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arXiv:astroph/0301002v2 24 Mar 2003
The CrossWavelet Transform and Analysis of Quasiperiodic
Behavior in the PearsonReadhead VLBI Survey Sources
Brandon C. Kelly, Philip A. Hughes, Hugh D. Aller, and Margo F. Aller
Astronomy Department, University of Michigan, Ann Arbor, MI 481091090
bckelly@umich.edu, hughes,hugh,margo@astro.lsa.umich.edu
ABSTRACT
We introduce an algorithm for applying a crosswavelet transform to analysis of quasiperiodic
variations in a timeseries, and introduce significance tests for the technique. We apply a contin
uous wavelet transform and the crosswavelet algorithm to the PearsonReadhead VLBI survey
sources using data obtained from the University of Michigan 26m parabloid at observing fre
quencies of 14.5, 8.0, and 4.8 GHz. Thirty of the sixtytwo sources were chosen to have sufficient
data for analysis, having at least 100 data points for a given timeseries. 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 KelvinHelmholtz
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 “shockinjet” model, in which the longer peri
odicity is linked with an otherwise quiescent jet,
and the shorter with a passing shockwave.
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 timeaveraged 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 timeaveraged wavelet power spectrum peaks.
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In order to use all of the available information in
the transform, we examine the crosswavelettrans
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 crosswavelet analysis
was used on the OJ94 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 PearsonReadhead (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 signaltonoise 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 26meter parabloid.
sixtyfive 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).
Sixtytwo 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 poorlysampled
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 CrossWavelet 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 timeaveraged 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 crosswavelet transform.
Given two time series fa(t) and fm(t), we can
construct the crosswavelet 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 crosswavelet (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 crosswavelet for
two sinusoids oscillating with the nearly same fre
quency.
3.3.Using the CrossWavelet 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
2t−t′
l
2]dt ≈
?∞
−∞
e−[(ω−k
l)it±1
2t−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,
˜fm2, 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 crosswavelet 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 ˜fc2/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 lag1 autoregressive (AR(1)) process (Hughes,
Aller, & Aller 1992), given by
xn= αxn−1+ zn, (17)
where α is the assumed lag1 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 chisquare distributed with one degree
of freedom, we expect the wavelet power ˜f2to
be chisquare 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 timeaveraged 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 scaleaveraged
wavelet power as
˜f2
L(t′
i) =
j2
?
j=j1
???˜f(lj,t′
i)
???
2
lj
. (22)
The scaleaveraged 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 scaleaveraged 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
=
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Nl is the number of dilations averaged over, and
δj0is the decorrelation distance in scale. For the
Morlet wavelet of k = 6, δj0= 0.60. The δj de
scribes how the dilations lj are constructed; it is
common to construct them as fractional powers
of two as it allows more emphasis on the smaller
scales. The factor Lavg/Lmidcorrects for the loss
of degrees of freedom from dividing the wavelet
power spectrum by scale in Equation (22).
4.2.Tests for the Cross Transform
Using the results of Torrence & Compo (1998)
from the previous section, we can derive signifi
cance tests for the crosswavelet transform. The
power of the crosswavelet˜fccan be written as
???˜fc(lj,t′
and is distributed as
i)
???
2
=
???˜fa(lj,t′
i)˜f∗
m(lj,t′
i)
???
2
=
???˜fa(lj,t′
i)
???
2???˜fm(lj,t′
i)
???
2
soid can be seen in Figure 2.
The results of this section are only valid when
the crosswaveletconsists of one source that has an
assumed background spectrum that is chisquare
distributed. If one is crossing two sources, f1(t)
and f2(t), that have assumed background spectra,
P1 and P2, that are chisquare distributed, then
the crosswavelet is distributed as
,
(26)
???˜fa(lj,t′
where the degrees of freedom ν is two. We can also
define the crosswavelet global power spectrum,
i)
???
2???˜fm(lj,t′
i)
???
2
σ2
⇒ Pj
???˜fm(lj,t′
i)
???
2χ2
ν
ν,
(27)
˜f2
CG(lj) =
1
Nj
i′
j
?
i=ij
???˜fc(lj,t′
i)
???
2
.(28)
Noting that the power of the continuous coeffi
cients for the type of mock signal given in Equa
tion (7) is independent of the translation t′, the
timeaveraged form of Equation (13), Fc, then be
comes
¯ Fc(τm) =
J
?
j=1
˜f2
CG(lj)
lj
=
J
?
j=1
˜f2
AG(lj)
???˜fm(lj)
???
(29)
2
lj
,
where˜f2
