Methods and apparatus to replicate panelists using a local minimum solution of an integer least squares problem

1. An apparatus comprising:

a data organizer to:

determine estimated aggregate panelist data based on attribute data corresponding to a seed panel; and

determine weight estimates to replicate respective seed panelists in the seed panel; and

a weight determiner to:

determine weight adjustments based on a ratio of (A) a first dot product of (i) the attribute data to (ii) a difference between the estimated aggregate panelist data and aggregate seed panelist data, and (B) a second dot product of the attribute data to the attribute data;

iteratively adjust the weight estimates based on the weight adjustments to generate final weight estimates; and

replicate respective ones of the seed panelists based on the final weight estimates to generate a synthetic audience representative of return path data reported by a plurality of media devices.

2. The apparatus of claim 1 , wherein when the weight estimates are known, the data organizer is to determine the estimated aggregate panelist data by combining attributes of seed panelists of the seed panel according to the weight estimates.

3. The apparatus of claim 1 , wherein when the weight estimates are not known, the data organizer is to:

determine a minimum error based on an equal distribution of weights; and

set the weight estimates to be the minimum error.

4. The apparatus of claim 1 , wherein the weight determiner is to:

access one of the weight estimates corresponding to a first one of the seed panelists;

determine a first weight adjustment for the first one of the seed panelists;

determine an adjusted first one of the weight estimates for the first one of the seed panelists; and

when the adjusted first one of the weight estimates is not substantially equal to the first one of the weight estimates:

update (A) the first one of the weight estimates based on the adjusted first one of the weight estimates and (B) the estimated aggregate panelist data based on a difference between the first one of the weight estimates and the adjusted first one of the weight estimates; and

continue to adjust the first one of the weight estimates.

5. The apparatus of claim 4 , wherein when the first one of the weight estimates is substantially equal to the adjusted first one of the weight estimates, the weight determiner is to iteratively adjust a second one of the weight estimates for a second one of the seed panelists.

6. The apparatus of claim 4 , wherein the weight determiner is to determine the first weight adjustment for the first one of the seed panelists based on a comparison of an aggregated data error and attributes of the first one of the seed panelists.

7. The apparatus of claim 6 , wherein the aggregated data error corresponds to a difference between the estimated aggregate panelist data and the input aggregate panelist data.

8. A method comprising:

determining, by executing an instruction with a processor, estimated aggregate panelist data based on attribute data corresponding to a seed panel;

determining, by executing an instruction with the processor, weight estimates to replicate respective seed panelists in the seed panel;

determining, by executing an instruction with the processor, weight adjustments based on a ratio of (A) a first dot product of (i) the attribute data to (ii) a difference between the estimated aggregate panelist data and aggregate seed panelist data, and (B) a second dot product of the attribute data to the attribute data;

generating, by executing an instruction with the processor, final weight estimates by iteratively adjusting the weight estimates based on the weight adjustments; and

replicating, by executing an instruction with the processor, respective ones of the seed panelists based on the final weight estimates to generate a synthetic audience representative of return path data reported by a plurality of media devices.

9. The method of claim 8 , wherein when the weight estimates are known, the determining of the estimated aggregate panelist data includes combining attributes of seed panelists of the seed panel based on the weight estimates.

10. The method of claim 8 , wherein when the weight estimates are known, the determining of the estimated aggregate panelist data includes:

determining a minimum error based on an equal distribution of weights; and

setting the weight estimates to be the minimum error.

11. The method of claim 8 , wherein the iteratively adjusting of the weight estimates includes:

accessing a first weight estimate corresponding to a first seed panelist;

determining a first weight adjustment for the first seed panelist;

determining an adjusted weight estimate for the first seed panelist; and

when the adjusted weight estimate is not substantially equal to the first weight estimate:

updating (A) the first weight estimate based on the adjusted weight estimate and (B) the estimated aggregate panelist data based on a difference between the first weight estimate and the adjusted weight estimate; and

continuing to adjust the first weight estimate.

12. The method of claim 11 , further including, when the first weight estimate is substantially equal to the adjusted weight estimate, iteratively adjusting a second weight estimate for a second panelist.

13. The method of claim 11 , wherein the determining of the first weight adjustment for the first seed panelist is based on a comparison of an aggregated data error and attributes of the first seed panelist.

14. The method of claim 13 , wherein the aggregated data error corresponds to a difference between the estimated aggregate panelist data and the input aggregate panelist data.

15. A tangible computer readable storage medium comprising instructions which, when executed, cause a machine to at least:

determine weight estimates to replicate respective seed panelists in a seed panel;

determine weight adjustments based on a ratio of (A) a first dot product of (i) a difference between estimated aggregate panelist data and aggregate seed panelist data to (ii) attribute data corresponding to the seed panel, and (B) a second dot product of the attribute data to the attribute data;

iteratively adjust the weight estimates based on comparisons of the estimated aggregate panelist data as determined based on the weight adjustments to generate final weight estimates; and

replicate respective ones of the seed panelists based on the final weight estimates to generate a synthetic audience representative of return path data reported by a plurality of media devices.

16. The computer readable storage medium of claim 15 , wherein the instructions cause the machine to determine the estimated aggregate panelist data by combining attributes of seed panelists of the seed panel according to the weight estimates when the weight estimates are known.

17. The computer readable storage medium of claim 15 , wherein when the weight estimates are not known, the instructions cause the machine to:

determine a minimum error based on an equal distribution of weights; and

set the weight estimates to be the minimum error.

18. The computer readable storage medium of claim 15 , wherein the instructions cause the machine to:

access a first one of the weight estimates corresponding to a first one of the seed panelists;

determine a first weight adjustment for the first one of the seed panelists;

determine an adjusted first one of the weight estimates for the first one of the seed panelists; and

when the adjusted first one of the weight estimates is not substantially equal to the first one of the weight estimates:

update (A) the first one of the weight estimates based on the adjusted first one of the weight estimates and (B) the estimated aggregate panelist data based on a difference between the first one of the weight estimates and the adjusted first one of the weight estimates; and

continue to adjust the first one of the weight estimates.