The Cross-Wavelet Transform and Analysis of Quasiperiodic Behavior in the Pearson-Readhead VLBI Survey Sources

Abstract

We introduce an algorithm for applying a cross-wavelet transform to analysis of quasiperiodic variations in a time-series, and introduce significance tests for the technique. We apply a continuous wavelet transform and the cross-wavelet algorithm to the Pearson-Readhead VLBI survey sources using data obtained from the University of Michigan 26-m parabloid at observing frequencies of 14.5, 8.0, and 4.8 GHz. Thirty of the sixty-two sources were chosen to have sufficient data for analysis, having at least 100 data points for a given time-series. Of these thirty sources, a little more than half exhibited evidence for quasiperiodic behavior in at least one observing frequency, with a mean characteristic period of 2.4 yr and standard deviation of 1.3 yr. We find that out of the thirty sources, there were about four time scales for every ten time series, and about half of those sources showing quasiperiodic behavior repeated the behavior in at least one other observing frequency. Comment: Revised version, accepted by ApJ. 17 pages, 13 figures, color figures included as gifs, seperate from the text. The addition of statistical significance tests has resulted in modifying the technique and results, but the broad conclusion remain the same. A high resolution version may be found at http://www.astro.lsa.umich.edu/obs/radiotel/prcwdata.html

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arXiv:astro-ph/0301002v2 24 Mar 2003
The Cross-Wavelet Transform and Analysis of Quasiperiodic
Behavior in the Pearson-Readhead VLBI Survey Sources
Brandon C. Kelly, Philip A. Hughes, Hugh D. Aller, and Margo F. Aller
Astronomy Department, University of Michigan, Ann Arbor, MI 48109-1090
bckelly@umich.edu, hughes,hugh,margo@astro.lsa.umich.edu
ABSTRACT
We introduce an algorithm for applying a cross-wavelet transform to analysis of quasiperiodic
variations in a time-series, and introduce significance tests for the technique. We apply a contin-
uous wavelet transform and the cross-wavelet algorithm to the Pearson-Readhead VLBI survey
sources using data obtained from the University of Michigan 26-m parabloid at observing fre-
quencies of 14.5, 8.0, and 4.8 GHz. Thirty of the sixty-two sources were chosen to have sufficient
data for analysis, having at least 100 data points for a given time-series. Of these thirty sources,
a little more than half exhibited evidence for quasiperiodic behavior in at least one observing
frequency, with a mean characteristic period of 2.4 yr and standard deviation of 1.3 yr. We find
that out of the thirty sources, there were about four time scales for every ten time series, and
about half of those sources showing quasiperiodic behavior repeated the behavior in at least one
other observing frequency.
Subject headings: galaxies: active — methods: data analysis — radio continuum: galaxies
1.INTRODUCTION
It is well accepted that centimeter waveband
emission from AGNs is associated with a jet of
synchotron plasma, the accretion structure and
immediate environment of a supermassive black
hole contributing broad band emission from the
infrared to the gamma ray spectrum. Temporal
variations are observed in the radio flux, and a
number of processes have been proposed to explain
this, such as an accretion rate that may change
with time, an accretion disk that exhibits insta-
bility, an outflow that may be Kelvin-Helmholtz
unstable, or an outflow that may interact with am-
bient inhomogeneities. The temporal variations in
the radio flux motivate a search for characteristic
time scales, and, if found, would lend insight into
the mechanisms causing the variations.
In practice, searching for possible quasiperiodic
behavior masked by a stochastic component can
be rather difficult, especially in the context of ir-
regular time sampling. One promising technique
is to perform a continuous wavelet transform on
the signal, and map out the coefficients in wavelet
space. This was done on data for the BL Lac OJ
287, resulting in evidence for periods in the radio
spectral region of ∼ 1.66 yr, and ∼ 1.12 yr domi-
nating in the 1980s (Hughes, Aller, & Aller 1998).
The results led Hughes, Aller, & Aller (1998) to
propose that the results can best be explained by
a “shock-in-jet” model, in which the longer peri-
odicity is linked with an otherwise quiescent jet,
and the shorter with a passing shock-wave.
