOSA periodogram, which are more
accurate than Markov chain Monte Carlo (MCMC) confidence levels and, below
some threshold for the number of data points, less costly in computing time.
We then estimate the amplitude of the periodic component with least-squares
methods, and derive an approximate proportionality between the squared
amplitude and the periodogram. This proportionality leads to a new extension
for the periodogram: the weighted WOSA periodogram, which we recommend for
most frequency analyses with irregularly sampled data. The estimated signal
amplitude also permits filtering in a frequency band. Our results
generalise
and unify methods developed in the fields of geosciences, engineering,
astronomy and astrophysics. They also constitute the starting point for an
extension to the continuous wavelet transform developed in a companion
article . All the methods presented in this paper are
available to the reader in the Python package WAVEPAL.
Introduction
In many areas of geophysics, one has to deal with irregularly sampled time
series. However, most state-of-the-art tools for the frequency analysis
are designed to work with regularly sampled data. Classical methods include
the discrete Fourier transform (DFT), jointly with the Welch overlapping
segment averaging (WOSA) method, developed by , or the
multitaper method, designed in and .
Given the excellent results they provide, it is tempting to interpolate the
data and simply apply these techniques. Unfortunately, interpolation may
seriously affect the analysis with unpredictable consequences for the
scientific interpretation p. 224.
In order to deal with non-interpolated astronomical data,
and proposed what is now known as the Lomb–Scargle
periodogram (denoted here LS periodogram). The LS periodogram is at the basis
of many algorithms proposed in the literature, in particular in astronomy,
e.g. in , , or , and
in geophysics, e.g. in , ,
, , or .
More specifically, in climate and paleoclimate, the time series are often
very noisy, exhibit a trend, and potentially carry a wide range of periodic
components (see e.g. Fig. ). Considering all these properties,
we design in this work an operator for the frequency analysis generalising
the LS periodogram. The latter was built to analyse data which can be modelled
as a periodic component plus noise. Since the periodic component may not
necessarily oscillate around zero, and
extended the LS periodogram, proposing an operator that is
suitable to analyse data which can be modelled as a periodic component plus a
constant trend plus noise. Their operator is designed to take into account
the correlation between the constant trend and the periodic component, and is
now a classic tool for analysing astronomical irregularly spaced time series.
In climate and paleoclimate, the periodic component may oscillate around a
more complex trend than just a constant. This is why, in this work, we extend
the previous result by proposing an operator that is suitable to analyse data
which can be modelled as a periodic component plus a polynomial trend plus
noise. Our operator is also designed to take into account the correlation
between the trend and the periodic component. Our extended LS periodogram is,
however, not sufficient to deal with very noisy data sets, and it also
exhibits spectral leakage, like the DFT. In the world of regularly sampled
and very noisy time series, smoothing techniques can be applied to
reduce the variance of the periodogram, after tapering the time series in
order to alleviate spectral leakage see. One of them is
the WOSA method , which consists of segmenting the time
series into overlapping segments, tapering them, taking the periodogram on
each segment, and finally taking the average of all the periodograms. This
technique was transferred to the world of irregularly sampled time series in
the work of , where they apply the classical LS periodogram
to each tapered segment, and take the average. In this article, we generalise
their work by applying the tapered WOSA method to our extended LS
periodogram. Moreover, we show that it is preferable to weight the
periodogram of each WOSA segment before taking the average in order to get a
reliable representation of the squared amplitude of the periodic component.
This leads us to define the weighted WOSA periodogram, which we
recommend for most frequency analyses.
The periodogram is often accompanied by a test of significance for the
spectral peaks, which relies on the choice of an additive background noise.
Two traditional background noises are used in practice. The first one is the
Gaussian white noise, which has a flat power spectral density, and which is a
common choice with astronomical data sets, e.g. in
or . The second one is the Gaussian red noise or
Ornstein–Uhlenbeck process, for which the power spectral density is a
Lorentzian function centred at frequency zero, and which is a common choice
with (palaeo-)climate time series, e.g. those in or
. Arguments in favour of a Gaussian red noise as the
background stochastic process for climate time series are given in
Hasselmann's influential paper . Other background
noises are also found in geophysics, often under the form of an
autoregressive-moving-average (ARMA) process seep. 60, for an
extensive list. In this work, we consider a general class of
background noises, which are the continuous autoregressive-moving-average
(CARMA) processes, defined in Sect. . A
CARMA(p,q)
process is the extension of an ARMA(p,q) process to a continuous time
Sect. 11.5. Gaussian white noise and Gaussian red
noise are particular cases of a Gaussian CARMA process, i.e. they are a
CARMA(0,0) process and a CARMA(1,0) process, respectively. Recent advances now
allow for accurate estimation of the parameters of an irregularly sampled CARMA
process from one of its samples see.
Estimating the percentiles of the distribution of the weighted WOSA
periodogram of an irregularly sampled CARMA process is the core of this
paper. This gives the confidence levels for performing tests of significance
at every frequency, i.e. test if the null hypothesis – the time series is a
purely stochastic CARMA process – can be rejected (with some percentage of
confidence) or not. We aim at developing a very general approach. Let us
enumerate some key points.
Estimation of CARMA parameters is performed in a Bayesian framework and relies on state-of-the-art algorithms provided by .
In the special case of a white noise, we provide an analytical solution.
Based on 1, we provide confidence levels computed with Markov chain Monte Carlo (MCMC) methods, that fully take into account the uncertainty
of the parameters of the CARMA process, because we work with a distribution of values for the CARMA parameters instead of a unique set of values.
Alternatively to 2, if we opt for the traditional choice of a unique set of values for the parameters of the CARMA background noise, we develop
a theory providing analytical confidence levels. Compared to a MCMC-based approach, the analytical method is more accurate and, if the
number of data points is not too high, quicker to compute, especially at high confidence levels, e.g. 99 or 99.9 %. Computing high levels
of confidence is required in some studies, for example in paleoceanography .
Confidence levels are provided for any possible choice of the overlapping factor for the WOSA method, extending the traditional 50 % overlapping choice .
Under the case of a white noise background, without WOSA segmentation and without tapering, we define the F periodogram as an
alternative to the periodogram. It has the advantage of not requiring any parameter to be estimated.
Finally, we note that spectral power and estimated squared amplitude are no
longer the same thing if the time series is irregularly sampled. Both
quantities may be of physical interest. We estimate the amplitude of the
periodic component with least-squares methods, and derive an approximate
proportionality between the squared amplitude and the periodogram, from which
we deduce the weights for the weighted WOSA periodogram. The estimated signal
amplitude also gives access to filtering in a frequency band.
The paper is organised as follows. In Sect. , we introduce the notations and
recall some basics of algebra. In Sect. , we
define the model for the data and write the background noise term into a
suitable mathematical form. Section starts
with some reminders about the Lomb–Scargle periodogram and then extends it to
take into account the trend, and a second extension deals with the WOSA
tapered case. In Sect. , we
remind the reader that significance testing is nothing but a statistical hypothesis
testing. Under the null hypothesis, we estimate the parameters of the CARMA
process and estimate the distribution of the WOSA periodogram, either with
Monte Carlo methods or analytically. In the case of a white noise background,
we define the F periodogram as an alternative to the periodogram. Section aims at computing the amplitude of the periodic
component of the signal, and the difference between the squared amplitude and
the periodogram is explained. Sections
and are based on the results of Sect. . There, we propose a third extension for the LS
periodogram and show how to perform filtering. Section
presents an example of analysis on a palaeoceanographic time series. Finally,
a Python package named WAVEPAL is available to the reader and is presented in
Sect. .
Notations and mathematical backgroundNotations
Let us introduce the notations for the time series. The measurements X1,X2,…,XN are done at the times t1,t2,…,tN respectively, and
we assume there is no error in the measurements or in the times. They
are cast into vectors belonging to RN:
|t〉=t1t2⋮tNand|X〉=X1X2⋮XN.
We use here the bra–ket notation, which is common in physics. In
RN, the transpose of |a〉 is 〈a|, i.e. 〈a|′=|a〉, and in CN, 〈a| is the conjugate
transpose of |a〉, i.e. 〈a|∗=|a〉. The inner product
of |a〉 and |b〉 is 〈a|b〉.
Let A be a (m,n) matrix
and B be
a (n,m) matrix. If A is real, A′ denotes its transpose, and if A is complex, A∗
denotes its conjugate transpose. The trace of AB is denoted by tr(AB) and we have tr(AB)=tr(BA).
Let |Y〉 be a vector in RN and A be a (M,N) matrix. The notations A|Y〉 and |AY〉 refer to the same
vector.
We use the terminology Gaussian white noise or simply white noise for a (multivariate) Gaussian random variable with constant mean and covariance matrix σ2I.
|Z〉 always denotes a standard multivariate Gaussian white noise, i.e.|Z〉=dN(0,I),where =d means “is equal in distribution” and I is
the identity matrix.
A sequence of independent and identically distributed random variables is denoted by “iid”.
Orthogonal projections in RN
The orthogonal projection on a vector space spanned by the m linearly
independent vectors |a1〉, ..., |am〉 in RN for
some m∈N0 (m≤N) is
Psp‾{|a1〉,…,|am〉}=V(V′V)-1V′,
where sp‾{|a1〉,…,|am〉} is the closed
span of those m vectors, i.e. the set of all the linear combinations
between them. V is a (N,m) matrix defined by
V=|||a1〉…|am〉||.
Like for any orthogonal projection, we have the following equalities:
Psp‾{|a1〉,…,|am〉}=Psp‾{|a1〉,…,|am〉}′=Psp‾{|a1〉,…,|am〉}2.
The m linearly independent vectors |a1〉, ..., |am〉 may be
orthonormalised by a Gram–Schmidt procedure, leading to m orthonormal
vectors |b1〉, ..., |bm〉, and the orthogonal projection may
then be rewritten as
Psp‾{|a1〉,…,|am〉}=Psp‾{|b1〉,…,|bm〉}=∑k=1m|bk〉〈bk|.
Under that form, we see that the above projection has m eigenvalues equal
to 1 and (N-m) eigenvalues equal to 0.
Let |c1〉, ..., |cq〉 be q linearly independent vectors in
RN, with q≤m, and such that
sp‾{|c1〉,…,|cq〉}⊆sp‾{|a1〉,…,|am〉}. Then (Psp‾{|a1〉,…,|am〉}-Psp‾{|c1〉,…,|cq〉}) is an orthogonal projection on
sp‾{|c1〉,…,|cq〉}∩sp‾{|a1〉,…,|am〉}⟂, and
Psp‾{|a1〉,…,|am〉}Psp‾{|c1〉,…,|cq〉}=Psp‾{|c1〉,…,|cq〉}Psp‾{|a1〉,…,|am〉}7=Psp‾{|c1〉,…,|cq〉}.
Moreover, for any vector |Y〉∈RN, we have
||(Psp‾{|a1〉,…,|am〉}-Psp‾{|c1〉,…,|cq〉})|Y〉||28=||Psp‾{|a1〉,…,|am〉}|Y〉||2-||Psp‾{|c1〉,…,|cq〉}|Y〉||2.
We recommend the book of for more details.
Quantifying the irregularity of the sampling
The biggest time step for which t1, ..., tN are a subsample of a
regularly sampled time series is the greatest common divisor
The GCD
is usually defined on the integers, but we can extend it to rational numbers.
In practice, t1, ..., tN come from measurements with a finite precision
and are thus rational numbers.
(GCD) of all the time steps of |t〉.
In formulas,
ΔtGCD=GCDΔt1,…,ΔtN-1,
where
Δtk=tk+1-tk∀k∈{1,…,N-1},
and
∀k∈{1,…,N},∃m∈Zs.t.tk=mΔtGCD,
where Z denotes the space of integer numbers. Quantifying the
irregularity of the sampling is then straightforward. We define
rt=100(N-1)ΔtGCDtN-t1.
This ratio is between 0 and 100 %, the latter value being reached
with regularly sampled time series.
The model for the dataDefinition
A suitable and general enough model to analyse the periodicity at frequency
f=Ω2π is
|X〉=|Trend〉+EωcosΩ|t〉+ϕω+|Noise〉13=|Trend〉+Aω|cΩ〉+Bω|sΩ〉+|Noise〉,
with Aω=Eωcos(ϕω),
Bω=-Eωsin(ϕω), and
Eω2=Aω2+Bω2. The terms |cΩ〉 and
|sΩ〉 are defined componentwise, i.e.
|cΩ〉=cos(Ω|t〉)=[cos(Ωt1),…,cos(ΩtN)]′ and |sΩ〉=sin(Ω|t〉)=[sin(Ωt1),…,sin(ΩtN)]′. We have added the subscript ω to differentiate between the probed frequency, ω, and the data frequency,
Ω. Indeed, the periodogram (defined in Sect. ), the amplitude periodogram (Sect. ) and the weighted
WOSA periodogram (Sect. ) do not necessarily probe the
signal at its true frequency Ω.
