Methods and apparatus to estimate census-level audience sizes and duration data across dimensions and/or demographics

A system comprising:

• memory;
• programmable circuitry;
• and
• instructions to cause the programmable circuitry to:
• select at least one constraint equation corresponding to one or more events, the one or more events corresponding to first audience members and second audience members, the first audience members to include the second audience members;
• solve the at least one constraint equation to determine one or more constraint parameter values, the one or more constraint parameter values corresponding to the one or more events;
• and
• determine, based on one or more intermediate equations and the one or more constraint parameter values, an event audience size and an event duration corresponding to (a) the first audience members and (b) a combination of the one or more events and one or more demographics.

A non-transitory computer readable medium comprising instructions that, when executed, cause programmable circuitry to at least:

• select at least one constraint equation corresponding to one or more events, the one or more events corresponding to first audience members and second audience members, the first audience members to include the second audience members;
• solve the at least one constraint equation to determine one or more constraint parameter values, the one or more constraint parameter values corresponding to the one or more events;
• and
• determine, based on one or more intermediate equations and the one or more constraint parameter values, an event audience size and an event duration corresponding to (a) the first audience members and (b) a combination of the one or more events and one or more demographics.