trum for the actual signal fa(ti). This distribution
can be modeled as
AG(lj) is the global wavelet power spec
¯Fc(τm) ⇒
J
?
j=1
Pj
???˜fm(lj)
???
2
lj
?χ2
ν′
j
ν′
j
?
,(30)
where the degrees of freedom ν′
jare modeled as
ν′
j= νj+ νl,(31)
for νj and νl given in the previous section. This
allows us to smooth the wavelet power in both
time and scale, thus increasing the degrees of free
dom, and to receive information of the charac
teristic time scale(s) of our time series from the
mock coefficients˜fmrather than from the global
wavelet spectrum. As mentioned in § 3.3, Equa
tion (29) allows us to interpret the crosswavelet
of a time series and an analyzing signal as a fil
ter in wavelet space, where the coefficients of the
continuous transform for the analyzing signal fil
ter the global wavelet spectrum of our time series,
˜f2
AG. Averaging the cross transform in scale then
allows us to find the analyzing signal with which
the time series is best correlated over the entire
range of time and scale. A plot of¯ Fcfor a sinu
???˜f1(l,t)˜f∗
2(l,t)
???
σ1σ2
⇒Zν(p)
ν
?
P1P2, (32)
where σ1 and σ2 are the respective standard de
viations, the degrees of freedom ν is two for com
plex wavelets, and Zν(p) is the confidence level
for a given probability p for the square root of the
product of two chisquare distributions (Torrence
& Compo 1998). One can find the confidence level
Zνby inverting the integral p =?Zν
22−ν
Γ2(ν/2)zν−1K0(z),
0
fν(z)dz. The
probability distribution is given by
fν(z) =
(33)
where z is the random variable, Γ is the Gamma
function, and K0(z) is the modified Bessel func
tion of order zero.
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5.ALGORITHM
ING THE CROSS TRANSFORM TO
ANALYZE QUASIPERIODIC BEHAV
IOR AND SIMULATIONS
FOR IMPLEMENT
5.1.Algorithm for Analysis of Quasiperi
odic Behavior
When using real data, we have no a priori
knowledge, if any, of the periodicity for the par
ticular signal of interest, and require an algorithm
that will allow us to find it. The results of § 3.3 and
§ 4 will serve as a guide in developing this. First,
we subtract the mean from the signal and perform
the continuous wavelet transform given by Equa
tion (1), only multiplying by a factor of
allow proper normalization and replacing the in
tegral with a sum. Then, we divide the continuous
coefficients by the standard deviation of the time
series to assure that the expectation value of ˜f2
for white noise is unity, and cross them with the
coefficients for a number of mock (analyzing) sig
nals, each of which has a period τn= τn−1+ δτ.
For each of these, we average over the relevant
region in wavelet space as in Equation (29), and
look for extrema. The extrema of¯ Fc(τn) then cor
respond to characteristic time scales, related by
τa= 0.973˜ τn, where ˜ τnis the period of the mock
signal corresponding to the extremum.
The nth mock signal is given by
√δt to
fn(t) = Ancos
?2π
τnt + φn
?
,(34)
where we use the subscript n instead of m on the
parameters to emphasize that they are for the nth
mock signal. We choose a value of An such that
the global wavelet spectrum of the analyzing sig
nal is normalized so that its maximum value is
unity. This provides a convenient normalization
among the analyzing signals, assuring that they
all have the same power.
mock signals are used to analyze the actual time
series, resulting in a step size of δτ ∼ 0.045 for
most sources (this varies slightly because not all
of the sources have the same time window). We
set φnequal to zero, as ˜fc2is independent of the
phase difference between the signals. Many of the
signals analyzed in this paper have typical values
for the time range on the order of a = 80 yr and
b = 101 yr, which arise from using the year 1900
as a baseline. The dilation values are chosen such
In this analysis, 100
that 2δt < l < Ndataδt/3 yr, which allows us to
be conservative when omitting edge effects and to
admit the approximations used in Equations (8)
and (13).
When plotting the coefficients, those that fell
within the cone of influence at any given dilation
were set to a constant value, thus masking the
edge effects. Values of lj are shown logarithmi
cally, so as to allow more sensitivity in the dilation
region corresponding to shorter mock periods. It
is helpful to examine the global wavelet spectrum
of the time series (Eq.[20]), and to compare with
the crosswavelet results, so we include it in our
analysis. We assume a background spectrum Pjof
the form AR(1) (Eq.[18]), and estimate the lag1
autocorrelation coefficient by calculating the lag1
and lag2 autocorrelations, α1 and α2. The lag
1 autocorrelation coefficient is then estimated as
α = (α1+√α2)/2. The background spectrum Pj
then allows us to compute confidence levels using
the results of § 4. We plot contours on the continu
ous and crosswavelet transforms corresponding to
the 90% confidence level for an AR(1) process. We
also plot 90% and 75% correlated noise confidence
levels on the plots of the global wavelet spectrum
f2
as well as 90% confidence levels for white noise
(Pj = 1). It should be noted that by confidence
level, we mean that this given percentage of as
sumed background (i.e., noise) processes will pro