The background noiseDefinition of a CARMA process
We follow here the definitions and conventions of , and
technical details can be found in Sect. 11.5.
The background noise term, |Noise〉, considered in this paper
is a zero-mean stationary Gaussian CARMA process sampled at the times of |t〉. As explained in the
following, it generalises traditional background noises used in geophysics.
A CARMA(p,q) process is simply the extension of an ARMA(p,q) process to a
continuous time
A CARMA(p,q) process sampled at the times of an
infinite regularly sampled time series is an ARMA(p,q) process.
. A zero-mean
CARMA(p,q) process y(t) is the solution of the following stochastic
differential equation:
dpy(t)dtp+αp-1dp-1y(t)dtp-1+…+α0y(t)14=βqdqϵ(t)dtq+βq-1dq-1ϵ(t)dtq-1+…+ϵ(t),
where ϵ(t) is a continuous-time white noise process with zero mean
and variance σ2. It is defined from the standard Brownian motion
B(t) through the following formula:
σdB(t)=ϵ(t)dt.
The parameters α0, ... , αp-1 are the autoregressive
coefficients, and the parameters β1, ..., βq are the moving
average coefficients; αp=β0=1 by definition. When p>0, the
process is stationary only if q<p and the roots r1,…,rp of
∑k=0pαkzk=0
have negative real parts. Strictly speaking, the derivatives of the Brownian
motion dkBdt, k>0, do not exist, and we
therefore interpret Eq. () as being equivalent to
the following measurement and state equations:
y(t)=〈b|w(t)〉,
and
d|w(t)〉=A|w(t)〉dt+dB(t)|e〉,
where |b〉=[β0,β1,…,βq,0,…,0]′ is a vector of
length p, |e〉=[0,0,…,0,σ]′, and
A=010…0001…0⋮⋮⋮⋱⋮000…1-α0-α1-α2…-αp-1.
Equation () is nothing else but an Itô differential equation for the state vector |w(t)〉.
In practice, only CARMA processes of low order are useful in our framework,
typically, (p,q)=(0,0), (1,0), (2,0), (2,1), since at a higher order,
they often exhibit dominant spectral peaks see, which is
not what we want as a model for the spectral background. Indeed, on the basis
of our model, Eq. (), it is desirable that the spectral peaks
come from the deterministic cosine and sine components. We now consider two
useful particular cases of a CARMA process before analysing the general case.
Gaussian white noise
When p=0 and q=0, the process reduces to a white noise, normally
distributed with zero mean and variance σ2. The
|Noise〉 term in Eq. () is then simply
|Noise〉=σ|Z〉=K|Z〉,
with K=σI.
Gaussian red noise
When p=1 and q=0, the CARMA(1,0) or CAR(1) process is an
Ornstein–Uhlenbeck process or red noise , which is quite
of interest in geophysical and other applications . Since
we work with a discrete time series, it is necessary to find the solution of
Eq. () at t1, ..., tN. This is done by
integrating that equation between consecutive times, i.e. from ti-1 to
ti∀i∈{2,…,N}. The components of the
|Noise〉 vector are then as follows:
y(t1)=dN0,σ22α,21y(ti)=ρiy(ti-1)+ηi,∀i∈{2,…,N},
where
ρi=exp-α(ti-ti-1)and22ηi=dN0,σ22α(1-ρi2).
See and p. 343 for more details.
The requirement on stationarity, Eq. (),
imposes α>0. The generated time series has a constant mean equal to
zero and a constant variance equal to σ22α. The
|Noise〉 term in Eq. () can also be written
under a matrix form:
|Noise〉=K|Z〉,
where K is a (N,N) lower triangular matrix whose elements are
Ki,j=σ22α1-ρj2exp-α(ti-tj),∀j≤i,
where we define ρ1=0. This matrix form is used in Sect. .
Note that, if the time series is regularly sampled, ρ is a constant and
Eq. () becomes the equation of a finite-length AR(1)
process, which is stationary since α>0 implies ρ<1.
The general Gaussian CARMA noise
The solution of Eq. () at the time tn (n=2,…,N), that we denote by yn, is
yn=〈b|wn〉,25where |wn〉=exp(A(tn-tn-1))|wn-1〉+|ηn〉,
where |ηn〉 follows a multivariate normal distribution with zero mean
and covariance matrix Cn given by
Cn=∫0tn-tn-1dtexp(At)|e〉〈e|exp(A′t).
The above formula requires the computation of matrix exponentials and
numerical integration. This can be avoided by diagonalising matrix A, with
A=UDU-1. D is a diagonal matrix with diagonal elements given by the
roots of Eq. ():
Dkk=rk,∀k∈1,…,p,
and U is a Vandermonde matrix, with
Ulk=rkl-1∀l,k∈1,…,p.
Now, by defining |w̃n〉=U-1|wn〉, we get
29ayn=〈b|U|w̃n〉,29bwhere |w̃n〉=Λn|w̃n-1〉+|η̃n〉.
The matrix exponential exp(A(tn-tn-1)) has been transformed into
Λn=U-1exp(A(tn-tn-1))U, which is simply a diagonal matrix
with elements Λnkk=exp(rk(tn-tn-1)). The covariance
matrix of |η̃n〉, that we write Σn, also takes a
relatively simple form:
Σnkl=-σ2κkκl∗rk+rl∗1-exp(rk+rl∗)(tn-tn-1),30∀k,l∈{1,…,p},
which is a Hermitian matrix, and where |κ〉 is the last column of
U-1. The initial condition y1 is determined by imposing stationarity,
which is fulfilled only if |w1〉 has a zero mean and a covariance
matrix V whose elements are
Vkl=-σ2∑m=1prmk-1(-rm)l-12Re{rm}∏s=1,s≠mp(rs-rm)(rs∗+rm),31∀k,l∈{1,…,p}.
Stationarity implies that the process y(t) has a zero mean and variance
〈b|V|b〉∀t. All the above formulas and how to get
them can be found in , and Sect. 11.5.2.
Generation of a CARMA(p,q) process can be performed with the Kalman filter
since Eqs. () and () are nothing
but the state and measurement equations, respectively seefor more
details. Alternatively, |y〉 can be written under a matrix
form as in Eq. (). Matrix formalism is useful in Sect. . Let us start with Eq. ():
|w̃n〉=Λn|w̃n-1〉+U-1|ηn〉.
The covariance matrix of |ηn〉, Cn=UΣU∗, is of course
real symmetric and positive semi-definite. We thus have the following Schur
decomposition:
Cn=QnQn′,
where Qn is a real matrix. Consequently,
|w̃n〉=Λn|w̃n-1〉+U-1Qn|ϵn〉=ΛnΛn-1|w̃n-2〉+ΛnU-1Qn-1|ϵn-1〉+U-1Qn|ϵn〉=…34=∑i=2n∏l=i+1nΛlU-1Qi|ϵi〉+∏l=2nΛl|w̃1〉,
where |ϵ1〉, ..., |ϵn〉 are iid standard
Gaussian white noises. The product of the Λ's can be simplified. Its
diagonal elements are as follows:
(Yin)jj:=∏l=i+1nΛljj=exprj(tn-ti).
As stated above, |w1〉 follows a multivariate normal distribution
with zero mean and covariance matrix V. We can use again the Schur
decomposition to write V=WW′, where W is a real matrix, yielding
|w̃n〉=∑i=2nYinU-1Qi|ϵi〉+Y1nU-1W|ϵ1〉36=∑i=1nPin|ϵi〉,
with P1n=Y1nU-1W and Pin=YinU-1Qi for i>1. The CARMA
process at time tn is then given by
yn=〈b|U|w̃n〉37=∑i=1n〈b|U|Pin|ϵi〉.
Finally, the |Noise〉 term in Eq. () is
|Noise〉=|y〉=〈b|U|P11〈0|……〈0|〈b|U|P12〈b|U|P22〈0|…〈0|⋱⋱〈b|U|P1N〈b|U|P2N……〈b|U|PNN|ϵ1〉|ϵ2〉⋮|ϵN〉38=K|Z〉,
where K is a (N,N×p) real matrix and |Z〉 has a length
N×p. Matrix K is triangular if p=1, which is the particular case
treated in Sect. .
The trend
The model for the trend must be as general as possible and compatible with a
formalism based on orthogonal projections (see Sect. ). This is the reason we choose a
polynomial trend of some degree m:
|Trend〉=∑k=0mγk|tk〉,where |tk〉=[t1k,…,tNk]′,
where |tk〉 is defined componentwise, i.e. |tk〉=[t1k,…,tNk]′. Whether or not to consider the presence of a trend in the model for the
data is left to the user, given that we can always interpret a polynomial
trend of low order as a very low-frequency oscillation.
Periodogram and relativesLomb–Scargle periodogram
Consider the orthogonal projection of the data |X〉 onto the vector
space spanned by the vectors cosine and sine, defined by
|cω〉=cos(ω|t〉) and
|sω〉=sin(ω|t〉). The periodogram at the frequency
f=ω2π is defined as the squared norm of that projection:
||Psp‾{|cω〉,|sω〉}|X〉||2.
When the time series is regularly sampled with a constant time step Δt, and if we only consider the Fourier angular frequencies,
ωk=2πkNΔt (k=0, ..., N-1), the periodogram
defined above is equal to the squared modulus of the DFT of real signals.
Now, rescale |cω〉 and |sω〉 such that they are
orthonormal. This can be done by defining
|cω♯〉=cosω|t〉-βωΣi=1Ncos2ωti-βω,41|sω♯〉=sinω|t〉-βωΣi=1Nsin2ωti-βω,
where βω is the solution of
tan2βω=Σi=1Nsin2ωtiΣi=1Ncos2ωti.
The spanned vector space naturally remains unchanged (see Fig. ). These formulas are nothing but the Lomb–Scargle
formulas Eq. 10. The periodogram is now
||Psp‾{|cω〉,|sω〉}|X〉||2=〈cω♯|X〉2+〈sω♯|X〉2.
Note that, for any signal |X〉∈RN,
0≤||Psp‾{|cω〉,|sω〉}|X〉||2〈X|X〉≤1,
and this is equal to 1 if |X〉=A|cω〉+B|sω〉.
Some properties of the LS periodogram are presented in Appendix . Here and for the rest of the article, the
frequency f=ω/2π is considered as a continuous variable.
Periodogram and mean
The LS periodogram applies well to data which can be modelled as
|X〉=Aω|cΩ〉+Bω|sΩ〉+|Noise〉.
However, the periodic components may not necessarily oscillate around zero,
and a better model is
|X〉=μ|t0〉+Aω|cΩ〉+Bω|sΩ〉+|Noise〉,
where |t0〉=[1,1,…,1]′. Subtracting the average of the data is
then often done before applying the LS periodogram. But that mere operation
implicitly assumes that 〈t0|cΩ〉=〈t0|sΩ〉=0, which is not necessarily the case. In other words,
the data average is not necessarily equal to μ, the process mean. Figure a illustrates that fact. Note that this
discrepancy occurs in regularly sampled data as well, at non-Fourier
frequencies, i.e. when NΔt is not a multiple of the probing period.
See Fig. b.
Schematic view of the linear rescaling in RN leading to
the Lomb–Scargle formulas. In yellow is drawn a subset of
sp‾{|cω〉,|sω〉}. A span is
invariant under linear combinations of its vectors. The dashed line
corresponds to the minimal Euclidean distance between the data |X〉
and
sp‾{|cω〉,|sω〉}.
Signal average and sampling. (a) The continuous signal is
in dashed blue and it is irregularly sampled at red dots. The continuous
signal oscillates around 1 (blue line), which does not correspond to the
average of the sampled signal (red line). (b) Same as
panel (a) with a regularly sampled
signal.
In order to deal with the mean in a suitable way, we define the periodogram
as
||(Psp‾{|t0〉,|cω〉,|sω〉}-Psp‾{|t0〉})|X〉||2.
Formula () is taken from ,
or ; equivalence between them is
shown in Appendix .
[Psp‾{|t0〉,|cω〉,|sω〉}-Psp‾{|t0〉}]
is also an orthogonal projection. A simple example will justify the
principle. Consider the following purely deterministic mono-periodic signal
with N data points:
|Y〉=μ|t0〉+A|cω〉+B|sω〉=V3|Φ〉,
with
V3=||||t0〉|cω〉|sω〉|||,
and
|Φ〉=μAB.
