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Astronomical Data Analysis Software and Systems VII
ASP Conference Series, Vol. 145, 1998
R. Albrecht, R. N. Hook and H. A. Bushouse, e
Ö Copyright 1998 Astronomical Society of the Pacific. All rights reserved.
ds.
Methodology of Time Delay Change Determination for
Uneven Data Sets
V.L.Oknyanskij
Sternberg Astronomical Ins., Universitetskij Prospekt 13, Moscow,
119899, Russia
Abstract. At the previous ADASS Conference (Oknyanskij, 1997a)
we considered and used a new algorithm for time­delay investigations in
the case when the time delay was a linear function of time and the echo
response intensity was a power­law function of the time delay. We applied
this method to investigate optical­to­radio delay in the double quasar
0957+561 (generally accepted to be a case of gravitational lensing). It
was found in this way that the radio variations (5 MHz) followed the
optical ones, but that the time delay was a linear function of time with
the mean value being about 2370 days and with the rate of increase
V # 110 days/year.
Here we use Monte­Carlo simulations to estimate the significance
of the results. We estimate (with the 95% confidence level) that the
probability of getting the same result by chance (if the real V value was
equal to 0 days/year) is less then 5%. We also show that the method
can also determine the actual rate of increase V a of the time delay in
artificial light curves, which have the same data spacing, power spectrum
and noise level as real ones.
We briefly consider some other possible fields for using the method.
1. Introduction
At the previous ADASS Conference (Oknyanskij, 1997a, see also Oknyanskij
1997b) we considered and used a new algorithm for time­delay investigations in
the case when the time delay was a linear function of time and the echo response
intensity was a power­law function of the time delay. We applied this method
to investigate optical­to­radio delay in the double quasar 0957+561 (generally
accepted to be a case of gravitational lensing). It was found in this way that the
radio variations (5 MHz) followed the optical ones, but that the time delay was
a linear function of time with the mean value being about 2370 days and with
the rate of increase V # 110 days/year.
The cross­correlation function for this best fit is shown in Figure 1 together
with the cross­correlation function for the data without any corrections for pos­
sible variability of the time delay value. The maximum of the cross­correlation
for the last case (if V = 0 days/year) is less then 0.5. So we can note that our
fit explains the real data significantly better then the simple model with some
constant time delay. Meanwhile Monte­Carlo simulations are needed to estimate
488

Time Delay Change Determination for Uneven Data Sets 489
Figure 1. Cross­correlation functions for combined radio and optical
light curves. The solid curve shows the cross­correlation function with
correction of the data for the best fits parameters: V = 110 days/year
and # = 0.7 (Oknyanskij, 1997a,b). The dashed curve shows the sim­
ple cross­correlation function for the data without any correction i.e.,
taking V=0 days/year.
the significance of our result. The methodology of these estimations is briefly
explained below.
2. Monte­Carlo Simulations
The Monte­Carlo method is a powerful tool that can be used to obtain a
distribution­independent test in any hypothesis­testing situation. It is particu­
larly useful for our task, because it is very di#cult or practically impossible to
use some parametric way to test some null against the alternative hypothesis.
We used the Monte­Carlo method to estimate the significance of the result about
optical­to­radio delay (Oknyanskij & Beskin, 1993) obtained on the basis of ob­
servational data in the years 1979­1990 . With 99% confidence level we rejected
the possibility that the high correlation could be obtained if the optical and ra­
dio variations were really independent. In a subsequent paper we will estimate
the probability of obtaining our result by chance (Oknyanskij 1997a,b) applying
the new method to observational data obtained in the years 1979­1994. Here we
assume that the optical and radio data are correlated and estimate the signifi­
cance of the result about possible variability of the time delay value. So here,
under H 0 , optical and radio data are correlated but there is some constant time
delay and the absolute maximum of the correlation coe#cient obtained for some
V # V t > 0 days/year reflects random variation. We take V t = 110 days/year
as the test value. When H 1 is true, the time delay is not some constant, but

