Persistent Misconceptions in Autocorrelation Estimation

The non-parametric sample autocovariance function, estimated as mean lagged products of random observations, is the Fourier transform of the periodogram. Hence, the quality of the sample autocovariance as a representation of stochastic data is as poor as that of a raw periodogram. The lagged product estimate is not based on any efficient estimation principle. However, 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 data with unknown characteristics. The spectral accuracy of the selected model is better than the accuracy of all variants of periodograms. Also 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 non-parametric mean-lagged-product estimates. More important, the estimated time series parameters define the autocovariance as a complete function, for all lags together

I. Introduction

A historical perspective of spectral estimation described how the first spectral estimates in the early 19^th Century used glass prisms to reveal spectral lines [1]. It would still last a long time before Wiener [2] and Khintchine [3] described the relation between the autocorrelation function and the spectrum of a stationary stochastic process. Wiener could use only a few lagged product autocovariances to estimate the spectrum. The rediscovery of the FFT algorithm was a major computational breakthrough [4]. This was foliowed by the maximum entropy spectral analysis of Burg [5] with autoregressive (AR) models. AR models still required order selection, for which Akaike developed the famous AIC criterion [6].