The projection at ω is
Psp‾{|t0〉,|cω〉,|sω〉}-Psp‾{|t0〉}|Y〉=I-Psp‾{|t0〉}Psp‾{|t0〉,|cω〉,|sω〉}|Y〉=I-Psp‾{|t0〉}V3|Φ〉=|Y〉-Psp‾{|t0〉}|Y〉51=A|cω〉+B|sω〉-〈t0|cω〉〈t0|t0〉A|t0〉-〈t0|sω〉〈t0|t0〉B|t0〉.
We see that it is invariant with respect to μ, and we find back the
signal minus its average. We thus have
||(Psp‾{|t0〉,|cω〉,|sω〉}-Psp‾{|t0〉})|Y〉||2=NVar(|Y〉),
where
Var(|Y〉)=∑i=1NYi2/N-∑i=1NYi2/N2.
This is a result similar to what we get with regularly sampled data and the
DFT
If we have |Y〉=μ|t0〉+A|eω〉, where
|eω〉=exp(i2πω|t〉) and ω is a Fourier
frequency, then
||DFTω(|Y〉)||2=||Psp‾{|eω〉}|Y〉||2=N||A||2=NVar(|Y〉).
Var is here the biased variance, which is defined as the squared norm of the
signal minus its average value, and divided by N.
.
Now, we do a Gram–Schmidt orthonormalisation like in
in order to simplify Formula (). To this end, we
define the three orthonormal vectors
|h0〉=|t0〉/|||t0〉||, |h1〉 and |h2〉
satisfying
sp‾{|t0〉,|cω〉,|sω〉}=sp‾{|h0〉,|h1〉,|h2〉}.
Consequently,
Psp‾{|t0〉,|cω〉,|sω〉}-Psp‾{|t0〉}=|h1〉〈h1|+|h2〉〈h2|,
and
||Psp‾{|t0〉,|cω〉,|sω〉}-Psp‾{|t0〉}|X〉||255=〈h1|X〉2+〈h2|X〉2.
Note that, for any signal |X〉∈RN, we have
≤||Psp‾{|t0〉,|cω〉,|sω〉}-Psp‾{|t0〉}|X〉||2NVar(|X〉)≤1,
and this is equal to 1 for a signal given by
|X〉=μ|t0〉+A|cω〉+B|sω〉.
Periodogram and a polynomial trend
If we want to work with the full model, Eq. (), which has a polynomial
trend of degree m, we can naturally extend the result of Sect. and work with
||Psp‾{|t0〉,|t1〉,…,|tm〉,|cω〉,|sω〉}-Psp‾{|t0〉,|t1〉,…,|tm〉}|X〉||257=〈hm+1|X〉2+〈hm+2|X〉2,
where |hm+1〉 and |hm+2〉 are determined from a
Gram–Schmidt orthonormalisation starting with the orthonormalisation of
|t0〉, ..., |tm〉.
It may happen that, for large m, the correlation matrix in the formula of
orthogonal projection is singular. In that case, two options, less optimal,
are possible: reduce the degree m, or perform the detrending before the
spectral analysis, for example with a moving average.
Similarly to Sect. , we have, for any signal
|X〉∈RN,
0≤||Psp‾{|t0〉,|t1〉,…,|tm〉,|cω〉,|sω〉}-Psp‾{|t0〉,|t1〉,…,|tm〉}|X〉||2|||X〉-Psp‾{|t0〉,…,|tm〉}|X〉||2≤1,
and this is equal to 1 for a signal given by
|X〉=∑k=0mγk|tk〉+A|cω〉+B|sω〉.
Finally, we have a result similar to Eq. (), in the sense that the
projection given in Eq. () is
invariant with respect to the parameters of the trend (but it naturally
depends on the choice of the degree m).
Tapering the periodogram
A finite-length signal can be seen as an infinite-length signal multiplied by
a rectangular window. This implies, among other things, that a mono-periodic signal
will have a periodogram characterised by a peak of finite width, possibly with
large side lobes, instead of a Dirac delta function. This is called
spectral leakage.
The phenomenon has been deeply studied in the case of regularly sampled data.
Leakage may be controlled by choosing alternatives to the default rectangular
window. This is called windowing or taperingseefor an extensive list of windows. They all share the
property of vanishing at the borders of the time series.
In the case of irregularly sampled data, building windows for controlling the
leakage is a much more challenging task. Even with the default rectangular
window, leakage is very irregular and is data and frequency dependent, due to the
long-range correlations in frequency between the vectors on which we do the
projection. To our knowledge, no general and stable solution for that issue
is available in the literature. We thus recommend using the default
rectangular window, i.e. do no tapering, if rt, defined in Eq. (), is small, and use simple windows, like the sin2 or the
Gaussian window, for moderately irregularly sampled data (rt greater than
80 or 90 %). With tapering, Formula () becomes
||Psp‾{|t0〉,|t1〉,…,|tm〉,|Gcω〉,|Gsω〉}-Psp‾{|t0〉,|t1〉,…,|tm〉}|X〉||2,
where G is a frequency-independent diagonal matrix, which is used to weight
the sine and cosine vectors. For example, with a sin2 window, also called
Hanning window, we have
Gkk=sin2π(tk-t1)tN-t1∀k∈{1,…,N}.
Smoothing the periodogram with the WOSA methodThe consistency problem
Besides spectral leakage, another issue with the periodogram is consistency.
Indeed, for regularly sampled time series, the periodogram is known not to be
a consistent estimator of the true spectrum as the number of data points
tends to infinity seechap. 10. Another view of the
problem is that the periodogram remains very noisy regardless of the number of
data points we have at our disposal. Smoothing procedures are therefore
applied to reduce the variance of the periodogram. The drawback of any
smoothing procedure is naturally a decrease of the frequency resolution.
Among the smoothing methods available in the literature, two are
traditionally used: multitaper methods (MTMs), developed by
and , and the Welch overlapping segment
averaging (WOSA) method . See for a
unified view.
Multitaper methods are certainly not generalisable to the case of irregularly
sampled data, except in very specific cases that are not of interest in
geophysics, like in , which deals with band-limited
signals, useful in the field of the telecommunications, or ,
which considers regularly sampled time series with some gaps, useful for time
series with a ratio rt, defined in Eq. (), close to 100. We will
then use the WOSA method applied to the LS periodogram, like in
and , or to its relatives (Formulas , , or
the most general ).
Principle of the WOSA methodTrendless time series
The time series is divided into overlapping segments. The tapered LS
periodogram is computed on every segment, and the WOSA periodogram is the
average of all these tapered periodograms. This technique relies on the fact
that the signal is stationary, as always in spectral
analysis
Basically, the spectrum cannot be defined without
that hypothesis. See the Wiener–Khinchin theorem, e.g. in chap. 4
. The length of the segments and the overlapping factor
need to be chosen depending on how much we want to reduce the variance of the
noise. As a general rule, shortening the segments will decrease the frequency
resolution. Consequently, there is always a trade-off between the frequency
resolution and the variance reduction.
For regularly sampled data, each segment of fixed length has the same number
of data points. In the irregularly sampled case, it is not the case any more
and we have two options.
Take segments with a fixed number of points and thus a variable length. In the non-tapered case, the periodogram
on each segment provides deterministic peaks (coming from the deterministic sine–cosine components) with more or less the
same height. But variable length segments will give deterministic peaks of variable width.
Take segments of fixed length but with a variable number of data points. The periodogram on each segment provides
deterministic peaks with more or less the same width, except if there is a big gap at the beginning or at the end of the
segment, such that its effective length is reduced. But they will have variable height since the number of data points is not constant.
We judge it is better to have peaks with similar width on each segment when
averaging the periodograms in a frequency band. Consequently, we recommend
the second option. An example of WOSA segmentation is shown in Fig. a.
Time series with a trend
The only difference with the previous case is that, for each segment, we
consider the projection on |t0〉, ..., |tm〉 jointly with the
tapered cosine and sine components. Formula () is
applied to each segment with |Gcω〉 and |Gsω〉
localised on the WOSA segment, but |t0〉, ..., |tm〉 are
taken on the full length of the time series, because the trend is the one of
the whole time series.
The WOSA periodogram in formulas
Two parameters are required: the length of WOSA segments, D, and the
overlapping factor, β∈[0,1[; β=0 when there is no overlap. We
denote by Q the number of WOSA segments, which is equal to
Q=tN-t1-D(1-β)D+1,
where ⌊⌋ is the floor function. Because of the rounding, D
must be adjusted afterwards:
D=tN-t11+(1-β)(Q-1).
Define τq to be the starting time of the qth segment
(q∈{1,…,Q}). Note that τq is not necessarily equal to one of
the components of |t〉. It follows that
τq=t1+(1-β)(q-1)D,q=1,…,Q.
The WOSA periodogram is then
||PWOSA(ω)|X〉||2=1Q∑q=1Q||Psp‾{|t0〉,|t1〉,…,|tm〉,|Gqcω,q〉,|Gqsω,q〉}-Psp‾{|t0〉,|t1〉,…,|tm〉}|X〉||264=1Q∑q=1Q〈X|Psp‾{|t0〉,|t1〉,…,|tm〉,|Gqcω,q〉,|Gqsω,q〉}-Psp‾{|t0〉,|t1〉,…,|tm〉}|X〉.
Note that the sum of these orthogonal projections is no longer an
orthogonal projection. |Gqcω,q〉 and
|Gqsω,q〉 are the tapered cosine and sine on the
qth segment. For example, with the Hanning (sin2) window,
|Gqcω,q〉k=gq(tk)cosω(tk-τq),65|Gqsω,q〉k=gq(tk)sinω(tk-τq),
where
gq(tk)=sin2π(tk-τq)Dif 0≤tk-τq≤D,0otherwise.
It may be shown that sp‾{|t0〉,|t1〉,…,|tm〉,|Gqcω,q〉,|Gqsω,q〉} is
invariant with the variable τq appearing in the cosine and sine terms,
so that we can impose τq=0∀q inside the cosine and sine terms.
In Formula (), for each orthogonal projection, we apply
a Gram–Schmidt orthonormalisation (similarly to Sect. ):
||PWOSA(ω)|X〉||2=1Q∑q=1Q〈X|h1,q(ω)〉〈h1,q(ω)|X〉67+〈X|h2,q(ω)〉〈h2,q(ω)|X〉,
where, for each q, |h1,q(ω)〉 and |h2,q(ω)〉
are orthonormal. We are now able to write the WOSA periodogram under a simple
matrix form:
||PWOSA(ω)|X〉||2=〈X|MωMω′|X〉,
where
Mω=1Q|||||h1,1(ω)〉|h2,1(ω)〉…|h1,Q(ω)〉|h2,Q(ω)〉||||.
Practical considerations
First, note that the Gram–Schmidt orthonormalisation process requires at
least m+3 data points. WOSA segments with less than m+3 points must
therefore be ignored in the average of the periodograms.
Second, as we want to get deterministic peaks with more or less the same
width on every segment, a WOSA segment is kept in the average if the data
cover some percentage of its length D, namely,
qth segment kept if: 100tq,2-tq,1D≥C,
where tq,1 and tq,2 are the times of the first and last data points
inside in the qth segment, and C is the coverage factor. Its default
value in WAVEPAL is 90 %.
Third, the frequency range on the qth segment is bounded by these two
frequencies:
fmin=1tq,2-tq,1andfmax=12Δt‾q.
The maximal period (1/fmin) corresponds to the effective length
on the segment. The maximal frequency in the case of regularly sampled data
must be the Nyquist frequency, fmax=1/2Δt. For
irregularly sampled data, different choices for Δt‾q are
possible. As suggested in Appendix , an option
is Δt‾q=ΔtGCD,q, but this choice is
insufficient to avoid pseudo-aliasing issues. Imagine for example a
regularly sampled time series with 1000 data points and Δt=1. Add
one point at the end with the last time step being 0.1. The resulting
irregularly sampled time series will thus have ΔtGCD=0.1.
If we take fmax=5, it is obvious that some kind of aliasing
will occur between f=0.5 and fmax. This it what we call
pseudo-aliasing. A much better choice in this case is of course
fmax=0.5. Section 5 of provides
further discussions on this topic.
