Correlated Noise

The reason that our formulae represent only lower bounds on the errors is that our assumptions about the properties of the noise may be false. In particular, we expect that in general the errors in the observed magnitudes will be correlated in time, due to transparency variations in the Earth's atmosphere, for instance. Schwarzenberg-Czerny (1991) shows how the error formulae may be modified in this case, and we re-derive his result below. In section 2, the assumption of uncorrelated noise was expressed by the relation $\langle \Delta m_i \Delta m_j \rangle = 0$ for $i \neq j$. For correlated noise this expression can be non-zero if i and j are close enough to one another, i.e., within a ``correlation length''. One simple choice for the correlation function would be

$$\langle \Delta m_i \Delta m_j \rangle = \left\{ \begin{arra... ... 0 & \mbox{\hspace{1em} otherwise} \end{array} \right. ,$$

where D is an estimate of the number of consecutive data points which are correlated. Instead of the above, we choose the following form for the correlation function of the magnitude fluctuations:

$$\langle \Delta m_i \Delta m_j \rangle = \sigma^2(m) \,\, e^{-\frac{(t_i-t_j)^2}{(\Delta t D/2)^2} }$$

If we use this expression for the correlation to compute $\sigma^2(\omega) \equiv \langle \omega^2 \rangle$, then, after some manipulations and making the same approximations as before, we find that

$$\sigma^2(\omega) = \sigma^2_0(\omega) \cdot A(\omega,D),$$

where $\sigma^2_0(\omega)$ is the variance of $\omega$ in the uncorrelated case as given by equation 9, and the function A is

$$A(\omega,D) = D \, \frac{\sqrt{\pi}}{2} \hspace{0.3em} e... ...t( \frac{D \Delta t \omega}{4} \right)^2 } \hspace{0.1em} .$$ (12)

If we take these correlations into account for the errors in phase and amplitude, we find that the variance of these quantities is also multiplied by the factor $A(\omega,D)$. For correlations in time which are much shorter than the period of the signal, i.e., $D \omega \Delta t \ll 1$, we have $A \approx D$. This is the result found by Schwarzenberg-Czerny (1991), in his equation 25, for example (this is sometimes referred to as ``red noise''). In the other limit, $D \omega \Delta t \gg 1$, the function A approaches zero and the errors vanish. This occurs because the errors in magnitude have become completely correlated in time and now just represent a constant offset to all the data points, which produces no errors in the least squares fits. Finally, we note that A is a maximum when $D \Delta t = 2 \sqrt{2}/\omega \approx 0.5 \, P$, i.e., when the correlation time is of order the period of the signal, which, of course, is to be expected. In this case, the value of A is $A_{\rm max} \approx 0.24 \, P/\Delta t$. For observations of pulsating white dwarfs, the periods may be as high as 1000 sec and the sampling may be every 10 sec, so that $A_{\rm max} \approx 24$. Thus, the error estimates derived in the second section could be too low by a factor of $\sqrt{A} \approx 5$ if the errors are correlated in the way we have assumed. To summarize, the effect of correlations in the noise is in general to increase the magnitude of the errors. For correlations in time which are short compared to the period of the signal, the result is to multiply the previously obtained variances by D, the number of consecutive data points which are correlated. Equivalently, this means multiplying the equations for the errors themselves, equations 4, 8, and 9, by a factor of $\sqrt{D}$. As Schwarzenberg-Czerny (1991) points out, it is possible to obtain an estimate for D from a fit to the data, either visually, or by calculating the autocorrelation function of the residuals and fitting an appropriate function (such as a Gaussian) of width D to it.