acf

Auto Correlation Function fast method (Application).

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

[n]Xronos> acf

[n]Xronos> acf [series1]

[n]Xronos> acf[/qualifier1/qual2/... etc]  [series1]

More on acf

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 in acf, 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, before the auto correlation function is calculated, gaps and rejected newbins are replaced with zeroes.

The auto correlation error bars can only be calculated directly from the standard deviation of the average of the autocorrelations 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 autocorrelation errors in each interval by propagating the newbins statistical errors use the slow algorithm autocorrelation application acs). 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, by setting global parameter number 10 to 1.

Normalisations

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

• =0 autocorrelations are normalised by dividing by the number of good newbins in each interval, i.e. they are autocovariances.
• =1 (d/f); autocorrelations are normalised by dividing by the number of good newbins and the variance of newbins in each interval. Since the variance contains also the variance expected from (Poisson) noise etc. in newbins , the values of the autocorrelation for non zero time delays will be underestimated. The autocorrelation at zero time delay is =1 by definition.
• =2 non-zero (zero) delay autocorrelations are normalised by dividing by the number of good newbins and the excess variance (variance) of newbins in each interval. Since the effects due to the variance expected from Poisson noise in newbins are now removed, the autocorrelation 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. In this case in order to avoid a meaningless negative normalisation, acf adopts automatically normalisation number 1.
• =3 non-zero (zero) delay autocorrelations are normalised by dividing by the excess variance (variance) of newbins in each frame after averaging autocovariances. This is meant to avoid problems arising with normalisation number 2 when one or more intervals in a frame have negative excess variance. Problems arise again when, due to statistical fluctuations, the excess variance of a frame is negative. In that case, in order to avoid a meaningless negative normalisation, acf adopts automatically normalisation number 1. Note that a correction of the data error for instrument dead time effects might also be necessary; see e.g. EXOSAT ME light curves.