NPGNonlinear Processes in GeophysicsNPGNonlin. Processes Geophys.1607-7946Copernicus PublicationsGöttingen, Germany10.5194/npg-25-175-2018A general theory on frequency and time–frequency analysis of irregularly sampled time series based on projection methods – Part 2: Extension to time–frequency analysisLenoirGuillaumeguillaume.lenoir@hotmail.comCrucifixMichelhttps://orcid.org/0000-0002-3437-4911Georges Lemaître Centre for Earth and Climate Research, Earth and Life Institute, Université catholique de Louvain, 1348, Louvain-la-Neuve, BelgiumBelgian National Fund of Scientific Research, rue d'Egmont, 5, 1000 Brussels, BelgiumGuillaume Lenoir (guillaume.lenoir@hotmail.com)5March201825117520016June20174July20174December20176December2017This work is licensed under the Creative Commons Attribution 4.0 International License. To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/This article is available from https://npg.copernicus.org/articles/25/175/2018/npg-25-175-2018.htmlThe full text article is available as a PDF file from https://npg.copernicus.org/articles/25/175/2018/npg-25-175-2018.pdf
Geophysical time series are sometimes sampled
irregularly along the time axis. The situation is particularly frequent in
palaeoclimatology. Yet, there is so far no general framework for handling the
continuous wavelet transform when the time sampling is irregular.
Here we provide such a framework. To this end, we define the scalogram as the
continuous-wavelet-transform equivalent of the extended Lomb–Scargle
periodogram defined in Part 1 of this study . The signal
being analysed is modelled as the sum of a locally periodic component in the
time–frequency plane, a polynomial trend, and a background noise. The mother
wavelet adopted here is the Morlet wavelet classically used in geophysical
applications. The background noise model is a stationary Gaussian continuous
autoregressive-moving-average (CARMA) process, which is more general than the
traditional Gaussian white and red noise processes. The scalogram is smoothed
by averaging over neighbouring times in order to reduce its variance. The
Shannon–Nyquist exclusion zone is however defined as the area corrupted by
local aliasing issues. The local amplitude in the time–frequency plane is
then estimated with least-squares methods. We also derive an approximate
formula linking the squared amplitude and the scalogram. Based on this
property, we define a new analysis tool: the weighted smoothed scalogram,
which we recommend for most analyses. The estimated signal amplitude also
gives access to band and ridge filtering. Finally, we design a test of
significance for the weighted smoothed scalogram against the stationary
Gaussian CARMA background noise, and provide algorithms for computing
confidence levels, either analytically or with Monte Carlo Markov chain
methods. All the analysis tools presented in this article are available to
the reader in the Python package WAVEPAL.
Introduction
The continuous wavelet transform (CWT) is widely used for the time–frequency
analysis of geophysical time series, mainly through its scalogram, which is
the squared modulus of the CWT. The CWT relies on a probing function, called
the mother wavelet. A common choice for the mother wavelet is the Morlet
wavelet , which is well suited for the analysis of
signals whose components have a time-varying frequency and/or a time-varying
amplitude. The scalogram is then often smoothed to reduce its variance, and
significance testing against a stationary Gaussian white or red noise is
commonly applied. State-of-the-art references in
climate for the analysis of regularly sampled time series include
, who provide the basis for the subsequent works;
, who define a smoothing method for the scalogram (which
is a particular case of the wavelet coherency developed in there); and
and , who build more reliable
significance tests for the smoothed scalogram. A non-exhaustive list of the
applications, in climatology, of the scalogram of the CWT with the Morlet
wavelet, includes the following.
Studies in climate and weather: analysis of the El Niño Southern Oscillation in , analysis of the Arctic
oscillation in , or the analysis of daily precipitation in the Alps in .
Studies in palaeoclimate: analysis of the astronomical forcing in , analysis of the
mid-Pleistocene transition
in , or the analysis of the equatorial Pacific thermocline over the last eight glacial periods in .
Most of these studies use the algorithms provided by the papers cited above
or similar algorithms, and all of them require the data to be regularly
spaced. However, it may happen that the time series be intrinsically
irregularly sampled (this actually happens in some of the above examples) and
it is then interpolated on a regularly spaced grid in order to apply the
algorithms of the CWT and its scalogram. But the interpolation procedure may
seriously affect the analysis with unpredictable consequences for the
scientific interpretation, especially when performing significance testing.
This is illustrated in Appendix .
A solution to this problem was addressed by Foster in a series of articles
which share a common thread with our two papers, in the sense that it first
generalises the Lomb–Scargle periodogram, based on orthogonal projection
methods, in and , and then extends the
formalism to the continuous wavelet transform, in ,
allowing the time series not to be interpolated. Foster's formulas were motivated
by the astronomical study of the light curves of variable stars, which are
unevenly sampled time series with large gaps. The methods presented in this
article are influenced by Foster's theory and it is shown in
Appendix that most of its formulas can actually be
deduced from our general framework. The main limitations of Foster's theory
are the following (see Appendix for detailed
explanations): significance testing is only performed for the white noise
background case, it only deals with the unsmoothed scalogram, and the areas
in the time–frequency plane corrupted by aliasing are underestimated. It
therefore suffers from a limited interest in geophysical
applications
An application of Foster's formulas on palaeoclimate
data is found in .
. Another study tackling the problem of
the estimation of the scalogram of irregularly sampled time series, without
interpolating the data, is given in . In there, the
authors propose an estimator of the amplitude of the signal, locally in the
time–frequency plane, but no significance testing is performed. The other
limitations of their algorithm are basically the same as for Foster's
formulas. This is detailed in Appendix .
In this article, we extend the analysis tools that we derived in the first
part of this study in the case of the frequency analysis
of irregularly sampled time series. They are based on a similar model, which
is a locally periodic component in the time–frequency plane, plus a
polynomial trend, plus a stationary Gaussian continuous
autoregressive-moving-average (CARMA) process. Let us sketch the main points
of the article. First, the taper of the periodogram, derived in
Sect. 4.4, is chosen here to be a time-dependent
Gaussian function with a variance depending on the scale in order to define
the Morlet wavelet-based scalogram. This is detailed in Sect.
of this work. Second, the scalogram is smoothed in order to reduce its
variance, by averaging over neighbouring times. To this end, we apply the
same formula as in . This is explained in
Sect. . Third, in
Sect. , we estimate the amplitude of the locally
periodic component, extending the results obtained in Sect. 6 of
Part 1, and define, in
Sect. of this article, the weighted
smoothed scalogram as the time–frequency analogue of the weighted WOSA
periodogram defined in the first part of this study .
Fourth, we define in Sect. the
Shannon–Nyquist exclusion zone (SNEZ) as the area of the time–frequency
plane which must be excluded from the analysis because of the local aliasing
issues. Fifth, we design a test of significance for the weighted smoothed
scalogram, against the stationary Gaussian CARMA background noise. This is
based on the theory developed in Sect. 5 of Part 1. More specifically, we
define a null and an alternative hypothesis, and estimate the distribution of
the weighted smoothed scalogram under the null hypothesis, either
analytically, conserving the first moments of the distribution, or with
Markov Chain Monte Carlo (MCMC) methods. The latter approach allows the
uncertainty in the parameters of the CARMA background process to be fully
taken into account. This is presented in
Sect. . Sixth, we provide, in
Sect. , formulas for filtering
the signal in a band delimited by two scales, or with the ridges, which are
the lines going through the maxima of the estimated amplitude, in the
time–frequency plane. Ridge filtering is based on state-of-the-art
algorithms provided in and
(https://github.com/jonathanlilly/jLab). Seventh, we define in
Sect. the global scalogram as the time-averaged
weighted smoothed scalogram, resulting in a periodogram-like analysis tool
with a frequency-varying bandwidth. Eighth, we illustrate, in
Sect. , the theory on the same palaeoclimate
data set as in our first article . Finally, a Python
package named WAVEPAL is available to the reader and is presented in
Sect. . Before tackling the problem of irregularly
sampled time series, the paper starts with the theory of the CWT applied to
continuous-time signals. This gives the bases for the subsequent
developments.
Most of the mathematical concepts and notations are introduced in the first
part of this study , and the reader is invited to revise
them. Throughout this article, we will denote the equations of the preceding
paper by, for example, “Eq. (I,30)”, meaning “the equation (30) of
Part 1”, and will refer to the paper itself by “Part 1”.
The continuous wavelet transform of continuous-time processesThe continuous wavelet transform and its scalogram
The mathematical background to Fourier analysis is given in
Appendix . Let S denote the
Schwartz space. The continuous wavelet transform of a function
x∈S(R) is
Sx(τ,a)=〈ψτ,a|x〉,
where ψτ,a∈S(R) is defined by
ψτ,a(t)=c(a)ψt-τa.
