Abstract: Methods, apparatus, systems and articles of manufacture are disclosed that estimate the second frequency moment. An example apparatus includes a hashing generator to input a first audience member identifier and a second audience member identifier of the media impression data through a hashing function, an output of the hashing generator being a first bit-value representation of the first audience member identifier and a second bit-value representation of the second audience member identifier, a vector generator to increment first and second values of corresponding first and second positions in a vector based on the first and second bit-value representations, respectively, a second frequency moment generator to estimate the second frequency moment of the media impression data using the vector, and a comparator to, using the second frequency moment, determine whether a computer can complete a query request of the media impression data based on query processing constraints.

Abstract: A disclosed example apparatus includes a communication interface to: access impression count data corresponding to a plurality of platforms; and access deduplicated total audience size data; an arithmetic logic unit to: generate a total impressions count by aggregating the impression count data corresponding to the plurality of platforms; generate an estimated per-platform impression count by dividing the total impressions count by a number of the platforms, a solver controller to: instruct a numerical solver to utilize the number of the platforms, the deduplicated total audience size data, and the estimated per-platform impression count to estimate the deduplicated audience size of a first platform of the platforms; and store the deduplicated audience size of the first platform in memory.

Abstract: Methods, apparatus, systems, and articles of manufacture to perform computer-based monitoring of audiences of network-based media using information theory to estimate intermediate level unions are disclosed. An example apparatus to determine a deduplicated, census-based audience metric of media includes panel union calculator circuitry to calculate a threshold statistic corresponding to an intermediate union of a panel hierarchy, and census union calculator circuitry to calculate a deduplicated audience value corresponding to the intermediate union of the census hierarchy based on the threshold statistic.

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: Methods, and apparatus are disclosed to estimate a deduplicated audience of a partitioned audience of media presentations. An example apparatus includes interface circuitry, instructions in the apparatus, and processor circuitry to execute the instructions to at least: identify one or more nodes of a graph structure selected for estimation of a deduplicated audience, and estimate a value indicative of the deduplicated audience across a first selected node of the graph structure and a second selected node of the graph structure.

Abstract: Methods and apparatus for determining a unique audience exposed to media while reducing memory resources of a computing device are disclosed herein. An example apparatus includes memory; instructions in the apparatus; and processor circuitry to execute the instruction to, 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; generate a product by multiplying a count of the impressions by a square of a count of registered users of a database proprietor exposed to the media; generate a quotient by dividing the product by a count of demographic impressions logged by the database proprietor; and determine a unique audience size based on a square root of the quotient.

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: Example technical solutions to determine census-level audience metrics across demographics are disclosed. Disclosed example apparatus include a distribution parameter solver to initialize distribution parameter values for a probability of an individual within a demographic being (i) included in a subscriber audience for the demographic and (ii) having a first impression duration, the subscriber audience having a first subscriber audience size. Disclosed example apparatus also include a divergence parameter solver to determine divergence parameter values between (i) the first subscriber audience size and the first impression duration and (ii) a census-level audience size and a second impression duration based on the initialized distribution parameter values.

Abstract: Methods, apparatus, systems and articles of manufacture to estimate and deduplicate audiences are disclosed herein. An example apparatus includes a controller to determine a subunion of at least first and second marginal audiences of media based on of panel data and census data, the panel data including a panel impression count and a panel audience size, and the census data including a census impression count, an audience size calculator to determine a census audience size of the at least the first and second marginal audiences based on the panel impression count and the panel audience size and determine a subunion census audience size, the subunion census audience size corresponding to an overlap between the at least the first and second marginal audiences; and a report generator to generate a report including the census audience size and the subunion census audience size.

Abstract: Methods and apparatus to estimate census level audience sizes, impression counts, and duration data are disclosed. Example instructions cause one or more processor to at least set up a system of equations based on first census data; select a census parameter value based on a constraint for census parameter selection, the constraint based on media access represented in panel data; iteratively solve for second census data that satisfies the system of equations using the panel data and the census parameter value, the second census data including an audience size value, an impression count value, and a duration value for a media item accessed by anonymous audience members in a demographic group; and generate a report including the second census data.

Abstract: An example apparatus includes an audience size calculator circuitry to determine a first census-level audience size, and an impression count calculator circuitry to determine a first census-level impression count.

