= Autocorrelation
{title2=$\rho(h)$}
{wiki}
= ACF
{c}
{synonym}
= Autocorrelation function
{synonym}
The <autocorrelation function> of a real <weakly stationary process> with positive <variance> is its normalized <autocovariance>:
$$
\rho(h)=\frac{\gamma(h)}{\gamma(0)}=\operatorname{Corr}(X_{t+h},X_t).
$$
It is dimensionless, with $\rho(0)=1$, $\rho(-h)=\rho(h)$, and $|\rho(h)|\le1$ by the <Cauchy-Schwarz inequality>. It is unchanged by replacing $X_t$ with $aX_t+b$ for fixed $a\ne0$. This makes <autocorrelation> useful when comparing records measured on different scales. The <ACF> is a property of the model; the <sample autocorrelation function> estimates it and will generally not reproduce theoretical zeros exactly.
An <AR(1)> with coefficient $\phi=0.7$ has <autocorrelations> $0.7,0.49,0.343$ at lags one through three. With $\phi=-0.7$ these become $-0.7,0.49,-0.343$: the same diminishing magnitude, alternating in sign. An <MA(1)> with $\theta=0.7$ instead has lag-one <autocorrelation> $0.7/1.49=70/149\simeq0.4698$ and zero at every higher lag. Equal numerical coefficients in an <autoregressive model> and a <moving-average model> do not mean equal dependence. See https://www.stat.berkeley.edu/~bartlett/courses/153-fall2010/lectures/3.pdf[Bartlett, lecture 3: theoretical ACFs and prediction].
\Image[/autoregressive-moving-average-correlation.png]
{title=Theoretical autocorrelation: AR(1) with coefficients 0.7 and -0.7, compared with MA(1) with coefficient 0.7}
The bars show exact model <autocorrelations>, not estimates from the illustrated <sample paths> in <time series>. The <AR(1)> retains a diminishing trace of a disturbance at arbitrarily large lags; the <MA(1)> stops sharing driving noise after lag one. A zero <autocorrelation> means <uncorrelated random variables>, and does not generally imply <independent random variables>. For a <Gaussian process> it does imply independence of the two coordinates.
There is a direct prediction interpretation. Among affine predictions based on just $X_t$, the best <linear predictor> of $X_{t+h}$ is $\mu+\rho(h)(X_t-\mu)$, with <mean squared prediction error> $\gamma(0)[1-\rho(h)^2]$. Thus negative <autocorrelation> is also useful predictive information. A prediction using the whole observed past can do better; <best linear prediction from a finite past> describes that problem. Interpreting the affine prediction as a <conditional expectation> requires additional assumptions, such as a joint <normal distribution>.
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