duce behavior less than what is seen in the wavelet
transforms. For the time series analyzed here, the
90% confidence levels for correlated noise mark
where 90% of AR(1) processes will produce values
less than the 90% values, and likewise for the 75%
confidence levels and the 90% confidence levels for
white noise.
To satisfy the admissibility condition (4) we
choose a value of k = 6. In reality, Equation (4)
is not formally satisfied for the Morlet Wavelet of
Equation (1). However if k ≥ 5, the admissibil
ity condition is satisfied to within the accuracy of
computer algorithms using single precision arith
metic.
AGand the averaged crosswavelet power¯ Fc(τn),
5.2.Simulations
To test our technique, we ran several simula
tions. First, to investigate the ability of our tech
nique to detect a purely periodic signal in the pres
7
Page 8
ence of noise we used a signal of the form
fa(t) = 2sin(πt + 1.1) + Nn(t), (35)
where N is an amplitude factor and n(t) a noise
function that generates white noise. We show the
continuous and cross transforms for N = 10 in Fig
ure 3. The cross wavelet plot is that for a mock
signal of period τn= 2.059 yr. Plots of the global
wavelet spectrum and¯ Fc for this simulation are
shown in Figure 5a. As can be seen in Figure 5a,
the period is easily recovered using the technique
with noise of amplitude fivetimes that of the am
plitude of the sinusoid (signaltonoise of 0.2). Be
cause signaltonoise values as low as S/N = 0.2
are never realized, our technique is is valid for the
time series studied here.
In addition, we also applied the algorithm to
a sinusoid that changes amplitude and phase over
the time window. The transforms are shown in
Figure 4, and the power spectra in Figure 5b. A
change in amplitude affects the amplitude of the
wavelet coefficients, as would be expected. The ef
fects of a discontinuous change in the phase φacan
be seen as disturbing the structure of the trans
forms in these regions. The technique gives a char
acteristic period of 2.0 yr, showing that a sudden
change in the phase of a signal does not effect the
technique.
In practice, we are using this technique to inves
tigate quasiperiodic behavior and to give a char
acteristic time scale for a time series, rather than
attempting to recover a periodic signal buried be
neath noise (although this certainly may be done).
To illustrate our technique in the case of pure
noise, we applied the crosswavelet technique to
Gaussian white noise and correlated noise. The
wavelet transforms for a white noise signal are
shown in Figure 6. As can be seen, temporary
quasiperiodic structure can arise for white noise,
however the signal is clearly distinguished from
more coherent signals by the complex structure
throughout the plot, particularly in the low dila
tion (high frequency) region. Plots of the global
wavelet power spectrum and the averaged cross
power¯ Fcare shown in Figure 8a. Although there
are distinct peaks in the global spectrum at small
scales, the average power of the crosswavelet,¯ Fc,
does not show distinct peaks at these scales. We
interpret this as being the result of smoothing in
scale as well as time when calculating ¯ Fc; the
Global Power Spectrum for 2*COS(Pi*t+1.1)
12345
Fourier Period (yr)
0
10
20
30
40
Global Power Spectrum / Variance
90% Corr
75% Corr
90% White
Average Cross Power for 2*COS(Pi*t+1.1)
12345
Analyzing Period (yr)
0
50
100
150
200
250
Average Cross Power / Variance
90% Corr
75% Corr
90% White
Fig. 2.— Global wavelet power spectrum˜f2
and the average cross power¯ Fc(τn) for a sinusoid
of period τa= 2.0 yr. Also shown are the confi
dence levels for white and correlated noise, assum
ing a lag1 autocorrelation of α = 0.911362.
AG(lj)
Global Power Spectrum for A(t)*COS(Pi*t+Phi(t))
12345
Fourier Period (yr)
0
5
10
15
20
25
30
Global Power Spectrum / Variance
90% Corr
75% Corr
90% White
Average Cross Power for A(t)*COS(Pi*t+Phi(t))
12345
Analyzing Period (yr)
0
50
100
150
200
250
Average Cross Power / Variance
90% Corr
75% Corr
90% White
Global Power Spectrum for 2*COS(Pi*t+1.1)+10*n(t)
12345
Fourier Period (yr)
0
2
4
6
8
Global Power Spectrum / Variance
90% Corr
75% Corr
90% White
Average Cross Power for 2*COS(Pi*t+1.1)+N*n(t)
12345
Analyzing Period (yr)
0
20
40
60
80
Average Cross Power / Variance
90% Corr
75% Corr
90% White
Fig. 5.— Global power spectrum˜f2
erage cross power¯ Fc(τn) for the sinusoids of Fig
ure 3 (top two panels) and Figure 4 (bottom two
panels). Confidence levels for correlated and white
noise are shown.
AG(lj) and av
8
Page 9
global wavelet spectrum is averaged in time for
a given dilation lj, and can result in peaks even
in the case of nonstationary power, as is seen in
the continuous plot for white noise. However, by
crossing with an analyzing signal, and averaging
in dilation, we can smooth out spurious extrema,
as is seen in the plot of¯ Fc.
To illustrate the technique for correlated noise,
we used a test signal of the type AR(1) with lag1
autocorrelation α = 0.9. Many of the PR sources
had lag1 autocorrelations of α ∼ 0.9, and we
found it helpful to simulate an AR(1) process of
α similar to what we infer for the data. Figure 7
shows the transforms for this signal. As expected,