Here, ψ is called the mother wavelet, τ∈R is the
translation time, a∈R0+ is the scale, and
c(a)∼am with m∈Q. We can write the CWT as a convolution
product,
Sx(τ,a)=ψa♯⋆x(τ),
where
ψa♯(t)=c(a)ψ-ta‾,
in which ⋅‾ denotes the complex conjugate. From the
convolution theorem,
Sx^(ω,a)=2πψa♯^(ω)x^(ω)=2πac(a)ψ^(aω)‾x^(ω),
and Sx(τ,a) is then obtained by taking the inverse Fourier transform.
|Sx(τ,a)|2 gives the local power in the
time–scale plane, and is
called the scalogram by analogy with the periodogram.
The wavelet power spectrum
The wavelet power spectrum (WPS) of a continuous-time stochastic process
{x(t)}t∈R is defined by the following see:
WPSx(τ,a)=E{|Sx(τ,a)|2},
where the expectation is taken over the samples of the stochastic process. A
simple example is the WPS of a real-valued stationary white noise. Define
{η(t)}t∈R satisfying the following covariance property:
E{η(t)η(t′)}=σ2δ(t-t′).
Its WPS is then
WPSη(τ,a)=ac(a)2||ψ||2σ2.
The Morlet wavelet as the mother wavelet
In this article, we choose the mother wavelet ψ to be the Morlet wavelet
:
ψ(t)=π-1/4σ0-1/2exp(iω0t)-exp-ω02σ02/2exp-t2/2σ02.
This mother wavelet is a complex plane wave weighted by a Gaussian, to which
is added a correction term to make it admissible
The
admissibility criteria is required for ψ to be a wavelet p. 5.
, i.e. satisfying
∫-∞+∞dω|ψ^(ω)|2|ω|-1<∞.
This correction term is negligible
We have
exp(-(5.5)2)=7.288×10-14 and exp(-(5.5)2/2)=2.700×10-7.
for
σ0ω0≥5.5. If this inequality is satisfied, and with the
variable change a′=a/ω0, the CWT with the Morlet wavelet is
S(τ,a′)=c(a′)∫-∞+∞dtexp-i(t-τ)a′exp-(t-τ)22σ02ω02a′2x(t),
where c(a′)∼(a′)m, m∈Q, and c(a′) holds all the
multiplicative constants. Without loss of generality, we impose σ0=1
and assume that
ω0≥5.5
is fulfilled in the following of this article. Therefore,
S(τ,a)=c(a)∫-∞+∞dtexp-i(t-τ)aexp-(t-τ)22ω02a2x(t)=(ψa♯⋆x)(τ),
where
ψa♯(t)=c(a)exp(it/a)exp-t2/2ω02a2.
Under this form, interpreting Eq. () is straightforward:
the CWT is the inner product between the signal x and a Gaussian wave
packet centred in τ=t, of period 2πa, and with numerical
support
The length of the support of the Gaussian may be
approximated by 6 times its standard deviation.
of length 6ω0a.
As the scale increases (decreases), the support becomes wider (
narrower).
On the parameter c(a)
There are two common choices for c(a)seeSect. 3.
The first one is c(a) proportional to 1/a,
c(a)∼1a,
and gives a constant L2 norm for ψτ,a, namely
||ψτ,a||=||ψ||. This implies that the wavelet power spectrum of
a white noise is flat, as we can see in Eq. (). The
second choice is
c(a)∼1a,
which gives a constant L1 norm for ψτ,a and, most
importantly, gives the same maximal power for sines of the same amplitude and
with different frequencies. Indeed, from the Fourier transform of
ψa♯,
ψa♯^(ω)=c(a)aω0exp-ω02(ωa-1)22,
and applying Eq. (), we must require c(a)a to be constant to
have the maxima of the scalogram of a sum sine waves (all with the same
amplitude but with different frequencies) invariant with the scale.
The parameter ω0 and the time–frequency resolution
The parameter ω0 controls the time–frequency resolution, as it can be
seen from the standard deviations of the Gaussian weights in
ψa♯, Eq. (), and in its Fourier
transform, Eq. (). The standard deviations are equal
to ω0a and 1/ω0a respectively. Consequently, for a fixed
scale, increasing (decreasing) the value of ω0 will generate a
CWT with a better (worse) frequency resolution and a worse (better) time resolution. This property is of primary importance for the
applications to time series, as illustrated in
Sect. . Note that, for any time–frequency
transform, there is always a trade-off between time and frequency
localisation according to the Fourier uncertainty principle. The Morlet
wavelet exhibits the best trade-off, thanks to its Gaussian shape. We provide
further details on this topic in
Appendix .
Scale-to-period conversion
The Morlet wavelet is often used to detect the periodicities in a signal, and
it is therefore suitable to convert scales a into periods T. In practice, take a signal x(t)=Aexp(iωt)=Aexp(i2πt/T). Its scalogram writes
|S(τ,a)|2=2πAc(a)2ω02a2exp-ω02(ωa-1)2,
and is independent of τ. Scale-to-period conversion is performed with
the value of the scale for which |S(τ,a)|2 is at its maximum (as a function of
a). We find the following:
T=2πaif c(a)∼1/a,4πω0aω0+ω02+2if c(a)∼1/a.
For a fixed scale, and while ω0≥5.5, the difference between both
never exceeds 2 %.
Reconstruction with the amplitude ridges
Reconstruction of a signal can be performed with the CWT along the
amplitude ridges
Another type of ridge is the phase ridge, defined in , but we consider only the amplitude
ridges in this study since they are easier to generalise to irregularly time
series. A comparison of both the amplitude and phase ridges is found in
.
, which are the lines going through
the maxima of the scalogram. Indeed, take the signal x(t)=Aexp(iωt)
and c(a)∼1/a. Its scalogram is maximum at a=1/ω (from Eq. ) and we can therefore easily recover the
amplitude A at each time τ, going through the ridge a(τ)=1/ω
in the scalogram, on which we have |S(τ,1/ω)|=αA∀τ, where α∈R is a multiplicative constant. Jointly with
the amplitude, the full signal x(t) can be exactly recovered from the CWT
along the ridge.
This can be extended to signals with slowly varying amplitude and phase
see, namely,
x(t)=A(t)exp(iϕ(t)),such thatdϕdt≫1AdAdt,
for
which the CWT taken along the ridge, i.e. at the maxima of its modulus, can
approximately reconstruct x(t). The inequality in
Eq. (), called the asymptoticity
condition, means that the instantaneous frequency inside the wave packet must
be much smaller than the frequency of the amplitude of the wave packet. The
analysis can be further extended to a sum of asymptotic signals plus noise,
which can be detected by multiple ridges . When considering a real signal like x(t)=A(t)cos(ϕ(t)), we
have to work with its analytic counterpart, which is built from the
Fourier transform of x, x^, for which we impose
x^(ω<0)=0 and then take the inverse Fourier transform.
Analyticity ensures that the phase and amplitude of a signal are uniquely
determined seeand the references therein for more
details. State-of-the-art algorithms for ridge detection are
developed in and are available for use in the package
jLab (https://github.com/jonathanlilly/jLab), in which the
ridge-finding algorithm is general enough to be applied to various mother
wavelets, such as the Morlet wavelet.
By construction, ridge filtering is well-adapted for filtering a
multi-periodic signal, even if it is plunged in a noisy environment
. In such conditions, it outperforms the techniques based
on the Hilbert transform. As mentioned in p. 4135,
“[...] the Hilbert transform can lead to disastrous results as the amplitude
and phase will then reflect the aggregate properties of the multi-component
signal”.
Writing the scalogram under the formalism of orthogonal projections
Finally, we mention that the scalogram can be written under the formalism of
orthonormal projections. Indeed, defining
yτ,a(t)=π-1/4ω0aexpi(t-τ)aexp-(t-τ)22ω02a2,
which has a unit norm, the scalogram can be formulated as
|S(τ,a)|2=γ(a)|〈yτ,a|x〉|2=γ(a)|||yτ,a〉〈yτ,a|〈yτ,a|yτ,a〉|x〉||2,
where γ(a)=α (α∈R) if c(a)∼1/a, or
γ(a)∼1/a if c(a)∼1/a.
The continuous wavelet transform of irregularly sampled time seriesThe model for the data
We consider the same model as in Part 1:
|X〉=|Trend〉+Eτ,acos(Ω|t〉+ϕτ,a)+|Noise〉=|Trend〉+Aτ,a|cΩ〉+Bτ,a|sΩ〉+|Noise〉,
where |X〉=[X1,…,XN]′ and is real, |t〉=[t1,…,tN]′, Aτ,a=Eτ,acos(ϕτ,a), Bτ,a=-Eτ,asin(ϕτ,a), Eτ,a2=Aτ,a2+Bτ,a2, |cΩ〉=[cos(Ωt1),…,cos(ΩtN)]′ and |sΩ〉=[sin(Ωt1),…,sin(ΩtN)]′.