The continuous wavelet transform detects
quasiperiodic behavior through visual examina-
tion of the map over translation and dilation, and
assigns a characteristic time scale by looking for
peaks in the time-averaged wavelet power (see
§ 4.1). The scales that the peaks occur at cor-
respond to Fourier periods, from which a charac-
teristic time scale may be deduced. Although the
continuous transform is effective, when assigning
a characteristic time scale it does not fully include
information of how a time series varies in dilation;
the time scale corresponds to those dilations where
the time-averaged wavelet power spectrum peaks.
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In order to use all of the available information in
the transform, we examine the cross-wavelettrans-
form, in which the coefficients for the continuous
transform of a signal are multiplied by the complex
conjugate of the coefficients of another signal. The
results are then mapped out in wavelet space and
analyzed for correlation. A cross-wavelet analysis
was used on the OJ-94 project light curve (Lehto
1999), and has seen application in other areas of
science as well (Pancheva et al. 2000; Kyprianou
& Staszewski 1999). We examine this method in
greater depth, as well as a technique for finding
the characteristic period of a signal and apply
this to the Pearson-Readhead (PR) VLBI survey
sources using data from the University of Michi-
gan Radio Astronomy Observatory (UMRAO) ob-
served at three frequencies. The PR group is well
suited for analyzing quasiperiodicity, due to the
high signal-to-noise in the data for most of these
sources (Aller, Aller, & Hughes 2002), as well as
investigating quasiperiodic variations (QPVs) as
a function of optical source type, since it contains
representatives from optical classes QSO, galaxy,
and BL Lac. In addition, the time base of the
UMRAO data is around twenty years for most
sources, allowing possible quasiperiodic events of
as long as four years to be observed for at least
four to five cycles. Using UMRAO data, evidence
has been found for periodic behavior in the cen-
timeter waveband emission for the BL Lac object
0235+164 (Roy et al. 2000). However, there have
not been any convincing indications of periodicity
for PR sources (Aller, Aller, & Hughes 2002).
2.THE DATA
The data for this study were acquired as part
of the UMRAO program, using the University of
Michigan 26-meter parabloid.
sixty-five PR sources are within the declination
limits of the parabloid, and these sources have
been observed at least trimonthly at 14.5, 8.0, and
4.8 GHz since the fall of 1984, with occasional gaps
due to poor weather and positions too near to the
sun to be observed. The observing technique and
reduction procedures used are those described in
Aller et al. (1985) and Aller et al. (2002), with the
latter including a rescaling of the data for the pur-
pose of conforming with the flux system of Ott et
al. (1994).
Sixty-two of the
3. WAVELET TRANSFORMS
3.1.The Continuous Wavelet Transform
The continuous wavelet transform involves de-
composing a signal f(t) into a number of trans-
lated and dilated wavelets. The main idea behind
this is to take a “mother” wavelet, translate and
dilate it, convolve it with the function of interest,
and map out the coefficients in “wavelet space”,
spanned by translation and dilation. Periodic be-
havior then shows up as a pattern spanning all
translations at a given dilation, and this redun-
dancy in the wavelet space makes detection of peri-
odic behavior rather easy. The wavelet transform
preserves temporal locality, which is an advantage
over Fourier analysis. For instance, power associ-
ated with irregular sampling does not contribute
to the coefficients as in Fourier analysis, which
is extremely helpful when using poorly-sampled
data.
There are several common types of mother
wavelets. In this analysis we use a Morlet wavelet
of kψ= 6, given by
ψMorlet= π−1/4e−ikψte−|t2|/2,(1)
with coefficients
˜f (l,t′) =
?
R
f (t)ψ∗
lt′ (t)dt(2)
and
ψlt′ (t) =
1
√lψ
?t − t′
l
?
, l ∈ R+, t ∈ R.(3)
The continuous transform must also satisfy an ad-
missibility condition, requiring zero mean:
?
