In reading the paper On Time Series Analysis of Public Health and Biomedical Data (subscription required for the PDF), described in the last post, I was introduced to an interpretation of a time series that I was not familiar with, that is, in terms of a stochastic process. I remember being told in my course on time series analysis that a stochastic process and time series were synonymous for our purposes—although more may have been said at the time, which I’ve since forgotten—but there’s obviously more to it than that.
A time series is a single observation of a possibly infinite collection of time series. In other words the time series itself can be viewed as a random variable, and within that time series is a realization of a collection of random variables ordered in time. The possibly infinite sequence of random variables ordered in time is a stochastic process. When we consider a stochastic process we are concerned with the probability model for the individual random variables, and also combinations of them.
A will to be independent
Making inferences from a time series is making inferences from a single realization of a stochastic process, that is, a single observation at each time. The idea of stationarity—that statistical properties of the time series do not depend on time—is used to develop probability based theory specific to time series analysis. Basically, the relative time difference between variables in a stationary time series will change the probability distribution, whereas time shifting will not. The assumption of stationarity implies that the dependence between variables decreases with increasing time separation (which leads to a discussion of “auto”-correlation, which I won’t describe here). Therefore, more (nearly independent) information will be accumulated the longer a series is followed.
Longitudinal data are repeated measures for short periods, resulting in many realizations of short time series. We assume that the time series are independent, and that repeated observations lead to zero correlation with increasing time separation (which is stronger than stationarity, especially for short time series). In this case we look to increase the number of time series instead of the number of observations for a single time series. This is a topic onto itself (one I’m not currently familiar with), and therefore only mentioned briefly in the paper. What I found particularly interesting was the idea of bootstrapping a time series (based on splitting a time series into several shorter pieces), something I’ll need to look into further.