Abstract: Methods and apparatus for determining a unique audience exposed to media while reducing memory resources of a computing device are disclosed herein. An example apparatus includes means for logging a plurality of impressions based on impression requests from a plurality of client devices, the plurality of impressions corresponding to media accessed at the client devices; means for obtaining counts, the obtaining means to: obtain a count of demographic impressions logged by a database proprietor; and obtain a count of registered users of the database proprietor exposed to the media; and means for determining a unique audience size by: multiplying a count of the plurality of impressions by a square of the count of the registered users to generate a product; dividing the product by the count of the demographic impressions to generate a quotient; and determining the unique audience size based on a square root of the quotient.

Abstract: Disclosed examples to estimate audience sizes of media include a coefficient generator to determine coefficient values for a polynomial based on normalized weighted sums of variances, a normalized weighted sum of covariances, and cardinalities corresponding to a first plurality of vectors of counts from a first database proprietor and a second plurality of vectors of counts from a second database proprietor, a real roots solver to determine a real root value of the polynomial, the real root value indicative of a number of audience members represented in the first plurality of vectors of counts that are also represented in the second plurality of vectors of counts, and an audience size generator to determine the audience size based on the real root value and the cardinalities of the first plurality of vectors of counts and the second plurality of vectors of counts.

Abstract: Methods, apparatus, systems and articles of manufacture are disclosed that estimate the second frequency moment. An example apparatus includes a hashing generator to input a first audience member identifier and a second audience member identifier of the media impression data through a hashing function, an output of the hashing generator being a first bit-value representation of the first audience member identifier and a second bit-value representation of the second audience member identifier, a vector generator to increment first and second values of corresponding first and second positions in a vector based on the first and second bit-value representations, respectively, a second frequency moment generator to estimate the second frequency moment of the media impression data using the vector, and a comparator to, using the second frequency moment, determine whether a computer can complete a query request of the media impression data based on query processing constraints.

Abstract: Methods, apparatus, systems, and articles of manufacture to estimate cardinality of users represented in arbitrarily distributed bloom filter arrays are disclosed. An apparatus includes a communications interface to obtain a first Bloom filter array generated by a first computer of a first database proprietor and obtain a second Bloom filter array generated by a second computer of a second database proprietor. The apparatus includes a cardinality analyzer to estimate a total number of unique users included in a union of corresponding ones of the first and second users represented in the first and second Bloom filter arrays regardless of whether the first and second users are mapped, based on a first distribution or a second distribution different than the first distribution, to the corresponding first and second Bloom filter arrays.

Abstract: Methods and apparatus for determining a unique audience exposed to media while reducing memory resources of a computing device are disclosed herein. An example apparatus includes means for logging a plurality of impressions based on impression requests from a plurality of client devices, the plurality of impressions corresponding to media accessed at the client devices; means for obtaining counts, the obtaining means to: obtain a count of demographic impressions logged by a database proprietor; and obtain a count of registered users of the database proprietor exposed to the media; and means for determining a unique audience size by: multiplying a count of the plurality of impressions by a square of the count of the registered users to generate a product; dividing the product by the count of the demographic impressions to generate a quotient; and determining the unique audience size based on a square root of the quotient.

Abstract: An example apparatus includes an association controller to generate a tree structure association for a total audience size that accessed a plurality of media items, the tree structure association including a first node representative of a first media item accessed by first audience members of the total audience size and a second node representative of a second media item accessed by second audience members of the total audience size, a matrix generator to generate a matrix by selecting a sum of probabilities value corresponding to the tree structure association, the sum of probabilities value representative of a probability of the first audience members accessing the first media item and storing the sum of probabilities value in an element of the matrix, and a commercial solver to estimate a deduplicated audience size of the total audience size using the matrix.

Abstract: Methods and apparatus for determining a unique audience exposed to media while reducing memory resources of a computing device are disclosed herein. Example instructions 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 count of demographic impressions logged by a database proprietor; obtain a count of registered users of the database proprietor exposed to the media; and execute a process to determine a unique audience size by multiplying a count of the plurality of impressions by a square of the count of the registered users to generate a product; dividing the product by the count of the demographic impressions to generate a quotient; and determining the unique audience size based on a square root of the quotient.

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 for determining a unique audience exposed to media while reducing memory resources of a computing device are disclosed herein. Example instructions 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 count of demographic impressions logged by a database proprietor; obtain a count of registered users of the database proprietor exposed to the media; and execute a process to determine a unique audience size by multiplying a count of the plurality of impressions by a square of the count of the registered users to generate a product; dividing the product by the count of the demographic impressions to generate a quotient; and determining the unique audience size based on a square root of the quotient.