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ASP Conference Series, Vol. 145, 1998

Editors: R. Albrecht, R. N. Hook and H. A. Bushouse

**V.L.Oknyanskij**

Sternberg Astronomical Ins., Universitetskij Prospekt 13, Moscow,
119899, Russia

Here we use Monte-Carlo 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
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.

At the previous ADASS Conference (Oknyanskij, 1997a, see also Oknyanskij 1997b) we considered and used a new algorithm for time-delay investigations in the case when the time delay was a linear function of time and the echo response intensity was a power-law function of the time delay. We applied this method to investigate optical-to-radio 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 days/year.

The cross-correlation
function for this best fit is shown in Figure 1 together with the
cross-correlation function for the data without any corrections for possible
variability of the time delay value. The maximum of the cross-correlation
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 Monte-Carlo simulations are needed
to estimate the significance of our result. The methodology of these
estimations
is briefly explained below.

The idea of our Monte-Carlo 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 ( = 2370 days) and about
the same maximum values of cross-correlation 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 (see Figure 2) that yield a value bigger or equal to **V** provides
an estimate of the value. When ,standard error of the estimated
value can be approximated by well-known formula
(see Robbins and Van Ryzin 1975).
An approximate 95% confidence interval for the true *p* value can be written as
. As it is seen from Figure 2 - . The approximate standard error of this value is about .We can write the 95% confidence interval for and conclude with a 95% confidence level that 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

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
Monte-Carlo simulation as it is explained in (1) and (2), but the actual
value is 110 days/year. Then we obtain the histogram
(Figure 3), which shows the distribution of obtained 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 ' 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 value,
since the distribution in the Figure 3 is quite narrow.
Using this histogram we approximately estimate the standard error
15 days/year (for **V** value, which has been
found for Q 0956+561).

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 power-law 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 110 days/year (if actual
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.

Finally, we can briefly note some other fields where the method may be used:

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 different AGNs might have coincident recurrent patterns in their light curves. However it has been found that the time-scales 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 different time scales. It is possible to use our method with some enhancements to investigate this possibility. Some Monte-Carlo estimations of the significance would be also very useful. The probability that these common recurrent patterns in AGNs occur by chance should be estimated.

Fernandes, R.C., Terlevich R. & Aretxaga I. 1997, MNRAS, 289, 318

Oknyanskij, V.L., & Beskin, G.M. 1993, in: Gravitational
Lenses in the Universe: Proceedings of the 31st Liege International
Astrophysical Colloquium, eds. J.Surdej *at al.* (Liege, Belgium:
Universite de Liege, Institut d'Astrophysique), 65

Oknyanskij, V.L. 1997a, in Astronomical Data Analysis Software and Systems VI, ASP Conf. Ser., Vol. 125, eds. Gareth Hunt and 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

© Copyright 1998 Astronomical Society of the Pacific, 390 Ashton Avenue, San Francisco, California 94112, USA

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