To be clear about what the bounds should be on the mean, consider a daily mean computed from four 6-hourly mean samples centered at 3Z, 9Z, 15Z, and 21Z. If the 6-hour means have bounds 0-6Z, 6-12Z, 12-18Z, and 18-24Z, then we want the daily mean to extend from the beginning of the interval represented by the first sample (i.e., from 0Z) to the end of the last sample (i.e., to 24Z, or 0Z of the next day). So for a daily mean for the first month of this year, the bounds would be 2024-01-01 0:00:00 and 2024-01-02 0:00:00, while the coordinate value would be 2024-01-01 12:00:00.

Also, in the above example, one could simply average the four 6- hour time means because they were fully contained within a single day (and fully covered all the hours of the day). If the 6-hourly time-mean samples were centered instead on 0Z, 6Z, 12Z, and 18Z, then to form a daily mean extending from 0-24Z, you would need to compute the mean as (.5x0 + x1 + x2 + x3 + .5*x4)/4. That is each sample should be weighted by the time interval overlapping the daily time-interval of interest.

@pochedls Can you update with edge cases from meeting.

In my comment on @tomvothecoder's PR, I said I thought we could take the upper and lower bound for each group. Say you wanted a JJA average and the bounds for each month were at the start and the end of the month (e.g., June had bounds of 7/1 00:00 and 8/1 00:00:

...     |     M     |     J     |     J     |     A     |     S     |     ...     |
     ...            |                 JJA               |                 ...

Then you would simply take the June lower bound (6/1 00:00) as the lower JJA bound (and take the August upper bound: 8/1 00:00 as the upper bound for JJA).

This won't work if each month's bounds do not line up with the group averaging frequency. For example, if you had pentad data of 5-day intervals:

... |A |B |C |D |E |F |G |H |I |J |K |L |M |N |O |P |Q |R |S |T |U |V |W |X |Y |Z |  ...
     ...            |                 JJA               |                 ...

In this case, pentad F and N are partially in the JJA season. F could have bounds like ("2020-06-28 00:00:00", "2020-07-02 00:00:00"). Taking the lower bound (2020-06-28 00:00:00) would be incorrect and the bound should be set to 2020-07-01 00:00.

An algorithm to do this is not immediately obvious to me.

When you compute the monthly (or seasonal) mean from data representing 5-day means, you would need to:

1. find all pentads with upper and/or lower bounds falling within the target period.
2. consider the first contributing pentad and if its lower bound is less than the beginning of the target period (e.g., it spans days from the preceding month to the target month), reset it to the beginning of the target period (e.g., the beginning of the month).
3. consider the last contributing pentad and if its upper bound is greater than the end of the target period (e.g., it extends beyond the target period to the next month), reset it to the end of the target period (e.g., the end of the month).
4. define weights for each sample as the time interval it spans, i.e., from the beginning (possibly now adjusted) time-bound to the end (possibly adjusted) time-bound.
5. Compute the weighted time mean from the pentads with those weights (pentads falling fully within the target period would be equally weighted, but the pentads at the beginning and end of the period might be down-weighted).
6. For the resulting monthly (or seasonal) weighted means, define its time bounds with lower bound set to the (possibly now adjusted) beginning bound of the first sample contributing to the mean, and the upper bound set to the (possibly now adjusted) end bound of the last sample contributing to the mean.

Although @taylor13's comment is seemingly off-topic (this issue is on generating bounds), he is probably right that the averaging functionality may need to be refactored as a related issue. I think @taylor13's points are documented in this issue.