We have added subscripts (τ,a) since all the subsequent analyses will be
done in the time–scale plane. The trend is a polynomial of degree m,
|Trend〉=∑k=0mγk|tk〉,
and the background noise term, |Noise〉, is a zero-mean
stationary Gaussian CARMA process sampled at the times of |t〉, as
defined in Sect. 3.2 of Part 1. As stated in Part 1, considering or not 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.
The scalogram
When applying the CWT to finite discrete time series, a choice for the
discretisation must be made. In the influential paper of
, which deals with regularly sampled time series, the
expression under the form of a convolution product in the Fourier space,
Eq. (), is conserved, and computed with the discrete Fourier
transform (DFT) of the data. The CWT is then the inverse DFT of the
convolution product. Unfortunately, we cannot extend the convolution
theorem
The convolution theorem for continuous-time functions is
given in Appendix , and its counterpart for
regularly sampled time series is given in p. 74.
to
irregularly spaced time series and we cannot therefore follow the same
computational procedure as in . Alternatively, we can
conserve the squared norm of the orthogonal projection,
Eq. (). The advantage of such a
formalism is that it can be applied to irregularly sampled time series, as
shown in Part 1. Similarly to Part 1, we work with cosines and sines instead
of working with complex exponentials. Very little difference is observed
between both choices. Based on the results of
Sect. , our Morlet wavelet scalogram for
irregularly sampled time series is therefore
||Psp‾{|Gτ,acτ,a〉,|Gτ,asτ,a〉}|X〉||2,
where Gτ,a is a diagonal matrix with diagonal elements
Gτ,aii=exp-(ti-τ)22ω02a2,∀i∈{1,…,N},
and
|cτ,a〉=cos(|t〉-τ)/a,|sτ,a〉=sin(|t〉-τ)/a,
are vectors of length N. We can impose τ=0 into the cosine and sine
terms, since
sp‾{|Gτ,acτ,a〉,|Gτ,asτ,a〉}
is invariant with respect to the variable τ appearing in the cosine and
sine, and the scalogram becomes
||Psp‾{|Gτ,aca〉,|Gτ,asa〉}|X〉||2.
In the following, the notations |Gτ,aca〉
and Gτ,a|ca〉 refer to the same vector. Our wavelet
scalogram is similar to the tapered periodogram defined in Sect. 4.4 of
Part 1, and its properties and generalisations will therefore be similar as
well. In particular, the variables a and τ are considered as
continuous variables, similarly to the continuous frequency variable of
Part 1.
(a) The regularly sampled (RS) time series, with Δt=1 ka, which represents the caloric summer insolation at 65∘ N to
which is added a realisation of a Gaussian red noise process with
α=0.1 and σ equal to half of the standard deviation of the
original time series (these parameters are defined in Sect. 3.2.3 of Part 1).
The red dots are obtained from randomly removing 75 % of the data points
of the RS time series, resulting in an irregularly sampled (IS) time series
with 500 data points. Panels (b), (c) and (d)
compare of the scalograms with ω0=10, jointly with their 95 %
analytical confidence levels against a red noise background.
(b) Scalogram of the RS time series computed with the classical
approach. (c) Scalogram of the RS time series computed with WAVEPAL.
(d) Scalogram of the IS time series computed with WAVEPAL. The black
zone, called the Shannon–Nyquist exclusion zone (SNEZ) and defined in
Sect. , is the area where the sampling is
not sufficient to probe the lowest periods. In panels (b),
(c) and (d), the two lateral shaded areas are the
half-cones of influence (see Sect. ), and the bottom
shaded area is the refinement of the SNEZ (defined in
Sect. ). The bounds of the colour scale in the
panels (b), (c) and (d) are the extrema of the
scalogram in panel (c) over the non-shaded area in order to make a
meaningful visual comparison. Technical details about the computation of the
scalogram and its confidence levels are given in
Sects. and
.
When the time series is regularly sampled, the scalogram, given by Eq. (), is extremely close to what is obtained with the
traditional approach based on the convolution theorem, e.g. in
, or . This is
illustrated with a regularly sampled time series representing a noisy version
of the caloric summer insolation
It is computed from the R package
PALINSOL (https://CRAN.R-project.org/package=palinsol) from the
formulas of and the data of .
at
65∘ N, in Fig. b and c. We
also show the effect of randomly removing a large amount of data points, with
the resulting time series being irregularly sampled. This is illustrated in
Fig. d and we observe that our algorithms still do a good
job at estimating significant regions, although there are some artefacts and
the power tends to be overestimated.
Equation () reduces to the Lomb–Scargle periodogram,
defined in Eq. (I,40), if the weight Gτ,a is replaced by the identity matrix. Similarly to the Lomb–Scargle
periodogram, we rescale |Gτ,aca〉 and
|Gτ,asa〉 such that they are orthonormal. This can
be done by defining
where βτ,a is the solution of
tan(2βτ,a)=Σi=1NGτ,aii2sin(2ti/a)Σi=1NGτ,aii2cos(2ti/a).
The scalogram is then
||Psp‾{|Gτ,aca〉,|Gτ,asa〉}|X〉||2=〈Gτ,aca♯|X〉2+〈Gτ,asa♯|X〉2.
Scalogram and trend
Analogously to Sect. 4.3 of Part 1, we extend the scalogram to take into
account the presence of a polynomial trend of degree m in the data. Indeed,
the scalogram defined in Sect. applies well to data which can
be modelled as
|X〉=Aτ,a|cΩ〉+Bτ,a|sΩ〉+|Noise〉.
If we want to work with the full model, Eq. (), holding a
polynomial trend of degree m, we define a new scalogram as
||Psp‾{|t0〉,|t1〉,…,|tm〉,|Gτ,aca〉,|Gτ,asa〉}-Psp‾{|t0〉,|t1〉,…,|tm〉}|X〉||2,
which is invariant with respect to the parameters of the trend. This is the
analogue of Eq. (I,57).
Smoothing the scalogram
The scalogram suffers from the same inconsistency issue as the periodogram,
in the sense that it remains very noisy regardless of the number of data
points we have at our disposal
The scalogram often looks
smooth because neighbouring points in the time–frequency plane are
strongly correlated, but it nevertheless remains inconsistent see the
discussion inSect. 4.2.
. Smoothing techniques must
therefore be applied, and we proceed like in Part 1, extending the formulas
used with regularly sampled time series. Note that the disadvantage of any
smoothing procedure is that the resolution (in time, frequency or both,
depending on the smoothing choice) is reduced. Consequently, there is always
a trade-off between variance reduction and resolution.
Smoothing is traditionally performed by averaging the scalogram over
neighbouring points in the time–scale plane, either by averaging over times
followed by averaging over scales , or
simply by averaging over time . In this work, we apply the
latter technique because, even for very simple signals like
|X〉=sin(ω|t〉), the correlations in the scalogram between
neighbouring scales, for a fixed time, are highly irregular when the time
series is irregularly sampled, unlike the correlations between neighbouring
times, for a fixed scale, which are driven by the Gaussian shape of the wave
packets |Gτ,aca〉 and |Gτ,asa〉. Smoothing
over time must be carried out in accordance with the length of the support of
the wave packets, which is proportional to the scale and to parameter
ω0 (Eq. ). This choice also implies that the
number of oscillations over which smoothing is performed is constant
throughout the time–scale plane. This results from Eq. ().
We adopt here the smoothing procedure of for which they
derived analytical asymptotic results in the case of regularly sampled time
series. The averaging window is a square window with a length proportional to
the scale. Our smoothed scalogram is
||Psmoothed(τ,a)|X〉||2=12γω0a∫τ-γω0aτ+γω0adτ′||Psp‾{|t0〉,|t1〉,…,|tm〉,|Gτ′,aca〉,|Gτ′,asa〉}-Psp‾{|t0〉,|t1〉,…,|tm〉}|X〉||2,
in which γ is called the smoothing coefficient. Appendix provides further details on the practical
implementation of the bounds of integration.
The amplitude scalogramDefinition
We want to estimate the amplitude
Eτ,a=Aτ,a2+Bτ,a2 of our model, Eq. (), at a given point (τ,a) in the time–scale plane. The
estimation of Eτ,a2 is called the amplitude scalogram and
is denoted by E^τ,a2. We start with a trendless signal and
derive an approximate formula linking the amplitude scalogram and the
scalogram.