In practice,
Δt‾q=max∑k=1NGqk,kΔtcktr(Gq),∑k=1N-1Hqk,kΔtktr(Hq),
where
Δtk=tk+1-tk∀k∈{1,…N-1},Δtck=tk+1-tk-12∀k∈{2,…N-1},Δtc1=t2-t1,73ΔtcN=tN-tN-1,
and Hq is a diagonal matrix with
Hqk,k=taper at time tk+tk+12,k∈{1,…,N-1}
appears to work well. More justification and an example are provided in
Part 2 of this study Sect. 3.8, where it is shown that
such a formula can handle aliasing issues in the case of time series with
large gaps. Matrix Hq is similar to matrix Gq,
defined in Sect. , but with elements taken at
(tk+tk+1)/2 instead of tk. Quantity Δt‾q is
equal to the maximum between the average time step and the average central
time step if there is no tapering (Gq=Hq=I)
and is equal to Δt in the regularly sampled case. These restrictions
on the frequency bounds imply that the total number of WOSA segments, Q, in
Formula (), is not the same for all the frequencies.
This is illustrated in Fig. b.
Fourth, in order to have a reliable average of the periodograms, we only
represent the periodogram at the frequencies for which the number of WOSA
segments is above some threshold. In WAVEPAL, default value for the threshold
at frequency f is
threshold: min{10,max{f}Q(f)},
where Q(f) is the number of WOSA segments at frequency f. It means that
frequency f belongs to the range of frequencies of the WOSA periodogram if
Q(f) is greater than or equal to the threshold.
Significance testing with the periodogramHypothesis testing
Significance testing allows us to test for the presence of periodic
components in the signal. It is mathematically expressed as a hypothesis
testing seechap. 10. Taking our model, Eq. (), the null hypothesis is
H0:Aω=Bω=0.
Therefore, |X〉=|Trend〉+|Noise〉. The
alternative hypothesis is
H1:Aω and Bω are not both zero.
The decision of accepting or rejecting the null hypothesis is based on the
periodogram evaluated at ω, whose general formula is given in Eq. (). The test is performed independently for each
frequency (pointwise testing). Concretely, for each frequency, we
compute the distribution of the periodogram under the null hypothesis, and
then see if the data periodogram at that frequency is above or below
a given percentile (e.g. the 95th) of that distribution. The
percentile is called level of confidence. If the data periodogram is
above the Xth percentile of the reference distribution, we
reject the null hypothesis with X % of confidence. The level of significance is equal to (100-X) %, e.g. a 95 % confidence level is
equivalent to a 5 % significance level. Hypothesis testing is, for this
reason, often called significance testing. See Fig. c and d for an
illustration on paleoclimate data. We recommend chap. 6 for more details on the methodology.
To perform significance testing, we thus need
to estimate the parameters of the process under the null hypothesis (this is studied in
Sect. );
to estimate the distribution of the periodogram under the null hypothesis (this is studied in Sect. ).
Estimation of the parameters under the null hypothesisIntroduction
Under the null hypothesis, the signal is
|X〉=|Trend〉+|Noise〉, and we thus need
to estimate the parameters of the trend and those of the zero-mean CARMA
process. The best statistical approach is to estimate them jointly, and
marginalise over the parameters of the trend, since the periodogram is
invariant with respect to these parameters, according to Sect. . But this would imply very involved
computations that are way beyond the scope of this work. We are thus forced
to a compromise and proceed as follows: data are detrended,
|Xdet〉=|X〉-Psp‾{|t0〉,|t1〉,…,|tm〉}|X〉, and then we estimate the parameters of the
CARMA process, based on the model μ|t0〉+|Noise〉,
where |Noise〉 is a zero-mean stationary Gaussian CARMA
process sampled at the times of |t〉.
Estimation of CARMA parameters is done in a Bayesian framework. We analyse
separately the case of the white noise, which is done analytically, and the
case of CARMA(p,q) processes with p≥1, for which Markov chain
Monte Carlo (MCMC) methods are required. Bayesian analysis provides a
posterior distribution of the parameters based on priors.
Gaussian white noise
We want to estimate the two parameters of the white noise, the mean μ and
the variance σ2. According to the Bayes theorem,
Π(μ,σ2|D)=ΠD|μ,σ2Πμ,σ2Π(D)78∼ΠD|μ,σ2Πμ,σ2,
where Π is the probability density function (PDF) and D is the detrended
data Xdet,1,…,Xdet,N. Based on the PDF of a
multivariate white noise, the likelihood function is
Π(D|μ,σ2)79=12πσ2Nexp-∑i=1NXdet,i-μ22σ2.
We take Jeffreys priors for μ and
σ2:
Π(μ,σ2)=Π(μ)Π(σ2),80with Π(μ)∼1 and Π(σ2)∼1σ2.
Jeffreys priors are non-informative and invariant under reparametrisation. Note that Π(σ2) is log-uniform.
Since we do not actually need to estimate μ (see Sect. and Formula ), we marginalise over that variable,
Π(σ2|D)=∫-∞+∞dμΠ(μ,σ2|D)∼1σ2∫-∞+∞dμΠ(D|μ,σ2)∼1σ212πσ2Nexp-∑i=1NXdet,i22σ2∫-∞+∞dμexp(-(aμ2+2bμ))81∼1σ212πσ2Nexp-∑i=1NXdet,i22σ2πaexpb2a,
with a=N/2σ2 and b=-∑i=1NXdet,i/2σ2.
Rearranging terms gives
Π(σ2|D)∼1σ2N+12exp-1βσ2,
with β=2/Nσ^2, where σ^2= is the (biased)
variance of the detrended
data
σ^2=1N∑i=1NXdet,i2-1N∑i=1NXdet,i2
.
With the variable change y=1/σ2, we have
Π(y|D)∼yN-32exp(-y/β),
which is nothing but a gamma distribution:
1σ2=dγN-12,2Nσ^2.
Note that the mean of the distribution in Eq. () is equal to
(N-1)/(Nσ^2), which is the usual unbiased estimator of
1/σ2. Finally, the PDF of σ2 is at its maximum at
σmax2=NN+1σ^2.
This is obtained from the derivative of Eq. ().
Gaussian CARMA(p,q) noise with p≥1
For other cases than the white noise, provide robust
algorithms to estimate the posterior distribution of the CARMA parameters and
of the parameter μ of an irregularly sampled, purely stochastic, time
series, which can be modelled as a CARMA process. Those algorithms are based
on Bayesian inference and MCMC methods. In particular we recommend reading
Sects. 3.3 and 3.6 of for a discussion on the choice of the
priors and for computational considerations, respectively. That paper is
accompanied by a Python and C++ package called CARMA pack. Some
outputs of CARMA pack are shown in Sect. .
Estimation of the distribution of the periodogram under the null hypothesisWorking with a trendless stochastic process
Under the null hypothesis, the signal is
|X〉=|Trend〉+|Noise〉=∑k=0mγk|tk〉+|Noise〉.
The WOSA periodogram, Eq. (), is invariant with
respect to the parameters of the trend, so that we can pose γk=0 for all k
and |X〉 reduces to a zero-mean CARMA process.
Monte Carlo approach
For each frequency, we need the distribution of the WOSA periodogram, Eq. (), where |X〉 is now a CARMA process for
which we know the distribution of its parameters, from Sect. . With
Monte Carlo methods, we are thus able to estimate any percentile of the
distribution of the periodogram. If |X〉 is a zero-mean white noise,
|X〉 is sampled from a standard normal distribution multiplied by the
square root of the variance, whose inverse is sampled from the gamma
distribution (Eq. ). If |X〉
is a CARMA(p,q) process with p≥1, |X〉 is generated with the
Kalman filter (from the CARMA pack – see Sect. ). An example of
confidence levels is shown in Fig. d.
We are thus able to estimate confidence levels for the WOSA periodogram,
taking into account the uncertainty in the parameters of the background
noise.
Analytical approach
If we consider constant CARMA parameters, we show in this section that
analytical confidence levels can be computed, even in the very tail of the
distribution of the periodogram of the background noise. An example is given
in Fig. c. The advantage of the analytical approach
is twofold.
It provides confidence levels converging to the exact solution, as the number of conserved moments increases (see below). From a certain number of conserved
moments, we can consider that convergence is numerically reached (see Fig. ). Such an approach is particularly interesting
for high confidence levels, as illustrated in Fig. c with the 99.9 % confidence level,
for which a MCMC approach would require a huge number of samples to get a satisfactory accuracy.
As a consequence, for a given percentile, computing time is usually shorter with the analytical method than with the MCMC method. We note, however, that the
MCMC approach generally needs less computing time when the number of data points becomes large, as shown in Appendix .
First approximation
If the marginal posterior distribution of each CARMA parameter is unimodal,
we take the parameter value at the maximum of its PDF (white noise case, see
Eq. ), or the median parameter
For CARMA
processes with p>0 and q≥0, the marginal posterior distribution is
obtained by MCMC methods, and determining the maximum of the PDF thus
requires some post-processing, such as smoothing the distribution. A simple
alternative is to take the median.
(other cases). Note that
multi-modality
tends to appear more frequently for CARMA processes of high order. Working
with a unique set of parameters allows us to find an analytical formula for
the distribution of the WOSA periodogram. Considering the matrix forms of the
CARMA noise (Eq. or )
and the WOSA periodogram (Eq. ), we demonstrate the
following theorem.
The WOSA periodogram, defined in Eq. (), under the
null hypothesis (), is||PWOSA(ω)|X〉||2=d∑k=12Q(ω)λk(ω)χ1k2,where |X〉=∑k=0mγk|tk〉+K|Z〉, K is the
CARMA matrix defined in Eq. () or
(), and Q(ω) is the number of WOSA
segments at ω.
χ112, ..., χ12Q(ω)2 are iid chi-square
distributions with 1 degree of freedom, and λ1(ω), ...,
λ2Q(ω)(ω) are the eigenvalues of
Mω′KK′Mω and are non-negative. Matrix Mω is
defined in Eq. ().
Proof. Since the WOSA periodogram, Eq. (), is invariant with
respect to the parameters of the trend, we pose them as equal to zero and
consider the zero-mean CARMA process
|X〉=K|Z〉.
The periodogram is thus
||PWOSA(ω)|X〉||2=〈Z|K′MωMω′K|Z〉=〈γ|γ〉,
with |γ〉=Mω′K|Z〉. Since |Z〉 is a standard
multivariate normal distribution, we have
|γ〉=dN(0,Mω′KK′Mω).Mω′KK′Mω is a (2Q(ω),2Q(ω)) real symmetric
positive semi-definite matrix. We can thus diagonalise it:
∃ an orthogonal matrix U s.t. U′Mω′KK′MωU=D,
with D being a diagonal matrix with the 2Q(ω) non-negative
eigenvalues of Mω′KK′Mω. We now have
U′|γ〉=dN(0,D),
and
||PWOSA(ω)|X〉||2=〈γ|γ〉=〈γ|UU′|γ〉91=〈Z|DD|Z〉=d∑k=12Q(ω)λk(ω)χ1k2,
where the χ1k2 distributions are iid.
The pseudo-spectrum is defined as the expected value of the
periodogram distribution:
S^(ω)=∑k=12Q(ω)λk(ω)=trMω′KK′Mω.
The difference between the pseudo-spectrum and the traditional
spectrum is explained in Appendix .
If the background noise is white, we have K=σI and this
implies that
tr(Mω′KK′Mω)=tr(Mω′Mω)σ2=tr(MωMω′)σ2=2σ2,
such that the pseudo-spectrum is
S^(ω)=2σ2,
and is thus flat. This is a well-known result of the LS periodogram
, generalised here to more evolved periodograms. Moreover,
if there is no WOSA segmentation (Q(ω)=1∀ω), the
periodogram is exactly chi-square distributed with 2 degrees of freedom:
||Psp‾{|t0〉,|t1〉,…,|tm〉,|Gcω〉,|Gsω〉}-Psp‾{|t0〉,|t1〉,…,|tm〉}σ|Z〉||294=dσ2χ112+σ2χ122=dσ2χ22,
which is also a generalisation of a well-known result of the LS periodogram .
The variance of the distribution of the periodogram, Eq. (), is equal to
2∑k=12Q(ω)λk2(ω)=2||Mω′KK′Mω||F2,
where ||⋅||F is the Frobenius norm. As expected, it decreases with
Q, as illustrated in Fig. .
Going back to Eq. (), it is well-known that a linear
combination of (independent) χ2 distributions is not analytically
solvable. Fortunately, excellent approximations are available in
, allowing Monte Carlo methods to be avoided.
Second approximation
We approximate the linear combination of independent chi-square
distributions, conserving its first d moments. When d→∞, the
approximation converges to the exact distribution. In practice, estimation of
a percentile is already very good with a very few moments, as illustrated in
Fig. . Let us proceed step by step by
increasing the number of conserved moments. Define
X=∑k=12Q(ω)λk(ω)χ1k2.
