Abstract: Methods, apparatus, systems, and articles of manufacture to estimate cardinality through ordered statistics are disclosed. In an example, an apparatus includes processor circuitry to selects a sample dataset from a first reference dataset of media assets and partitions the sample dataset into m mutually exclusive subsets of approximately equal size. The processor circuitry then estimates a ratio of a sample weighted average and empirical cumulative distribution of an approximately largest order statistic from at least one of the m subsets and generates an estimate of a total cardinality of the first reference dataset by multiplying the ratio by approximately m.

Abstract: Methods, apparatus, systems, and articles of manufacture to virtually estimate cardinality with global registers are disclosed. An example apparatus includes processor circuitry to s assign subsets of a sample dataset to a shared global register array, the shared global register array having a first number of registers, the sample dataset selected from a reference dataset of media assets; identify a virtual register array from the shared global register array that includes data elements associated with a label value, the virtual register array including a second number of registers less than the first number of registers; determine a maximum rank value of the label value across the virtual register array; and calculate a cardinality estimate of the label value across the virtual register array based on the second number of registers and the maximum rank value.

Abstract: Methods, apparatus, systems, and articles of manufacture are disclosed to determine unique audience size via clustered data. An example apparatus includes memory, machine readable instructions, and processor circuitry to at least one of execute or instantiate the machine readable instructions. The example processor circuitry is to form a first matrix identifying matching and non-matching instances of first values of respective demographics of an audience of media and second values of the respective demographics. Also, the processor circuitry is to normalize the first matrix to generate an account sharing adjustment matrix. Additionally, the processor circuitry is to apply the account sharing adjustment matrix to a vector of one or more counts of impressions of the media to correct for one or more members of the audience that share a user account registered with a database proprietor, the one or more counts of the impressions corresponding to the respective demographics.

Abstract: Methods, apparatus, systems, and articles of manufacture to estimate cardinality through ordered statistics are disclosed. In an example, an apparatus includes processor circuitry to selects a sample dataset from a first reference dataset of media assets and partitions the sample dataset into m mutually exclusive subsets of approximately equal size. The processor circuitry then estimates a ratio of a sample weighted average and empirical cumulative distribution of an approximately largest order statistic from at least one of the m subsets and generates an estimate of a total cardinality of the first reference dataset by multiplying the ratio by approximately m.

Abstract: Disclosed examples access data from a database, the data stored across multiple registers of the database; determine (a) a maximum rank for each of the multiple registers and (b) a maximum rank count for each of the multiple registers; determine a frequency distribution based on the maximum ranks and the maximum rank counts; and generate a report including at least one of the frequency distribution, the maximum ranks, or the maximum rank counts.

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, apparatus, systems, and articles of manufacture are disclosed to determine unique audience size via clustered data. An example apparatus includes memory, machine readable instructions, and processor circuitry to at least one of execute or instantiate the machine readable instructions. The example processor circuitry is to form a first matrix identifying matching and non-matching instances of first values of respective demographics of an audience of media and second values of the respective demographics. Also, the processor circuitry is to normalize the first matrix to generate an account sharing adjustment matrix. Additionally, the processor circuitry is to apply the account sharing adjustment matrix to a vector of one or more counts of impressions of the media to correct for one or more members of the audience that share a user account registered with a database proprietor, the one or more counts of the impressions corresponding to the respective demographics.

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 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: 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: 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: 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 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, apparatus, systems, and articles of manufacture to virtually estimate cardinality with global registers are disclosed. An example apparatus includes processor circuitry to s assign subsets of a sample dataset to a shared global register array, the shared global register array having a first number of registers, the sample dataset selected from a reference dataset of media assets; identify a virtual register array from the shared global register array that includes data elements associated with a label value, the virtual register array including a second number of registers less than the first number of registers; determine a maximum rank value of the label value across the virtual register array; and calculate a cardinality estimate of the label value across the virtual register array based on the second number of registers and the maximum rank value.

Abstract: Disclosed examples determine a unique audience exposed to media while reducing memory resources of a computing device. Example instructions program programmable circuitry to log a plurality of impressions corresponding to media accessed at client devices; generate a first value based on a count of the impressions and a count of users of a database proprietor that accessed the media; generate a second value based on the first value and a count of demographic impressions logged by the database proprietor; and determine a unique audience size based on the second value.

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: Methods and apparatus to estimate census-level audience sizes and duration data across dimensions and/or demographics are disclosed. An example apparatus includes constraint equation control circuitry to select at least one constraint equation corresponding to one or more events, the one or more events corresponding to first audience members and second audience members, the first audience members to include the second audience members, constraint parameter estimation circuitry to solve the at least one constraint equation to determine one or more constraint parameter values, the one or more constraint parameter values corresponding to the one or more events, and output generation circuitry to determine, based on one or more intermediate equations and the one or more constraint parameter values, an event audience size and an event duration corresponding to (a) the first audience members and (b) a combination of the one or more events and one or more demographics.

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, 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: 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.