Trendless signal
Formula (I,118) is applied with the left-hand-side term changed to encompass
wavelet formalism. A^ and B^ are determined by
projecting the data onto the tapered cosine and sine:
Psp‾{|Gτ,aca〉,|Gτ,asa〉}|X〉=A^τ,a|cω〉+B^τ,a|sω〉=Vω2|Φ^τ,a〉,
where the taper Gτ,a is defined in Eq. (),
Vω2=|||cω〉|sω〉||, and |Φ^τ,a〉=A^τ,aB^τ,a.
Conversion from the angular frequency ω to the scale a is performed
with the formula ω=1/a (justification is given in
Sect. ). Using the same development as in
Sect. 6.2.2 of Part 1, we obtain
|Φ^τ,a〉=(Va2′Gτ,aVa2)-1Va2′Gτ,a|X〉.
The amplitude scalogram is then
E^τ,a2=|||Φτ,a^〉||2.
The approximations made in Sect. 6.2.2 of Part 1 are valid in this work, and
applying Eq. (I,128) to our case gives an approximate formula linking the
scalogram and the amplitude scalogram, namely
E^τ,a2≈2trGτ,a2trGτ,a2||Psp‾{|Gτ,aca〉,|Gτ,asa〉}|X〉||2.
Let us compare this equation with its continuous counterpart, Eq. (), in which the weight must be
γ(a)∼1/a to get an estimation of the local squared amplitude, as
explained in Sect. . The comparison is made by analysing
the weight of the right-hand-side term of Eq. () in the
continuous limit:
1Δt‾2trGτ,a2trGτ,a2⟶2∫-∞+∞dtexp-(t-τ)2ω02a2∫-∞+∞dtexp-(t-τ)22ω02a22=1πω0a,
where Δt‾ is the average time step. This is proportional to
1/a and it is therefore consistent with the continuous case.
Signal with a trend
Formula (I,129) is applied with the left-hand-side term changed to encompass
wavelet formalism:
Psp‾{|t0〉,|t1〉,…,|tm〉,|Gτ,aca〉,|Gτ,asa〉}|X〉=∑k=0mγk^|tk〉+A^|cω〉+B^|sω〉=Vωm+3|Φ^〉,
where
Vωm+3=|||||t0〉…|tm〉|cω〉|sω〉||||,
and
|Φ^〉=γ0^⋮γm^A^B^.
Conversion from the angular frequency ω to the scale a is performed
with the formula ω=1/a (justification is given in
Sect. ). Using the same development as in
Sect. 6.3 of Part 1, we obtain
|Φ^τ,a〉=Wτ,am+3′Vam+3-1Wτ,am+3′|X〉,
where Wτ,am+3 is identical to Vam+3 except in the last two
columns, where the cosine and sine are tapered by Gτ,a. This gives
A^τ,a=Φ^τ,a(m+2),B^τ,a=Φ^τ,a(m+3),
where Φ^τ,a(m+2) and Φ^τ,a(m+3) are
the two last components of vector |Φ^τ,a〉. The
amplitude scalogram is then
E^τ,a2=A^τ,a2+B^τ,a2.
With smoothing
Like in Part 1, estimating the amplitude is more robust against noise when a
smoothing procedure is performed. We apply to the squared amplitude,
Eq. (), the same kind of smoothing as for the
scalogram (see Eq. ), giving
E^τ,a2=12γω0a∫τ-γω0aτ+γω0adτ′Φ^τ,a(m+2)2+Φ^τ,a(m+3)2.
Appendix provides further details on the
practical implementation of the bounds of integration.
The weighted smoothed scalogram
The weighted smoothed scalogram is the analogue of the weighted WOSA periodogram, defined in Sect. 7 of Part 1, and its objectives are the same,
i.e. to keep the advantages of both the amplitude scalogram and the
scalogram, namely,
provide an estimation of the squared amplitude of a signal, locally in the time–frequency plane, by weighting the scalogram like in
Eq. ();
conserve the advantage of the formalism of orthogonal projections in order to avoid the matrix inversions required for the computation of the
amplitude scalogram (see, for example, Eq. , relying on Eq. which requires a matrix inversion).
The last item is useful for building confidence levels when performing a test
of significance (see Sect. ).
The disadvantage of weighting the smoothed scalogram is that it no longer
provides a flat pseudo-wavelet power spectrum for a white noise
signal (see
Sect. ),
analogously to its frequency counterpart (see Sect. 7 of Part 1). The
weighted smoothed scalogram is derived from Eq. (),
in which the integrand is weighted by the right-hand-side weight of
Eq. (), namely,
||Psmoothed(τ,a)|X〉||2=12γω0a∫τ-γω0aτ+γω0adτ′2tr(Gτ′,a2)tr(Gτ′,a)2||Psp‾{|t0〉,|t1〉,…,|tm〉,|Gτ′,aca〉,|Gτ′,asa〉}-Psp‾{|t0〉,|t1〉,…,|tm〉}|X〉||2.
Appendix provides further details on the
practical implementation of the bounds of integration. We recommend the use
of the weighted smoothed scalogram in most time–frequency analyses under
irregular sampling.
Cone of influence
When the wave packets |Gτ,aca〉 and |Gτ,asa〉
intersect the borders of the time series, a part of their support can stand
after the last point of the time series, or before the first point of the
time series. Consequently, one has to remove two half-cones from the area
under analysis. From Eq. (), the support of the wave
packets is approximately equal to 2βω0a, so that the excluded
areas are given by
{τ,a}such that|τ-t1|≤βω0a and |τ-tN|≤βω0a,
with β=3 (conservative choice) or β=2choice
in. We recommend the conservative choice. When smoothing is
performed, Eq. () becomes
{τ,a}such that|τ-t1|≤(β+γ)ω0aand|τ-tN|≤(β+γ)ω0a,
where γ controls the smoothing length; see Eqs. (), () and
().
This has another implication: the maximal scale available by the analysis is
amax=tN-t12(β+γ)ω0.
Aliasing and Shannon–Nyquist exclusion zone (SNEZ)
(a) The time series |X〉=sin(2π|t〉/0.01)
and (b) its time step, with the vertical axis in log-scale. The time
vector |t〉 is taken from a real palaeoclimate time series
(Liviu Giosan, WHOI, personal communication,
2017).
When probing the irregularly sampled time series with the wavelet packet, it
may happen that the period of the oscillation inside the packet, 2πa, is
too low compared to the local time step in the time series, therefore causing
aliasing issues according to the Shannon–Nyquist theorem, locally in the
time–scale plane. As stated in Part 1, this issue also happens with the WOSA
periodogram. We adapt formulas (I,72), (I,73) and (I,74) and define the local
time step as
Δt‾τ,a=max{Δt‾Gτ,a,Δt‾Hτ,a},
where
Δt‾Gτ,a=∑k=1NGτ,ak,kΔtcktr(Gτ,a),Δt‾Hτ,a=∑k=1N-1Hτ,ak,kΔtktr(Hτ,a),Δtk=tk+1-tk,∀k∈{1,…N-1},Δtck=tk+1-tk-12,∀k∈{2,…N-1},Δtc1=t2-t1,ΔtcN=tN-tN-1,
and Hτ,a is a diagonal matrix with
Hτ,ak,k=exp-tk+tk+12-τ22ω02a2,k∈{1,…,N-1}.
We then apply the Shannon–Nyquist theorem to this local time step, namely:
Compute the scalogram at (τ,a) if a≥aSNEZ(τ),
where
aSNEZ(τ) is the largest solution of a=Δt‾τ,aπ.
We call Shannon–Nyquist exclusion zone, the area in the
scalogram that does not satisfy Eq. () and
which is therefore delimited by aSNEZ. Note that matrix
Hτ,a is similar to matrix Gτ,a, defined in Eq. (), but with elements taken at (tk+tk+1)/2 instead
of tk. Quantity Δt‾τ,a is equal to the maximum
between the average weighted time step and the average weighted central time
step.
Weighted (unsmoothed) scalogram of the time series presented in Fig. a. (a) No correction for aliasing.
(b) Corrected with Δt‾τ,a=Δt‾Gτ,a. (c) Corrected with Δt‾τ,a=Δt‾Hτ,a. (d) Corrected with
Δt‾τ,a=max{Δt‾Gτ,a,Δt‾Hτ,a}.