Analytical variance of the WOSA periodogram for a Gaussian red noise
with σ=2 and α=1/20 (see Sect. for the
definition of a red noise) for different values of Q. The frequency range
is chosen such that, for each curve, Q(ω) is constant all along. The
red noise is built on the irregularly sampled times of ODP1148 core (see
Sect. ).
1-moment approximation
We require the expected
value of the process to be conserved, which is satisfied with the following
approximation:
X≈d12Q(ω)∑k=12Q(ω)λk(ω)χ2Q(ω)2,
or, equivalently,
X≈d12Q(ω)S^(ω)χ2Q(ω)2.
2-moment approximation
The approximate distribution of the linear combination of the chi-square
distributions must have two parameters, and we conserve the expected value
and variance. A chi-square distribution with M degrees of freedom provides
a good fit:
X≈dgχM2.
Equating the expected values and variances gives
M=tr(A)2||A||F2 and g=||A||F2tr(A),
where A=Mω′KK′Mω and ||A||F2 is the squared Frobenius
norm of matrix A, i.e. the sum of its squared eigenvalues. Note that
gχM2=dγM/2,2g, where 2g is the scale
parameter of the gamma distribution, which motivates the following d-moment
approximation.
The d-moment approximation
We apply here the formulas presented in . Let fX be the
PDF of X. This distribution is approximated by the PDF of a
dth degree gamma-polynomial distribution:
fX(x)≈γα,β(x)∑i=0dξixi,x≥0,
where the parameters α and β are estimated with the 2-moment
approximation detailed above, and ξ0, ..., ξd are the solution of
ξ0ξ1⋮ξd=η(0)η(1)…η(d-1)η(d)η(1)η(2)…η(d)η(d+1)⋮⋮⋮⋮⋮η(d)η(d+1)…η(2d-1)η(2d)-11μ(1)⋮μ(d).
Here, μ(1), ..., μ(d) are the exact first d moments of X and can be
computed analytically by recurrence see Eq. 5 of, and
η(h) is the hth moment of the gamma distribution,
η(h)=βhΓ(α+h)/Γ(α). The approximate cumulative
distribution function (CDF) of X, evaluated at c0, is then
FX(c0)≈1Γ(α)∑i=0dξiβiγi+α,c0/β,c0>0,
where γ(s,x) is the lower incomplete gamma function:
γ(s,x)=∫0xdtts-1exp(-t).
After all that chain of calculus, we reached our objective, that is, the
estimation of a confidence level for the WOSA periodogram. It is given by the
solution c0 of
1Γ(α)∑i=0dξiβiγi+α,c0/β-p=0,
for some p value p, e.g. p=0.95 for a 95 % confidence level.
The gamma-polynomial approximation can be extended to the
generalised gamma-polynomial approximation. The latter is based on
the generalised gamma distribution and is defined in
Appendix . It gives
percentiles that usually converge faster than those given by the
gamma-polynomial approximation. However, we observed
that the generalised gamma-polynomial approximation is quite sensitive to the
quality of the first guess for the three parameters of the generalised gamma
distribution (see
Appendix ). We thus
recommend the use of the gamma-polynomial approximation as a first choice.
Both options are available in WAVEPAL.
Finally, we mention that there exists an alternative expression to the above
development, in terms of Laguerre polynomials see.
It has the advantage of not requiring the matrix inversion in Eq. (), the latter possibly being singular at large values of the
degree d. However, we have not found any improvement on the stability or
computing time using that approach.
The F periodogram for the white noise background
We have shown in Eq. () that the periodogram of a
Gaussian white noise is exactly chi-square distributed if there is no WOSA
segmentation. Significance testing against a white noise requires the
estimation of the white noise variance after having detrended the data.
Knowing that a F distribution is the ratio of independent chi-square
distributions, it is possible to get rid of the detrending and variance
estimation and deal with a well-known distribution, by working with
(N-m-3)||Psp‾{|t0〉,|t1〉,…,|tm〉,|cω〉,|sω〉}-Psp‾{|t0〉,|t1〉,…,|tm〉}|X〉||22||I-Psp‾{|t0〉,|t1〉,…,|tm〉,|cω〉,|sω〉}|X〉||2.
We call it the F periodogram. We already know that the numerator is
invariant with respect to the parameters of the trend of the signal. It is
clear that the denominator is invariant with respect to the parameters of the
trend as well as with respect to the amplitudes of the periodic components
(only the |Noise〉 term remains when applying it to Eq. ). Moreover, that ratio is invariant with respect to the
variance of the signal. Last but not least, the orthogonal projections in the
numerator, [Psp‾{|t0〉,|t1〉,…,|tm〉,|cω〉,|sω〉}-Psp‾{|t0〉,|t1〉,…,|tm〉}], and in the denominator,
[I-Psp‾{|t0〉,|t1〉,…,|tm〉,|cω〉,|sω〉}], are done on
spaces that are orthogonal to each other. Consequently, if we consider the
null hypothesis () with a white noise, the numerator and
the denominator follow independent chi-square distributions, and
(N-m-3)||(Psp‾{|t0〉,|t1〉,…,|tm〉,|cω〉,|sω〉}-Psp‾{|t0〉,|t1〉,…,|tm〉})|X〉||22||(I-Psp‾{|t0〉,|t1〉,…,|tm〉,|cω〉,|sω〉})|X〉||2=d(N-m-3)χ222χN-m-32105=dF(2,N-m-3),
where
|X〉=d∑k=0mγk|tk〉+Nμ,σ2106=d|Trend〉+Nμ,σ2,
and where F(2,N-m-3) is the Fisher–Snedecor distribution with parameters
2 and N-m-3. In conclusion, the F periodogram can be an alternative to
the periodogram when performing significance testing. It has the advantage of
not requiring any parameter to be estimated and applies under the following
conditions.
The background noise is assumed to be white.
There is no WOSA segmentation.
There is no tapering.
The F periodogram is available in WAVEPAL under the above requirements.
With a WOSA segmentation, projections at the numerator and at the denominator
are not performed any more on orthogonal spaces, and this cannot therefore be
applied.
The above results are a generalisation of formulas in
and . See Appendix for
additional details.
The amplitude periodogramDefinition
Going back to Eq. (), we now look for the amplitude
Eω=Aω2+Bω2 at a given frequency
f=ω2π. The estimation of Eω2 is called the
amplitude periodogram and is denoted by E^ω2. We
estimate Aω and Bω with a least-squares approach. We
start with a trendless signal, and will show that the amplitude periodogram
and the periodogram are approximately proportional.
Trendless signalNo tapering
The estimated amplitudes we look for, A^ω and
B^ω, are the solution of
A^ω,B^ω=argmin{(A,B)∈R2}|||X〉-A|cω〉+B|sω〉||2.
Since we look for the minimal distance, the solution is given by the
orthogonal projection onto the vector space spanned by |cω〉
and |sω〉, namely
Psp‾{|cω〉,|sω〉}|X〉=A^ω|cω〉+B^ω|sω〉.
Let us develop this equation:
Vω2(Vω2′Vω2)-1Vω2′|X〉=Vω2|Φ^ω〉,
where
Vω2=|||cω〉|sω〉|| and |Φ^ω〉=A^ωB^ω,
and we find the well-known expression for the solution of a least-squares
problem:
|Φ^ω〉=Vω2′Vω2-1Vω2′|X〉.
Finally,
E^ω=|||Φ^ω〉||.
In the regularly sampled case, at the Fourier frequencies, the amplitude
periodogram is proportional to the periodogram, with a factor 2/N (or a
factor 1/N at ω=0 and ω=π/Δt; the projection being
done on the single cosine at those frequencies). It is no longer the case
with irregularly sampled time series, and the proportionality is only
approximate:
E^ω2≈2N||Psp‾{|cω〉,|sω〉}|X〉||2.
To prove the above formula, rewrite the model (Eq. ) at
Ω=ω:
114|X〉=Eωcosω|t〉+ϕω-βω+βω+|Noise〉=Aωcos(ω|t〉-βω)+Bωsinω|t〉-βω+|Noise〉,
where βω is defined in Eq. () and makes the
phase-lagged sine and cosine orthogonal. Aω and Bω no
longer have the same expressions as in Eq. (), but we still
have Eω2=Aω2+Bω2. We can rewrite Eq. () but this time with Vω2 holding the above
phase-lagged sine and cosine. We now make use of the approximation stated in
p. 449:
∑i=1Ncos2ωti-βω≈N2and115∑i=1Nsin2ωti-βω≈N2.
Note that the sum of both is exactly equal to N. Equation () is then obtained, observing that
Vω2′Vω2≈N2I. Basic
trigonometry gives the following equalities for the relative error of the
above approximations:
∑i=1Ncos2ωti-βω-N/2N/2=∑i=1Nsin2ωti-βω-N/2N/2116=∑i=1Ncos2ωti-βωN,
so that the two approximations of Eq. () reduce to only one:
∑i=1Ncos2ωti-βωN≈0.
The quality of this approximation is illustrated in Fig. .
With tapering
Like with the periodogram, leakage also appears in the amplitude periodogram.
Consequently, it may be better to work with the projection on tapered cosine
and sine if the data are not too irregularly sampled, as explained in
Sect. . Consideration of the tapered case is
also an important mathematical prerequisite for an extension to the
continuous wavelet transform. This is developed in Part 2 of this study
.
A^ω and B^ω are determined by
projecting the data onto tapered cosine and sine:
Psp‾{|Gcω〉,|Gsω〉}|X〉=A^ω|cω〉+B^ω|sω〉.
Developing the equation gives
|Φ^ω〉=Vω2′GVω2-1Vω2′G|X〉,
and
E^ω=|||Φ^ω〉||,
where Vω2 is defined in Sect. and
G is defined in Sect. .
Illustration of the quality of the approximations
(a) Eq. (),
(b) Eq. () and
(c) Eq. (). In blue: no tapering
(square taper); in green: sin2 taper; in red: Gaussian taper. The
approximation () is thus in blue in
panel (a) or (b). Each panel represents the left-hand side
of the equation, multiplied by 100, to express percentage. This indicates how
small the numerator is compared to the denominator. The time vector
|t〉 comes from the ODP1148 core (see Sect. ) for
which ΔtGCD=1kyr.
Note that the approach we follow does not correspond to the classical least-squares problem as above since, in Eq. (), the
cosine and sine are tapered only on the left-hand side of the equality.
However, one can reconstruct a signal from its projection coefficients with
a different function than the one which is used to determine those coefficients
seeEq. II.8, p. 15, in which the similarity to
Vω2|Φ^ω〉=Vω2(Vω2′GVω2)-1Vω2′G|X〉
is evident. Note that
Vω2(Vω2′GVω2)-1Vω2′G is a
projection, since it is idempotent, but the projection is not orthogonal,
because it is not symmetric.
Similarly to the non-tapered case, we now determine an approximate
proportionality between the amplitude periodogram and the tapered
periodogram. We start with the model (Eq. ) evaluated at
Ω=ω and written under the following form
|X〉=Aωcosω|t〉-βω+Bωsinω|t〉-βω121+|Noise〉,
where βω is introduced such that 〈Gcω|Gsω〉=0, or equivalently, such that
Vω2′G2Vω2 is diagonal. A little development gives
the formula for determining βω:
tan(2βω)=∑i=1NGii2sin2ωti∑i=1NGii2cos2ωti,
which is a generalisation of Eq. (). We now make use of the
following approximations:
123a∑i=1NGiicos2(ωti-βω)tr(G)≈0,123b∑i=1NGii2cos2(ωti-βω)tr(G2)≈0,
which are similar to the approximation made in Eq. ().
That implies, with no extra approximation, the following formulas:
∑i=1NGiicos2ωti-βω≈tr(G)2,124∑i=1NGiisin2ωti-βω≈tr(G)2,
and
∑i=1NGii2cos2ωti-βω≈tr(G2)2,125∑i=1NGii2sin2ωti-βω≈tr(G2)2.
Note that in Eqs. () and (), the sum of the two
members is conserved and we find back Eq. () when
G=I. Moreover, we approximate the following sum:
∑i=1NGiicosωti-βωsinωti-βωtr(G)/2≈0,
so that Vω2′GVω2 is diagonal. The quality of these
approximations is illustrated in Fig. .
Putting all of this together gives
Vω2′GVω2≈tr(G)2I,and127Vω2′G2Vω2≈tr(G2)2I,
from which we deduce
E^ω2≈2tr(G2)tr(G)2||Psp‾{|Gcω〉,|Gsω〉}|X〉||2.
