Abstract: Methods, systems, and apparatus for calibrating data using relaxed benchmarks constraints are described. An example apparatus for generating a unique solution when calibrating data via a calibration model having relaxed benchmark constraints includes processor circuitry to execute computer-readable instructions. The computer-readable instructions are to execute the calibration model based on matrix data, a target loss function, a weight loss function, and a budget parameter. The computer-readable instructions are to determine calibrated weights resulting from execution of the calibration model. The computer-readable instructions are to incorporate a stability parameter into the calibration model in response to determining that the calibrated weights do not provide a unique solution for the executed calibration model. The stability parameter is to reduce an influence of the budget parameter on the calibration model to enable the generation of a unique solution.

Abstract: Methods, apparatus, systems, and articles of manufacture are disclosed to to estimate population reach from marginal ratings. An example apparatus includes an interface to access marginal ratings of the media for a recorded audience, marginal ratings of the media for a population audience, and reach of the media for the recorded audience; a reach determiner to iteratively estimate a reach of the media for the population audience based on the marginal ratings for the recorded audience, the marginal ratings for the population audience, and the reach for the recorded audience; and the interface to output the estimated reach for the population audience.

Abstract: Example methods, apparatus, systems, and articles of manufacture are disclosed to estimate population reach for different unions based on marginal ratings. An example apparatus includes memory and processor circuitry to determine a population reach estimate of a union of time intervals for which media ratings data is available, the population reach estimate based on a pseudo universe estimate of a population audience corresponding to the union of the time intervals; determine a pseudo universe estimate of a recorded audience corresponding to the union of the time intervals; determine the pseudo universe estimate of the population audience based on the pseudo universe estimate of the recorded audience; and in response to a decision to update the pseudo universe estimate of the population audience to reduce an error of the population reach estimate output the population reach estimate of the union when the error of the population reach estimate satisfies a threshold.

Abstract: A disclosed example includes determining a plurality of cross-demographic total census parameter values based on ratios of third-party impression counts and third-party audience sizes, selecting a first cross-demographic total census parameter value as an upper bound constraint from the plurality of cross-demographic total census parameter values based on comparisons between ones of the plurality of cross-demographic total census parameter values, selecting a second cross-demographic total census parameter value from the plurality of cross-demographic total census parameter values based on the second cross-demographic total census parameter value being less than the upper bound constraint, determining a first census-level audience size based on the second cross-demographic total census parameter value, and determining a first census-level impression count based on the second cross-demographic total census parameter value.

Abstract: Processor systems to estimate audience sizes and impression counts for different frequency intervals are disclosed. An example processor system includes a memory management unit (MMU) to assign requests from computing devices indicative of accesses to media to a first block of memory and to assign user-identified impression data corresponding to user-identified impressions to a second block of memory. The processor system including an arithmetic logic unit (ALU) to determine multipliers relating a first probability distribution for the user-identified impressions to a second probability distribution for census impressions. The multipliers based on probability constraints defined by weighted probabilities associated with the second probability distribution, where different ones of the probabilities weighted based on estimated populations for corresponding ones of the different demographics.

Abstract: An example apparatus includes at least one memory, instructions in the apparatus, and processor circuitry to execute the instructions to generate a reference demographic impression distribution based on first impressions of logged impressions at a first server corresponding to first client devices, access an inaccurate demographic impression distribution based on second impressions corresponding to second client devices, determine an estimated demographic impression distribution based on an inaccurate demographic impression distribution, the estimated demographic impression distribution representative of the second impressions distributed across the different demographic segments, determine a matrix based on a reference demographic impression distribution and Lagrange multiplier values, determine an error indicator value based on the matrix, generate, in response to the error indicator value satisfying a threshold, an accuracy-improved demographic impression distribution, and store, in the at least one memory

Abstract: Methods, systems, and apparatus for calibrating data using relaxed benchmarks constraints are described. An example apparatus for generating a unique solution when calibrating data via a calibration model having relaxed benchmark constraints includes a calibration engine to execute the calibration model based on a target loss function, a weight loss function, and a budget parameter. The example apparatus further includes a calibrated weights determiner to determine calibrated weights resulting from execution of the calibration model. The example apparatus further includes a calibration model validator to incorporate a stability parameter into the calibration model in response to determining that the calibrated weights do not provide a unique solution for the executed calibration model. The stability parameter is to reduce an influence of the budget parameter on the calibration model to enable the generation of a unique solution.

