Abstract: Methods and apparatus to estimate cardinality of users represented across multiple bloom filter arrays are disclosed. Examples includes processor circuitry to execute and/or instantiate instructions to generate a first composite Bloom filter array based on first and second Bloom filter arrays. The processor circuitry is to generate a final composite Bloom filter array based on the first composite Bloom filter array and a third Bloom filter array. Different ones of the first, second, and third Bloom filter arrays representative of different sets of users who accessed media. The first, second, and third Bloom filter arrays including differential privacy noise. The processor circuitry to estimate a cardinality of a union of the first, second, and third Bloom filter arrays based on the final composite Bloom filter array.

Abstract: Methods and apparatus to estimate cardinality across multiple datasets represented using Bloom filter arrays are disclosed. Disclosed examples include processor circuitry to execute and/or instantiate instructions to determine an inclusion-exclusion expression that defines an audience size for a user group of interest. Terms in the inclusion-exclusion expression corresponding to either a first cardinality of a first one of at least three Bloom filter arrays or a second cardinality of a union of two or more of the Bloom filter arrays. Different ones of the Bloom filter arrays representative of different sets of users who accessed media. The at least one processor further to estimate, based on the inclusion-exclusion expression, the audience size of the user group of interest.

Abstract: Methods and systems to estimate audience measurement metrics based on users represented in Bloom filter arrays are disclosed. An apparatus includes a communications interface to receive a first Bloom filter array from a first computer of a first database proprietor. The first Bloom filter array is representative of first users who accessed media. The first users are registered with the first database proprietor. The first Bloom filter array includes a first array of first elements. Values of respective ones of the first elements are either a 0 or a 1 based on whether quantities of the first users allocated to the respective ones of the first elements are even or odd. The apparatus further includes a Bloom filter array analyzer to estimate a first cardinality for the first Bloom filter array. The first cardinality is indicative of a total number of the first users who accessed the media.

Abstract: Methods, apparatus, systems, and articles of manufacture to estimate audience measurement metrics based on users represented in Bloom filter arrays are disclosed. An apparatus includes a communications interface to receive a first Bloom filter array from a first computer of a first database proprietor. The first Bloom filter array is representative of first users who accessed media. The first users are registered with the first database proprietor. The first Bloom filter array includes a first array of first elements. Values of respective ones of the first elements are either a 0 or a 1 based on whether quantities of the first users allocated to the respective ones of the first elements are even or odd. The apparatus further includes a Bloom filter array analyzer to estimate a first cardinality for the first Bloom filter array. The first cardinality is indicative of a total number of the first users who accessed the media.

Abstract: Methods and apparatus to estimate cardinality of users represented across multiple bloom filter arrays are disclosed. Examples includes processor circuitry to execute and/or instantiate instructions to generate a first composite Bloom filter array based on first and second Bloom filter arrays. The processor circuitry is to generate a final composite Bloom filter array based on the first composite Bloom filter array and a third Bloom filter array. Different ones of the first, second, and third Bloom filter arrays representative of different sets of users who accessed media. The first, second, and third Bloom filter arrays including differential privacy noise. The processor circuitry to estimate a cardinality of a union of the first, second, and third Bloom filter arrays based on the final composite Bloom filter array.

Abstract: Methods, apparatus, systems, and articles of manufacture to estimate cardinality of users represented in arbitrarily distributed bloom filter arrays are disclosed. A system includes a communication interface to: access a first Bloom filter array representative of first entries in a first database, the first entries allocated to ones of first elements in the first Bloom filter array based on a non-uniform distribution of outputs of a hash function applied to the first entries, and access a second Bloom filter array representative of second entries in a second database. The system also includes machine readable instructions to cause one or more processors to estimate a cardinality of a union of the first and second entries based on the non-uniform distribution of the outputs of the hash function.

Abstract: A disclosed example includes at least one memory, programmable circuitry, and instructions to cause the programmable circuitry to increment a first value corresponding to a first position in a vector based on a first bit value representation, the first bit value representation corresponding to a first audience member identifier; increment a second value corresponding to a second position in the vector based on a second bit value representation, the second bit value representation corresponding to a second audience member identifier; estimate the second frequency moment of the media impression data using the vector; determine the variance of the second frequency moment; and based on the second frequency moment, schedule a query of the media impression data to be processed by a computer at a future time.

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: A disclosed example includes at least one memory, programmable circuitry, and instructions to cause the programmable circuitry to increment a first value corresponding to a first position in a vector based on a first bit value representation, the first bit value representation corresponding to a first audience member identifier; increment a second value corresponding to a second position in the vector based on a second bit value representation, the second bit value representation corresponding to a second audience member identifier; estimate the second frequency moment of the media impression data using the vector; determine the variance of the second frequency moment; and based on the second frequency moment, schedule a query of the media impression data to be processed by a computer at a future time.