R
ψ(t)dt = 0.(4)
It should be noted that l corresponds to dilations,
and t′refers to translations. Generally the pa-
rameter kψremains fixed throughout the analysis,
and for simplicity will hereafter be just k. Fig-
ure 1a shows the continuous wavelet transform for
a sinusoidal signal. The periodic behavior is eas-
ily revealed by the redundancy in the plot. For
the interested reader, more detailed information
on the continuous wavelet transform may be found
in Farge (1992).
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3.2.The Cross-Wavelet Transform
The continuous wavelet transform is effective
for examining how a time series varies in time and
scale, but fails to include how it varies over a range
of scales when assigning a period that best charac-
terizes it. After identifying that a periodic pattern
exists in what can be potentially noisy and poorly
sampled data, one finds the dilation which char-
acterizes the period from the time-averaged data
(i.e., the wavelet power spectrum), and from that
dilation the period is calculated. For a quasiperi-
odic signal there is no unique dilation; there is a
need for a method of directly measuring a char-
acteristic time scale, or scales, that includes infor-
mation on how a source varies in dilation, and for
this we shall examine the cross-wavelet transform.
Given two time series fa(t) and fm(t), we can
construct the cross-wavelet transform
˜fc(l,t′) =˜fa(l,t′) ·˜f∗
where˜fa and˜fm are given by Equation (2). In
this analysis fa(t) is the source signal and fm(t)
is a sinusoidal mock signal, although one can use
any two time series thought to be correlated.
Because we are looking for quasiperiodic behav-
ior, first assume an ideal signal of the form
m(l,t′),(5)
fa(t) = Aaei(ωat+φa), −∞ < t < ∞, (6)
where ωa= 2π/τafor period τa. Anticipating that
it will be productive to cross the actual signal with
a signal of a similar form, we try a mock signal
fm(t) = Amei(ωmt+φm), −∞ < t < ∞. (7)
Each of these signals will have continuous wavelet
coefficients given by Equation (2):
˜f (l,t′) = A
?
2lπ1/2e−1
2(lω−k)2ei(ωt′+φ). (8)
The cross-wavelet (Eq.[2]) for these two signals be-
comes
˜fc(l,t′) = 2√πAaAmei(φa−φm)le−k2×
e−1
2[(ω2
a+ω2
m)l2−2kl(ωa+ωm)]e(ωa−ωm)it′. (9)
Defining
A≡
≡
≡
≡
≡
2√πAaAme−k2
ω2
m
ωa+ ωm
ωa− ωm
φa− φm,
η
γ
a+ ω2
β
φ
Equation (9) then becomes
˜fc(l,t′) = Ale−1
2(ηl2−2kγl)ei(βt′+φ),(10)
which is easily interpreted. From this equation
we note two important properties of this cross-
wavelet—first, it has the form of a Gaussian in
the dilation coordinate l, and second, it is sinu-
soidal in the translation coordinate t′with fre-
quency given by the difference in the frequencies
of the actual and the mock signal (i.e., the beat
frequency). When these two frequencies are equal,
the translation dependence is lost, and the cross-
wavelet reduces to a Gaussian in the dilation co-
ordinate. Figure 1b, shows the cross-wavelet for
two sinusoids oscillating with the nearly same fre-
quency.
3.3.Using the Cross-Wavelet Transform
to Analyze Periodicity
Motivated by the results from the previous sec-
tion, we can develop an algorithm to find the pe-
riod which best characterizes a signal, i.e., the
characteristic time scale. Because we are dealing
with real signals, noise will be present, and we ex-
plore the role of noise in § 4 and § 5. In addition,
the values of t and t′do not extend to infinity,
but over the range t = a to t = b. Because of the
finiteness of the range of t, the range of dilations
becomes lmin< l < lmax. We can still use Equa-
tion (8) as an approximation when our range of l
and t′is such that the wavelets ψlt′ (t) contribute
negligibly outside of a < t < b, and
?b
a
e−[(ω−k
l)it±1
2|t−t′
l
|2]dt ≈
?∞
−∞
e−[(ω−k
l)it±1
2|t−t′
l
|2]dt,
(11)
Returning to Equation (10), we define the mod-
ulus as
???˜fc(l,t′)
???
