library(tswge) library(ggplot2) acfdf <- function(vec){ vacf <- acf(vec, plot = F) with(vacf, data.frame(lag,acf)) } ggacf <- function(vec){ ac <- acfdf(vec) ggplot(data = ac, aes(x = lag, y = acf))+ geom_hline(aes(yintercept = 0)) + geom_segment(mapping = aes(xend = lag , yend = 0)) } tplot <- function(vec){ df <- data.frame("X" = vec, "t" = seq_along(vec)) ggplot(data = df, aes(x = t, y = X)) + geom_line() }
Two events are independent in a time series if the probability that an event at time t occurs in no way depends on the ocurrence of any event in the past or affects any event in the future. Mathematically, this is written as:
$$\mathrm{Independence: }P \left(x_{t+1}|X_t\right) = P\left(X_{t+1}\right)$$
If two events are independent, their corellation is 0 That is if $X_t$ and $X_{t+k}$ are independent, $\rho_{x_{t},x_{t+k}}=0$
Corollary: If the correlation between two variables is not zero, then they are not independent
In other words if $\rho_{x_{t},x_{t+k}} \neq 0$, they are not independent.
In time series we look at the autocorrelation between $X_t$ and $X_{t+1}$ etc (with t and t+1 it is lag 1 autocorrelation)
For example, visually, let us define a vector, $Y5$ and take its autocorrelation:
Y5 <- c(5.1,5.2,5.5,5.3,5.1,4.8,4.5,4.4,4.6,4.6,4.8,5.2,5.4,5.6,5.4,5.3,5.1,5.1,4.8,4.7,4.5,4.3,4.6,4.8,4.9,5.2,5.4,5.6,5.5,5.5) tplot(Y5) + ggthemes::theme_few() ggacf(Y5) + ggthemes::theme_few()
Now let us look at autocorrelation of independent data
xs = gen.arma.wge(n = 250)
xs = gen.arma.wge(n = 250)
Let's check it out
tplot(xs) + ggthemes::theme_few() ggacf(xs) + ggthemes::theme_few()
We see that the autocorrelation is more or less zero
Add the following code to your website.
For more information on customizing the embed code, read Embedding Snippets.