Methods and apparatus for estimating total unique audiences

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.

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.