2
= A2l2e−(ηl2−2kγl).(12)
Integrating over the region of wavelet space we
mapped out, weighting by the scale, and approxi-
mating the integral over l out to infinity, we arrive
at
Fc(τm)=
?lmax
A2(b − a)kγ
lmin
?b
a
???˜fc(l,t′)
?π
???
2
l
dt′dl
≈
ηηe
k2γ2
η .(13)
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We have empirically found that this function at-
tains it maximum value at τa = 0.973τm for a
value of k = 6. This result is not surprising, as
one would expect the power of the cross coeffi-
cients to reach their maximum when the periods
for the two signals are almost equal. One can think
of the mock signal as acting like a filter in wavelet
space; however, the power of the mock coefficients,
|˜fm|2, are slightly asymmetric about the peak (see
Eq.[8]). The weighted total power of the cross co-
efficients, Fc, does not attain its maximum exactly
at τm= τabecause of this slight asymmetry.
To avoid confusion, one must remember that
previously when the integration was over [a,b] we
integrated over the original coordinate t and, be-
cause the wavelets have compact support in t, we
approximated the integral by taking the limits to
be [−∞,∞]. Here, however, the integral is over
the translation coordinate t′, so we must restrict
the limits to [a,b]. In addition, the approximation
of the integral over l is valid so long as˜fc(l,t′)/l
does not peak in the dilation coordinate too close
to lminor lmax. For our purposes, lmin= 2δt and
lmax= Ndataδt/3, where Ndata is the number of
data points and δt is the time spacing. Typical val-
ues of b−a for the PR Survey Sources are roughly
20 years, and we require at least Ndata= 100 data
points for a reliable cross-wavelet analysis, which
results in 0 < lmin ≤ 0.4 yr and lmax < 12 yr.
We search for periods within 0.5 < τ < (b − a)/4
years. We have inspected |˜fc|2/l graphically to
assure that the approximation is valid for periods
τa≥ 0.7 yr for all sources having more than 100
data points.
In practice, one must worry about edge effects
as the time series is finite. Because the convo-
lution of Equation (2) introduces power into the
wavelet coefficients˜f from the discontinuity at the
edges of the time series, there is a region where
the wavelet coefficients will be contaminated from
edge effects. This region is known as the cone of
influence. The wavelet power associated with edge
effects becomes negligible for translations t′far-
ther than√2l from the edge (Torrence & Compo
1998)
One more useful quantity is the point˜l where
˜fc(l,t′) peaks in the dilation coordinate. This is
given by
˜l =k (ωa+ ωm) +?(k2+ 4)(ω2
a+ ω2
m)
m) + 2k2ωaωm
2(ω2
a+ ω2
.
(14)
When ωm= ωa, this becomes
˜l =k +√k2+ 2
2ωa
,(15)
or,
τa=
4π˜l
k +√k2+ 2, τa=2π
ωa.(16)
This equation shows the relationship between a
scale l and the corresponding Fourier period, and
can also be arrived at by inputing a sinusoid for
f(t) in Equation (2). For clarification, we will use
the term ‘Fourier period’ when referring to the
period associated with a certain scale l, and the
terms ‘analyzing period’ or ‘mock period’ when re-
ferring to the period τmassociated with the mock
signal.
4.SIGNIFICANCE TESTS
4.1.Tests for the Continuous Transform
Wavelet analysis has historically suffered from a
lack of statistical significance tests. In Torrence &
Compo (1998) an excellent discussion of statistical
significance for the continuous wavelet transform
is given and supported by Monte Carlo results,
and we summarize the relevant points. When im-
plementing the wavelet transform, one must use
sums and discrete points rather than the theoreti-
cal treatment given earlier, and we switch to using
these.