there is more activity at larger scales (lower fre
quencies). Coherent activity does appear in the
case of this type of correlated noise, however the
power is not stationary and can not be character
ized by a particular scale. The global power spec
trum and¯ Fccan be seen in Figure 8b. Peaks ap
pear at similar periods for both, however the plot
of¯Fcshows that the peaks are less significant than
those seen in the global power spectrum. The rea
son for this can be seen in the continuous plot; the
power appears to split into two time scales around
t′= 90. When we cross the continuous transform
with our analyzing signal transform, and average
over both translation and dilation, we see that the
time scale of ∼ 3.8 yr is not as significant as it
appears in the global wavelet spectrum.
We can see from the simulations, as well as
from the results of applying the transforms to the
PR sources, that the average cross power,¯ Fc(τn),
places more emphasis on the smaller scales and pe
riods than the global power spectrum. This is to
be expected, as the amplitude of the global power
spectrum for the time series of interest,˜f2
has a dependence on the scale lj (see Eq.[8]). If
we do not normalize the global power spectrum
of the mock signal, ˜fm2, such that its maximum
value is unity for all periods τn, then the ampli
tude of ˜fm2will also have a scale dependence.
This would result in the amplitude of¯Fc being
dependent on the scale, lj as well (note that¯ Fc
would not be dependent on l2
by the scale in Eq.[29]). However, normalizing the
mock power spectrum ˜fm2so that it its maxi
mum value is unity removes the scale dependence
in ˜fm2, and thus in¯Fc.
In addition, we applied the crosswavelet trans
AG(lj),
jbecause we divide
Global Power Spectrum for Correlated Noise
12345
Fourier Period (yr)
0
5
10
15
20
Global Power Spectrum / Variance
90% Corr
75% Corr
90% White
Average Cross Power for Correlated Noise
12345
Analyzing Period (yr)
0
20
40
60
80
Average Cross Power / Variance
90% Corr
75% Corr
90% White
Global Power Spectrum for White Noise
12345
Fourier Period (yr)
0.0
0.5
1.0
1.5
2.0
2.5
Global Power Spectrum / Variance
90% White
75% White
Average Cross Power for White Noise
12345
Analyzing Period (yr)
0
20
40
60
Average Cross Power / Variance
90% White
75% White
Fig. 8.— Global power spectrum˜f2
erage cross power¯ Fc(τn) for the white noise signal
of Figure 6 (top two panels) and the correlated
noise signal of Figure 7 (bottom two panels). Con
fidence levels for correlated and white noise are
shown.
AG(lj) and av
9
Page 10
form technique to the BL Lacs OJ 287 and AO
0235+164, as previous analysis has shown evi
dence for periodicity in these sources. For OJ 287
the technique detected a characteristic time scale
of ∼ 1.6 yr, with a shorter time scale of ∼ 1.1 yr,
confirming results found earlier by Hughes, Aller,
& Aller (1998). The crosswavelet analysis also
confirmed earlier results for A0 0235+164, giving
time scales of ∼ 1.9 and ∼ 3.3 yr, which is in
good agreement with the 3.61 yr found by Roy et
al. (2000) using a LombScargle periodogram. We
were not able to confirm the longer periods found
by Roy et al. (2000), as they fell outside of our con
dition of at least four cycles in the timewindow.
6.THE DATA ANALYSIS
The results of using the algorithm described
in § 5.1 to search UMRAO data on the Pearson
Readhead survey sources are given in Table 1. Ob
servations that we considered to have insufficient
data to give a reliable analysis of quasiperiodicity
(about half) are denoted by D. Such observations
were excluded because we required at least 100
data points for a given time series, as we assumed
that 100 points are needed to adequately define
its character. When analyzing the light curves,
we used the entire time window, which varied be
tween observing frequencies and sources. Since we
are using a finite step size δτ ∼ 0.045 yr, we do
not expect to use an analyzing signal with period
τn= τa, but rather with period τn≈ τa. However,
because we are analyzing quasiperiodic behavior
we do not expect the sources to have a welldefined
period (i.e., a spike in Fourier space), but rather a
characteristic time scale. The error resulting from
the finite step size of 0.045 yr is irrelevant in this
context, as there is no sense found in assigning an
extremely precise value for a quasiperiodic source.
We only record time scales at or above the 90%
confidence level.
Many of the sources that exhibited quasiperi
odic variations had characteristic time scales be
tween one and four years, with an average time
scale for all quasiperiodic sources of 2.4 yr and a
standard deviation of 1.3 yr. This, of course, is
to be expected as our observing interval is around
twenty years, and we are only interested in varia
tions with at least four possible repetitions across
the interval. Table 2 shows statistics with respect
to source type and Figure 9 shows histograms for
the number of characteristic periods found with re
spect to source type, binned every 0.25 yr. Out of
thirty total sources with sufficient data, eighteen
were quasars, four were galaxies, and eight were
BL Lacs. A little over half of the sources showed
evidence for quasiperiodicity in at least one ob