Finally, we mention that the above relation is also approximate in the
case of regularly sampled time series.
Signal with a trend
We now work with the full model (Eq. ) including the trend. Our aim
is again to find the amplitude Eω, or, equivalently Aω
and Bω. We proceed in the same way as in Sect. :
Psp‾{|t0〉,|t1〉,…,|tm〉,|Gcω〉,|Gsω〉}|X〉129=∑k=0mγk^|tk〉+A^ω|cω〉+B^ω|sω〉=Vωm+3|Φ^ω〉,
where
Vωm+3=|||||t0〉…|tm〉|cω〉|sω〉||||,
and
|Φ^ω〉=γ0^⋮γm^A^ωB^ω.
We can write Psp‾{|t0〉,|t1〉,…,|tm〉,|Gcω〉,|Gsω〉}=Wωm+3(Wωm+3′Wωm+3)-1Wωm+3′,
where Wωm+3 is identical to Vωm+3 except in the
last two columns, where the cosine and sine are tapered by G. We thus
obtain
|Φ^ω〉=Wωm+3′Vωm+3-1Wωm+3′|X〉,
and
E^ω2=A^ω2+B^ω2=Φ^ω(m+2)2+Φ^ω(m+3)2,
where Φ^ω(m+2) and Φ^ω(m+3) are
the two last components of vector |Φ^ω〉.
With WOSA
The signal being stationary, we can estimate the amplitude on overlapping
segments and take the average. That gives a better estimation, more robust
against the background noise, but it has the disadvantage of widening the
peaks and thus reducing the resolution in frequency. We simply take Eq. (), apply it to each segment
We
remind the reader
that the vectors |tk〉 associated to the trend are taken on the whole
time series. Only the (tapered) cosine and sine are taken on the WOSA
segment.
, and compute the average. We have
E^ω2=1Q(ω)∑q=1Q(ω)Φ^q,ω(m+2)2+Φ^q,ω(m+3)2.
Amplitude periodogram or periodogram?
So far, we have studied in detail the periodogram and its confidence levels
as well as the estimated amplitude. Of course, confidence levels can also be
determined for the amplitude, with Monte Carlo simulations, or with an
analytical approximation similar to Sect. .
In the regularly sampled case, at Fourier frequencies, the cosine and sine
vectors are orthogonal, so that, in the non-tapered case and with a constant
trend, there is no difference between the periodogram and the amplitude
periodogram, up to a multiplicative constant. Even with WOSA segmentation,
the number of data points being identical on each segment, that
multiplicative constant remains invariant.
In the irregularly sampled case, choosing one or the other depends on what
one wants to conserve. On the one hand, the periodogram conserves the flatness of the white
noise pseudo-spectrum (see Eq. ) and can
therefore be of interest to study the background noise of the time series.
On the other hand, the amplitude
periodogram gives direct access to the estimated signal amplitude. Another
criteria to take into account is the computing time. Indeed, the amplitude
periodogram requires matrix inversions (or, equivalently, resolution of
linear systems) and is then slower to compute, while the periodogram allows orthogonal projections to be dealt with and is computationally more
efficient. Finally, we mention that, with a trendless signal, the difference
between both is rather explicit (see Eq. ):
Periodogram: ||A^ω|cω〉+B^ω|sω〉||2
versus
Amplitude periodogram: A^ω2+B^ω2.
This is variance (multiplied by the number of data points) versus squared
amplitude. A compromise between the amplitude periodogram and the periodogram
is the weighted periodogram, which is defined in the next section.
The weighted WOSA periodogram
Taking into account the approximate linearity between the amplitude
periodogram and the tapered periodogram, Eq. (), a possibility is to perform the
frequency analysis with a weighted version of the WOSA periodogram. On each
WOSA segment, the periodogram is weighted by
wq=2tr(Gq2)/tr(Gq)2,q=1,…,Q(ω). The
advantage of the weighted WOSA periodogram is to provide deterministic peaks
(coming from Aω|cω〉+Bω|sω〉) of
more or less equal power on all the WOSA segments, thus alleviating the issue
stated in Sect. . The disadvantage is that the
pseudo-spectrum of a white noise is not flat any more (Eq. is not valid any more, except when Q=1).
Working with the weighted version is done by modifying matrix Mω,
Eq. (), which is now
137Mω=1Q(ω)||||w1|h1,1(ω)〉w1|h2,1(ω)〉…wQ(ω)|h1,Q(ω)(ω)〉wQ(ω)|h2,Q(ω)(ω)〉||||.
Note that the weights wq are the same on each segment when the time series
is regularly sampled, so that the whole WOSA periodogram is, in that case,
just multiplied by a constant, and the pseudo-spectrum of a white noise is
flat. We observed that the weighted periodogram is often very close to the
amplitude periodogram, like in the example presented in Fig. . We thus recommend the use of the weighted
WOSA periodogram in most analyses.
When filtering is to be performed, the
amplitude periodogram must be computed as well. This is the topic of the next
section.
Filtering
We want to reconstruct the deterministic periodic part,
A^ω|cω〉+B^ω|sω〉
of our model (Eq. ) evaluated at Ω=ω. From Eq. (), we can extract
A^ω=Φ^ω(m+2) and
B^ω=Φ^ω(m+3), and reconstruction at a
single frequency is therefore direct. Reconstruction on a frequency range can
be done by summing
A^ω|cω〉+B^ω|sω〉
over ω.
Note that, in theory, reconstruction could be done segment by segment, using
the WOSA method. But, in practice, we observe that it does not give good
results with stationary signals. Of course, if the signal is not stationary,
reconstruction segment by segment is a clever choice, but, with such signals,
it is better to use more appropriate tools such as the wavelet transform. See
the second part of this study , in which some examples of
filtering are given.
Application on palaeoceanographic data
The time series we use to illustrate the theoretical results is the benthic
foraminiferal δ18O record from that holds 608
data points with distinct ages and covers the last 6 million years. An
example of frequency analysis is described below.
Preliminary analysis
We first look at the sampling; ΔtGCD=1 kyr, and
rt=10.13 %. Following the recommendation of Sect. , we therefore use the default rectangular
window taper. The sampling and its distribution are drawn in Fig. . We then choose the degree of the polynomial trend to be
m=7; see Fig. . This choice for m is justified by a
sensitivity analysis performed in Sect. . We
remind the reader that the time series is not detrended before estimating the spectral
power of the data, but it is detrended before estimating the confidence
levels.
CARMA(p,q) background noise analysis
We choose the order of the background noise CARMA process. We opt for the
traditional red noise background , p=1 and q=0.
Note that we observe similar confidence levels with other choices (see the
sensitivity analysis in Sect. ). We then
estimate the parameters of the stationary CARMA process (here, a red noise)
on the detrended data. This is done with the algorithm provided by
(see Sect. ). Quality of the fit is
analysed in Fig. a, c and e. Figure a analyses the residuals. If the detrended data are a
Gaussian red noise, the residuals must be distributed as a Gaussian white
noise. We see that the distribution is indeed close to a Gaussian. Figure c shows the autocorrelation function (ACF) of the residuals.
If the residuals are a Gaussian white noise sequence, they must be
uncorrelated at any lag. We can therefore arrange the residuals on a regular
grid with a unit step and then take the classical ACF, which can only be
applied to regularly sampled data. Figure c is consistent with
the assumption that the residuals are uncorrelated. Figure e
shows the ACF of the squared residuals. If the residuals are a Gaussian white
noise sequence, the squared residuals are a white noise sequence (which is
not Gaussian any more) and must therefore be uncorrelated at any lag.
Deviations from the confidence grey zone indicate that the variance is
changing with time and the signal is therefore not stationary. This is
actually what is happening with our time series. Changes in variance are
already visible on the raw time series (Fig. ). Remember that,
at this stage, we are within the world of the null hypothesis, Eq. (), and slight violation of the goodness of fit may be
due to the presence of additive periodic deterministic components, that is
the alternative hypothesis.
The age step, (tk-tk-1),∀k∈2,…,N, and its
distribution.
The time series and its 7th-degree polynomial
trend.
CARMA(1,0) background noise analysis. Panels (a),
(c) and (e) assess the fit. (a) Standardised
residuals. (c) ACF of the residuals. (e) ACF of the squared
residuals. The lag refers to an arbitrary scale on which the data are
regularly spaced with a unit step. The grey portion is the 95 %
confidence region. Panels (b), (d) and (f) show
the samples of the MCMC and the posterior marginal distributions (top panel),
jointly with the ACF of the MCMC samples (bottom panel). (b) Mean.
(d) Standard deviation of the white noise term.
Panel (f) shows log(α), where α is defined in
Sect. .
Frequency analysis. (a) The time series, in blue, and the
WOSA segments, in red. (b) Number of WOSA segments per frequency.
(c, d) Weighted WOSA periodogram and the confidence levels (CL) at
95 and 99.9 %. Analytical CL (Anal. CL) are computed with the median
parameters of the red noise process. In panel (c), the MCMC CL are
computed from the MCMC red noise time series, all generated with the median
red noise parameters. In panel (d), the MCMC CL are computed from
the MCMC red noise time series, generated with stochastic parameters, that
are taken from the joint posterior distribution of the parameters of the red
noise process.
The marginal posterior distributions of the CARMA parameters are shown in
Fig. b, d and f, jointly with
the ACF of the MCMC samples. Each distribution is unimodal, and we may
therefore use the analytical approach of Sect. to estimate the confidence levels.
Based on the ACFs of the MCMC samples of the three parameters, we skim off
the initial joint distribution of the parameters to make their samples almost
uncorrelated. In this example, we pick up 1231 samples among the 16 000 initial ones. This number of 1231 samples results from the fact that we
impose an ACF which is less than 0.2 for each marginal
distribution
As explained in Sect. , these
1231 samples are then used to compute the median parameters, producing the
analytical confidence levels of Fig. c and
d and the MCMC confidence levels of Fig. c. The MCMC confidence levels of Fig. d are computed from 50 000 samples of the
parameters, after skimming off a distribution with much more samples.
.
Frequency analysis
We compute the weighted WOSA periodogram of Sect. . The frequency range is automatically
determined from the results of Sect. . The length of the WOSA
segments depends on the required frequency resolution. Here we choose
segments of about 600 kyr and a 75 % overlapping. The WOSA
segmentation is presented in Fig. a.
The weighted WOSA periodogram and its 95 and 99.9 % confidence
levels are presented in Fig. c and
d. Both figures display the analytical confidence
levels, which are computed with the median parameters of the red noise
process (that is, the median of 1231 samples of the distributions shown in
Fig. b, d and f) and a 12-moment
gamma-polynomial approximation (Sect. ). We can check for the convergence
of the gamma-polynomial approximation, at some frequencies. This is presented
in Fig. . Figure c also shows the MCMC confidence levels, computed
from 50 000 red noise time series, all generated with the median red noise
parameters. As we can see in Fig. c, the matching
between the analytical and MCMC confidence levels is excellent, also in the
very tail of the distribution, at the 99.9 % confidence level. We can go
a step further and take into account the uncertainty in the CARMA parameters,
as explained in Sect. . Figure d
presents the MCMC confidence levels that are computed from 50 000 red noise
time series, generated with stochastic parameters, that are taken from the
joint posterior distribution of the parameters of the red noise process. The
number of WOSA segments per frequency, denoted by Q(f) in Sects. to , is in
Fig. b, and provides an indication of the noise
damping per frequency. Indeed, the variability due to the background noise is
increasingly damped as the number of WOSA segments grows.
At six particular frequencies, check for the convergence of the
analytical percentiles.
We also compute the amplitude periodogram, Eq. (),
which is actually very close to the weighted periodogram, as shown in Fig. . Similar results are obtained using other
tapers (not shown). This illustrates the quality of the approximations made
in Sect. . Note that the estimation of the amplitude
Eω of the model (Eq. ) is always biased by the background
noise (we observe in Fig. that the peaks
emerge from a baseline which is well above zero).
Sensitivity analysis for the degree of the polynomial trend
We show in Fig. that the degree m of the
polynomial trend, taken between 5 and 10, does not substantially influence
the WOSA periodogram. Below m=5, the trend no longer fits the
data correctly (from a mere visual inspection), while above m=10, spurious
oscillations may appear.
Comparison between the amplitude periodogram (= squared amplitude)
and the weighted periodogram. The green curve is the same as the black curve
of Fig. c and
d.
