# Unit 5 Autocorrelation Concepts and Notation

## 5.1 Theoretical concept of \(\rho\)

\[\rho = \frac{E\left[\left(X-\mu_X\right)\left(Y-\mu_Y\right)\right]}{\sigma_X \sigma_Y} = \frac{\mathrm{covariance between X+y}}{\sigma_X \sigma_Y}\]

In a stationary time series, we have autocorrelation and autocovariance

## 5.2 autocovariance

\[\gamma_k = E\left[\left(X_t - \mu\right)\left(X_{t+k} - \mu\right)\right]\]

## 5.3 autocorrelation

\[\rho_k =\frac{E\left[\left(X_t - \mu\right)\left(X_{t+k} - \mu\right)\right]}{\sigma_{X_t} \sigma_{X_{t+k}}} = \frac{E\left[\left(X_t - \mu\right)\left(X_{t+k} - \mu\right)\right]}{\sigma^2_X} = \frac{\gamma_k}{\sigma_X^2}\]

This works because of constant variance and constant mean!

Note that in a stationary time series:

\[\sigma_X^2 = 1E[(X_t - \mu)^2] = E[(X_t -\mu )(X_t - \mu)] = \gamma_0\]

Therefore:

\[\rho_k = \frac{\gamma_k}{\gamma_0}\]

## 5.4 Stationary Covariance

Correlation is not affected by where, only by how far apart, that is: \[ \rho_h = \mathrm{Cor} \left(X_t, X_{t+h}\right)\]

Let us try it out with this little snippet:

```
ggsplitacf <- function(vec) {
h1 <- vec[1:(length(vec)/2)]
h2 <- vec[(length(vec)/2):length(vec)]
first <- ggacf(vec) + ggthemes::theme_few()
second <- ggacf(h1) + ggthemes::theme_few()
third <- ggacf(h2) + ggthemes::theme_few()
cowplot::plot_grid(first, second, third, labels = c("original", "first half",
"second half"), nrow = 2, align = "v")
}
```

`Realize = gen.arma.wge(500, 0.95, 0, sn = 784)`

Lets check out the ACF

`ggsplitacf(Realize)`

This looks pretty stationary to me! Let’s try with some sample data:

```
data("noctula")
tplot(noctula) + ggthemes::theme_few()
ggsplitacf(noctula)
```

So in this case our mean probably looks constant, our variance could be constant as well, however the ACFs are clearly different. Let’s look at a different dataset:

```
data(lavon)
tplot(lavon) + ggthemes::theme_few()
ggsplitacf(lavon)
```

In this case, our mean looks maybe to be constant around 495, our variance does not appear to be constant, and our ACF looks pretty good, however I would say overall this probably is not stationary, due to the variance and slightly off ACF.