Methods and apparatus for estimating total unique audiences

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

based on impression requests from a plurality of client devices via a network, log a plurality of impressions corresponding to media accessed at the client devices;

obtain a count of demographic impressions logged by a database proprietor;

obtain a count of registered users of the database proprietor exposed to the media; and

execute a process to determine a unique audience size by:

multiplying a count of the plurality of impressions by a square of the count of the registered users to generate a product;

dividing the product by the count of the demographic impressions to generate a quotient; and

determining the unique audience size based on a square root of the quotient.

2. The non-transitory computer readable storage medium of claim 1 , wherein the determining of the unique audience size based on the square root of the quotient is to generate the unique audience size by the machine when the unique audience size is not available from a server of the database proprietor.

3. The non-transitory computer readable storage medium of claim 1 , wherein the process is a first process, the instructions are further to cause the machine to select the first process to determine the unique audience instead of a second process to determine the unique audience, the first process to cause more resource-efficient operation of a computer than the second process by utilizing less processor resources and memory resources than the second process.

4. The non-transitory computer readable storage medium of claim 1 , wherein the registered users of the database proprietor exposed to the media correspond to the demographic impressions and to at least some of the plurality of impressions.

5. The non-transitory computer readable storage medium of claim 1 , wherein the instructions are further to cause the machine to generate a report indicating the unique audience.

6. The non-transitory computer readable storage medium of claim 1 , wherein the instructions are further to cause the machine to credit the media associated with the plurality of impressions based on the unique audience size.

7. The non-transitory computer readable storage medium of claim 1 , wherein the instructions are further to cause the machine to determine a frequency distribution for the plurality of impressions based on the unique audience size by:

dividing the unique audience size by the count of the plurality of impressions to determine a second quotient; and

calculating a geometric distribution based on the second quotient.

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

based on impression requests from a plurality of client devices via a network, log a plurality of impressions corresponding to media accessed at the client devices;

obtain a first count of first demographic impressions corresponding to the media accessed via a first website and logged by a first database proprietor, the first demographic impressions corresponding to first registered users of the first database proprietor;

obtain a second count of second demographic impressions corresponding to the media accessed via a second website and identified by a second database proprietor, the second demographic impressions corresponding to second registered users of the second database proprietor;

obtain a first count of the first registered users exposed to the media;

obtain a second count of the second registered users exposed to the media;

generate a constraint matrix and a constraint vector based on the first count of the first demographic impressions and the second count of the second demographic impressions, the constraint vector representative of a plurality of ratios of constraints to a count of the plurality of impressions; and

determine a first unique audience size exposed to the media via the first website, a second unique audience size exposed to the media via the second website, and a total unique audience size exposed to the media via the first and second websites based on the constraint matrix and the constraint vector.

9. The non-transitory computer readable storage medium of claim 8 , wherein the determining of the first unique audience size, the second unique audience size, and the total unique audience size based on the constraint matrix and the constraint vector is to generate the first unique audience size, the second unique audience size, and the total unique audience size when the first unique audience size, the second unique audience size, and the total unique audience size are not available from a server of the database proprietor.

10. The non-transitory computer readable storage medium of claim 8 , wherein the total unique audience size is a count of unique audience members across the first unique audience and the second unique audience.

11. The non-transitory computer readable storage medium of claim 8 , wherein the constraint matrix is a first constraint matrix and the constraint vector is a first constraint vector, and the instructions cause the machine to determine the first unique audience size, the second unique audience size, and the total unique audience size by:

generating the first constraint matrix and the first constraint vector based on the first count of the first demographic impressions, the first count of the first registered users, the second count of the second demographic impressions, and the second count of the second registered users;

determining impression characteristics associated with the first constraint matrix and the first constraint vector, the impression characteristics including maximized values based on the first constraint matrix and the first constraint vector; and

generating a second constraint matrix and a second constraint vector based on the impression characteristics, the first and second constraint matrices and the first and second constraint vectors corresponding to audience characteristics, the audience characteristics including maximized values based on the second constraint matrix and the second constraint vector, the first unique audience size, the second unique audience size, and the total unique audience size being based on the audience characteristics.

12. The non-transitory computer readable storage medium of claim 11 , wherein the instructions cause the machine to determine a frequency distribution based on the impression characteristics.

