The auto-correlation function analysis

The basic concept of the correlation analysis consists in searching for corresponding features, whether simultaneous or shifted by a certain time lag, in the light curves of two signals. The cross-correlation function CF at different time lagsτfor two light curves x(t) and y(t) is given by:

CF(τ)= R∞

−∞(x(t)−¯x)(y(t+τ)−¯y) dt R∞

−∞(x(t)−¯x)(y(t)−¯y)dt . (9.7)

The characteristic time scales of the variations 125

Figure 9.6: Auto-correlation function of the 22 GHz light curve as obtained with the discrete (circles) and interpolated (continuous line) approaches. τ0 as calculated with different definitions are shown (horizontal continuous lines):τ_HWHMandτ_AC=0superimposed to the AC plot andτ⁽²⁾_AC=0.6andτ⁽²⁾_l=2.25on the left side of the AC.

As in real cases the available data sets are not continuously sampled but limited to a number N of data points, the cross-correlation function for two real light curves x_νand y_νis given by:

CF(τ)= PN

i=1[xν(ti)−x¯ν][yν(ti+τ)−y¯ν]

(N−1) σ_xνσ_yν , (9.8)

whereσ_xνandσ_yνare the standard deviations of the two light curves. The main issue connected to the correlation analysis is the sampling of the data sets which is usually far from being regular, as it would be required to apply Eq. (9.8). Different solutions have been proposed to solve this problem. One possibility is to linearly interpolate one of the two curves in order to use the values of the interpolated function at the times t0+τ, ...,tN+τ, where t0, ...,tNare the times sampled by the other curve (Gaskell

& Sparke 1986). This is the so-called interpolated correlation function method (ICF). Edelson &

Krolik (1988) proposed an alternative option consisting in a rebinning of the light curves on time intervalsδt around eachτ. Therefore the value of the correlation function at a certainτcorresponds to an average over the range [τ−δt/2, τ+δt/2]. This method is known as the discrete correlation analysis (DCF).

Both methods are known to have strengths and weaknesses depending on the data and their sam-pling (White & Peterson 1994). Poor or peculiar samsam-plings can even produce misleading or spurious results. In order to reduce this risk and obtain more reliable results, we decide to apply both methods, using the improvements proposed by Gaskell & Peterson (1987) and White & Peterson (1994). For the ICF, this consists in averaging the correlation function found interpolating the first light curve with that obtained interpolating the second one (Gaskell & Peterson 1987). For the DCF, for each time lag we compute the mean and the variance using only the points that contribute to the calculation of the

Figure 9.7: Auto-correlation function of the H light curve as obtained with the discrete (circles and dashed line) and interpolated (continuous line) approaches. The narrow spike at small lags is a measurement of the noise of the light curve.

corresponding correlation coefficient (White & Peterson 1994).

The auto-correlation function AC is a particular case of the cross-correlation one, where the two light curves x(t) and y(t) are the same one:

AC(τ)= R∞

−∞(x(t)−¯x)(x(t+τ)−¯x) dt R∞

−∞(x(t)−¯x)²dt . (9.9)

Due to its definition, the auto-correlation function has always its maximum at 0 lag and develops symmetrically at positive and negative lags, decreasing with different patterns towards lower correla-tion values (Fig. 9.6). The auto-correlacorrela-tion funccorrela-tion might present in some cases a thin spike around τ = 0 (Fig. 9.7) due to the scattering in the data points of the light curve. We renormalise all the auto-correlation functions to their value at 0.05 years in order to remove this effect.

τ₀definition

The different shapes of the AC function contain information about the relative importance of the different sampled time scales and they can be parametrised in different ways to define a characteristic time scaleτ₀of the variations of the light curve. A few possibilities are to calculateτ₀as:

• the smallerτfor which AC(τ)=0 (≡τ_AC=0)

• the half width at the half maximum (HWHM) of the AC function (≡τ_HWHM)

• the second moment of the correlation peak (Fromerth & Melia 2000), calculated in the interval [−τ_k,+τ_k], either with fixed extremes for all the light curves (≡τ⁽²⁾_l=τ

k) or at a fixed level AC_k of

The characteristic time scales of the variations 127 the correlation coefficient, implying possibly differentτ_k for each light curve (≡ τ⁽²⁾_AC=AC

k). In this case the definition of the characteristic time scale is:

τ⁽²⁾= R_τk

−τkAC(τ) τ²dτ Rτk

−τkAC(τ) dτ . (9.10)

Even though other definitions are possible (e.g. Hovatta et al. 2007), we limit the comparison to those listed above, excluding the data sets with less than 40 data points, for which the AC cannot be reliably calculated. The absolute value ofτ₀ naturally depends on the definition that is chosen, as in the example auto-correlation shown in Fig. 9.6. τ⁽²⁾_l=τ

k and τ⁽²⁾_AC=AC

k also depends on the choice of the integration extremes, as it can be seen in Fig. 9.8. To study this dependence, we use 16 different, equally spaced,τ_k in the interval [0.25,4] and a AC_kvalue of 0.6 for the computation of the characteristic time scale as second moment of the correlation peak. Due to the sometimes sparse or peculiar sampling of the light curves,τ₀ can also depend on whether the DCF or the ICF method is applied. In Fig. 9.8 three examples representative of howτ⁽²⁾_l=τ

k varies with the integration extremes and the two methods are shown. For the 15 GHz data set (top panel), there are only small differences between the results of the ICF and DCF andτ⁽²⁾_l=τ

k smoothly increases with the integration limit. For the 5 keV light curve (centre panel), there is an irregular increase of τ⁽²⁾_l=τ

k and different values are found with the DCF and ICF methods. The data sets showing a similar behaviour should therefore be excluded. For the 22 GHz light curve (bottom panel),τ⁽²⁾_l=τ

k increases up toτ_k =2.25 and then reaches a plateau with both the ICF and DCF methods due to the fact that the correlation coefficient becomes negative atτ_k = 2.25, therefore making unusable this definition above this value. Other light curves show the same behaviour but at differentτ_k (down toτ_k ≤1). It is therefore very difficult to choose aτ_k for which the calculation ofτ⁽²⁾_l=τ

k is meaningful for most of the light curves presented here. For this reason we do not consider this method suitable for our purposes.

Figure 9.8: Dependence of theτ⁽²⁾_l=τ

k values on the choice of the integration extremeτk and of the discrete (circles) or interpolated (triangles) correlation methods for the 15 GHz, 5 keV and 22 GHz light curves.

At this point, we select one definition of τ₀ for the following of our analysis, assuming that the most stable method is that minimizing the differences between the results obtained with the DCF and ICF methods. When looking at the average of the parameter ^|τ

DCF 0 −τ^ICF₀ |

τ^DCF₀ +τ^ICF₀ for the differentτ₀definitions it appears that forτ_AC=0 the differences are on average the smallest. As differences in the ICF and DCFτ₀are anyway present, for each light curve the range included between the ICF and DCF values is our best estimate of the ACτ₀.