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I have a dataset that includes observations at different time points. There are multiple observations at the same time points and the time points are not evenly spaced.

Now, I would like to test whether the data are random or autocorrelated in time. Can I call such a dataset as a time series? If so, what kind test methods I can try to use for such an irregularly spaced time series?

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While it's apparently some form of time series, if there are multiple observations at a single point, it might be a little unusual to call it a time series.

One difficulty I see is how do I relate (say) 2 observations at time $\tau$ to 3 observations at time $\tau-s$? What's the model? Does the expectation at time $\tau$ relate to the average at time $\tau-s$ or what?

With single-observation-at-a-time, there's certainly ways to write continuous-time autocorrelated models.

A common example is the Ornstein-Uhlenbeck process. See also the Vasicek model, a particular example of its use.

It's possible to write an unevenly-sampled observational model based on the O-U process as something of the form (if $s,t$ are consecutive observation times):

$\hspace{1cm}(y_t-\mu) = \phi^{t-s} (y_s-\mu) + n_{t-s}\,,\quad$ where $\:n_u\sim N(0,\frac{\sigma^2}{2\phi} (1-e^{-2\phi u}))$

(I think!), which when $t-s=1$ (i.e. the usual regular-time interval situation) corresponds to an AR(1).

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