Methods and apparatus to estimate de-duplicated unknown total audience sizes based on partial information of known audiences
Example methods, apparatus, systems, and articles of manufacture are disclosed to estimate de-duplicated unknown total audience sizes based on partial information of known audiences. An example apparatus includes an association controller to generate a tree structure association corresponding to a union of first and second margins of time; and one or more commercial solvers to: perform parallel computations on a processor to determine multipliers by solving equations corresponding to the tree structure association, the multipliers corresponding to the first total audience size for the union, the second total audience size for the first margin, and the third total audience size for the second margin; and determine an estimate for the third total audience size for the second margin of time based on the multipliers.
1. An apparatus comprising:
an association controller to generate a tree structure association corresponding to a union of first audience members of a first media segment of media and second audience members of a second media segment of the media, wherein the union is representative of a known first total audience size of a first demographic group from a plurality of demographic groups exposed to the media, a second total audience size is known for the first demographic group exposed to the first media segment at a first time, and a third total audience size is unknown for the first demographic group exposed to the second media segment at a second time; and
one or more commercial solvers to:
perform parallel computations on a processor to determine multipliers by solving equations corresponding to the tree structure association, the multipliers corresponding to the first total audience size for the union, the second total audience size for the first media segment, and the third total audience size for the second media segment; and
determine an estimate for the third total audience size for the second media segment corresponding to the second time based on the multipliers, at least one of the association controller or the one or more commercial solvers includes hardware.
2. The apparatus of claim 1 , wherein the association controller is to generate the tree structure association by:
storing a first identifier in association with the first media segment, the second media segment with a second identifier, and a third media segment with a third identifier; and
storing at least one of corresponding descendant nodes or corresponding ancestor nodes in association with the first media segment, the second media segment, and the union.
3. The apparatus of claim 1 , wherein the equations include a first equation corresponding to the first media segment, a second equation corresponding to the second media segment, and a third equation corresponding to the union.
4. The apparatus of claim 3 , wherein the one or more commercial solvers are to solve for a first multiplier of the first equation and a second multiplier of the second equation in parallel.
5. The apparatus of claim 1 , wherein the one or more commercial solvers are to solve the equations iteratively.
6. The apparatus of claim 1 , wherein the multipliers include a first multiplier corresponding to the first media segment, a second multiplier corresponding to the second media segment, and a third multiplier corresponding to the union, the one or more commercial solvers to:
set the first multiplier equal to a first variable multiplier;
set the second multiplier equal to a constant value; and
set the third multiplier equal to a second variable multiplier, the solving of the equations resulting in a determination of a first value for the first variable multiplier and a second value of the second variable multiplier.
7. The apparatus of claim 6 , wherein the constant value is at least one of 1 or 0.
8. The apparatus of claim 1 , wherein the union is a first union, the tree structure association corresponds to a second union of third audience members of a third media segment of the media corresponding to a third time and fourth audience members of a fourth media segment corresponding to a fourth time, and a fourth total audience size corresponding to the first demographic group exposed to the media for a third union, the third union including the first and second unions, the third union being a highest ancestor in the tree structure association.
9. The apparatus of claim 8 , wherein the one or more commercial solvers are to determine the estimate of the third total audience size for the second media segment corresponding to the second time based on a difference between the fourth total audience size of the media and a fifth total audience size of the media at the third union if the first union is removed from the tree structure association.
10. The apparatus of claim 1 , wherein the one or more commercial solvers are to solve the equations analytically to facilitate the parallel computations faster than solving without the parallel computations.
11. A non-transitory computer readable storage medium comprising instructions which, when executed, cause a machine to at least:
generate a tree structure association corresponding to a union of first audience members of a first media segment of media and second audience members of a second media segment of the media, wherein the union is representative of a known first total audience size of a first demographic group from a plurality of demographic groups exposed to the media, a second total audience size is known for the first demographic group exposed to the first media segment at a first time, and a third total audience size is unknown for the first demographic group exposed to the second media segment at a second time;
perform parallel computations on a processor to determine multipliers by solving equations corresponding to the tree structure association, the multipliers corresponding to the first total audience size for the union, the second total audience size for the first media segment, and the third total audience size for the second media segment; and
determine an estimate for the third total audience size for the second media segment corresponding to the second time based on the multipliers.
12. The computer readable storage medium of claim 11 , wherein the instructions cause the machine to generate the tree structure association by:
storing a first identifier in association with the first media segment, the second media segment with a second identifier, and a third media segment with a third identifier; and
storing at least one of corresponding descendant nodes or corresponding ancestor nodes in association with the first media segment, the second media segment, and the union.
13. The computer readable storage medium of claim 11 , wherein the equations include a first equation corresponding to the first media segment, a second equation corresponding to the second media segment, and a third equation corresponding to the union.
14. The computer readable storage medium of claim 13 , wherein the instructions cause the machine to solve for a first multiplier of the first equation and a second multiplier of the second equation in parallel.
