of Z-transform of a discrete signal. While the brute force algorithm is order , several efficient algorithms exist which can compute the autocorrelation in order . For example, the Wiener–Khinchin theorem allows computing the autocorrelation from the raw data with two fast Fourier transforms (FFT): where IFFT denotes the inverse fast Fourier transform. The asterisk denotes complex conjugate. Alternatively, a multiple correlation can be performed by using brute force calculation for low values, and then progressively binning the data with a logarithmic density to compute higher values, resulting in the same efficiency, but with lower memory requirements. Estimation For a discrete process with known mean and variance for which we observe observations , an estimate of the autocorrelation coefficient may be obtained as for any positive integer . When the true mean and variance are known, this estimate is unbiased. If the true mean and variance of the process are not known there are several possibilities: If and are replaced by the standard formulae for sample mean and sample variance, then this is a biased estimate. A periodogram-based estimate replaces in the above formula with . This estimate is always biased; however, it usually has a smaller mean squared error. Other possibilities derive from treating the two portions of data and separately and calculating separate sample means and/or sample variances for use in defining the estimate. The advantage of estimates of the last type is that the set of estimated autocorrelations, as a function of , then form a function which is a valid autocorrelation in the sense that it is possible to define a theoretical process having exactly that autocorrelation. Other estimates can suffer from the problem that, if they are used to calculate the variance of a linear combination of the 's, the variance calculated may turn out to be negative. Regression analysis In regression analysis using time series data, autocorrelation in a variable of interest is typically modeled either with an autoregressive model (AR), a moving average model (MA), their combination as an autoregressive-moving-average model (ARMA), or an extension of the latter called an autoregressive integrated moving average model (ARIMA). With multiple interrelated data series, vector autoregression (VAR) or its extensions are used. In ordinary least squares (OLS), the adequacy of a model specification can be checked in part by establishing whether there is autocorrelation of the regression residuals. Problematic autocorrelation of the errors, which themselves are