METHODS AND APPARATUS TO DETERMINE WEIGHTS FOR PANELISTS IN LARGE SCALE PROBLEMS
Methods and apparatus to solve for a minimum variance solution for a constrained weight problem in a scalable and memory efficient manner are disclosed. An example instructions cause a machine to at least store a constraint matrix and a population matrix into a memory, the constraint matrix representing demographics associated with a plurality of panelists in a population, the population matrix representing second demographics of the population; determine a linear system based on a set of constraints including the constraint matrix and the population matrix; determine a solution to the linear system by determining an inverse of a first matrix of the linear system; and output a resultant set of weights based on the solution, the resultant set of weights satisfying the set of constraints.
This patent arises from a continuation of U.S. patent application Ser. No. 16/459,261, which was filed on Jul. 1, 2019, which is a continuation of U.S. patent application Ser. No. 14/865,639, which was filed on Sep. 25, 2015. U.S. patent application Ser. No. 16/459,261 and U.S. patent application Ser. No. 14/865,639 are hereby incorporated herein by reference in their entirety. Priority to U.S. patent application Ser. No. 16/459,261 and U.S. patent application Ser. No. 14/865,639 are hereby claimed.
FIELD OF THE DISCLOSUREThis disclosure related generally to audience measurement and, more particularly, to methods and apparatus to determine weights for panelists in large scale problems.
BACKGROUNDContent providers and advertisers hire panelist to yield a representative sample of a desired demographic group of a population of users. Calibrating the representative sample is a technique used to improve estimates and reduce cost of having to construct a larger sample to achieve the same accuracy. Conventional calibration involves finding weights for the representative sample that match the population constraints while attempting to minimize variance (e.g., increasing the precision) of the sample. However, as the number of constraints increases, conventional calibration techniques require more memory than is available on a computer. Therefore, conventional calibration techniques are not scalable for medium-to-large scale problems.
The figures are not to scale. Wherever possible, the same reference numbers will be used throughout the drawing(s) and accompanying written description to refer to the same or like parts
DETAILED DESCRIPTIONDetermining the size and specific demographics of a media viewing audience helps media content providers and distributors schedule media programming and determine a price for advertising during the programming. In addition, accurate estimates of media viewing demographics enable advertisers to target certain types and sizes of audiences. To collect these demographics, an audience measurement company enlists a plurality of media viewers (e.g., panelists) to cooperate in an audience measurement study (e.g., a panel) for a predefined length of time. The viewing habits and demographic data associated with the panelists are collected and used to statistically represent the total media viewing audience (e.g., a population). In some examples, weights are applied to each panelist such that the demographics of the weighted panelists match the demographics of the total population. However, there are many different solutions to match panelist data to population data. An optimal solution may be any solution that optimizes any selected criteria to determine a unique solution. In some examples, an optimal solution may be a unique solution that minimizes the sum of squares, sum of values, etc. In some examples a unique solution may not exist. Examples disclosed herein determine a unique solution that optimizes minimum variance (e.g. maximizing effective sampling size) based on a set of auxiliary constraints (e.g., weights satisfying the population, weights being non-negative, weights satisfying preset boundaries, etc.).
Data determined for a sample population of panelists may have demographic statistics corresponding to demographic statistics of a target population to which the data is to be applied. For example some rating campaigns (e.g., such as audience measurement campaigns, market research campaigns, etc.) divide a sample population into several different demographic categories (e.g., such as categories based on gender, age, income, location, mobile phone usage, etc.) and an individual in the population may belong to multiple of theses demographic categories. A given ratings campaign may then involve determining viewing data (e.g., media exposure viewing data and/or other audience measurement data, market research ratings data, etc.) for individual demographic categories or combination of such demographic categories for a sample population, and applying the viewing data to different (e.g., larger) target population. However, in order to accurately represent the target population weights are determined and applied to each individual in the sample population to satisfy all auxiliary constraints.