13. The non-transitory computer readable storage medium of claim 11 , wherein the impression characteristics include probabilities representing likelihoods of different numbers of people exposed to the media via at least one of the first website or the second website.

14. The non-transitory computer readable storage medium of claim 11 , wherein the audience characteristics include probabilities representing likelihoods of different sizes of unique audiences corresponding to at least one of the first website or the second website.

15. The non-transitory computer readable storage medium of claim 8 , wherein the first database proprietor is the second database proprietor.

16. The non-transitory computer readable storage medium of claim 8 , wherein the instructions cause the machine to generate a report indicating at least one of the first unique audience size, the second unique audience size, or the total unique audience size.

17. The non-transitory computer readable storage medium of claim 8 , wherein the instructions cause the machine to credit the media associated with the plurality of impressions based on at least one of the first unique audience size, the second unique audience size, or the total unique audience size.

18. The non-transitory computer readable storage medium of claim 8 , wherein the constraint matrix includes constraints represented in sub-matrices, the constraints including at least one of a first size of a first audience exposed to the media via the first website, a second size of a second audience exposed to the media via the second website, a third size of a total audience exposed to the media via the first and second websites, a fourth size of a universe audience via any web site, a first count of first impressions corresponding to the media accessed via the first website, and a second count of second impressions corresponding to the media accessed via the second web site.

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

based on impression requests from a plurality of client devices via a network, log a plurality of impressions corresponding to media accessed at the client devices;

obtain a first count of first demographic impressions corresponding to the media accessed via a first website and logged by a first database proprietor, the first demographic impressions corresponding to first registered users of the first database proprietor;

obtain a second count of second demographic impressions corresponding to the media accessed via a second website and logged by a second database proprietor, the second demographic impressions corresponding to second registered users of the second database proprietor;

obtain a first count of the first registered users exposed to the media;

obtain a second count of the second registered users exposed to the media;

determine a first enumeration table, the first enumeration table including first values based on a system of non-linear equations associated with the first count of the first demographic impressions, the first count of the first registered users, the second count of the second demographic impressions and the second count of the second registered users;

determine a second enumeration table including second values based on a second system of non-linear equations associated with the first values calculated in the first enumeration table and the plurality of impressions; and

determine a first unique audience size of the media accessed via the first website, a second unique audience size of the media accessed via the second website, and a total unique audience size of the media accessed via the first and second websites using second expressions solved based on the second values in the second enumeration table.

20. The non-transitory computer readable storage medium of claim 19 , wherein the determining of the first unique audience size, the second unique audience size, and the total unique audience size using second expressions solved based on the second values in the second enumeration table is to generate the first unique audience size, the second unique audience size, and the total unique audience size when the first unique audience size, the second unique audience size, and the total unique audience size are not available from a server of the database proprietor.

21. The non-transitory computer readable storage medium of claim 19 , wherein the first enumeration table is an estimated distribution of the plurality of impressions across the first registered users and the second registered users.

22. The non-transitory computer readable storage medium of claim 19 , wherein the second enumeration table is an estimated distribution of the plurality of impressions across the total unique audience.

23. The non-transitory computer readable storage medium of claim 22 , wherein the instructions cause the machine to determine a frequency distribution of impressions for the total unique audience size based on the second enumeration table.

24. The non-transitory computer readable storage medium of claim 19 , wherein the second enumeration table is based on population constraints, the population constraints representative of:

a third count of the first demographic impressions corresponding to the media accessed via the first website, the third count of the first demographic impressions including the first count of the first demographic impressions; and

a fourth count of the second demographic impressions corresponding to the media accessed via the second website, the fourth count of the second demographic impressions including the second count of the second demographic impressions.

25. The non-transitory computer readable storage medium of claim 19 , wherein the first database proprietor is the second database proprietor.

26. The non-transitory computer readable storage medium of claim 19 , wherein the instructions cause the machine to generate a report indicating at least one of the first unique audience size, the second unique audience size, or the total unique audience size.

27. The non-transitory computer readable storage medium of claim 19 , wherein the instructions cause the machine to credit the media associated with the plurality of impressions based on at least one of the first unique audience size, the second unique audience size, or the total unique audience size.