Methods and apparatus to improve reach calculation efficiency

An apparatus to improve an efficiency of determining a published reach, comprising:

• a market data evaluator to identify a negative binomial distribution feasibility region corresponding to a candidate gross rating point (GRP) value;
• a maximum entropy engine to:
• in response to the market data evaluator identifying the negative binomial distribution feasibility region is associated with a number of samples below a threshold:
• generate a maximum entropy distribution, based on estimating a sample distribution of marketing data, the maximum entropy distribution constrained by a first GRP value and a first reach value, the first GRP value corresponding to the sample distribution of marketing data, the first reach value based on the first GRP value;
• a maximum entropy constraint manager to generate a geometric distribution based on estimating a minimum cross entropy of (a) the maximum entropy distribution and (b) the sample distribution of marketing data;
• a minimum cross entropy constraint manager to constrain the minimum cross entropy with the candidate GRP value of the sample distribution of marketing data, the candidate GRP value based on an advertising campaign increase quantity;
• and
• a conserved quantity engine to:
• determine a model of the candidate GRP value, the model based on an assistance value corresponding to the candidate GRP value;
• and
• reduce a computational burden associated with determining the published reach of the sample distribution of marketing data by generating closed-loop conserved quantity expressions of the geometric distribution based on the model of the candidate GRP value.

A tangible computer readable storage medium comprising instructions to improve an efficiency of determining a published reach that, when executed, causes a processor to, at least:

• identify a negative binomial distribution feasibility region corresponding to a candidate gross rating point (GRP) value;
• in response to identifying the negative binomial distribution feasibility region is associated with a number of samples below a threshold, estimate a sample distribution of marketing data to generate a maximum entropy distribution, the maximum entropy distribution being constrained by a first GRP value and a first reach value, the first GRP value empirically measured, the first GRP value corresponding to the sample distribution of marketing data, the first reach value based on the first GRP value;
• generate a geometric distribution based on estimating a minimum cross entropy of (a) the maximum entropy distribution and (b) the sample distribution of marketing data, the minimum cross entropy being constrained by the candidate GRP value of the sample distribution of marketing data, the candidate GRP value based on an advertising campaign increase quantity;
• determine a model of the candidate GRP value, the model based on an assistance value corresponding to the candidate GRP value;
• and
• reduce a computational burden associated with determining the published reach of the sample distribution of marketing data by generating closed-loop conserved quantity expressions of the geometric distribution based on the model of the candidate GRP value.