serving frequency.Column six shows the ratio
of recorded time scales to individual time series,
which gives an idea of the average number of time
scales per source. For instance, if three quasars
had sufficient data, and these quasars had eight
individual time series between them over all the
observing frequencies, with five of these individ
ual time series exhibiting QPVs, then the ratio of
recorded characteristic periods to individual time
series would be 5/8 for quasars. A value of 0.33333
would mean that on average most of the sources
for this particular type showed evidence for one
characteristic time scale over the three observing
frequencies, and two time scales over three observ
ing frequencies for a value of 0.66667. As can be
seen from the table, on average there was about
a little more than one characteristic period ob
served for every three observing frequencies for all
sources. The last column gives the ratio of sources
with quasiperiodic behavior in more than one ob
serving frequency to the total number of quasiperi
odic sources. Roughly half of the quasiperiodic
sources exhibit quasiperiodicity across more than
one observing interval. It appears that quasars
exhibit characteristic time scales more frequently
than BL Lacs, as about twothirds of the QSOs
showed QPVs, while only about onthird of the
BL Lacs did; however, all of the BL Lacs showing
QPVs repeated the behavior in more than one ob
serving frequency, while only half of the quasars
did. This may well reflect the fact that the BL
Lacs of the PR survey sources generally have flat
spectra (Aller et al. 2002).
does not appear to be a distinction among opti
cal classes regarding the length of the time scales.
Among sources showing QPVs in more than one
observing frequency, the characteristic time scales
were similar among the different observing fre
quencies. However, it is difficult to perform a com
plete statistical analysis of quasiperiodicity with
respect to source type, as only 30 sources were
analyzed. In addition, there does not appear to
be a distinct relationship between time scales and
In addition, there
10
Page 11
observing frequency; characteristic time scales do
not consistently lengthen with increasing observ
ing frequency or vice versa. The plots of one par
ticularly promising source for periodic behavior,
the quasar 0804+499, can be seen in Figures 10
and 11 at an observing frequency of 4.8 GHz.
Comparing the results for the PR sources with
the results expected for correlated noise (§ 4), we
find it unlikely that most of the PR sources with
characteristic time scales are exhibiting behavior
that can be attributed to an AR(1) process. If
the time series for the PR sources were the result
of AR(1) processes, we would expect the ratio of
recorded time scales to time series, Qt/T, to be
∼ 0.1, as we record characteristic time scales at or
above the 90% confidence level. However, as can
be seen in Table 2, Qt/T = 0.42.
The lack of uniform quasiperiodic behavior
across observing frequencies is unexpected for
some of the sources with flat spectra. We have
explored the light curves for these sources and
conclude that, while generally flat, they do exhibit
behavior that is different across the three frequen
cies. Localized activity is sufficient to remove or
significantly weaken evidence for quasiperiodic be
havior during that time interval, and if quasiperi
odic behavior was noted at earlier or later times,
it often did not span enough cycles (around four)
to warrant recording a characteristic time scale.
Typically such events are not sufficient to affect
the averagespectral index, howeverthey can result
in changes in the smallscale structure of the light
curves which is detected by the wavelet technique.
As an example, Figure 12 shows the continuous
transforms for 1823+568 at 8.0 GHz and 14.5
GHz, and the wavelet power spectra are shown
in Figure 13. Although the light curves appear
similar, the continuous transforms show behavior
that is different between the two observing fre
quencies. Comparison with the power spectra and
the continuous plots explains the discrepancy; the
transform for the 8.0 GHz curve exhibits a tran
sient time scale of ∼ 2.0 yr, seen after ∼ 1993,
whereas the 14.5 GHz curve exhibits a longer time
scale of ∼ 3.3 yr that persists throughout the time
window. We address the lack of common char
acteristic time scales with respect to observing
frequency further in § 7.
In addition, we also note that for the 14.5 GHz
data for 1823+568 the global wavelet spectrum
Quasars
0123456
Time Scale (yr)
0
2
4
6
8
31 Without a Time Scale
Galaxies
0123456
Time Scale (yr)
0
2
4
6
8
6 Without a Time Scale
BL Lacs
0123456
Time Scale (yr)
0
2
4
6
8
17 Without a Time Scale
All Time Series
0123456
Time Scale (yr)
0
2
4
6
8
54 Without a Time Scale
Fig.
statistics. Time scales are binned every 0.25 years.
9.— Histograms for the quasiperiodicity
Fig. 11.— Global power spectrum˜f2
the average cross power¯ Fc(τn) for 0804+499 at
4.8 GHz. The three peaks corresponding to the
three time scales seen in Figure 10a are seen in
both plots. The ∼ 2.7 yr time scale is the most
prevalent, as expected from visual inspection of