Abstract: Example methods, apparatus, systems, and articles of manufacture are disclosed to estimate population reach for different unions based on marginal ratings. An example apparatus includes a reach determiner to determine a population reach estimate of a union of time intervals for which media ratings data is available, the population reach estimate based on a pseudo universe estimate of a population audience corresponding to the union of the time intervals; a pseudo universe determiner to determine a pseudo universe estimate of a recorded audience corresponding to the union of the time intervals; determine the pseudo universe estimate of the population audience based on the pseudo universe estimate of the recorded audience; and iteratively update the pseudo universe estimate of the population audience to reduce an error; and a consistency checker to adjust the population reach estimate, the reach determiner to output the population reach estimate of the union.

Abstract: Example methods, apparatus, and articles of manufacture are disclosed to estimate population reach. An example apparatus includes processor circuitry to determine first multipliers corresponding to a panelist impression count and panelist audience size totals of at least one of a first margin of media, a second margin of the media, or a union of the first margin and the second margin, the first margin, the second margin, and the union included in a tree association; concurrently determine second multipliers using the tree association and the first multipliers; determine third multipliers corresponding to a total audience size exposed to the media at at least one of the first margin, the second margin, or the union based on the tree association using database proprietor impression totals; and determine, based on the third multipliers, an estimate for the population reach of the media for at least one of the first margin, the second margin, or the union.

Abstract: An example apparatus to determine census-based audience metrics of media across multiple demographics includes an audience size determiner to determine a first census-level audience size based on a third-party audience size, a third-party impression count, and a first cross-demographic total census parameter value; an impression counts determiner to determine a first census-level impression count based on the third-party audience size, the third-party impression count, and the first cross demographic total census parameter value; and a verifier to determine whether the first cross-demographic total census parameter value satisfies a constraint based on the first census-level impression count; and when the constraint is not satisfied the verifier to: (a) discard the first cross-demographic total census parameter value, and (b) select a second cross-demographic total census parameter value, the second cross-demographic total census parameter value to determine a second census-level audience size and a second ce

Abstract: Methods and apparatus to determine reach with time dependent weights are disclosed. An example method disclosed herein includes determining a first subset of panelists exposed to media at a first time; determining a second subset of the panelists exposed to the media at a second time; applying a first plurality of weights to the first subset of panelists to generate first weighted exposures; applying a second plurality of weights to the second subset of the panelists to generate second weighted exposures; for a first panelist in the first and second subsets, determining a first cumulative weighted exposure by determining which of the first and second weighted exposures is largest for each of the first panelist; for a second panelist in one of the first and second subsets, determining a second cumulative weighted exposure by based on the weight applied to the second panelist; and determining a reach of the media for the first and second times by combining the first and second cumulative weighted exposures.

Abstract: An example apparatus includes a matrix processor in circuit with a probability generator to determine a first matrix representative of element-wise multiplication between a constraint matrix and a first transpose matrix of the estimated demographic impression distribution, the constraint matrix based on the reference demographic impression distribution and determine a second matrix by multiplying the first matrix with a second transpose matrix of the constraint matrix. The apparatus further includes an error determiner in circuit with the matrix processor, the error determiner to determine an error indicator value based on the second matrix, the error indicator value indicative of an error associated with the estimated demographic impression distribution, and a probability generator to generate, in response to the error indicator value satisfying a threshold, an accuracy-improved demographic impression distribution.