15. The computer readable storage medium of claim 11 , wherein the instructions cause the machine to solve the equations iteratively.
16. The computer readable storage medium of claim 11 , wherein the multipliers include a first multiplier corresponding to the first media segment, a second multiplier corresponding to the second media segment, and a third multiplier corresponding to the union, the instructions to cause the machine to:
set the first multiplier equal to a first variable multiplier;
set the second multiplier equal to a constant value; and
set the third multiplier equal to a second variable multiplier, the solving of the equations resulting in a determination of a first value for the first variable multiplier and a second value of the second variable multiplier.
17. The computer readable storage medium of claim 16 , wherein the constant value is at least one of 1 or 0.
18. The computer readable storage medium of claim 11 , wherein the union is a first union, the tree structure association corresponds to a second union of third audience members of a third media segment of the media corresponding to a third time and fourth audience members of a fourth media segment corresponding to a fourth time, and a fourth total audience size corresponding to the first demographic group exposed to the media for a third union, the third union including the first and second unions, the third union being a highest ancestor in the tree structure association.
19. The computer readable storage medium of claim 18 , wherein the instructions cause the machine to determine the estimate of the third total audience size for the second media segment corresponding to the second time based on a difference between the fourth total audience size of the media and a fifth total audience size of the media at the third union if the first union is removed from the tree structure association.
20. A method comprising:
generating a tree structure association corresponding to a union of first audience members of a first media segment of media and second audience members of a second media segment of the media, wherein the union is representative of a known first total audience size of a first demographic group from a plurality of demographic groups exposed to the media, a second total audience size is known for the first demographic group exposed to the first media segment at a first time, and a third total audience size is unknown for the first demographic group exposed to the second media segment at a second time; and
performing parallel computations on a processor to determine multipliers by solving equations corresponding to the tree structure association, the multipliers corresponding to the first total audience size for the union, the second total audience size for the first media segment, and the third total audience size for the second media segment; and
determining an estimate for the third total audience size for the second media segment corresponding to the second time based on the multipliers.
21. An apparatus comprising:
memory;
instructions in the apparatus;
processor circuitry to execute the instruction to:
generate a tree structure association corresponding to a union of first audience members of a first media segment of media and second audience members of a second media segment of the media, wherein the union is representative of a known first total audience size of a first demographic group from a plurality of demographic groups exposed to media, a second total audience size is known for the first demographic group exposed to the first media segment at a first time, and a third total audience size is unknown for the first demographic group exposed to the second media segment at a second time; and
perform parallel computations on a processor to determine multipliers by solving equations corresponding to the tree structure association, the multipliers corresponding to the first total audience size for the union, the second total audience size for the first media segment, and the third total audience size for the second media segment; and
determine an estimate for the third total audience size for the second media segment corresponding to the second time based on the multipliers.
22. The apparatus of claim 21 , wherein the processor circuitry is to generate the tree structure association by:
storing a first identifier in association with the first media segment;
storing the second media segment with a second identifier;
storing a third media segment with a third identifier; and
storing at least one of corresponding descendant nodes or corresponding ancestor nodes in association with the first media segment, the second media segment, and the union.
23. The apparatus of claim 21 , wherein the equations include a first equation corresponding to the first media segment, a second equation corresponding to the second media segment, and a third equation corresponding to the union.
24. The apparatus of claim 23 , wherein the processor circuitry is to solve for a first multiplier of the first equation and a second multiplier of the second equation in parallel.
25. The apparatus of claim 21 , wherein the processor circuitry is to solve the equations iteratively.
26. The apparatus of claim 21 , wherein the multipliers include a first multiplier corresponding to the first media segment, a second multiplier corresponding to the second media segment, and a third multiplier corresponding to the union, the processor circuitry to:
set the first multiplier equal to a first variable multiplier;
set the second multiplier equal to a constant value; and
set the third multiplier equal to a second variable multiplier, the solving of the equations resulting in a determination of a first value for the first variable multiplier and a second value of the second variable multiplier.
27. The apparatus of claim 26 , wherein the constant value is at least one of 1 or 0.
28. The apparatus of claim 21 , wherein the union is a first union, the tree structure association corresponds to a second union of third audience members of a third media segment of the media corresponding to a third time and fourth audience members of a fourth media segment corresponding to a fourth time, and a fourth total audience size corresponding to the first demographic group exposed to the media for a third union, the third union including the first and second unions, the third union being a highest ancestor in the tree structure association.
29. The apparatus of claim 28 , wherein the processor circuitry is to determine the estimate of the third total audience size for the second media segment corresponding to the second time based on a difference between the fourth total audience size of the media and a fifth total audience size of the media at the third union if the first union is removed from the tree structure association.
30. The apparatus of claim 21 , wherein the processor circuitry is to solve the equations analytically to facilitate the parallel computations faster than solving without the parallel computations.