the continuous plot. Also shown are the confidence
levels for correlated and white noise.
AG(lj) and
11
Page 12
Fig. 13.— Global power spectrum˜f2
the average cross power¯ Fc(τn) for 1823+568 at
8.0 GHz (top two panels) and 14.5 GHz (bottom
two panels). Although the ∼ 2.0 yr time scale ac
tivity seen post1990 in the continuous transform
for the 8.0 GHz data is significant (see Figure 12a),
these plots show that the source does not globally
show any behavior in 8.0 GHz that is not incon
sistent with an AR(1) process. For the 14.5 GHz
data, power is shared over a broad range in scale in
both˜f2
smooths the two peaks seen in the global spectrum
into one distinct peak. The single time scale inter
pretation given by the cross technique is visually
reinforced in the continuous plot (see Figure 12b).
AG(lj) and
AGand¯ Fc(τn), however the cross transform
˜f2
in assigning a time scale. The global spectrum re
veals evidence for two time scales, one at ∼ 2.9 yr
and the other at ∼ 3.8 yr, while the average cross
power shows evidence for a time scale at ∼ 3.3
yr. Comparison between the two leads us to con
clude that the characteristic time scale seen in the
plot of¯ Fc is more accurate. The global wavelet
spectrum shows a broad spread of power between
the two time scales, where the time scales are in
ferred from the somewhat poorly resolved peaks.
The average cross power plot also shows a broad
spread of power, but with only one peak. This be
havior is similar to what was described in § 5.2 for
the case of white noise; because the crosswavelet
technique smooths in dilation as well as in trans
lation, and acquires information regarding time
scales from the analyzing time series, it smooths
the two poorly resolved peaks in the global wavelet
spectrum into one broad peak. Analyzing the con
tinuous plot, we see that the plot of¯ Fcbetter rep
resents the behavior seen in the continuous trans
form, as power appears to shared among a range
of scales of ∼ 1.0 yr, rather than two distinct time
scales separated by about a year. It is more ap
propriate to assign one characteristic time scale
in this case rather than two. Behavior similar to
what is seen in the 14.5 GHz data was seen in the
4.8 GHz data as well, but with less significance.
A comparison of the results from Table 1, with
those from the structure function analysis per
formed by Hughes, Aller, & Aller (1992) finds
broad agreement, in the sense that most of the
sources that we find to have characteristic time
scales also exhibited time scales in the structure
function analysis, and likewise for those sources
lacking time scales.After comparing the time
scales from the structure function analysis for
those sources that had time scales short enough to
meet our requirement of four repetitions over the
time window (i.e., shorter than 45 years for most
sources), we find that many of the time scales we
find are comparable to those found from the struc
ture function analysis. For instance, we deduce a
time scale of ∼ 2.6 yr for the source 0836+710 at
4.8 GHz, while the structure function finds that
the time series for this source is not correlated
above a time scale of 2.88 yr; also, we find a
characteristic time scale of ∼ 2.0 yr for the BL
Lac 1803+784, and the structure function analysis
AG(lj) and the average cross power¯ Fc(τn) differ
12
Page 13
gives a time scale of 1.86 yr. Although the struc
ture function analysis gives a measure of the time
scale above which variations appear to be uncorre
lated, which is not the same as the time scale that
we measure, we find that the results found from
the crosswavelettechnique agree with those found
the earlier structure function analysis of Hughes,
Aller, & Aller (1992).
Color postscript plots for the continuous and
crosswavelet transforms, as well as the plots for
the global wavelet spectrum and the average cross
power, are available for all sources from the UM
RAO website at
http://www.astro.lsa.umich.edu/
obs/radiotel/prcwdata.html.
7.CONCLUSIONS
We conclude that, complimented with the con
tinuous wavelet transform, the crosswavelet tech
nique can be an effective tool in the search for
quasiperiodicity of a time series. Comparing the
results for the PearsonReadhead survey sources
with that expected for a correlated noise process
of the form AR(1), we conclude that the observed
quasiperiodic behavior is unlikely to be the result
of an AR(1) process, as about 40% of the time
series for the PR sources had characteristic time
scales compared to about 10% expected for an
AR(1) process. The observed time scales may be
the result of a type of correlated noise that is not
AR(1), and it certainly may not even be station
ary noise. However, even if the QPVs arise from
correlated noise, it is meaningful to explore the
characteristic time scale of a time series, and such
results provide a useful diagnostic of the underly
ing variations.
After applying the crosswavelet algorithm to
the PearsonReadhead VLBI survey sources, anal
ysis revealed evidence for quasiperiodic variations
in ∼ 57% of the sources, as well as evidence that
∼ 67% of quasars, ∼ 38% of the BL Lacs, and 50%
of the galaxies have quasiperiodic behavior in at