Abstract: Methods, apparatus, systems and articles of manufacture (e.g., physical storage media) to estimate and use duplication factors for audience measurement are disclosed. Example apparatus disclosed herein are to access first marginal ratings values that represent respective portions of a first population associated with corresponding ones of a plurality of media segments, estimate the duplication factor based on a difference between a sum of the first marginal ratings values and a largest one of the first marginal ratings values, access second marginal ratings values that represent respective portions of a second population associated with corresponding ones of the media segments, and output, based on the duplication factor and the second marginal ratings values, a reach value for a union of the media segments, the reach value to represent a number of unique individuals of the second population associated with at least one of the media segments.

Abstract: Example methods, apparatus, and articles of manufacture are disclosed to estimate population reach. An example apparatus includes an association controller to generate a tree association corresponding to a union of a first margin of media and a second margin of the media; and one or more commercial solvers to determine first multipliers by solving first equations corresponding to panelist impressions and panelist audience totals, the second margin, or the union; perform first parallel computations with a processor to determine second multipliers to solve second equations corresponding to the tree association using the first multipliers; discard the first multipliers; perform second parallel computations to determine third multipliers by solving third equations corresponding to the tree association using database proprietor impression totals; and determine an estimate for a population reach of the media for at least one of the first margin, the second margin, or the union based on the third multipliers.

Abstract: Methods, apparatus, systems and articles of manufacture are disclosed to estimate an audience population. An example apparatus includes at least one memory; instructions in the apparatus; and processor circuitry to execute the instructions to: determine whether respective ones of respondents are associated with a characteristic; detect unique instances of the respective ones of the respondents; in response to the respective ones of the respondents being associated with the characteristic, increase a sample capture count by one; in response to detecting the unique instances of the respective ones of the respondents exhibiting the characteristic, increase a unique capture count by one; determine a seed population estimate based on the unique capture count; and determine a population estimate having the characteristic based on the sample capture count, the unique capture count, the seed population estimate, and a number of available samples.

Abstract: Methods, apparatus, systems and articles of manufacture are disclosed to estimate an audience population.

Abstract: Methods and apparatus to determine reach with time dependent weights are disclosed. An example method disclosed herein includes determining a first subset of panelists exposed to media at a first time; determining a second subset of the panelists exposed to the media at a second time; applying a first plurality of weights to the first subset of panelists to generate first weighted exposures; applying a second plurality of weights to the second subset of the panelists to generate second weighted exposures; for a first panelist in the first and second subsets, determining a first cumulative weighted exposure by determining which of the first and second weighted exposures is largest for each of the first panelist; for a second panelist in one of the first and second subsets, determining a second cumulative weighted exposure by based on the weight applied to the second panelist; and determining a reach of the media for the first and second times by combining the first and second cumulative weighted exposures.

Abstract: Methods, apparatus, systems, and articles of manufacture are disclosed to estimate population reach from marginals. An example apparatus includes memory including computer readable instructions; and a processor to execute the instructions to: iteratively converge on an output estimate of a pseudo universe estimate of a recorded audience of first media based on (A) a recorded reach for the recorded audience of the first media and (B) first marginal ratings for the recorded audience of the first media; determine a panel representation value based on the pseudo universe estimate of the recorded audience of the first media; and iteratively converge on an output estimate of a final reach of second media for a population audience based on the panel representation value and second marginal ratings for the population audience of the second media.

Abstract: Methods, apparatus, systems and articles of manufacture are disclosed to estimate an audience population.

Abstract: Methods, apparatus, systems and articles of manufacture are disclosed to improve multi-demographic modeling efficiency. An example apparatus includes a feature set aggregator to segregate training data based on feature sets of interest, and to identify households that participate in at least one of the feature sets of interest, a class enumerator to reduce multi-demographic model iterations by enumerating demographic combinations for the identified households, the enumerated demographic combinations including a single identifier to represent a combination of two or more demographic categories, and a modeling engine to generate training coefficients associated with respective ones of the enumerated demographic combinations.