ccf

Cross Correlation Function fast method (Application).

Application ccf calculates the cross correlation function of a time series by using a FFT algorithm. It requires two input series. It is run in the Question/Answer user interface by typing:

[n]Xronos> ccf

[n]Xronos> ccf [series1] [series2]

[n]Xronos> ccf[/qualifier1/qual2/... etc]  [series1] [series2]

More on ccf

The newbin time must correspond to an integer multiple of the maximum bin time. The default newbin time is either the maximum bin time (if fewer than 512 newbins are expected) or the integer multiple of it which gives a single interval with 512 newbins at most. Due to the FFT algorithm used in ccf, the number of newbins per interval must be a power of 2. The average count rate in each interval is subtracted from all newbins, and gaps and rejected newbins are replaced with zeroes, before the cross correlation function is calculated.

The time delays refer to the 1st series: therefore a cross correlation function peaking for positive time delays indicates that variations in the 2nd series lag those in the 1st series.

The cross correlation error bars can only be calculated directly from the standard deviation of the average of the cross correlations in each delay bin from different intervals, if the specified number of intervals per frame is larger than the value of global parameter number 9 (default =5); otherwise error bars are set =0. (To calculate the cross correlation errors in each interval by propagating the newbins statistical errors use application ccs). Error bars are plotted by default.

The effects introduced by windows, data gaps etc. can be studied by analysing the exposure profile of the time series. This is done by setting global parameter number 10 to 1. By setting global parameter number 12 to 1, a strict simultaneousness between the two series can be imposed: any rejected newbin or gap found in one series will cause the corresponding newbin of the other series to be rejected. This is useful to check whether artificial time delay between the two series are introduced by a non strict simultaneousness of the two series.

Note that no result rebinning is allowed by ccf.

Normalisations

The analysis normalisation flag, specified by global parameter number 11, has the following meaning for application ccf:

• =0 cross correlations are normalised by dividing by the sqrt of the product of the number of good newbins of the two series in each interval, i.e. they are cross covariances.
• =1 (d/f); cross correlations are normalised as in the case =0 and by dividing also by the sqrt of the product of the newbin variances of the two series in each interval. Since the variance contains also the variance expected from (Poisson) noise etc. in newbins, all values of the cross correlation are underestimated.
• =2 cross correlations are normalized as in the case =0 and by dividing also by the sqrt of the product of the newbin excess variances of the two series in each interval. Since the effects due to the variance expected from the Poisson noise in newbins are now removed, the cross correlation is now normalised correctly (note that a correction of the data errors for instrument dead time effects might also be necessary; see e.g. EXOSAT ME light curves). Problems arise when, due to statistical fluctuations, the excess variance of an interval is negative for any of the two series. In this case in order to avoid a meaningless negative normalisation, ccf adopts automatically normalisation number 1.
• =3 cross correlations are normalized by dividing by the sqrt of the product of the avg. newbin excess variances of the two series in each frame after averaging cross covariances. This is meant to avoid problems arising with normalisation number 2 when one or more intervals in a frame have a negative excess variance, for any of the two series. Problems arise again when, due to statistical fluctuations, the excess variance of a frame is negative for any of the two series. In this case, in order to avoid a meaningless negative normalisation, ccf adopts automatically normalisation number 1. Note that a correction of the data errors for instrument dead time effects might also be necessary; see e.g. EXOSAT ME light curves.