The Quality of Lagged Products and Autoregressive Yule–Walker Models as Autocorrelation Estimates

The sample autocorrelation function is defined by the mean lagged products (LPs) of random observations. It is the inverse Fourier transform of the raw periodogram. Both contain the same information, and the quality of the full-length sample autocorrelation to represent random data is as poor as that of a raw periodogram. The autoregressive (AR) Yule-Walker method uses LP autocorrelation estimates to compute AR parameters as a parametric model for the autocorrelation. The order of the AR model can be taken as the full LP length, or it can be determined with an order selection criterion. However, the autocorrelation function can more accurately be estimated with a general parametric time-series method. This parametric estimate of the autocorrelation function always has better accuracy than the LP estimates. The LP autocorrelation function is as long as the observation window, but parametric estimates will eventually die out. They allow an objective answer to the question of how long the autocorrelation function really is.

I. Introduction

The Processing of random signals became a useful computational tool after the rediscovery of the fast Fourier transform (FFT) algorithm [1]. The square of the absolute value of the FFT of the signal is the periodogram [2]. This algorithm enabled the routine Fourier analysis of large sets of stochastic data. The inverse Fourier transform of the periodogram is the lagged-product (LP) autocovariance estimate. It is computationally efficient to calculate the autocorrelation function as the inverse transform of the periodogram. Like the periodogram for spectra, it has been treated in the literature as if it is the obvious estimator for the autocorrelation function for random data [2]–[6].