Some conventional techniques to determine unique solution (e.g., unique set of weights) with minimum variance involve creating a linear system based on the constraints and solving the linear system using conventional linear algebra calculations. Such convention techniques include a predictor-corrector method. In such examples, a set of weights is arbitrarily selected and two linear systems (e.g., the predictor and the corrector linear system) are created to a) predict an adjustment to the selected set of weights and b) adjust the prediction to abide by constraints. During an iteration the selected weights are predicted by solving the predictor linear system and then corrected by inputting the predictor solutions into the corrector linear system. Once a set of weights applied to the linear systems, the selected weights are adjusted based on a solution from the corrector linear system. In such examples, the process is repeated such that each time the resultant set of weights are applied to the linear systems they are adjust to a set of weights closer to an optimal solution. The set of weights converges to a unique solution (e.g., the optimal solution). However, a number of elements in the linear systems dramatically increases as the number of unknowns (e.g., demographic categories and/or number of panelists) increases; thereby, increasing the amount of memory needed to solve the linear system. In fact, modern computer systems do not have enough available memory to determine a unique solution for some medium-to-large scale problems. Thus, conventional techniques of solving linear equations based on medium sized problems are not scalable or memory efficient.
Unlike such conventional techniques, example methods and apparatus disclosed herein implement technical solutions to address technical problems associated with finding a minimum variance solution for a constrained weight problem in a scalable and memory efficient matter. More specifically, example technical solutions disclosed herein eliminate the need to store the potentially large linear systems associated with the unique solution by solving the linear systems analytically such that the amount of computer memory required to solve the linear system is dramatically reduced. Technical solutions allow for acquiring optimal weights (e.g., weights that satisfy specific conditions and minimize variance) that could previously not be acquired via a computer. In some examples, the size reduction of a matrix using example methods and apparatus disclosed herein is on the order of 4,500 times less in size than the conventional techniques. This magnitude not only allows for larger systems to be calculated via a computer than conventional techniques, but also faster calculation of the optimal weights.
An example medium-scaled problem may include 10,000 sample panelists with 100 constraints (e.g., different demographics) from a total population. In order to determine a unique set of weights for each of the 10,000 panelists that represent the total population such that each weight is non negative and variance is minimized, linear systems must be stored into memory including a matrix with over 404 million entries. A large-scaled problem may include 100,000 sample panelists from the total population with 100 constraints. In such examples, a linear system must be stored into memory including a matrix with over 4×1010 entries, far exceeding the memory available by modern computer systems. Example methods and apparatus disclosed herein solve medium-scaled to large-scaled problems that cannot be solved using conventional techniques. Such methods and apparatus determine a set of weights that represent the total population wherein the number of entries for a matrix that needs to be stored into memory is no larger than the number of inputs. For example, the number of entries needed to be stored into memory for the medium-scaled problem (e.g., 10,000 panelists with 100 constraints) is 1 million (e.g., about 1/404 the number of entries using conventional methods) and the number of entries needed to be stored into memory for the large-scaled problem (e.g., 100,000 panelist with 100 constraints) is 10 million (e.g., about 1/4000 the number of entries using conventional methods).
Variance is a statistical measurement of a data set that is indirectly correlated to effective sample size (e.g., as variance is decreased, the effective sample size is increased). As the effective sample size increases, the precision of the sample increases. Increased precision leads to less expensive and more accurate estimations. Thus, samples with small variance are useful for making accurate estimations. Given some data ‘x’ and some frequency ‘f’, the variance is computed using the following formula:
For some matrix H where:
Then,
Therefore, in order to minimize variance for a set of weights one would need to minimize x for Var(X)=xT Hx, where H, as defined above, is conventionally known as the Hessian matrix. As shown above, variance is expressed in quadratic form. In this manner, the solution that minimizes the quadratic equation is the same solution that minimizes variance. Although the Hessian, H, may be arbitrary, examples disclosed herein specifically define H such that xT Hx equals the variance.
Example methods, apparatus, and articles of manufacture disclosed herein determine a solution to a linear system to find an optimal set of weights based on a set of constraints without storing the linear system into memory. The linear system is solved analytically using various techniques to break the linear system into smaller problems to be solved separately, thus conserving memory. In this manner, each calculation used to solve the linear system involves matrices no larger than the dimensions of a constraint matrix based on the set of constraints. The various techniques involve repeated application of block-inversion formula and Sherman-Morrison formula.
