What is a time series?

In this chapter and the next few chapters, we will work with a type of data called time-series data. Up until this point, we have worked with independent data—that is, data consisting of samples that are not related. A time series is typically a measurement of the same sample taken over time, which makes the samples in this type of data related. There are many time series present around us every day. A few common examples of time series are daily temperature measurements, stock price ticks, and the heights of ocean tides. While a time series does not need to be measured at fixed intervals, in this book, we will primarily be concerned with measurements taken at fixed intervals, such as daily or every second.

Let’s look at some notation. In the following equation, we have a variable x that is repeatedly sampled over time. The subscripts enumerate the sample points (sample 1 through sample t), and the whole series of samples is denoted X. The subscript...