We now justify Formula () with an example. Consider the
function X(t)=sin(2πt/0.01), sampled on an irregular grid. This is
drawn in Fig. a. The time step is represented in
Fig. b. These two figures show that the time series
exhibits intervals where it is more or less regularly sampled, separated by
large gaps. The weighted (unsmoothed) scalogram is drawn in
Fig. a. We remind the reader that the weighted scalogram is
supposed to estimate the local squared amplitude in the time–frequency
plane. Since X(t) has an amplitude equal to 1, we expect that the maximal
power of the scalogram be equal to 1, along a scale corresponding to the
period of x, for all τ. Because of the large gaps in the time series,
extended regions corrupted by aliasing occur in Fig. a,
resulting in a maximal power for the scalogram which is much greater than
1. Figure b, c and d present the weighted scalogram
corrected by the SNEZ. In Fig. b the SNEZ is computed with
Δt‾τ,a=Δt‾Gτ,a. We
observe that it does a good job of rejecting the areas where aliasing occur,
although it is desirable that the black areas peak on higher scales. In
Fig. c, the SNEZ is computed with Δt‾τ,a=Δt‾Hτ,a. We observe that most
of the aliasing-related areas are rejected, although we would rather the
black areas be wider. Finally, the SNEZ computed with
Δt‾τ,a=max{Δt‾Gτ,a,Δt‾Hτ,a} is
drawn in Fig. d and we observe that it does a very
satisfactory job at rejecting the areas where aliasing occur.
The SNEZ is applied to all the analysis tools defined above. When smoothing
is to be applied, it is performed on the areas outside of the SNEZ, since the
scalogram is not computed in the SNEZ. In the neighbourhood of the SNEZ,
adjustments of the smoothing procedure are therefore necessary, as explained
in Appendix .
Scalogram of the time series |X〉=sin(2π|t〉/10),
where |t〉 has a piecewise constant time step. From t=0 to t=200,
Δt=4. From t=200 to t=400, Δt=3. From t=400 to t=600,
Δt=2. (a) Weighted (unsmoothed) scalogram. The black area is
the SNEZ. (b) Same as panel (a) with the addition of the
refinement of the SNEZ, which is the shaded area on the top of the SNEZ.
(c) Amplitude scalogram (unsmoothed). The black area is the SNEZ.
(d) Same as panel (c) with the addition of refinement of
the SNEZ, which is the shaded area on the top of the SNEZ. In the four panels,
the bounds of the colour scale are the extrema of the scalogram over the
non-shaded area. Thanks to the refinement of the SNEZ, the upper bound of the
colour scale is close to 1, which is the value of the squared amplitude of
the signal |X〉.
From the scale to the period
Scale-to-period conversion is performed in the continuous limit, with Eq. (). The first case of Eq. (), with c(a)∼1/a, corresponds to
estimators of the amplitude, and is then used for scale-to-period
conversion with the amplitude scalogram (all the formulas of
Sect. ) and for the weighted smoothed scalogram, Eq. (). The second case of Eq. (), with c(a)∼1/a, is used for
scale-to-period conversion with the unweighted scalogram, that is, the
formulas appearing in Sect. , and
.
Refining the Shannon–Nyquist exclusion zone
As illustrated in Fig. , the Shannon–Nyquist exclusion
zone may not to be sufficient to avoid all the patches due to aliasing,
because of the correlations between neighbouring scales in the scalogram. We
therefore extend the Shannon–Nyquist exclusion zone by considering the
continuous limit case for the simple periodic signal x(t)=exp(i2πt/TSNEZ(τ)), where TSNEZ(τ) is the period at
the border of the SNEZ, determined by Eqs. () and
(). Its scalogram is given in Eq. (). In order to make the correspondence with all
the above formulas, three cases are considered.
c(a)∼1/a: in this case, we have |S(τ,a)|2∼exp(-ω02(2πa/TSNEZ(τ)-1)2), and the standard
deviation for the scale is then
σa,1(τ)=TSNEZ(τ)/22πω0. The border
of the extended Shannon–Nyquist exclusion zone at time τ is
therefore on scale aSNEZ(τ)+βσa,1(τ), where
β is a coefficient estimating the half-support of Gaussian-shaped
functions (it is defined in Sect. ).
c(a)∼1/a and work with |S(τ,a)|: in this case, we have |S(τ,a)|∼exp(-ω02(2πa/TSNEZ(τ)-1)2/2),
and the standard deviation for the scale is then σa,2(τ)=TSNEZ(τ)/2πω0. The border of the extended Shannon–Nyquist
exclusion zone at time τ is therefore on scale aSNEZ(τ)+βσa,2(τ).
c(a)∼1/a: in this case, we have |S(τ,a)|2∼aexp(-ω02(2πa/TSNEZ(τ)-1)2). We know from
Eq. () that the scalogram is at its maximum on the scale aSNEZ(τ)=TSNEZ(τ)(ω0+ω02+2)/4πω0.
The pseudo-standard deviation is computed such that aexp(-ω02(2πa/TSNEZ(τ)-1)2) decreases from its maximum by the same
percentage as in case 1, namely, βσa,3(τ) is equal to the largest of the two solutions ofaexp-ω02(2πa/TSNEZ(τ)-1)2=aSNEZ(τ)exp-ω02(2πaSNEZ(τ)/TSNEZ(τ)-1)2exp-β2/2,
in which the unknown is a. The border of the extended Shannon–Nyquist
exclusion zone at time τ is therefore on scale
a=aSNEZ(τ)+βσa,3(τ).
Case 1 is used with formulas giving the squared amplitude
E^τ,a2 in Sect. and with the
weighted smoothed scalogram, Eq. (). The
(unsquared) amplitude E^τ,a can also be of interest, and
case 2 is therefore used. Case 3 is used with formulas arising in
Sect. , and
. Finally, note that the refinement of the SNEZ
is performed after the smoothing procedure, because an extension of the SNEZ
may result from the smoothing, as explained in
Appendix .
Discretising τ and a
With regularly sampled data, the discretised variable τ is usually equal
to |t〉like in, or a subset of |t〉
with regularly spaced elements. For irregularly spaced time series, we opt
for the same type of grid as in the regularly sampled case, i.e. a linear
regular grid, namely
τk=τ0+kΔτ,k∈{0,…,K},with τ0≥t1 and τK≤tN.
The scales are commonly discretised as fractional powers of two
, namely
aj=amin2jδj,j=0,…,J,
where
J=log2amax/amin/δj.
Here, amin is the minimum over τ of aSNEZ (defined
in Eq. ), and amax is defined in Eq. (). Discretisation as a power law comes from the geometry of
the wavelet transform, and is justified in
Appendix .
The integrals in Eqs. (),
() and () are
discretised with the rectangle method. In particular, the discretisation of
the integrals in Eqs. () and
() allows these formulas to be written as
finite-size matrices. To this end, we apply a Gram–Schmidt orthonormalisation
to the orthogonal projections, like in Eq. (I,67). This gives
||Psmoothed(τ,a)|X〉||2=〈X|Mτ,aMτ,a′|X〉,
which is the analogue of Eq. (I,68). Mτ,a is a matrix of
size (N,2ncol(τ,a)), ncol(τ,a)≥1, where
ncol is a non-trivial function depending on the scale and on the
closeness of (τ,a) with the SNEZ and with edges of the time–frequency
plane.
Significance testing with the scalogramHypothesis testing
We test for the presence of periodic components, locally in the
time–frequency plane. Significance testing is mathematically expressed as a
hypothesis testing. Taking our model, Eq. (), the null
hypothesis is
H0:Aτ,a=Bτ,a=0.
Therefore, |X〉=|Trend〉+|Noise〉. The
alternative hypothesis is
H1:Aτ,a and Bτ,a are not both zero.
The decision of accepting or rejecting the null hypothesis is based on the
scalogram (Eq. ), independently for each
couple (τ,a) (this is called pointwise testing). Concretely,
for each couple (τ,a), we compute the distribution of the scalogram
under the null hypothesis, and then see if the data scalogram at
(τ,a) is above or below a given percentile of that distribution. The
percentile is called level of confidence. If the data scalogram 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.
To perform significance testing, we thus need
to estimate the parameters of the process under the null hypothesis (this is studied in Sect. 5.2 of
Part 1);
to estimate the distribution of the scalogram under the null hypothesis (this is studied in Sect. below).
Finally, we mention that, for regularly sampled time series, the pointwise
significance test can be supplemented with an
area-wise significance
test, which takes into account the correlations between neighbouring points
in the time–frequency plane. This is introduced in and
studied in detail in . Applying this method to irregularly
sampled series is way beyond the scope of this work, since the correlations
between neighbouring points in the time–frequency plane are highly
irregular.
Estimation of the distribution of the scalogram under the null hypothesis
Analytical confidence levels in function of the number of conserved
moments, at six particular couples (τ,a), for the scalogram of the time
series presented in Sect. . Parameter γ
is equal to 0.5. (a) 95th percentile. (b) 99.9th
percentile. Slow convergence as well as numerical instabilities (spurious
peaks) at high numbers of conserved moments are observed. Convergence cannot
therefore be numerically guaranteed.
The results obtained for the periodogram in Sect. 5.3 of Part 1 are valid for
the scalogram, with minor changes that we detail below.
Monte Carlo approach: the same procedure as in Part 1 is applied to the (weighted) smoothed scalogram, Eq. ()
or (). We can thus estimate the confidence levels for the (weighted) smoothed scalogram taking into account the uncertainty in the parameters of the background noise.