The audience measurement entity enlists a plurality of media consumers (often called panelists) to cooperate in an audience measurement study (often called a panel) for a predefined length of time. The media consumption habits and demographic data associated with these enlisted media consumers is collected and used to statistically determine the size and demographics of the entire audience of the media presentation. In order to determine media exposure data for demographics related to a total population (e.g., universe) of users, a separate weight may be applied (e.g., multiplied) to each panelist such that after the weights are applied, the sample matches the population. For example, if there are 4 panelists (e.g., 2 male and 2 female) for a total population of 100 people (e.g., 40 males and 60 females), there are a variety of weights that can be applied to the panelist to represent the total population. For example:
Although there are many solutions, there is only one unique solution that has all non-negative entries that minimizes variance (e.g., the optimal solution). In order to determine the unique solution, multiple linear equations must be solved recursively until an arbitrary set of weights converges to a desired unique set of weights. The linear equations are based on the population, the panel (e.g., the panelists and the demographics), and the set of constraints (e.g., non-negative, bounded, minimum variance, etc.). However, as the number of panelists and demographics increases, the size of the linear systems required to determine the unique solution becomes larger. For medium-scaled to large-scaled problems the number of entries is too large to store into memory of modern computer systems.
Example methods, apparatus, and articles of manufacture disclosed herein determine a unique set of weights for a panel of panelists based on a set of constraints without storing linear systems into memory. The linear systems (e.g., a predictor linear system and a corrector linear system) are solved analytically using various techniques to break the linear systems into smaller problems to be solved separately, thus conserving memory. In this manner, each calculation used to solve the linear system involves matrices no larger than the dimensions of a constraint matrix based on the set of constraints. The various techniques involve repeated application of block-inversion formula and Sherman-Morrison formula. Once the solution to the predictor linear system is found the solution is applied to the corrector linear system to determine a set of adjustments, the set of adjustments are applied to an arbitrary set of weights and the process is repeated until the arbitrary set of weights converges to a unique solution. The unique solution (e.g., optimal weights) may be applied (e.g., multiplied) to the panel of panelists. Data from the weighted panelists are used to represent the total population.
An example method for determining a plurality of weights for a plurality of panelist representative of a population is described herein. Such disclosed examples include storing a constraint matrix and a population matrix into a memory, the constraint matrix representing first demographics associated with a plurality of panelists in a population, the population matrix representing second demographics of the population. Such disclosed examples include determining, via a processor, a linear system based on a set of constraints including the constraint matrix and the population matrix. Such disclosed examples include determining, via a processor, an inverse of a matrix of the linear system, the determination of the inverse including breaking down the matrix into blocks, the dimensions of the blocks being no larger than dimensions of the constraint matrix. Such disclosed examples include outputting a set of weights based on the inverse of the linear system, the set of weights satisfying the set of constraints.
An example method disclosed herein includes storing a constraint matrix and a population matrix into a memory, the population matrix representing a population related to data of the constraint matrix. Such disclosed examples include determining, via a processor, a linear system based on a set of constraints including the constraint matrix and the population matrix. Such disclosed examples include determining, via a processor, an inverse of a matrix of the linear system, the determination of the inverse including breaking down the matrix into blocks, the dimensions of the blocks being no larger than dimensions of the constraint matrix. Such disclosed examples include outputting a set of weights based on the inverse of the linear system, the set of weights satisfying the set of constraints.
