3 Statistical analysis

For the data analysis we use the modulation index m, which is related to the mean flux density <S> and the rms flux density variations $\sigma_S$ by

$$m {\rm [\%]} = 100 \cdot \frac{\sigma_S}{<S>}\cdot$$ (1)

It provides a measure of the strength of the variations observed, although it does not take into account the error of the individual measurement. As a criterion whether a source is variable or not, we perform a $\chi^2$ test with

$$\chi^2 = \sum_{i=1}^{N} \left( \frac{S_i - <S>}{\Delta S_i} \right)^2\, ,$$ (2)

where the S_i are the individual flux densities and $\Delta S_i$ their errors, e.g. Bevington & Robinson (1992). Actually, we determine the probability that a constant function is a good fit to the data. Only sources for which the probability that they can be fitted by a constant function is $\leq 0.1$%, are considered to be variable.

To compare the modulation indices at different wavelengths and epochs, and to obtain a quantity that is independent of the measurement noise, we define the variability amplitude for the variable sources as

$$Y {\rm [\%]} = 3 \sqrt{m^2-m_0^2}\, ,$$ (3)

where m₀ is the modulation index of a non-variable source (see Heeschen et al. 1987). The factor 3 is somewhat arbitrary, but makes Y to be of the order of the peak-to-peak amplitude of the variations.

For the polarized intensity the corresponding quantities are

$$m_P {\rm [\%]} = 100 \cdot \frac{\sigma_P}{<P>}\, ,$$ (4)

$$Y_P {\rm [\%]} = 3 \sqrt{m_P^2-m_{P,0}^2}\, ;$$ (5)

and for the polarization angle we use

$$m_\chi {\rm [^\circ]} = \sigma_\chi,$$ (6)

$$Y_\chi {\rm [^\circ]} = 3 \sqrt{\sigma_\chi^2 - \sigma_{\chi,0}^2}\, \cdot$$ (7)

We note that this description of the polarization variability is somewhat problematic because the polarization measurements are more strongly affected by systematic effects than the total intensity data, and because the statistical errors have a Gaussian distribution in the Stokes parameters Q and U, not in P and $\chi$. We have nevertheless chosen the above description, since the separation of the variability of P and $\chi$ is more relevant for a physical interpretation of the variations.

Structure functions can be used for the analysis of the characteristics of the variability, and to search for typical timescales and periodicities. The first order structure function $D(\tau)$ is defined as (see Simonetti et al. 1985):

$$D (\tau) = \; < (\,S(t) - S(t+\tau)\,)^2>_{\rm t}$$ (8)

with $<.>_{\rm t}$ being the mean over time again. Since our data are unevenly sampled, we derived the structure functions using an interpolation algorithm. For any given time lag $\tau$, the value of $S(t+\tau)$ is calculated by linear interpolation of the two adjacent data points. Each structure function was derived twice, firstly starting from the beginning of the time series, and secondly starting at the end and proceeding backwards. This provides a rough assessment of the errors caused by the interpolation process (but not of the systematic uncertainty due to the finite length of the observations).

A typical timescale in the lightcurve (i.e., the time between a minimum and a maximum or vice versa) is indicated by a maximum of the structure function, while a period in the lightcurve causes a minimum of the structure function (e.g. Heidt & Wagner 1996). Following the nomenclature of Heeschen et al. (1987), structure functions can be used to define variability types, depending on the observed timescale. A source whose structure function reaches a maximum within the observing period is called type II (i.e., a fast-varying source). In the case of a monotonically increasing structure function (i.e., when the source shows no typical timescale less than the total observing period), it is assigned type I. Since this definition is problematic for longer observing campaigns as the ones in February 1990 and October 1992, we define a variable source whose structure function reaches a maximum or plateau within $\tau \leq 2$ days as being of type II (cf. Wagner & Witzel 1995), otherwise type I. Non-variable sources are assigned type 0.