least one observing frequency. The sources were
observed to have a mean characteristic time scale
of 2.4 yr, with standard deviation of 1.3 yr.
Because the analyzed radio band variations for
the PR sources originate in parsecscale jets, per
turbations that propagate at c will lead to observ
able fluctuations if they span the flow, guarantee
ing time scales of order years. In addition, coher
ent perturbations that arise from the excitation
of certain modes of oscillation of the flow, which
could give rise to quasiperiodic behavior with time
scales comparable to the dynamical response time
of the flow (i.e., years), have been shown to arise
quite naturally (Hardee et al. 2001). We conclude
that our results are in good agreement with the
characteristic time scales that we would expect to
observe based on the nature of these objects.
There is a transition region where the jet
changes from optically thick to optically thin, with
optical depth τ = 1 (Cawthorne 1991). Because
this region varies with the observing frequency, we
are looking at a different physical location in the
jet at each observing frequency. It is likely that
the time scale of quasiperiodic variations is de
pendent on their location in the jet. Naively, we
would expect to probe larger scale regions with
longer characteristic time scales, the lower the
observing frequency chosen.
analysis certainly find no such correlation of time
scale and frequency.However, jets may accel
erate due to adiabatic expansion, or decelerate
due to entrainment, leading to a change in bulk
Lorentz factor with position.
vature is now known to be a common feature of
these flows. A change in flow speed and/or flow
orientation with respect to the observer can lead
to a significant change in Doppler factor, and thus
to the observed time scale of activity, masking any
simple trends. Furthermore, local jet properties
and ambient conditions play a major role in de
termining what modes of the flow exist, and may
be excited, and observations that probe different
physical scales might well reveal activity in one
frequency band but not in another.
Although the number of sources analyzed here
is in no way exhaustive, we see no reason why
such quasiperiodic behavior should be confined to
the PearsonReadhead survey sources, and we find
it likely that many active galactic nuclei exhibit
quasiperiodicity. Only four galaxies had sufficient
data for analysis, and so are tabulated but not
discussed. A statistical analysis on the results of
applying the crosswavelet technique to a greater
number of sources would lead to a more interest
ing and conclusive comparison of quasiperiodic be
havior and source type, as well as evidence for
quasiperiodicity in active galactic nuclei in gen
The results of our
In addition, cur
13
Page 14
eral. In addition, the crosswavelet transform may
also be a useful tool for analyzing the correlation of
a source between different observing frequencies.
We would like to thank the anonymous ref
eree for many helpful and insightful comments
that have contributed to significant improvement
of this manuscript.
REFERENCES
Aller, H. D., Aller, M. F., Latimer, G. E., &
Hodge, P. E. 1985, ApJS, 59, 513
Aller, H. D., Aller, M. F., Latimer, G.E., &
Hughes, P. A. 2003, in preparation
Aller, M. F., Aller, H. D., & Hughes, P. A. 2003,
in press at ApJ
Cawthorne, T. V. 1991, in Beams and Jets in As
trophysics, ed. P. A. Hughes (Cambridge: Cam
bridge Univ. Press), 187
Farge, M. 1992, Ann. Rev. Fluid Mech., 24, 395
Hardee, P.E., Hughes, P.A., Rosen, A., & Gomez,
E.A. 2001, ApJ, 555, 744
Hughes, P.A., Aller, H.D., & Aller, M.F. 1992,
ApJ, 396, 469
Hughes, P.A., Aller, H.D., & Aller, M.F. 1998,
ApJ, 503, 662
Kyprianou, A., & Staszewski, W.J. 1999, Journal
of Sound Vibration, 228, 199
Lehto, H. J. 1999, in ASP Conf. Ser. 159, BL Lac
Phenomenon, ed. L.O. Takalo & A. Sillanp¨ a¨ a
(San Francisco: ASP) 257
Ott, M., Witzel, A., Quirrenbach, A., Krichbaum,
T. P., Standke, K. J., Schalinski, C. J., & Hum
mel, C. A. 1994, A&A, 284, 331
Pancheva, D., Mukhtarov, P., Mitchell, N.J.,
Beard, A.G., & Muller, H.G. 2000, Annales
Geophysicae, 18, 1304
Percival, D.P. 1995, Biometrika, 82, 619
Roy, M., Papadakis, I.E., RamosCol´ on, E.,
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This 2column preprint was prepared with the AAS LATEX
macros v5.0.
14
Page 15
Table 1
Characteristic periods found using the crosswavelet algorithm on the
PearsonReadhead VLBI survey sources. All periods are in years.
Source Opt. Class4.8 GHzData 8.0 GHzData 14.5 GHzData
0016+731
3C 20
0108+388
DA 55
0153+744
0212+735
3C 66B
3C 83.1
3C 84
0404+768
3C 147
3C 153
OI 147
OI 318
3C 179
0804+499
3C 196
0814+425
0831+557
0836+710
0850+581
0859+470
3C 216
3C 219
4C 39.25
4C 20.24
M 82
0954+556
0954+658
3C 236
1031+567
3C 268.1
3C 280
4C 66.22
3C 295
3C 309.1
3C 330
4C 41.43
1633+382
3C 343
1637+574
3C 345
1642+690
MKn 501
1739+522
1749+701
1803+784
3C 371
1823+568
3C 380
3C 388
3C 390.3
1928+738
3C 401
OV 591
OW 673
3C 438
BL LAC
2229+391
3C 452
2351+456
DA 611
Q
G
G
Q
Q
Q
G
G
G
G
Q
G
G
Q
Q
Q
Q
BL
G
Q
Q
Q
Q
G
Q
Q
G
Q
BL
G
G
G
G
G
G
Q
G
Q
Q
Q
Q
Q
Q
BL
Q
BL
BL
BL
BL
Q
G
G
Q
G
Q
Q
G
BL
G
G
Q
G
 218170 3.3
D
D
2.3
D
4.1
D
D

