Methods and apparatus to estimate population reach from different marginal ratings and/or unions of marginal ratings based on impression data

1. An apparatus to estimate a reach of media, the apparatus comprising:

interface circuitry to obtain panelist impressions from devices and database proprietor impressions from a database proprietor via network communications;

memory to store the panelist impressions and the database proprietor impressions;

association controller circuitry to generate a tree association corresponding to a union of a first margin of media and a second margin of the media; and

first and second commercial solvers to be executed in parallel by processor circuitry to:

access the panelist impressions from the memory;

determine first multipliers by solving first equations corresponding to the panelist impressions and panelist audience totals of at least one of the first margin, the second margin, or the union;

perform first parallel computations to determine second multipliers by using the first parallel computations to concurrently solve second equations corresponding to the tree association using the first multipliers;

discard the first multipliers;

perform second parallel computations to determine third multipliers corresponding to a total audience exposed to the media at at least one of the first margin, the second margin, or the union by using the second parallel computations to concurrently solve third equations corresponding to the tree association using database proprietor impression totals; and

determine an estimate for a population reach of the media for at least one of the first margin, the second margin, or the union based on the third multipliers, the interface circuitry to output a report corresponding to the estimate for the population reach of the media.

2. The apparatus of claim 1 , wherein the association controller circuitry is to generate the tree association by:

tagging the first margin with a first identifier, the second margin with a second identifier, and a third margin with a third identifier; and

tagging the first margin, the second margin, and the union with at least one of corresponding descendants or corresponding ancestors.

3. The apparatus of claim 1 , wherein the second equations include a first equation corresponding to the first margin, a second equation corresponding to the second margin, and a third equation corresponding to the union.

4. The apparatus of claim 3 , wherein the first and second commercial solvers are to solve for a first multiplier of the first equation in the second equations and a second multiplier of the second equation of the second equations in parallel.

5. The apparatus of claim 1 , wherein the first and second commercial solvers are to solve the first equations iteratively.

6. The apparatus of claim 1 , wherein the union is a first union, the tree association corresponding to a second union of a third and fourth margin of time, the total audience of the media corresponding to a third union of the first and second unions.

7. The apparatus of claim 6 , wherein the first and second commercial solvers are to determine the estimate of the population reach of the first union based on a difference between a first percentage of the total audience and a second percentage of the total audience if the first union is removed from the tree association.

8. A non-transitory computer readable storage medium comprising instructions which, when executed, cause a machine to at least:

obtain panelist impressions from devices and database proprietor impressions from a database proprietor via network communications;

store panelist impressions and database proprietor impressions in memory, the panelist impressions and the database impressions obtained via network communications;

generate a tree association corresponding to a union of a first margin of media and a second margin of the media;

access the panelist impressions from the memory;

determine first multipliers by solving first equations corresponding to the panelist impressions and panelist audience totals of at least one of the first margin, the second margin, or the union;

perform first parallel computations using a first commercial solver and a second commercial solver executed in parallel to determine second multipliers by using the first parallel computations to concurrently solve second equations corresponding to the tree association using the first multipliers;

discard the first multipliers;

perform second parallel computations using the first commercial solver and the second commercial solver to determine third multipliers corresponding to a total audience exposed to the media at at least one of the first margin, the second margin, or the union by using the second parallel computations to concurrently solve third equations corresponding to the tree association using database proprietor impression totals;

determine an estimate for a population reach of the media for at least one of the first margin, the second margin, or the union based on the third multipliers; and

output a report corresponding to the estimate for the population reach of the media.

9. The computer readable storage medium of claim 8 , wherein the instructions cause the machine to generate the tree association by:

tagging the first margin with a first identifier, the second margin with a second identifier, and a third margin with a third identifier; and

tagging the first margin, the second margin, and the union with at least one of corresponding descendants or corresponding ancestors.

10. The computer readable storage medium of claim 8 , wherein the second equations include a first equation corresponding to the first margin, a second equation corresponding to the second margin, and a third equation corresponding to the union.

11. The computer readable storage medium of claim 10 , wherein the instructions cause the machine to solve for a first multiplier of the first equation in the second equations and a second multiplier of the second equation of the second equations in parallel.

12. The computer readable storage medium of claim 8 , wherein the instructions cause the machine to solve the first equations iteratively.

13. The computer readable storage medium of claim 8 , wherein the union is a first union, the tree association corresponding to a second union of a third and fourth margin of time, the total audience of the media corresponding to a third union of the first and second unions.

14. The computer readable storage medium of claim 13 , wherein the instructions cause the machine to determine the estimate of the population reach of the first union based on a difference between a first percentage of the total audience and a second percentage of the total audience if the first union is removed from the tree association.

15. A method to estimate a population reach of media, the method comprising:

obtaining panelist impressions from devices and database proprietor impressions from a database proprietor via network communications;

storing the panelist impressions and the database proprietor impressions in memory;

generating, by executing an instruction with at least one processor, a tree association corresponding to a union of a first margin of media and a second margin of the media;

accessing the panelist impressions from the memory;

determining, by executing an instruction with the at least one processor, first multipliers by solving first equations corresponding to the panelist impressions and panelist audience totals of at least one of the first margin, the second margin, or the union;

performing, by executing an instruction with the at least one processor, first parallel computations with a first commercial solver and a second commercial solver executed in parallel to determine second multipliers by using the first parallel computations to concurrently solve second equations corresponding to the tree association using the first multipliers;

discarding, by executing an instruction with the at least one processor, the first multipliers;

performing, by executing an instruction with the at least one processor, second parallel computations with the first commercial solver and the second commercial solver to determine third multipliers corresponding to a total audience exposed to the media at at least one of the first margin, the second margin, or the union by using the second parallel computations to concurrently solve third equations corresponding to the tree association using database proprietor impression totals;

determining, by executing an instruction with the at least one processor, an estimate for a population reach of the media for at least one of the first margin, the second margin, or the union based on the third multipliers; and

outputting a report corresponding to the estimate for the population reach of the media.

16. The method of claim 15 , wherein the generating of the tree association includes:

tagging the first margin with a first identifier, the second margin with a second identifier, and a third margin with a third identifier; and

tagging the first margin, the second margin, and the union with at least one of corresponding descendants or corresponding ancestors.

17. The method of claim 15 , wherein the second equations include a first equation corresponding to the first margin, a second equation corresponding to the second margin, and a third equation corresponding to the union.

18. The method of claim 17 , further including solving for a first multiplier of the first equation in the second equations and a second multiplier of the second equation of the second equations in parallel.

19. The method of claim 15 , further including solving the first equations iteratively.

20. The method of claim 15 , wherein the union is a first union, the tree association corresponding to a second union of a third and fourth margin of time, the total audience of the media corresponding to a third union of the first and second unions.