A model for analyzing a series, which is the sum of a deterministic trend series and a stationary noise series, is the so called random walk with drift model, given by

\[y_t = \delta+y_{t-1}+w_t,\]

for \(t = 1, 2,...\), with initial condition \(y_0 = 0\), where \(w_t\) is white noise. The constant \(\delta\) is called the drift, and if \(\delta = 0\), then it is simply called random walk.

The equation from above may be rewritten as

\[y_t = \delta t + \sum_{j=1}^t w_j\text{,}\]

for \(t = 1,2,...\).

If \(\delta = 0\), the value of the time series at time \(t\) is equal to the value of the series at time \(t-1\) plus a completely random movement determined by \(w_t\).

\[\Delta y_t = y_t-y_{t-1} = w_t\]

The \(w_{t}\) values can be interpreted as independent “shocks” which perturb the current state \(y_t\) in the amount of \(w_{t}\) to produce a new state \(y_{t+1}\).

The graph below shows a random walk and a random walk with drift (\(\delta = 0.25\)) generated with R.

set.seed(250)

## random walk
w <- rnorm(n = 200, mean = 0, sd = 1)
y <- cumsum(w)

## random walk with drift
wd <- w + .25
yd <- cumsum(wd)

## plotting
plot.ts(yd,
  ylim = c(-5, 55),
  main = expression(paste("Random walk with " * sigma[w] * " = 1")),
  col = "blue"
)

lines(y, col = "red")

legend("topleft",
  legend = c(
    expression(paste(delta * " = 0.25")),
    expression(paste(delta * " = 0"))
  ),
  col = c("blue", "red"),
  lty = 1
)

The autocorrelation function of this model type shows typically a very slow decay.

library(ggfortify)
library(gridExtra)

p1 <- autoplot(acf(y, plot = FALSE)) +
  ggtitle("Serial Correlation of a random walk")

p2 <- autoplot(acf(yd, plot = FALSE)) +
  ggtitle("Serial Correlation of a random walk with drift")
grid.arrange(p1, p2, ncol = 1)

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Citation

The E-Learning project SOGA-R was developed at the Department of Earth Sciences by Kai Hartmann, Joachim Krois and Annette Rudolph. You can reach us via mail by soga[at]zedat.fu-berlin.de.

You may use this project freely under the Creative Commons Attribution-ShareAlike 4.0 International License.

Please cite as follow: Hartmann, K., Krois, J., Rudolph, A. (2023): Statistics and Geodata Analysis using R (SOGA-R). Department of Earth Sciences, Freie Universitaet Berlin.