The uncertainty of measured autocorrelation functions

Two general methods are available for the routine estimation of an autocorrelation function: the average of lagged products of the observations or a parametric solution with a time series model. Theoretical arguments as well as experience with measured random data prove that the accuracy of the time series estimate is always better than the lagged product estimator. Furthermore, the root mean square error is a poor measure for the uncertainty of the estimated autocorrelation function. The objective quality of two estimated autocorrelation functions which have the same mean squared error can be completely different. A deviation in the autocorrelation function with a small mean square error can destroy the positive definite property that guarantees a valid non-negative spectrum after transformation. Very large and important relative errors in weak parts of the spectrum will only give a small contribution to the value of the absolute root mean square error of the autocorrelation function that belongs to it. The cepstral measure is the preferred time domain uncertainty measure for autocorrelation functions. That is related to the mean square error of the logarithm of spectral estimates; the logarithm gives the same relative influence to all parts of the spectral density. The uncertainty of the spectrum and hence of the logarithm of the spectrum is easily related to the uncertainty of the parameters of time series models. This gives the possibility to evaluate the cepstral uncertainty of the autocorrelation function as a function of the known uncertainty of the estimated parameters