490 Oknyanskij
some increasing function of time. We need to calculate the distribution of V
under the null hypothesis. The p(V) value is probability, under H 0 , of obtaining
some V # value at least as extreme as (i.e., bigger or equal to) the V. The smaller
the p(V) value the more likely we are to reject H 0 and accept H 1 . If p(V t ) is
less than the typical value of 0.05 then H 0 can be rejected.
The idea of our Monte­Carlo test is the following:
1. We produce m=500 pairs of the simulated light curves which have the
same power spectra, time spacing, signal/noise ratio as the real optical and radio
data, but with constant value of time delay (# or = 2370 days) and about the
same maximum values of cross­correlation functions.
2. We apply the same method and make all the steps the same as was
done for the real data (Oknyanskij, 1997a,b), but for each of m pairs of the
simulated light curves. The proportion p(V) of obtained V # (see Figure 2) that
yield a value bigger or equal to V provides an estimate of the p(V) value. When
m # 100, standard error of the estimated p(V) value can be approximated by
well­known formula # = [p(1 - p)/m] 1/2 (see Robbins and Van Ryzin 1975).
An approximate 95% confidence interval for the true p value can be written as
p ± 2#. As it is seen from Figure 2 ­ p(V t ) # 3%. The approximate standard
error of this value is about 0.8%. We can write the 95% confidence interval for
p(V t ) = (3 ± 1.6)% and conclude with a 95% confidence level that p(V t ) # 5%.
So the H 0 can be rejected, i.e., time delay is some increasing function of time.
For the first step we assume that it is a linear function of time and found V =
110 days/year, a value approximately equal to the true one.
3. To show that the method has real abilities to determinate a value of
of the time delay rate of increase in the light curves we again use Monte­Carlo
simulation as it is explained in (1) and (2), but the actual V a value is 110
days/year. Then we obtain the histogram (Figure 3), which shows the distribu­
tion of obtained V # values. It is clear that the distribution has some asymmetry,
which could be a reason for some small overestimation of V value, since the
mathematical expectation of mean V' is about 114 days/year. Meanwhile we
should note that the obtained histogram shows us the ability of the method to
get the approximate estimate of the actual V a value, since the distribution in
the Figure 3 is quite narrow. Using this histogram we approximately estimate
the standard error #(V) # 15 days/year (for V value, which has been found for
Q 0956+561).
3. Conclusion and Ideas for Possible Applications of the Method
We have found a time delay between the radio and optical flux variations of
Q 0957+561 using a new method (Oknyanskij, 1997a,b), which also allowed us
to investigate the possibilities that (1) there is some change of time delay that
is a linear function of time, and (2) the radio response function has power­law
dependence on the time delay value. Here we estimate the statistical significance
of the result and the ability of the method to find the actual value of V as well as
its accuracy. We show that with 95% confidence level the probability of getting
a value of V # 110 days/year (if actual V a would be equal to 0 days/year) is
less then 5%. We estimate that standard error of the V value (which has been
found for Q 0957+561) is about 15 days/year.

Time Delay Change Determination for Uneven Data Sets 491
Figure 2. p(V) is probability to get some value V # # V if the actual
V a = 0 days/year. All Monte­Carlo test p values are based on apply­
ing our method to 500 pairs of artificial light curves with the actual
parameters # or (t 0 ) = 2370 days and V a = 0 days/year.
Figure 3. Result of Monte­Carlo test (see Figure 2) with actual val­
ues of V a = 110 days/year and # or (t 0 ) = 2370 days. The histogram
shows distribution of obtained V # values.
Finally, we can briefly note some other fields where the method may be
used:

492 Oknyanskij
1. Time delay between continuum and line variability in AGNs may be a
function of time as well as the response function possibly being some function
of time. So our method can be useful for this case.
2. Recently, it has been suggested by Fernandes et. al (1997) that variability
of di#erent AGNs might have coincident recurrent patterns in their light curves.
However it has been found that the time­scales for these patterns in NGC 4151
and 5548 are about the same, there are a lot of reasons to expect that the
patterns in AGN light curves may be similar, but have di#erent time scales.
It is possible to use our method with some enhancements to investigate this
possibility. Some Monte­Carlo estimations of the significance would be also very
useful. The probability that these common recurrent patterns in AGNs occur
by chance should be estimated.
References
Fernandes, R.C., Terlevich R. & Aretxaga I. 1997, MNRAS, 289, 318
Oknyanskij, V.L., & Beskin, G.M. 1993, in: Gravitational Lenses in the Uni­
verse: Proceedings of the 31st Liege International Astrophysical Collo­
quium, eds. J.Surdej at al. (Liege, Belgium: Universite de Liege, Institut
d'Astrophysique), 65
Oknyanskij, V.L. 1997a, in ASP Conf. Ser., Vol. 125, Astronomical Data Anal­
ysis Software and Systems VI, ed. Gareth Hunt & H. E. Payne (San
Francisco: ASP), 162
Oknyanskij, V.L. 1997b, Ap&SS, 246, 299
Robbins, H. & Van Ruzin, J. 1975, Introduction to Statistic (Science Research
Associates, Chicago), 167