Analytical approach (with a unique set of CARMA parameters):
Theorem 1 of Part 1 can be applied to the (weighted) smoothed scalogram, as follows.
The (weighted) smoothed scalogram, defined in Eq. (), under the null hypothesis (), is
The symbol =d means “is
equal in distribution”.
||Psmoothed(τ,a)|X〉||2=d∑k=12ncol(τ,a)λk(τ,a)χ1k2,where |X〉=∑k=0mγk|tk〉+K|Z〉 and K is the
CARMA matrix defined in Eq. (I,20) or (I,38). The χ1k2
distributions are iid, and λ1(τ,a), ..., λ2ncol(τ,a)(τ,a) are the eigenvalues of
Mτ,a′KK′Mτ,a and are non-negative. Matrix Mτ,a is
defined in Eq. ().
The pseudo-wavelet power spectrum, WPS^, is the analogue of the pseudo-spectrum defined in Eq. (I,92).
It is defined as the expected value of the (weighted) smoothed scalogram distribution, namelyWPS^(τ,a)=∑k=12ncol(τ,a)λk(τ,a)=trMτ,a′KK′Mτ,a.
For a Gaussian white noise background with variance σ2, the unweighted pseudo-wavelet power spectrum is flat, and is
equal to 2σ2, for all (τ,a). Moreover, if the scalogram is not smoothed, it is exactly chi-square distributed with 2 degrees of freedom:||Psp‾{|t0〉,|t1〉,…,|tm〉,|Gτ,aca〉,|Gτ,asa〉}-Psp‾{|t0〉,|t1〉,…,|tm〉}σ|Z〉||2=dσ2χ22,
where |Z〉 is a standard Gaussian white noise.
The variance of the distribution of the (weighted) smoothed scalogram at (τ,a) is equal to 2||Mτ,a′KK′Mτ,a||F2, where ||⋅||F is the Frobenius norm.
We approximate the linear combination of the independent chi-square distributions, appearing in Eq. (),
by a gamma-polynomial distribution conserving its first d moments, based on
the theory developed in . The formulas are given in
Sect. 5.3.3 of Part 1.
We observe, however, that the convergence of the percentiles (as the number
of conserved moments grows) strongly depends on the smoothing coefficient
γ, defined in Eqs. () and
(). As a general rule, the larger
γ is, the faster the convergence. Moreover, it turns out that the
gamma-polynomial approximation becomes numerically unstable at large numbers
of conserved moments, because the matrix in Eq. (I,100) becomes singular.
Consequently, for relatively small values of γ, convergence cannot be
numerically guaranteed. This is illustrated in Fig. . In such cases, a simple 2-moment
approximation is therefore a reasonable choice since it is always numerically
stable, it is much quicker than with higher numbers of conserved moments from
a computational point of view, and it provides a satisfactory approximation.
A comparison between the computing times of the Monte Carlo approach and the analytical approach is presented in Appendix .
(a) The time series and its 7th-degree polynomial trend.
(b) The age step, [tk-tk-1], ∀k∈2,…,N, and
its distribution.
Filtering with the amplitude scalogramBand filtering
From Sect. , Eq. ()
gives A^τ,a and B^τ,a. We can therefore
reconstruct the signal
A^τ,a|ca〉+B^τ,a|sa〉 over
the whole time–scale plane, i.e. for all (τ,a). Band filtering is
performed by averaging the reconstructed signal between scales
ajmin and ajmax, namely
Xfilt(τ)=1jmax-jmin+1∑j=jminjmaxA^τ,aj|caj〉+B^τ,aj|saj〉,
where the discretised scale is defined in Eq. (). Such a
filtering procedure is a generalisation of the scale-averaged wavelet power of which deals with trendless regularly sampled
signals. Note that we use the formulas for which there is no smoothing.
Indeed, the smoothing procedure in Eq. () does
not give access to A^τ,a and B^τ,a (only
the sum of their squared values is available). An example of band filtering
is shown in Figs. and
.
Ridge filtering
Consider a signal |X〉=Ecos(ω|t〉+ϕ). We can easily
reconstruct the signal from the estimated amplitudes A^τ,a
and B^τ,a, given by Eq. (), taken at the
maximum of the scalogram, in this case at a=1/ω. More generally, we
can reconstruct more complex signals relying on the theory of the
amplitude ridges, developed for continuous-time signals
(Sect. ) and which can approximately be applied to
irregularly sampled time series. An example of ridge filtering is shown in
Figs. and .
The global scalogram
Analogously to the global wavelet spectrum of
for trendless regularly sampled time series, we define here the global
scalogram as the scalogram averaged over time. Technically, it is nothing but
the smoothed scalogram (Eqs. ,
or ) with
integration over the whole interval of the analysis time τ. We can write
the discretised global scalogram under a similar matrix form as in
Eq. (), and find the confidence levels
according to Sect. . Compared to
the periodogram defined in Part 1, which has a fixed bandwidth, the global
scalogram has a varying bandwidth with the frequency. From
Fig. of
Appendix , we deduce
that the global scalogram exhibits a frequency resolution that gets better
when the frequency decreases. Examples of global scalograms are given in
Sect. .
Weighted smoothed scalogram (left) and its global scalogram (right)
with ω0=5.5. The 95 % analytical confidence levels (green) and
95 % MCMC confidence levels (magenta), against a red noise background,
are also drawn. Note that the green and magenta contours are almost
superposed. The two lateral shaded areas are the half-cones of influence, the
bottom black area is the SNEZ, and the shaded area above the SNEZ is the
refinement of the SNEZ. There are also two lateral black areas, where the
scalogram is not computed, because of the fixed-length smoothing per scale.
The bounds of the colour scale are the extrema of the scalogram over the
non-shaded area. As we work with the weighted scalogram, the power is an
estimation of the local squared amplitude. Dashed lines at usual palaeoclimate
periods are also drawn.
Weighted smoothed scalogram (left) and its global scalogram (right)
with ω0=15. The 95 % analytical confidence levels (green) and
95 % MCMC confidence levels (magenta), against a red noise background,
are also drawn. Note that the green and magenta contours are almost
superposed.
Application on palaeoceanographic dataPreliminary analysis
The time series we use to illustrate the theoretical results is the benthic
foraminiferal δ18O record from , which holds
608 data points with distinct ages and covers the last 6 million years.
The choice of a CARMA(1,0) process as the background stationary noise, as
well as the choice of m=7 for the degree of the polynomial trend, are
justified in Sect. 9 of Part 1, in which the same data set is used as an
example of frequency analysis. The time series, its trend and its time step
are drawn in Fig. . We remind the
reader that the time series is not detrended before computing the scalogram
of the data, but it is detrended before estimating the confidence levels.
Time–frequency analysis
The weighted smoothed scalogram
(Sect. ) and its 95 % analytical
and MCMC confidence levels are presented in Fig.
with parameter ω0=5.5, and in Fig. with
parameter ω0=15. As explained in Sect. ,
increasing ω0 results in a better frequency resolution and a worse
time resolution. In our example, the scalogram with ω0=15 exhibits
more clearly the period band around 40 kyr and the changes in
amplitude along that band. The form of the SNEZ, which is the black region at
the bottom in Figs. and ,
follows from the sampling of the time series presented in Fig. b.
The unsmoothed estimated amplitude (which is the square root of the
amplitude scalogram, Eq. ), jointly with the filtering
band in the interval [35, 45] kyr (shaded). Black and white curves are the
ridges. They go through the local maxima of the amplitude scalogram. The
white ones are the ridges in the band [35, 45] kyr. Parameters are
ω0=15, β=3 and δj=0.01.
The parameters are γ=0.5 (smoothing coefficient), 2 is the number
of conserved moments in the gamma-polynomial approximation (see the
discussion in
Sect. ),
a fixed-length smoothing per scale (see
Appendix ), β=3 (half-support of a
standard Gaussian function exp(-x2/2)) and δj=0.05 (coefficient
for the scale resolution).
Filtered signal in the band [35, 45] kyr. (a) Band
filtering. (b) Ridge filtering. The red curve is the amplitude of
the filtered signal, which is only available with ridge
filtering.
Filtering
As explained in Sect. , band and
ridge filtering are performed on the unsmoothed amplitude scalogram. This is
illustrated in Fig. , with a filtering band in
the interval [35, 45] kyr and with the ridges. From the whole set
of the ridges, we select those in the band [35, 45] kyr in order
to make a comparison with the band filtering. Band and ridge filtered signals
are shown in Fig. . We can see that the
amplitude modulations in Fig. a and
b are consistent. Compared to band
filtering, the ridge filtering method has the advantage of representing the
signal for which the amplitude scalogram is locally maximal, and also allows
the time-varying amplitude of the filtered signal to be reconstructed (in red in Fig. b). The drawback is that it rarely
delivers a continuous reconstruction with climate data.
