In josephsdavid/tstest: tstest

Autocorrelation

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()
}

Independence

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:

$$\mathrm{Independence: }P \left(x_{t+1}|X_t\right) = P\left(X_{t+1}\right)$$

Serial Dependence / Autocorrelation

A definition

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.

Autocorrelation Plots

Dependent(ish) data

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()

Independent(ish) data!

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

josephsdavid/tstest documentation built on July 3, 2019, 4:48 p.m.