Is there a mathematical way or algorithm to take a week over week data set gathered daily and convert it to daily data?

Question

I have a large amount of data that was gathered every day. The data is totals for the past week.

2001-01-01 to 2001-01-07 : 27
2001-01-02 to 2001-01-08 : 27
2001-01-03 to 2001-01-09 : 28
2001-01-04 to 2001-01-10 : 26
2001-01-05 to 2001-01-11 : 20
2001-01-06 to 2001-01-12 : 13
2001-01-07 to 2001-01-13 : 14
2001-01-08 to 2001-01-14 : 16
...
...
2006-01-01 to 2006-01-07 : 28

My question is whether or not there is some algorithm or mathematical function for figuring out what the values for the individual days were since the data was gathered daily.

If there isn't, I also have the data tallied month over month.

e.g:

2001-01-01 to 2001-01-31 : 300
2001-02-01 to 2001-01-28 : 287
2001-03-01 to 2001-01-31 : 322
2001-04-01 to 2001-01-30 : 328
2001-05-01 to 2001-01-29 : 365

Is there any formula or algorithm to merge the two data sets to reconcile the daily values?

If this is not the place to ask this question or it should have tags added to help get it to the right audience, any help would be appreciated.

Answer 1

With only the weekly totals, it's impossible to recover the individual daily values, since there are six fewer variables than equations. For example, imagine you only had data from 2001-01-01 to 2001-01-09:

2001-01-01 to 2001-01-07 : 27
2001-01-02 to 2001-01-08 : 27
2001-01-03 to 2001-01-09 : 28

Then you have nine variables (for the nine days) but only three equations. There's no way to distinguish the days 01-03 through 01-07; you can change these 5 daily values without affecting the weekly totals, as long as you keep the sum of these 5 values the same.

As Robert Israel pointed out in a comment, it seems possible that the additional data of the monthly totals might allow you to determine the exact values for each day. Indeed, with five years of data, you get 60 more equations; it's not obvious to me that (or under what precise conditions) these 60 equations contain six independent equations... but it definitely seems plausible. Some brief experimentation I did in Matlab suggests that you can indeed get up to full rank (and thus recover all the daily values) by adding less than a year's worth of monthly totals.