Why Rolling 24-Hour Volume Cannot Identify Minute Volume Alone
Summary
The document asks whether minute-level trading volume can be recovered from observations of rolling 24-hour volume sampled roughly once per minute. Under the stated minute indexing, the change between consecutive rolling totals combines volume from the newest minute with volume from the minute leaving the 24-hour window. A single difference therefore does not isolate either contribution.
The accepted answer explains the identifiability problem by dividing successive rolling buckets into overlapping and non-overlapping components. Although adjacent observations impose relationships among those components, there are more unknown quantities than independent equations, leaving an underdetermined system. Consequently, the rolling series alone is insufficient to determine exact minute volumes; additional information or assumptions would be needed. The explanation is structural rather than a numerical reconstruction method, and the question’s approximate sampling interval may also complicate applying a strict minute-by-minute equation to real observations.
Key ideas
- The difference between adjacent rolling 24-hour totals combines new volume with volume dropping out of the window.
- That difference alone cannot distinguish the two minute-level contributions.
- Overlapping bucket components create more unknown quantities than independent equations.
- Exact minute-volume recovery requires additional information or assumptions.
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Full text
# rolling 24-hr time series volume data - how to back out minute level volume?
# rolling 24-hr time series volume data - how to back out minute level volume?
I have data (trading volume) that is tracked on a rolling 24-hour basis approximately 1 minute or so. Here are some sample timestamps that highlight why I say approximately:
I am trying to use this data to infer the volume in a given minute. However, I am confused how to distinguish volume from the prior minute and the minute 24 hours ago.
Let $t$ represent a minute. Let the variable I'm interested in backing out be $VolumeMin_{t}$, for any given minute $t$. Let $Volume24hr_{t}$ represent the 24 hr rolling trading volume data I currently have.
Then, using the difference between the two timestamps, I get the following:
$Volume24hr_{t+1} - Volume24hr_{t} = VolumeMin_{t} + VolumeMin_{t-1440}$
Thus, using the difference, I can only seem to determine the aggregate volume from those two minutes. Is it impossible to have enough degrees of freedom to isolate $VolumeMin_{t}$?
If that's the case any help would be appreciated in other ways I might be able to think about inferring $VolumeMin_{t}$. Thank you!
## Answer by Julie Taylor (score 0, accepted)
https://quant.stackexchange.com/a/69747
I don't think it's possible to determine unless we have additional information. Consider the first two volume buckets:
$$V_1 = V_{1A}+V_{1B}+V_{1C}$$ $$V_2 = V_{2A}+V_{2B}+V_{2C}$$
where $V_{iA}$ represent the cumulative volume 24 hours ago only in bucket $i$ and not in time bucket $i+1$ and $V_{iC}$ represent the newest cumulative volume only in bucket $i+1$ and not in time bucket $i$.
It's clear in the above setting that $V_{1B}+V_{1C}=V_{2A}+V_{2B}$, there are still too many variables than equations (an underdetermined system) so unless we have an additional clue, we can't solve this.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.