# Unit 6 Practical Autocorrelation

## 6.1 Estimation!

### 6.1.1$$\rho_k \rightarrow 0$$

For a stationary time series, if the autocorrelation approaches zero, then :

A Single realization lets us estimate mean, variance, and autovvariance!

Remember that $$\rho_k = \frac{\gamma_k}{\gamma_0}$$, and we use n-k pairs to calculate it (the summation)

#### 6.1.1.1 Mean

Just calculate the mean normally for this case.

#### 6.1.1.2 Variance

$\mathrm{Var}\left(\bar{X}\right) = \frac{\sigma^2}{n} \sum^{n-1}_{k = -(n-1)} \left( 1 - \frac{\mid{k}\mid}{n} \right)\rho_k$

$$\sigma^2$$ is calculated as normal, we will see rhok next!

remember !

Now it is time for some code!

library(glue)
##
## Attaching package: 'glue'
## The following object is masked from 'package:dplyr':
##
##     collapse
xbar <- function(xs) {
mean(xs)
}

ghat_zero <- function(xs) {
summand <- (xs - xbar(xs))^2
mean(summand)
}

ghat_one <- function(xs) {
lhs <- xs[1:(length(xs) - 1)] - xbar(xs)
rhs <- xs[2:length(xs)] - xbar(xs)
summand <- lhs * rhs
summate <- sum(summand)
summate/length(xs)

}
rhohat_zero <- 1
rhohat_one <- function(xs) {
ghat_one(xs)/ghat_zero(xs)
}
v <- c(76, 70, 66, 60, 70, 72, 76, 80)
xbar(v)
## [1] 71.25
ghat_zero(v)
## [1] 34.9375
ghat_one(v)
## [1] 14.74219
rhohat_one(v)
## [1] 0.4219589