What is time series data

Below is a video from Aric LaBarr’s youtube series explaing time series concepts in less than 5 minutes. In the video below he explains what time series data is - as opposed to cross sectional data. As he points out, we also some times have a combination which we can e.g. aggregate to cross sectional data or time series data.

Time series notation

It is quite common in statistics in general, but also in time series analysis, to distinguish between capital and non-capital letters. While Y_t denotes the stochastic variable Y at time t, y_t is the realization of said stochastic variable Y_t, i.e. an observed value. Y_t is a specific variable, while \{Y_t\} is the entire time series. While a stochastic variable has for instance an expectation, a variance, a dependence structure and a distribution, an observation is just a number.

Let \mu_t=\mu_Y(t)=\mathrm{E}(Y_t) denote the mean function of Y_t. Assuming that E(Y_t^2)<\infty, the covariance function of \{Y_t\} is denoted \gamma(r,s)=\gamma_Y(r,s)= \mathrm{Cov}(Y_r,Y_s)=\mathrm{E}(Y_r-\mu_r)(Y_s-\mu_s). If the covariance function does not depend on the specific values of r and s, but rather the distance between them h = |r-s|, we write \gamma(r,s)=\gamma(h). Likewise, if \mu_t does not depend on t, we write \mu_t=\mu. This will be used in the section about stationarity below. Note that the variance of Y_t is given by \gamma(t,t) = \mathrm{Cov}(Y_t,Y_t) = \mathrm{Var}(Y_t) or \gamma(0) in the case where the covariance does not depend on t.

tsibble

The tsibble object will be important in this course. Since we are working in the tidyverse, things will be easier if we commit to it. Converting your data to a time series tibble (tsibble) is essential. The tsibble extends the tidy tibble data frame by introducing temporal structure. Read the chapter 2.1 about tsibble objects in the textbook. Here you find several coding examples.