I believe that this is a statistics rather than a programming question, though I am tied to an R implementation and hope for a reply in kind.
I have data that constitutes several time series. I expect to be studying this data for some time to come, and would like to apply a wide range of time series methods, including ARIMA models, vector autoregression, fractionally integrated series, and others.
My data is of daily frequency and has, in essence, four observations a day: two at fixed times at the beginning and end of the day, respectively, and a maximum and minimum value between those points, with no associated time for each. Although it does not involve any additional measurements, the interval between the end-of-day and the beginning-of-day is also meaningful, with the change morning to night and the change night to morning measured by values of the same two series, offset by one.
In implementing this as an R time series that various tools can use, I face two problems, both with the extremal values. The first is that these two points are only partially ordered in time. They fall in the daytime interval between the beginning and end of the day, but there is no information in the data about which occurs first.
The second, related problem is in representing the frequency. The start and end of the day happen at specific identifiable moments in time, always the same. I have not been able to find any time series representation of the occurrence of the maximum and minimum that R time series routines will accept, and that accurately reflect the temporal process as I understand it. This is a different problem than the first, because that problem deals with representing data points without knowing the order in which they occur, while this one deals with combining data that occur at fixed times with data where the time of occurrence is known only down to an interval.
Am I just out of luck here? Do I have to reinvent time series from the bottom up before I can use any of the standard tools? Or switch to some non-parametric time series representation about which I know nearly nothing? Or is there a common work-around, or an alternative time series implementation that can handle problems of this sort and speak to other R time series routines in a way they will understand?