Abstract: Methods and apparatus to estimate audience sizes using deduplication based on vector of counts sketch data are disclosed. An example apparatus to determine an audience size for media based on vector of counts sketch data includes: a coefficient analyzer to determine coefficient values of a polynomial based on variances, a covariance, and cardinalities corresponding to a first vector of counts from a first database and a second vector of counts from a second database; an overlap analyzer to determine a real root of the polynomial, the real root corresponding to an estimate of an overlap between the first vector of counts and the second vector of counts; and a report generator to estimate the audience size based on the estimate of the overlap and the cardinalities of the first vector of counts and the second vector of counts.

Abstract: Methods and apparatus to estimate cardinality across multiple datasets represented using Bloom filter arrays are disclosed. Disclosed examples include processor circuitry to execute and/or instantiate instructions to determine an inclusion-exclusion expression that defines an audience size for a user group of interest. Terms in the inclusion-exclusion expression corresponding to either a first cardinality of a first one of at least three Bloom filter arrays or a second cardinality of a union of two or more of the Bloom filter arrays. Different ones of the Bloom filter arrays representative of different sets of users who accessed media. The at least one processor further to estimate, based on the inclusion-exclusion expression, the audience size of the user group of interest.

Abstract: Methods and apparatus to estimate audience sizes using deduplication based on binomial sketch data are disclosed. An apparatus includes processor circuitry to instantiate coefficient analyzer circuitry to determine coefficient values of a polynomial based on (i) variances in values in the first sketch data and second sketch data, (ii) a first cardinality of the first sketch data, and (iii) a second cardinality of the second sketch data. The processor circuitry to instantiate overlap analyzer circuitry to determine a real root of the polynomial, the real root corresponding to the quantity of the second subscribers that are duplicates of the first subscribers. The processor circuitry to instantiate report generator circuitry to estimate a deduplicated audience size based on the estimate of the quantity of the second subscribers that are duplicates of the first subscribers and the first and second cardinalities.

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 and apparatus to estimate audience sizes using deduplication based on binomial sketch data are disclosed. An apparatus to estimate an audience size for media based on binomial sketch data includes a coefficient analyzer to determine coefficient values of a polynomial based on variances, a covariance, and cardinalities corresponding to first binomial sketch data from a first database and second binomial sketch data from a second database. The apparatus also includes an overlap analyzer to determine a real root of the polynomial. The real root corresponds to an estimate of an overlap between the first binomial sketch data and the second binomial sketch data. The apparatus further includes a report generator to estimate the audience size based on the estimate of the overlap and the cardinalities of the first and second binomial sketch data.

Abstract: Methods and apparatus to estimate audience sizes and durations for media accesses are disclosed. An example apparatus at least one memory, instructions, and processor circuitry to execute the instructions to select constraint equations corresponding to one or more events, the one or more events corresponding to a first audience and a second audience, the first audience to include the second audience, and determine, based on the constraint equations and an auxiliary equation, event audience sizes corresponding to the first audience and to the one or more events.

Abstract: Methods, apparatus, systems, and articles of manufacture are disclosed for audience measurement analysis including at least one memory, instructions, and processor circuitry to execute the instructions to scale at least one of subscriber duration values or total census-level duration values to generate a scaled duration value that corresponds to an impression count of at least a threshold amount of duration time, determine cross-demographic total census parameter values based on the scaled duration value, determine a search space of possible cross-demographic total census parameter values, the search space to include the determined cross-demographic total census parameter values, determine census-level metrics based on the determined cross-demographic total census parameter values, the census-level metrics including demographic census-level audience size data and demographic census duration values, output a report of the census-level metrics.

Abstract: Methods and apparatus to determine census information of events.

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, apparatus, systems, and articles of manufacture to estimate audience measurement metrics based on users represented in Bloom filter arrays are disclosed. An apparatus includes a communications interface to receive a first Bloom filter array from a first computer of a first database proprietor. The first Bloom filter array is representative of first users who accessed media. The first users are registered with the first database proprietor. The first Bloom filter array includes a first array of first elements. Values of respective ones of the first elements are either a 0 or a 1 based on whether quantities of the first users allocated to the respective ones of the first elements are even or odd. The apparatus further includes a Bloom filter array analyzer to estimate a first cardinality for the first Bloom filter array. The first cardinality is indicative of a total number of the first users who accessed the media.

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.