Chapter 2 The Structure of Temporal Data

Time series data are

• Observations or measurements that are indexed according to time

• Instead of \(X_i\) we have \(X_t\).

Why does that make things different?

• The time index has a special ordering.

• Data measured over time are not exchangeable, which is what we often assume when data are indexed by \(i\).

• Time can also have its own special meaning, standing in for other unobserved variables.

To be clear, a key property of time series data, which distinguishes it from other types of data that are commonly analyzed, is we do not imagine that we can randomly permute the indices of the data and model the data with the same distribution. There is an ordering to the data. Further, the stronger assumption of the data being independent generally does not apply.

One interesting, and perhaps unnerving, feature of time series data is that the data in their raw form provide very little real information. The raw data are, in a sense, the most useless form of the data. As a result, plotting or summarizing the raw data often do not provide a lot of insight into what is going on or why it is happening. However, because the time index has such special meaning, we can use the time index to decompose the time series data into variation at different time scales. The formal approach to time scale analysis is sometimes called Fourier analysis or spectral analysis, but there are informal approaches that are also useful.

Another way to think about time series data is that a time series actually represents a mixture of time series that vary at different time scales. Part of the job of analyzing time series data is to

1. Pick apart the mixture of time scales and describe how they differ from each other

2. Determine which time scales are of interest, either based on empirical properties or based on the scientific problem at hand