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 Part 1, and
time–frequency analysis relies on the theory developed in this article. It
is available at https://github.com/guillaumelenoir/WAVEPAL.
Conclusions
We defined the scalogram as an extension of the generalised Lomb–Scargle
periodogram developed in Part 1. This analysis tool is well suited for
irregularly sampled time series which can be modelled as a locally periodic
component in the time–frequency plane, plus a polynomial trend, plus a
Gaussian CARMA stochastic process. In the particular case of trendless
regularly sampled times series, we showed that the unsmoothed scalogram gives the same results as with the
traditional algorithms such as in . A smoothing
procedure, by averaging over neighbouring points in time, was then applied to
the scalogram in order to reduce its variance. Besides, we derived estimators
of the amplitude of the locally periodic component, based on the general
results of Part 1, and proposed an approximate formula linking the scalogram
and the squared amplitude. The latter result is at the basis of the weighted
smoothed scalogram, which is the analysis tool that we recommend for most
time–frequency analyses. We then showed that local aliasing issues may occur
in the analysis tools previously derived, implying the delimitation of a
forbidden area for the analyses, called the Shannon–Nyquist exclusion zone.
Moreover, a test of significance for the scalogram was designed, similarly to
its counterpart for the frequency analysis developed in Part 1. Finally, the
classical filtering procedures, namely band and ridge filtering, were made
available for use with our operator of the estimated amplitude.
The Python code generating the figures of this article is
available in the Supplement.
Fourier analysis of functions
L2(R) is the space of measurable functions on
R with finite energy:
||f||L22=∫-∞+∞dt|f(t)|2<∞.
This defines the squared norm for such functions, that we denote simply by
||f||2 in Sect. . We provide the
L2 space with the usual inner product:
〈f|g〉L2=∫-∞+∞dtf(t)‾g(t),
which makes it a Hilbert space; f(t)‾ denotes the complex
conjugate of f(t), and 〈f|g〉L2 is denoted by
〈f|g〉 in Sect. .
Strictly speaking, the Fourier transform and the convolution product cannot
be defined on L2(R). We therefore restrict to the
Schwartz space, S(R), which is a subspace of
L2(R), and on which the Fourier transform and the
convolution product can be defined. The Schwartz space is defined as follows:
f∈S⟺f∈C∞,∀m,j∈N:|t|m|f(j)(t)|→0when |t|→∞, i.e. f and all itsderivatives are rapidly decreasing.
The Fourier transform of f∈S(R) is defined by
f^(ω)=12π∫-∞+∞dtf(t)exp(-iωt),
and f^ is also in S(R). The inverse Fourier
transform is
f(t)=12π∫-∞+∞dωf^(ω)exp(iωt).
Some properties of the Fourier transform are listed below.
Parseval–Plancherel identities: 〈f|g〉=〈f^|g^〉 and ||f||2=||f^||2.
Convolution theorem: [f⋆g]^(ω)=2πf^(ω)g^(ω), where the convolution
product between f and g is (f⋆g)(t)=∫-∞+∞dt′f(t-t′)g(t′).
Translation–modulation: the Fourier transform of f(t-b) is exp(-iωb)f^(ω).
Dilation: the Fourier transform of f(at), a≠0, is 1|a|f^(ω/a).
Fourier uncertainty principle for the Morlet wavelet
The Fourier uncertainty principle states that the temporal variance and the
frequency variance of a function f∈S(R) satisfy
σt2σω2≥14,
where
σt2=12π||f||2∫-∞+∞dt(t-u)2|f(t)|2,
and
σω2=12π||f||2∫-∞+∞dt(ω-ξ)2|f^(ω)|2.
Here, μ and ξ are the average time and average frequency and are defined
with the same densities as for the variances. For the Morlet wavelet, the
densities are |ψa♯(t)|2 (from Eq. )
and |ψa♯^(ω)|2 (from Eq. ), up to a normalising multiplicative factor. As
they are Gaussian functions, their variances are trivial, and we have
σt2σω2=ω02a2212ω02a2=14.
This is equal to the lower bound of the inequality, as expected for Gaussian
functions seep. 43, for additional details
In that book,
the Fourier uncertainty principle is called Heisenberg uncertainty theorem, but we will not use this misattribution in a non-quantum
context.
. It is in that sense that the Morlet wavelet is said
to be ideally localised.
Uncertainty boxes and scale discretisationTime–frequency resolution and uncertainty boxes
We saw in Appendix that the standard
deviations of the continuous-time density |ψa♯(t)|2 and
continuous-frequency density |ψa♯^(ω)|2 are
σt=ω0a/2 and σω=1/2ω0a
respectively. Moreover, the centre angular frequency of
|ψa♯^(ω)|2 is ω=1/a, from
Eq. (). With all these coefficients and
Eqs. () and (), we can draw rectangles,
that we call uncertainty boxesp. 109
In that book,
the author calls them Heisenberg boxes, but as previously mentioned,
this is a misattribution in a non-quantum context.
, in the
time–frequency plane indicating the energy spread around each couple
(t,ω), or equivalently, the time–frequency resolution at each couple
(t,ω). This is illustrated in Fig. . Note
that their area is equal to σtσω=1/2 and is therefore
constant.
Scale discretisation
Scale discretisation is naturally based on the geometry of the boxes. We can,
for example, require that the frequency component of the centre of mass of
the box corresponding to scale aj be at the frequency of the border of the
box corresponding to scale aj+1. This is illustrated in Fig. . We obtain
1aj=1aj+1+β2aj+1,
where β is defined in Fig. , giving
aj+1=2+β2aj,
and by recurrence,
aj+1=2+β2ja1,j∈N.
Multiplying β by a positive factor, γ, allows to control the
density of the discretised scales. With variable change δj=log2[(2+βγ)/2], we obtain
aj+1=2jδja1,δj>0,j∈N.
Uncertainty boxes for the Morlet wavelet, with
α=ω0/2 and
β=1/2ω0.
Example of rule for the discretisation of scales taking into account
the geometry of the uncertainty boxes.
β=1/2ω0>0.
Smoothing the (amplitude) scalogram: technical details
In the formulas of the smoothed (amplitude) scalogram, Eqs. (), () and
(), the integration is in principle
performed over the interval [τ-γω0a,τ+γω0a].
When this interval intersects the edges of the time–frequency plane or the
SNEZ, we are no longer able to integrate over the full interval. Two choices
are then possible.
Keep the length of integration equal to 2γω0a, and therefore exclude from the analysis some areas of the time–frequency
plane. This results in two excluded zones at the time borders of the scalogram and in an extension of the SNEZ.
Shorten the interval of integration in order to not exclude from the analysis any extra region of the time–frequency plane.
Both options are available in WAVEPAL and we recommend the first one in
order to keep a consistent degree of smoothing at each point (τ,a) in
the time–scale plane.
Other studies on the scalogram for irregularly sampled time seriesIntroduction
We review the only two rigorous studies that we have found in the literature
about the estimation of the scalogram for irregularly sampled time series.
Like our theory, they are based on the Lomb–Scargle periodogram, and define a
kind of scalogram of the CWT for the Morlet mother wavelet. However, these
theories are too restrictive to have an interest in geophysical applications.
Foster's theoryIntroduction
In this section, we derive and comment on the formulas published in
, and based on developments published in
. Foster's theory is restricted to
the case of the unsmoothed scalogram applied to signals with an additive
Gaussian white noise and a piecewise trend for which the shape is the
envelope of the Morlet wavelet. It also defines something similar to our
amplitude scalogram and generalises the F periodogram of Part 1. We show that
some of its formulas can be deduced from our general theory seefor
the original derivation of the formulas, which is rather different from our
approach. Foster's formulas are available for use in a Fortran
code provided by the American Association of Variable Star Observers (AAVSO);
see https://www.aavso.org/sites/default/files/software/wwz.tar.gz.
Foster's approximation and weighted inner products
Let us start with the approximation made in and used in
. Define U as equal to a full rank real matrix, whose
columns are the vectors generating the vector space on which we project the
data vector |X〉, the latter belonging to RN. Define G
as equal to a real diagonal square matrix of size N with positive elements.
Foster's approximation Eq. 7.9 writes
In
, the author works with tensor notations, so that the
equivalence is not direct.
U′G2U≈tr(G2)tr(G)U′GU.
