The commands COVAR/TSA serves for the calculation of the covariance and autocovariance functions. Pairs of signals with matching ACF functions may be analysed further with DELAY/TSA. Matching ACF functions may be obtained for some data after some massaging.
This command can also be used for the calculation of the autocovariance function (ACF) by simply using the same series for the two input data sets. Here one shifted series is used as a model for the other. The covariance statistic is used to evaluate the consistency of the two series.
The covariance statistics is akin to the power spectrum statistics and hence to the statistics (Sect. 12.3.2, Lomb, 1976, Scargle, 1982). The number of degrees of freedom varies among time lag bins. Thus, in order to facilitate the evaluation of the results, errors of the ACF are returned. The expected value of the ACF for pure noise is zero. The value returned for 0 lag corresponds to the correlation of nearby but not identical observations. This is so because the correlation of any observation with itself is ignored in the present algorithm, for numerical reasons. The correlation function for a lag identical to zero can be easily computed as the signal variance.
For input, individual measurements must be given with their variances. DELAY/TSA requires the smoothed ACF, common for the two series, to be supplied by the user in analytical form. The form of the ACF can be determined using COVAR/TSA and the MIDAS FIT package (Vol. A, Chapter 8). For this purpose, the ACF of both series should be the same. Often this can be achieved after some massaging of the data. To broaden the ACF, pass the series through a low pass filter. NORMALIZE/TSA may be used to normalize the variances and thus to normalize the ACF maxima. The ACF is passed to the command either via values of the parameters of one of the functions predefined within the TSA package or as the source code of a user-supplied FORTRAN function.
The method is quite new; it should be applied with some caution. Its only presently known practical test has been a consistency check of the results of independent analyses of optical and radio light curves of a pair of gravitationally lensed quasar images (Press et al., 1992). Not only shapes but also values of the ACF should match. This may be achieved by scaling the variances of the observations with NORMALIZE/TSA.