## A Conceptual Vocalubary of Time Series Analysis

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### A

#### autocorrelation function (ACF)

The autocorrelation function (ACF) is the normalized autocovariance function defined as $$\rho (\tau)=\frac{\gamma(\tau)}{\sqrt{\gamma(0).\gamma(0)}}=\frac{\gamma(\tau)}{\gamma(0)},$$where $$\tau = |s-t|$$ is the lag time, or the amount of time by which the signal has been shifted.
The ACF measures the linear predictability of the series at time $$t$$, say $$x_t$$ using only the value $$\tau$$. Using the Cauchy–Schwarz inequality which implies$${|\gamma(\tau)|}^2 \leq \gamma(0)^2$$ it can be shown easily that$$-1< \rho(\tau)<1.$$

#### autocovariance function

The autocovariance function is defined as the second moment product$$\gamma (s,t)=cov(x_s,x_t)=E[(x_s- \mu_s)(x_t- \mu_t)]$$ for all $$s$$, $$t$$. Note that for all time points  s and t. The autocovariance measures the linear dependence between two points on the same series observed at different times. Recall from classical statistic $$\gamma (s,t)=0$$, $$x_t$$  and $$x_s$$  are not linearly related, but there still may be some dependence structure between them. If, however, $$x_t$$  and $$x_s$$ are bivariate normal, $$\gamma (s,t)=0$$ ensures their independence. It is clear that, for     $$s = t$$, the autocovariance reduces to the (assumed finite) variance,$$\gamma (s,t)=E[(x_s- \mu_s)(x_t- \mu_t)^2]=var(x_t).$$
Example: The white noise series $$w_t$$ has $$E(w_t)=0$$ and $$\gamma_w (s,t)=cov(w_s,w_t)= \left\{ {\begin{array}{ccccccccccccccc}{\sigma^2_w,}&{{s} = t,}\\{0,}&{{s} \ne t.}\end{array} } \right.$$