rolling 24-hr time series volume data - how to back out minute level volume?

Question

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 1

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.