D

D
D
0.7
2.2
1.1
D

D

D
D

D

3.7
D


D
D
D
D
0.9
D
D
D
D

D
D
5.1
4.9

1.2



3.3
D
D


D
D
D
D
249
D
D

D

D
D

D

D
D


D


D

D
D

D

D
D
1.0
1.0
1.1
D

D
1.4
D
D

D

3.6
D


D
D
D
D

D
D
D
D

D
D



1.4

2.0


D
D
125
811566674
228 244263
6361617872
141173134
134
136
193
133
125
196
124
134
1772.7, 1.8, 1.2
D

D
2.6
D
D

D

D
D
D

D
D
D
D
1.3
D
D
D
D

D
D



1.4

2.0, 3.9

3.3
D
D
D

D
D
D
D
1.4
D
D

D
177462282
222166271
189228194
5541282
141
815
104
124
150
103
221167
168176147
244432 315
851
167
230
248
217
272
220
135
1529
331
525
348
357
300
451
241
927
278
302
383
256
419
307
275
3.4, 2.4

D
D
D
D
3.7
D
D
D
D
150
356
154
469315
85614413.5, 1.6, 0.7
D
D
3.8
D
1243
148167
Note.—D signifies insufficient data,  signifies no detected quasiperiodicity. The number of data
points are given only for those sources with sufficient data. If two time scales are given, the more
prominent one is listed first.
15
Page 16
Fig. 1.— (a) Continuous wavelet transform for a
sinusoid of period τa= 2.0 yr. (b) Crosswavelet
transform for the sinusoid with another sinusoid
of period τm = 2.059 yr. The contours on both
power (modulus) plots are for the 90% confidence
level for correlated noise, assuming a lag1 auto
correlation of α = 0.911362.
Fig. 3.— (a) Continuous wavelet transform for a
sinusoid of period τa = 2.0 yr with white noise
added, S/N = 0.2. (b) Crosswavelet for the sinu
soid of (a). The contours on both power plots are
for the 90% confidence level for correlated noise,
assuming a lag1 autocorrelation of α = 0.283620.
Fig. 4.— (a) Continuous wavelet transform for a
sinusoid of period τa= 2.0 yr and varying ampli
tude and phase. (b) Crosswavelet transform for
the sinusoid of (a). The contours on both power
plots are for the 90% confidence level for corre
lated noise, assuming a lag1 autocorrelation of
α = 0.894581.
Fig. 6.— (a) Continuous wavelet transform for a
Gaussian white noise signal. White noise is eas
ily recognizable by its erratic and complex struc
ture in wavelet space, particularly at small dila
tions. (b) Crosswavelet transform for the Gaus
sian white noise signal, corresponding to an ana
lyzing signal of period τn= 0.736 yr. The contours
on both power plots are for the 90% confidence
level for white noise.
Fig. 7.— (a) Continuous wavelet transform for a
correlated noise signal of the form AR(1), with lag
1 autocorrelation coefficient α = 0.9. (b) Cross
wavelet transform for the correlated noise signal.
The signal is crossed with a mock signal of period
τn= 3.901 yr. The contours on both power plots
are for the 90% confidence level for an AR(1) pro
cess of α = 0.9.
Fig. 10.— (a) Continuous wavelet transform for
0804+499 at 4.8 GHz. The periodic behavior is
easily seen as repetition in the real plot. The mod
ulus plot shows the significance of the behavior;
particularly notable is the ∼ 2.7 yr period. (b)
Crosswavelet transform for 0804+499 at 4.8 GHz
with an analyzing signal of period τn = 2.86 yr.
The data is well correlated with a sinusoid of this
period, as is evidenced by the uniformity in the
crosswavelet plots. The contours on both power
plots are for the 90% confidence level for corre
lated noise, assuming a lag1 autocorrelation of
α = 0.913.
Fig. 12.— (a) Continuous wavelet transform for
the BL Lac 1823+568 at 8.0 GHz. The contours
are the 90% confidence level for an AR(1) process
of α = 0.864. Note the significant behavior seen
after 1990. (b) Continuous wavelet transform for
1823+568 at 14.5 GHz. The contours are the 90%
confidence level for an AR(1) process of α = 0.987.
One can see that power is distributed over a broad
range in scale for this plot, centered about a dila
tion corresponding to a time scale of ∼ 3.3 yr. Al
though the two time series share similarities, there
are enough differences to account for the varying
behavior seen in the transforms.
16
Page 17
Table 2
Quasiperiodicity Statistics.
Type
1
Sources
2
Time Series
3
Mean Time Scale (yr)
4
Stnd. Dev. (yr)
5
Q/S
6
Qt/T
7
M/Q
8
Quasars
Galaxies
BL Lacs
All Types
18
4
8
30
51
10
24
85
2.3
2.0
2.3
2.4
1.4
1.1
1.1
1.3
0.67
0.5
0.38
0.57
0.43
0.4
0.42
0.42
0.5
0.5
1.0
0.59
Note.—Q/S is the ratio of sources exhibiting quasiperiodicity in at least one observing frequency to the total
number of sources. Qt/T is the ratio of the total number of time scales to the total number of time series. M/Q
is the ratio of sources that showed quasiperiodicity in more than one observing frequency to the total number of
quasiperiodic sources.
17
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