Historical Misconceptions in Autocorrelation Estimation

The sample autocovariance function, which is estimated as mean lagged products (LPs) of random observations, can also be obtained as the inverse Fourier transform of the periodogram of the data that are augmented with zeros. Hence, the quality of the sample autocovariance as a representation of stochastic data is the same as that of a raw periodogram. The LP estimate is not based on any efficient estimation principle for random data. The spectral density and the autocovariance function can be estimated much more accurately with parametric time series models. A recent development in time series analysis gives the possibility to select automatically the type and the order of the best time series model for any set of random observations. The spectral accuracy of the selected model is better than the accuracy of modified periodograms. In addition, the accuracy of the parametric estimate of the autocovariance function is the same or better for every individual lag than what can be achieved by the nonparametric mean-LP estimates. Perhaps even more important, the estimated time series parameters define the autocovariance as a complete function, for all lags together.

I. Introduction

The first spectral estimates in the early 19th century used glass prisms to reveal spectral lines [1]. It would last a long time before Wiener [2] and Khintchine [3] independently described the relation between the autocorrelation function and the spectrum of a stationary stochastic process. For computational reasons, Wiener could use only a few lagged-product (LP) autocovariances to calculate an estimate for the spectrum. The rediscovery of the fast Fourier transform (FFT) algorithm in [4] was a major computational breakthrough for periodogram analysis. The autocovariance estimation with LPs is connected with the periodogram by the inverse Fourier transform. The reduced computer effort enabled the routine Fourier analysis of extensive sets of random data with the nonparametric method. This was the reason the analysis with tapered and windowed periodograms has been the main practical tool for windowed spectral analysis and truncated autocovariance estimation for a long time. A problem with this nonparametric estimation is that it requires a subjective spectral or lag window. The optimal length and type of that window cannot be chosen objectively, independent of the unknown character of the data. The choice depends on preferences of the data analyst. It is remarkable that the raw periodogram has never been considered as a useful spectral estimate for random processes, while its counterpart that is obtained by the inverse Fourier transform, i.e., the mean LPs, is often considered as the natural autocovariance estimate.