The model of correlated noise most likely to
closely resemble the UMRAO data is the univari-
ate lag-1 autoregressive (AR(1)) process (Hughes,
Aller, & Aller 1992), given by
xn= αxn−1+ zn, (17)
where α is the assumed lag-1 autocorrelation and
zn is a random deviate taken from white noise
(note that this is a ‘first order’ process). The nor-
malized discrete Fourier power spectrum of this
model is
Pj=
1 − α2
1 + α2− 2αcos(2πδt/τj),(18)
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where τjis the associated Fourier period (Eq.[16])
of a scale lj. If the time series of interest is an
AR(1) process, then a slice in the modulus of the
continuous coefficients, |˜f(l,t′)|2, along dilation
at a constant translation (the local wavelet power
spectrum) should have have the form in Equation
(18). Note that a white noise process is given by
α = 0.
Using the theoretical background spectrum
given above, we can develop significance tests for
the continuous transform. The background spec-
trum is the mean power spectrum expected for the
assumed noise process (i.e., the ‘background’ pro-
cess), against which we wish to compare the actual
signal. Physical processes that are the result of
this background process will produce power spec-
tra that are normally distributed about this mean
background spectrum. Assuming that the values
in our time series f(t) are normally distributed,
and because the square of a normally distributed
variable is chi-square distributed with one degree
of freedom, we expect the wavelet power |˜f|2to
be chi-square distributed with two degrees of free-
dom.The additional degree of freedom comes
from the fact that the wavelet coefficients˜f are
complex for the Morlet wavelet. In addition, the
expectation value for the wavelet power of a white
noise time series is just the variance σ2; this expec-
tation value provides a convenient normalization.
Using this normalization, the distribution for the
local wavelet power spectrum is
???˜f(lj,t′
i)
???
2
σ2
⇒ Pjχ2
ν
ν,
(19)
where ‘⇒’ means ‘is distributed as’ and ν is the
degrees of freedom, in this case two. The indices
on the scale l run from j = 1,2,...,J, where J
is the number of scales, and the indices on the
translation t′run from i = 1,2,...,Ndata.
We can also define the time-averaged wavelet
power spectrum, or the global wavelet spectrum,
as
i′
?
where ij and i′
final translations t′
ioutside of the cone of influ-
ence at a given scale lj, and Nj is the number t′
outside the cone of influence at that scale. It has
˜f2
G(lj) =
1
Nj
j
i=ij
???˜f(lj,t′
i)
???
2
,(20)
jare the indices of the initial and
i
been shown that the global wavelet spectrum pro-
vides an efficient estimation of the true power of a
time series (Percival 1995). Averaging the wavelet
power spectrum as in Equation (20) increases the
significance of the peaks, as the degrees of free-
dom is increased beyond what is used in the local
wavelet power spectrum. However, because the
coefficients are correlated in both time and scale,
the degrees of freedom for the global wavelet spec-
trum are
?
νj= 2
1 +
?Njδt
ljδt0
?2
,(21)
where δt0 describes the decorrelation length in
time. For the Morlet wavelet of k = 6, δt0= 2.32.
The global wavelet spectrum of a sinusoid is shown
in Figure 2.
Alternatively, we can smooth the wavelet spec-
trum in scale.We define the scale-averaged
wavelet power as
˜f2
L(t′
i) =
j2
?
j=j1
???˜f(lj,t′
i)
???
2
lj
.(22)
The scale-averaged wavelet power can be inter-
preted as a time series of the average variance in
the band lj1≤ lj≤ lj2. This distribution can be
modeled as
˜f2
i)
σ2
where the scale-averaged theoretical spectrum¯P
is
j2
?
The degrees of freedom νl in Equation (24) are
modeled as
L(t′
⇒¯Pχ2
νl
νl
,(23)
¯P =
j=j1
Pj
lj
. (24)
νl=2NlLavg
Lmid
?
1 +
?Nlδj
δj0
?2
, (25)
where
Lavg
=


lmin20.5(j1+j2)δj
lmin2(j−1)δj, j = 1,2,...,J,
j2
?
j=j1
1
lj


−1
Lmid
=
lj
=
5