Note that, when U is a 2-column matrix holding a cosine vector and a sine
vector, the above approximation can also be obtained from Eq. (I,127). The
orthogonal projection on the span of GU thus becomes
Psp‾{GU}=GU(U′G2U)-1U′G≈tr(G)tr(G2)GU(U′GU)-1U′G,
and, for any pair of vectors |Y〉 and |W〉 in RN,
we have
〈Y|Psp‾{GU}|W〉≈Neff〈Y|GUtr(G)(U′GU)-1tr(G)U′G|W〉tr(G),
where Neff=tr(G)2tr(G2) is defined
in Eq. 7.7 and called the effective number of data points. We can actually rewrite the right-hand side of Eq. () as Neff〈Y|Psp‾{U}|W〉Weighted, where the
weighted inner product is defined by:
〈Y|W〉Weighted=〈Y|G|W〉tr(G),
for any |Y〉 and |W〉 in RN. The term 〈⋅|⋅〉Weighted satisfies the requirements of an
inner product since the elements of G are positive seep. 43. Foster's theory is developed in a vector space provided
with this weighted inner product.
WWT
Now, we derive Foster's scalogram from our theory. The diagonal elements of
the weight matrix G are as follows Eq. 5-3:
Gτ,ωkk=exp-cω2(tk-τ)2.
Correspondence with our weight matrix, defined in Eq. (),
is performed with the variable changes ω=1/a and c=1/2ω02.
Next, consider the formula of the unsmoothed scalogram, Eq. (), with a=1/ω, and transformed to
accommodate for a trend given by |Gτ,ωt0〉. This results
in
||Psp‾{|Gτ,ωt0〉,|Gτ,ωcω〉,|Gτ,ωsω〉}|X〉||2-||Psp‾{|Gτ,ωt0〉}|X〉||2.
We then make use of the approximation of Eq. () with
U=[|t0〉|cω〉|sω〉] for the first
projection and U=|t0〉 for the second projection, resulting in the
following formula:
Neff||Psp‾{|t0〉,|cω〉,|sω〉}|X〉||Weighted2-||Psp‾{|t0〉}|X〉||Weighted2,
for which we now work in a vector space provided with the weighted inner
product. If |X〉 is a zero-mean Gaussian white noise, Formula () follows exactly a chi-square distribution with
2 degrees of freedom multiplied by the variance of the white noise, namely
σ2χ22. Consequently, under the null hypothesis that the process
is a Gaussian white noise, the following expression,
WWT=Neff2σ2||Psp‾{|t0〉,|cω〉,|sω〉}|X〉||Weighted2-||Psp‾{|t0〉}|X〉||Weighted2,
approximately follows a chi-square distribution with 2 degrees of freedom
and expected value 1. Formula () is rigorously the
same
Note that Eq. (5-10) of , which is
prerequisite for the formula of the WWT, is
probably erroneous, making unclear the correspondence with our
Eq. (). However, the formula given here in
Eq. () is strictly the same as the WWT encoded at
https://www.aavso.org/sites/default/files/software/wwz.tar.gz.
as the
weighted wavelet transform (WWT) of , in which
the author estimates σ2 as
σ^2=NeffNeff-1〈X|X〉Weighted-〈t0|X〉Weighted2.
Significance testing against a Gaussian white noise can be therefore be performed with the WWT.
Below, we comment on the WWT and make a comparison with our formulas.
The WWT is built on the assumption that the time series holds a Gaussian-shaped trend centred at the probed translation
time τ, the support of which varies with the probed frequency. This is equivalent to a constant trend in the vector space
provided with the weighted inner product. This contrasts with our choice for the trend, Eq. (), which is independent of the analysis function.
The WWT under the null hypothesis is only approximately chi-square distributed, compared to Formula () which is exactly chi-squared-distributed.
The estimation of the variance of the white noise, σ^2, which is part of the WWT formula, depends on the sampling.
However, two realisations of a white noise are uncorrelated regardless of the
time step separating them, and the estimation of its variance should thus be
independent of the sampling, like in Sect. 5.2.2 of Part 1.
To our point of view, working with weighted inner products, approximations like in Eq. () and complicated
tensor notations see does not bring a simple and unified view of the problematic.
WWA
The weighted wavelet amplitude (WWA), defined in Eq. 5–14, is similar to
our amplitude scalogram defined in Eq. (). The former is obtained from the latter taking the
trend to be |Gτ,ωt0〉, where Gτ,ω is defined
in Sect. . For practical applications, we note that
computing the inverse of a matrix is needed for the computation of the WWA
(this is also the case for our amplitude scalogram). But Foster's theory
lacks an in-depth consideration of aliasing issues, and the WWA at some
points of the time–frequency plane may be numerically infinite due to the
occurrence of singular matrices caused by aliasing.
WWZ
Under the null hypothesis that the data |X〉 is a Gaussian white
noise, its squared norm in the vector space provided with the weighted inner
product is approximately chi-square distributed with Neff degrees
of freedom, as this follows from the 2-moment approximation of Sect. 5.3.3 of
Part 1, in which formula (I,98) is applied to matrix G.
Consequently, under the null hypothesis, the following formula,
(Neff-3)||Psp‾{|t0〉,|cω〉,|sω〉}|X〉||Weighted2-||Psp‾{|t0〉}|X〉||Weighted22||X||Weighted2-||Psp‾{|t0〉,|cω〉,|sω〉}|X〉||Weighted2,
is approximately equal to the Fisher-Snedecor distribution with 2 and
Neff-3 degrees of freedom. Formula () is defined
in Eq. 5–12 and called the weighted wavelet Z-transform (WWZ). It generalises the F periodogram that we defined in
Sect. 5.4 of Part 1.
Mathias et al. (2004)'s theory
present a formula similar to our amplitude scalogram in
the case of a trendless signal. The difference with our
Eq. () is that they work with a complex
exponential, exp(iω|t〉), instead of sine and cosine. Switching these terms in our
Eq. () and taking a=1/ω gives the Eq. (17)
of . The authors then approximate the Gaussian shape of
the Morlet wavelet by a function with a finite support. Based on that
approximation, they develop a fast algorithm for the computation of the
scalogram. Apart from this advantage, this study is in fact more restrictive
than Foster's theory, since it does not perform significance testing, a zero
trend is assumed, no smoothing procedures are considered, and it does not
tackle the problem of aliasing issues explained in
Sect. and .
Warning about interpolating the time series
δ18O signal from the GISP2 ice core ,
for which the first 11 kyr are removed. Raw (red dots) and interpolated
(blue line) time series, with (a)Δt=30 yr and
(b)Δt=300 yr.
Time step of the δ18O signal from the GISP2 ice
core.
Scalogram of the time series presented in
Fig. and the 95 % analytical confidence levels
against a red noise. (a) Raw time series. (b) Interpolated
time series with Δt=30 yr. (c) Interpolated time series
with Δt=300 yr.
This appendix compares the scalograms and their confidence levels in the case
of interpolated and non-interpolated time series. The time series we consider
is the δ18O signal from the GISP2 ice core ,
for which the first 11 kyr are removed in order to facilitate the
detrending procedure. The time series is drawn in
Fig. a and b, and its time step is given in
Fig. . The interpolated time series is
built on a time grid with Δt=30 yr (this is the smallest time
step of the raw time series) in Fig. a, or Δt=300 yr in Fig. b. The (unsmoothed) scalograms with
ω0=15 of the raw and interpolated time series are shown in
Fig. , jointly with the 95 % analytical
confidence levels against a red noise. We observe that significance testing
is strongly dependent on the interpolation procedure. This is because the
parameters of the red noise are badly estimated when the time series is
interpolated. Consequently, in general, we cannot rely on interpolated time
series to perform significance testing. In particular, we draw the attention
on the geological stacks (such as in , or
), which are composed of multiple interpolated time series
and averaged together. Significance testing or analysis of the background
noise for such time series may therefore be strongly biased.
Finally, we observe that, in this example, the power of the scalogram of the
data is weakly affected by the interpolation.
Computing time: analytical versus Monte Carlo significance levels
Computing times for generating the scalogram 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.
A comparison between the computing times, for generating the scalogram, with
the analytical and with the MCMC confidence levels, based on the hypothesis
of a red noise background, is presented in Fig. . The computing
times 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 2-moment
approximation. The number of samples for the MCMC approach is 10 000 for the
95th percentiles and 100 000 for the 99th
percentiles. The smoothing coefficient is γ=0.5, and 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.
.
With this parametrisation, and within this interval of the number of data
points, we see that the analytical approach is faster than the MCMC approach.
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. Note, however, that the 2-moment approximation, for the estimation
of the analytical confidence levels, is very fast from a computational point
of view. Increasing the number of conserved moments may considerably increase
the computational cost associated with the analytical approach. But this
configuration is rarely used in practice because it often results in
numerical instabilities and badly estimated percentiles, as explained in
Sect. .
The Supplement related to this article is available online at https://doi.org/10.5194/npg-25-175-2018-supplement.
The authors declare that they have no conflict of
interest.
Acknowledgements
The authors are very grateful to Jean-Pierre Antoine, 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
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