In previous chapters, we have studied series of events occurring to one or more individuals observed over time. In Chapters 2 and 3, I looked at simple durations (survival times) where only one transition event to an absorbing state occurs. In Chapter 4, I extended this to series of duration times between recurrent events. In Chapters 5 and 6, each event resulted in a switch from one of a relatively limited number of states to another, with possible dependence on the previous state(s). We also have seen that dependence may be induced indirectly by postulating the existence of hidden states influencing the process, as in Chapter 7. It is now time to consider a few other, more complex, possibilities.

Birth processes

In Chapter 5 and Section 6.1.3, I looked at state dependence for categorical state spaces. This gives rise to Markov chain models. Another simple way to introduce dependence for such state spaces is by means of some form of birth process (Section 1.2.5; see also the example in Section 7.3.3). Instead of conditioning on the previous state(s), one conditions on the number(s) of previous times the series was in some given state(s). For binary point processes, this is just the number of previous occurrences of the recurrent event. In contrast to Markov chains, this will necessarily result in a nonstationary process.

Birth or contagion

Because birth or contagion processes are nonstationary, special care must be taken in choosing the time origin.