Note that we do not apply here the Akaike information criterion (AIC)
. Indeed, defining a stochastic model for the trend and
estimating its likelihood is quite tedious in our case, since we work with
CARMA stochastic processes. Moreover, at this stage, we do not want to choose
yet between the orders of the CARMA process.
Sensitivity analysis for the order of the CARMA process
Figure displays the confidence
levels for various orders of the CARMA process: (p,q)=(0,0), (p,q)=(1,0),
(p,q)=(2,0) and (p,q)=(2,1). It is clear that the CARMA(0,0) (= white
noise) does not capture enough spectral variability to perform significance
testing and that using a CARMA(2,0) or a CARMA(2,1) is basically equivalent
to using a red noise.
(a) Trends of different degrees for the time series.
(b) Weighted WOSA periodograms for different degrees of the trend.
Each periodogram is normalised like in
Eq. () in order to make a
meaningful comparison.
The weighted WOSA periodogram and its 95 % confidence levels for
different orders (p,q) of the CARMA process. Note that the marginal
posterior distributions of some parameters of the CARMA(2,0) and CARMA(2,1)
processes are multimodal, so the analytical approach cannot be applied, and
MCMC confidence levels must therefore be
used.
WAVEPAL Python package
WAVEPAL is a package, written in Python 2.X, that performs frequency and
time–frequency analyses of irregularly sampled time series, significance
testing against a stationary Gaussian CARMA(p,q) process, and filtering.
Frequency analysis is based on the theory developed in this article, and
time–frequency analysis relies on the theory developed in Part 2 of this
study . It is available at
https://github.com/guillaumelenoir/WAVEPAL.
Conclusions
We proposed a general theory for the detection of the periodicities of
irregularly sampled time series. This is based on a general model for the
data, which is the sum of a polynomial trend, a periodic component and a
Gaussian CARMA stochastic process. In order to perform the frequency
analysis, we designed new algebraic operators that match the structure of our
model, as extensions of the Lomb–Scargle periodogram and the WOSA method. A
test of significance for the spectral peaks was designed as a hypothesis
testing, and we investigated in detail the estimation of the percentiles of
the distribution of our algebraic operators under the null hypothesis.
Finally, we showed that the least-squares estimation of the squared amplitude
of the periodic component and the periodogram are no longer proportional if
the time series is irregularly sampled. Approximate proportionality relations
were proposed and are at the basis of the weighted WOSA periodogram, which is
the analysis tool that we recommend for most frequency analyses. The general
approach presented in this paper allows an extension to the continuous
wavelet transform, which is developed in Part 2 of this study
.
Code availability
The Python code generating the figures of this article is
available in the Supplement.
Some properties of the Lomb–Scargle periodogram
We present some properties of the LS periodogram, defined in Sect. .
Periodicity of the periodogram
The LS periodogram and all its generalisations (e.g. Eq. ) exhibit a periodicity similar to the DFT of
regularly sampled real processes: the periodogram over the frequency range
]-1/2ΔtGCD,1/2ΔtGCD] repeats itself
periodically. Moreover, the periodogram at frequency -f is equal to the
periodogram at frequency +f. Consequently, we must work at most on the
frequency range [0,1/2ΔtGCD[ to avoid aliasing.
Total reconstruction
Integrating the orthogonal projection
Psp‾{|cω〉,|sω〉} between
frequency 0 and 1/2ΔtGCD does not give the identity
operator. We only have an approximate equality. Using Lomb's approximation,
given in Eq. (), and no extra approximation, some algebra
gives
∫0π/ΔtGCDdω|cω♯〉〈cω♯|+|sω♯〉〈sω♯|≈2πNΔtGCDI.
It is interesting to compare it with the integration of complex exponentials,
which gives exactly the identity operator:
∫-π/ΔtGCDπ/ΔtGCDdω|eω♯〉〈eω♯|=2πNΔtGCDI,
where
|eω♯〉=1Nexp(iω|t〉)=1N(|cω〉+i|sω〉).
The above formula may be interpreted as a form of Parseval's identity. That
property of exact reconstruction is, incidentally, at the basis of the
multitaper method chap. 4. With that property and the no less
interesting mathematical properties of the complex exponentials, it is
legitimate to ask why we would not work with the projection on a complex
exponential instead of a projection on cosine and sine. The main disadvantage
of working with exponentials is the loss of power in the negative
frequencies. Indeed, the trendless model (Eq. ) at Ω=ω
can be rewritten as
|X〉=Eωexp(i(ω|t〉+ϕω))+exp(-i(ω|t〉+ϕω))2+|Noise〉A3=Cω|eω〉+Dω|e-ω〉+|Noise〉,
where |eω〉=exp(iω|t〉). In the case of irregularly
sampled time series, we no longer have, in general, 〈eω|e-ω〉=0, so that some power is lost in the negative
frequencies when projecting on
sp‾{|eω〉}. We could then think about
performing the projection on
sp‾{|eω〉,|e-ω〉}, but this
does not lead to the identity operator when integrating from frequency
-1/2ΔtGCD to +1/2ΔtGCD.
Invariance under time translation
As stated in , the LS periodogram is invariant under time
translation.
Psp‾{|cω〉,|sω〉} is of
course invariant under such a transformation. The result can be generalised
to more evolved projections. Indeed,
[Psp‾{|t0〉,|t1〉,…,|tm〉,|cω〉,|sω〉}-Psp‾{|t0〉,|t1〉,…,|tm〉}] is also invariant under time translation, provided all
the powers of |t〉 from 0 to m are taken into account. That
projection is also invariant under time dilatation if the frequency is
contracted accordingly.
Periodogram and mean: equivalence between published formulas
We show here the equivalence between some published formulas, with notations
that are a mix between those of the cited articles and those of the present
one in order to facilitate the reading.
p. 335 work with
||Psp‾{|t0〉,|cω〉,|sω〉}-Psp‾{|t0〉}|X〉||2.
It is defined for regularly sampled time series, and is suitable for
irregularly sampled time series as well. That formula is the same as Eq. ().
considers irregularly sampled time series and defines
the intensity (p. 620) by
I(ω)=c12+c22,
where c1=〈f|h1〉 and c2=〈f|h2〉; |f〉
contains the measurements (this is |X〉 in the present article) and
|h1〉 and |h2〉 are exactly the same as in Eq. ().
I(ω) is thus equal to Eq. ().
deal with irregularly sampled time series and define (their Eq. 1, p. 65):
SP(ν)=〈X|F1,0(ν)|X〉B3=〈X|A(ν)[A(ν)′A(ν)]-1A(ν)′|X〉,
where ν denotes the frequency (ν=ω/2π) and A(ν) is a
(N,2) matrix whose first column is |cω〉-|t0〉〈t0|cω〉/N and second column is
|sω〉-|t0〉〈t0|sω〉/N. Equation () is nothing but the squared norm of the orthogonal
projection of the data |X〉 onto the span of those two vectors. By a
Gram–Schmidt orthonormalisation, it is easy to see that
sp‾{|cω〉-|t0〉〈t0|cω〉/N,|sω〉-|t0〉〈t0|sω〉/N}=sp‾{|h1〉,|h2〉},
where |h1〉 and |h2〉 are defined in Eq. (). We thus have the periodogram defined in
Eq. ().
On the pseudo-spectrum
We define the pseudo-spectrum as the expected value of the WOSA
periodogram under the null hypothesis (see Sect. ):
S^(ω)=E||PWOSA(ω)|X〉||2,
where |X〉=|Trend〉+|Noise〉, in which
|Noise〉 is a zero-mean stationary Gaussian CARMA process
sampled at the times of |t〉, and the expectation is taken on the
samples of the CARMA noise. With what we have seen in Sect. and , the
periodogram is either obtained with Monte Carlo methods or analytically with
some approximations. In the former case, S^(ω) is estimated
by taking the numerical average of the periodogram at each frequency. In the
latter case, an analytical formula for the pseudo-spectrum is available.
Indeed, the process under the null hypothesis is
|X〉=K|Z〉+∑k=0mγk|tk〉, where K is defined
in Eqs. () or (), and we
have
S^(ω)=∑k=12Q(ω)λk(ω)=trMω′KK′Mω,
where the different terms are defined in Theorem .
When dealing with a trendless signal, we can perform the WOSA on the
classical tapered periodogram, and the pseudo-spectrum becomes
S^(ω)=E||PWOSA(ω)|X〉||2C3=E∑q=1Q(ω)||Psp‾{|Gqcω,q〉,|Gqsω,q〉}|X〉||2.
In the case of regularly sampled data, Eq. ()
converges to the spectrumS(ω) as the number of data points
increases (up to a multiplicative factor Δt, the time step). See
where it is shown that
||PWOSA(ω)|X〉||2 is a mean-square-consistent and
asymptotically unbiased estimator of the spectrum. The spectrumS(ω), also called Fourier power spectrum, of a regularly
sampled zero-mean real stationary process |X〉 is defined by the
following
see
Sect. 10.3
In that book, the authors work with the projection on
complex exponentials,
|eω〉=|cω〉+i|sω〉, instead of a
projection on cosine and sine. But this is asymptotically the same since,
asymptotically, the cosine and sine are orthogonal at all the frequencies.
of:
S(ω)=ΔtlimN→∞E||Psp‾{|cω〉,|sω〉}|X〉||2.
Now, considering Eq. (), we thus have, for trendless
regularly sampled time series, the following 1-moment approximation:
||PWOSA(ω)|X〉||2≈d12QS(ω)χ2Q2.
With that approximation, the spectrum S(ω), which is well known for
some processes such as ARMA processes, gives access to the confidence levels.
The above formula is widely used in the literature on regularly sampled time
series in the case of one WOSA segment (Q=1), for which the 1-moment
approximation is good enough see, for instance, Eq. 17 in.
In the case of irregularly sampled data, the spectrum S(ω) can be
defined over the frequency range [-1/2ΔtGCD,1/2ΔtGCD[. This follows from the spectral representation theorem
chap. 4 applied to irregularly sampled time series.
But S^(ω) usually strongly differs from S(ω), except
in the white noise case where the spectrum is flat. Building estimators of
the spectrum S(ω) in the case of irregularly sampled time series
actually seems very challenging, as briefly discussed in Sect. .
The generalised gamma-polynomial distribution as an approximation for the linear combination of chi-square distributions
We extend the gamma-polynomial approximation of Sect. to the generalised
gamma-polynomial approximation. Both conserve the first d moments of the
distribution X. The generalised gamma-polynomial approximation is based on
the generalised gamma distribution, which has three parameters, such that the
prerequisite of a d-moment approximation is a 3-moment approximation with the
generalised gamma distribution.
3-moment approximation
We work with the generalised gamma distribution, which has three parameters,
X≈dγα,β,δ.
Its PDF is
fγ(x;α,β,δ)=δβαδΓ(α)xαδ-1exp-(x/β)δD2α,β,δ>0,
where Γ is the gamma function. It reduces to the gamma distribution
when δ=1. Its moments are
μ(k)=βkΓ(α+k/δ)Γ(α)k∈N.
Equating the first 3 moments (k=1,2,3) of the generalised gamma to the
first 3 moments of X gives α, β and δ. But, that
requires the zeros of a nonlinear 3-dimensional function to be found. We observed
that root-finding algorithms may be sensitive to the choice of the first
guess, and particular attention must therefore be dedicated to it.
In , it is shown that, if Y follows a generalised gamma
distribution, working with ln(Y) allows the parameters
α, β, δ to be easily found. Indeed, it only requires a root-finding for a
monotonic unidimensional function. Unfortunately, the distribution of the
logarithm of a linear combination of chi-square distributions is not known.
We thus use the 2-moment approximation, for which we can find the moments of
the logarithm of the distribution. Indeed, if we write
Y=dgχM2, in which g and M are determined from Eq. (), and Z=ln(Y), some calculus gives us
the cumulant generating function of Z:
K(t)=tln(2g)+lnΓ(M/2+t)-lnΓ(M/2),
from which we obtain the cumulants. The first three are
D5aκ(1)=ln(2g)+ψ0(M/2),D5bκ(2)=ψ1(M/2),D5cκ(3)=ψ2(M/2),
where ψi is the polygamma function (ψ0 is the digamma function).
From the cumulants, we have the expected value κ(1), the variance
κ(2) and the skewness κ(3)/κ(2)3/2. Applying Eq. (21)
of gives us the parameters α0, β0,
δ0 for Y, parameters that we then use as a first guess for the
generalised-gamma approximation of X.
