Editing Autocorrelation (section)

==Estimation==
For a [[Discrete signal|discrete]] process with known mean and variance for which we observe <math>n</math> observations <math>\{X_1,\,X_2,\,\ldots,\,X_n\}</math>, an estimate of the autocorrelation coefficient may be obtained as

<math display=block> \hat{R}(k)=\frac{1}{(n-k) \sigma^2} \sum_{t=1}^{n-k} (X_t-\mu)(X_{t+k}-\mu) </math>

for any positive integer <math>k<n</math>. When the true mean <math>\mu</math> and variance <math>\sigma^2</math> are known, this estimate is '''[[Biased estimator|unbiased]]'''. If the true mean and [[variance]] of the process are not known there are several possibilities:
* If <math>\mu</math> and <math>\sigma^2</math> are replaced by the standard formulae for sample mean and sample variance, then this is a '''[[Biased estimator|biased estimate]]'''.
* A [[periodogram]]-based estimate replaces <math>n-k</math> in the above formula with <math>n</math>. This estimate is always biased; however, it usually has a smaller [[mean squared error]].<ref>{{cite book |title=Spectral Analysis and Time Series |first=M. B. |last=Priestley |location=London, New York |publisher=Academic Press |year=1982 |isbn=978-0125649018 }}</ref><ref>{{cite book | last=Percival | first=Donald B. | author2=Andrew T. Walden | title=Spectral Analysis for Physical Applications: Multitaper and Conventional Univariate Techniques | url=https://archive.org/details/spectralanalysis00perc_105 | url-access=limited | year=1993 | publisher=Cambridge University Press | isbn=978-0-521-43541-3 | pages=[https://archive.org/details/spectralanalysis00perc_105/page/n217 190]–195}}</ref>
* Other possibilities derive from treating the two portions of data <math>\{X_1,\,X_2,\,\ldots,\,X_{n-k}\}</math> and <math>\{X_{k+1},\,X_{k+2},\,\ldots,\,X_n\}</math> separately and calculating separate sample means and/or sample variances for use in defining the estimate.{{Citation needed|date=May 2020}}<!--I really can't find a citation for that last type of estimator, even though I can verify numerically that it solves the negativity issue raise in doi:10.1080/00031305.1993.10475997-->

The advantage of estimates of the last type is that the set of estimated autocorrelations, as a function of <math>k</math>, 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 <math>X</math>'s, the variance calculated may turn out to be negative.<ref>{{Cite journal|last=Percival|first=Donald B.|date=1993|title=Three Curious Properties of the Sample Variance and Autocovariance for Stationary Processes with Unknown Mean|journal=The American Statistician|language=en|volume=47|issue=4|pages=274–276|doi=10.1080/00031305.1993.10475997}}</ref>