Analysis of irregularly sampled time series

Question

What is the difference between irregularly sampled time series and non-linear time series? Also, what are the best methods for the analysis of irregularly sampled time series? Are there any sample data sets for such time series?

Answer 1

A non-linear time series is a time series which is generated by a non-linear process. For an regularly sampled time series $x_0, …, x_n$, this corresponds to a non-linear dependence of $x_t$ on $x_{t-1}, x_{t-2}, …, x_{t-m}$. For irregularly sampled time series, there is no such correspondence.

As a result, there is no general relation between the linearity of a time series and the regularity of the sampling, as the first property is about what you observe and the second property is about how you observe. So, there are

• regularly sampled linear time series,
• irregularly sampled linear time series,
• regularly sampled non-linear time series,
• irregularly sampled non-linear time series.

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Irregularly sampled time series are a rather broad topic, so I can only give you some question you should ask yourself:

It is very crucial what determines your sampling times. They could be:

• comparably random, e.g., if you can only take samples, if nobody else claimes the laboratory and this is not mainly determined by schedules
• depending on a complex process, e.g., if you make astronomical observations and can only do so, if the weather is fine and at a specific time of the year
• strongly depend on the observed process, e.g., if you analyse seismical activity which is only recorded in the temporal vicinity of earth quakes

The latter two options may introduce a strong bias on your data and affect the results of your analysis, if you are not careful. In particular they may cause the time series to appear to be non-linear (if you ignore the sampling), even if the underlying process isn’t.