The d-moment approximation
We extend here the formulas
In , formulas are
given for the gamma-polynomial distribution, but as suggested by the authors,
they can easily be generalised to the generalised gamma-polynomial
distribution
presented in . Let fX be the PDF of X; fX is
approximated by the PDF of a dth degree generalised
gamma-polynomial distribution:
fX(x)≈γα,β,δ(x)∑i=0dξixi,x≥0,
where the parameters α, β and δ are estimated with the
above 3-moment approximation; ξ0, ..., ξd are the solution of
Eq. (), where
η(h)=βhΓ(α+h/δ)/Γ(α). The estimation of a
confidence level for the WOSA periodogram is then the solution c0 of
1Γ(α)∑i=0dξiβiγi/δ+α,(c0/β)δ-p=0,
for some p value p, e.g. p=0.95 for a 95 % confidence level. If we
pose δ=1, the generalised gamma-polynomial approximation
reduces to the gamma-polynomial approximation presented in Sect. .
Computing time: analytical versus Monte Carlo significance levels
A comparison between the computing times, for generating the WOSA
periodogram, with the analytical and with the MCMC significance levels, based
on the hypothesis of a red noise background, is presented in Fig. . They are expressed in function of the number of data points,
which are disposed on a regular time grid in order to make a meaningful
comparison. Confidence levels with the analytical approach are estimated with
a 10-moment approximation, and the number of samples for the MCMC approach is
10 000 for the 95th percentiles and 100 000 for the
99th percentiles. The other parameters are default parameters
of WAVEPAL. All the runs were performed on the same computer
CPU
type: SandyBridge 2.3 GHz. RAM: 64 GB.
.
We see that the analytical approach is faster than the MCMC approach as long
as the number of data points is below some threshold, the latter increasing
with the level of confidence. Indeed, the analytical approach delivers
computing times of the same order of magnitude regardless of the percentile
(the two blue curves in Fig. a and b are of the
same order of magnitude), unlike the MCMC approach, which must require more
samples as the level of confidence increases in order to keep a sufficient
accuracy. The difference between both computing times therefore increases as
the level of confidence increases.
Computing times for generating the WOSA periodogram with analytical
(blue) and MCMC (green) confidence levels, in function of the number of data
points (disposed on a regular time grid). Log–log scale. (a) 95th
percentiles. (b) 99th percentiles.
On the F periodogram
The formula of the F periodogram (Eq. ) is based on
pp. 335–336. In that book, the authors work with a
constant trend. We have generalised the formula in order to deal with a
polynomial trend.
A slightly different formula was published in p. 65, again
with a constant trend. The F periodogram is denoted by θF in their
paper. In the case of a generalisation to a polynomial trend, their formula
becomes
(N-2)||Psp‾{|t0〉,|t1〉,…,|tm〉,|cω〉,|sω〉}-Psp‾{|t0〉,|t1〉,…,|tm〉}|X〉||22||I-Psp‾{|t0〉,|t1〉,…,|tm〉,|cω〉,|sω〉}-Psp‾{|t0〉,|t1〉,…,|tm〉}|X〉||2,
but, unlike Eq. (), it has a denominator which is not
invariant with respect to the parameters of the trend.
The supplement related to this article is available online at: https://doi.org/10.5194/npg-25-145-2018-supplement.
Competing interests
The authors declare that they have no conflict of
interest.
Acknowledgements
The authors are very grateful to Reik Donner, Laurent Jacques, Lilian Vanderveken, and Samuel Nicolay, for
their comments on a preliminary version of the paper. This work is
supported by the Belgian Federal Science Policy Office under contract
BR/12/A2/STOCHCLIM. Guillaume Lenoir is currently supported by the FSR-FNRS grant PDR T.1056.15 (HOPES).
Edited by: Jinqiao Duan
Reviewed by: two anonymous referees
ReferencesAkaike, H.: A new look at the statistical model identification, IEEE T.
Automat. Contr., 19, 716–723, 10.1109/TAC.1974.1100705, 1974.Bretthorst, L.: Nonuniform Sampling: Bandwidth and Aliasing, in: AIP
Conference Proceedings – Bayesian Inference and Maximum Entropy Methods in
Science and Engineering, edited by: Rychert, J., Gary, E., and Smith, R.,
vol. 567, 1–28, Boise, Idaho, USA, 10.1063/1.1381847,
1999.
Brockwell, P. and Davis, R.: Time Series: Theory and Methods, Springer Series
in Statistics, Second edn., Springer, New York, USA, 1991.Brockwell, P. and Davis, R.: Introduction to Time Series and Forecasting,
Springer Texts in Statistics, Third edn., Springer International Publishing, 10.1007/978-3-319-29854-2, 2016.Bronez, T.: Spectral estimation of irregularly sampled multidimensional
processes by generalized prolate spheroidal sequences, IEEE T. Acoust.
Speech, 36, 1862–1873, 10.1109/29.9031, 1988.Ferraz-Mello, S.: Estimation of Periods from Unequally Spaced Observations,
Astron. J., 86, 619–624, 10.1086/112924, 1981.Fodor, I. and Stark, P.: Multitaper spectrum estimation for time series with
gaps, IEEE T. Signal Proces., 48, 3472–3483,
10.1109/78.887039, 2000.Ghil, M., Allen, M. R., Dettinger, M. D., Ide, K., Kondrashov, D., Mann, M. E.,
Robertson, A. W., Saunders, A., Tian, Y., Varadi, F., and Yiou, P.: Advanced
spectral methods for climatic time series, Rev. Geophys., 40, 1003,
10.1029/2000RG000092, 2002.Harris, F.: On the use of windows for harmonic analysis with the discrete
Fourier transform, Proceedings of the IEEE, 66, 51–83,
10.1109/PROC.1978.10837, 1978.Hasselmann, K.: Stochastic climate models Part I. Theory, Tellus, 28,
473–485, 10.1111/j.2153-3490.1976.tb00696.x, 1976.
Heck, A., Manfroid, J., and Mersch, G.: On period determination methods,
Astron. Astrophys. Sup., 59, 63–72, 1985.Jeffreys, H.: An Invariant Form for the Prior Probability in Estimation
Problems, P. Roy. Soc. Lond. A Mat., 186, 453–461,
10.1098/rspa.1946.0056,
1946.Jian, Z., Zhao, Q., Cheng, X., Wang, J., Wang, P., and Su, X.:
Pliocene-Pleistocene stable isotope and paleoceanographic changes in the
northern South China Sea, Palaeogeogr. Palaeocl., 193, 425–442,
10.1016/S0031-0182(03)00259-1, 2003.Jones, R. and Ackerson, L.: Serial correlation in unequally spaced longitudinal
data, Biometrika, 77, 721–731, 10.1093/biomet/77.4.721,
1990.Kelly, B., Becker, A., Sobolewska, M., Siemiginowska, A., and Uttley, P.:
Flexible and Scalable Methods for Quantifying Stochastic Variability in the
Era of Massive Time-domain Astronomical Data Sets, Astrophys. J.,
788, 33, 10.1088/0004-637X/788/1/33, 2014.Kemp, D.: Optimizing significance testing of astronomical forcing in
cyclostratigraphy, Paleoceanography, 31, 1516–1531, 10.1002/2016PA002963,
2016.Lenoir, G.: Time-frequency analysis of regularly and irregularly sampled time
series: Projection and multitaper methods, PhD thesis, Université
catholique de Louvain – Faculté des Sciences – Georges Lemaître
Centre for Earth and Climate Research, Louvain-la-Neuve, Belgium, available
at: https://dial.uclouvain.be/pr/boreal/object/boreal:191751 (last
access: 22 February 2018), 2017.Lenoir, G. and Crucifix, M.: A general theory on frequency and
time–frequency analysis of irregularly sampled time series based on
projection methods – Part 2: Extension to time–frequency analysis, Nonlin.
Processes Geophys., 25, 175–200, 10.5194/npg-25-175-2018, 2018.Lomb, N.: Least-squares frequency analysis of unequally spaced data,
Astrophys. Space Sci., 39, 447–462, 10.1007/BF00648343, 1976.Mortier, A., Faria, J. P., Correia, C. M., Santerne, A., and Santos, N. C.:
BGLS: A Bayesian formalism for the generalised Lomb-Scargle
periodogram, Astron. Astrophys., 573, A101,
10.1051/0004-6361/201424908, 2015.
Mudelsee, M.: Climate Time Series Analysis – Classical Statistical and
Bootstrap Methods, in: Atmospheric and Oceanographic Sciences
Library, vol. 42, Springer, Dordrecht, the Netherlands, 2010.Mudelsee, M., Scholz, D., Röthlisberger, R., Fleitmann, D., Mangini, A.,
and Wolff, E. W.: Climate spectrum estimation in the presence of timescale
errors, Nonlin. Processes Geophys., 16, 43–56,
10.5194/npg-16-43-2009, 2009.Pardo Igúzquiza, E. and Rodríguez Tovar, F.: Spectral and
cross-spectral analysis of uneven time series with the smoothed
Lomb-Scargle periodogram and Monte Carlo evaluation of statistical
significance, Comput. Geosci., 49, 207–216,
10.1016/j.cageo.2012.06.018, 2012.
Priestley, M.: Spectral Analysis and Time Series, Two Volumes Set, Probability
and Mathematical Statistics – A series of Monographs and Textbooks, Third edn., Academic
Press, London, UK, San Diego, USA, 1981.Provost, S.: Moment-Based Density Approximants, The Mathematica Journal, 9,
727–756, available at: http://www.mathematica-journal.com/issue/v9i4/DensityApproximants.html (last access: 22 February 2018), 2005.Provost, S., Ha, H.-T., and Sanjel, D.: On approximating the distribution of
indefinite quadratic forms, Statistics, 43, 597–609,
10.1080/02331880902732123, 2009.Rehfeld, K., Marwan, N., Heitzig, J., and Kurths, J.: Comparison of
correlation analysis techniques for irregularly sampled time series, Nonlin.
Processes Geophys., 18, 389–404, 10.5194/npg-18-389-2011,
2011.Riedel, K. and Sidorenko, A.: Minimum bias multiple taper spectral estimation,
IEEE T. Signal Proces., 43, 188–195,
10.1109/78.365298, 1995.Robinson, P.: Estimation of a time series model from unequally spaced data,
Stoch. Proc. Appl., 6, 9–24, 10.1016/0304-4149(77)90013-8, 1977.Scargle, J.: Studies in astronomical time series analysis II – Statistical
aspects of spectral analysis of unevenly spaced data, Astrophys.
J., 263, 835–853, 10.1086/160554, 1982.Schulz, M. and Mudelsee, M.: REDFIT: estimating red-noise spectra directly
from unevenly spaced paleoclimatic time series, Comput. Geosci., 28,
421–426, 10.1016/S0098-3004(01)00044-9, 2002.Schulz, M. and Stattegger, K.: SPECTRUM: spectral analysis of unevenly spaced
paleoclimatic time series, Comput. Geosci., 23, 929–945,
10.1016/S0098-3004(97)00087-3, 1997.Stacy, E. W. and Mihram, G. A.: Parameter Estimation for a Generalized Gamma
Distribution, Technometrics, 7, 349–358,
10.1080/00401706.1965.10490268, 1965.
Thomson, D.: Spectrum estimation and harmonic analysis, Proceedings of the
IEEE, 70, 1055–1096, 10.1109/PROC.1982.12433, 1982.Torrence, C. and Compo, G.: A Practical Guide to Wavelet Analysis, B. Am.
Meteorol. Soc., 79, 61–78,
10.1175/1520-0477(1998)079<0061:APGTWA>2.0.CO;2, 1998.
Torrésani, B.: Analyse continue par ondelettes, Savoirs actuels/Série
physique, CNRS Editions and EDP Sciences, Paris, France, 1995.Uhlenbeck, G. E. and Ornstein, L. S.: On the Theory of the Brownian Motion,
Phys. Rev., 36, 823–841, 10.1103/PhysRev.36.823, 1930.Vio, R., Andreani, P., and Biggs, A.: Unevenly-sampled signals: a general
formalism for the Lomb-Scargle periodogram, Astron. Astrophys.,
519, A85, 10.1051/0004-6361/201014079, 2010.Walden, A. T.: A unified view of multitaper multivariate spectral estimation,
Biometrika, 87, 767–788, 10.1093/biomet/87.4.767,
2000.Welch, P.: The use of fast Fourier transform for the estimation of power
spectra: A method based on time averaging over short, modified periodograms,
IEEE T. Acoust. Speech, 15, 70–73,
10.1109/TAU.1967.1161901, 1967.Zechmeister, M. and Kürster, M.: The generalised Lomb-Scargle
periodogram, Astron. Astrophys., 496, 577–584,
10.1051/0004-6361:200811296, 2009.