Turning to the figures,
The example total population 100 is a total population of users of a particular device (e.g., an audience). For example, the total population 100 may be a total population of television viewers, computing device users, mobile device users, radio listeners, Internet users, video game users, and/or any population of media users. Data, including demographic data, may be obtained and/or known for the example total population 100. The data from the example total population 100 is represented in the population data 106. The example population data 106 is obtained and stored. In some examples, population data may be obtained from a database proprietor that provides service to large numbers of subscribers. Such service may include, but is not limited to, cable television services, email services, social networking services, news media services, cloud storage services, streaming music services, streaming video services, satellite radio services, cellular services, video gaming services, online retail shopping services, credit monitoring services, etc. In some examples, the database proprietor maintains user account records corresponding to users registered for the media services provided by the database proprietors. The user account records may include demographic information (e.g., gender, age, income, location, education level, occupation, etc.). In some examples, however, media usage data may not be known for the total population 100. Alternatively, media usage data may not be released to the example AME 108. Additionally, although general demographics may be known (e.g., the number of 16-25 year olds, the number of males, the number of users whose income is less than $50,000), complex demographics (e.g., the number of 16-25 year old males making less than $50,000) may not be known or released to the example AME 108.
In order to determine the media usage behavior and/or complex demographics of the example total population 100, an example sample 102 may be used. The example sample 102 is a group (e.g., a panel) of monitored panelist within the total population 100. Data from the panelist may be acquired using local people meters, portable people meters, surveys, and/or any other means for obtaining data from a panelist. The data, including media usage data and detailed demographic data, of the example sample 102 are represented in the example panelist data 104. The example panelist data 104 is obtained, stored, and calibrated to represent the total population 100.
In the illustrated example, the AME 108 does not provide the media to the total population 100 and is a trusted (e.g., neutral) third party (e.g., The Nielsen Company, LLC) for providing accurate media access (e.g., exposure) statistics. The AME 108 establishes a panel of users (e.g., the example sample 102) who have agreed to provide their demographic information and to have their media exposure activities monitored. When an individual joins the sample 102 (e.g., a panel), the individual (e.g., panelist) provides detailed information concerning the person's identity and demographics (e.g., gender, age, ethnicity, income, home location, occupation, etc.) to the AME 108.
The example AME 108 includes the example collector 110 and the example sample calibrator 112. The example collector 110 collects the example panelist data 104 (e.g., media exposure data and demographic data from the sample 102) and the example population data 106 (e.g., demographics data from the total population 100). In some examples, the panelist data 104 and the population data 106 are transmitted to the example AME 108 via a network (e.g., a local area network, a wide area network, the Internet, a cloud, or any other type of communications network). The sample collector 110 sends the collected panelist data 104 and population data 106 to the example sample calibrator 112. The example sample calibrator 112 calibrates the panelist data by determining a set of weights that, when applied to the panelist data in the sample 102, optimally represent the total population 100, as further described in
The example panelist data receiver 200 receives the example panelist data 104 stored in, or received by, the example AME 108 of
Once the example panelist demographics determiner 202 divides the panelist into the demographic buckets for each demographic, the constraint determiner 204 creates a constraint matrix based the panelists and demographic data. The dimensions of the constraint matrix are (c, n) where c is the number of demographic constraints (e.g., Male, Female, low income, medium income, high income, etc.) and n is the panelists in the sample panel. Once the example constraint determiner 204 creates the constraint matrix, the constraint matrix is stored into example memory 212.
The example population data receiver 206 receives the example population data 106 stored in the example AME 108 of
Once the example population demographics determiner 208 determines population totals for the demographic buckets for each demographic, the constrained population determiner 210 creates a constrained population matrix based on the totals. The dimensions of the constrained population matrix are (c, 1) where c is the number of constraints (e.g., based on the number of demographic buckets). Once the example constrained population determiner 210 creates the constrained population matrix, the constrained population matrix is stored into the example memory 212.
The example weight determiner 214 gathers the constraint matrix and the constrained population matrix from the example memory 212 to determine and output a set of optimal weights that, when applied to the panelists of the example sample 102, accurately match the example total population 100 while minimizing variance. Additionally, the weight determiner 214 may receive various other constraints including an equality constraint (e.g., an equation where the product of the constraint matrix and an unknown weight matrix equals the constrained population matrix) a non-negative constraint (e.g., the weights need to be positive), and/or a boundary constraint (e.g., the weights are bounded above and/or below a threshold amount). The example weight determiner 214 determines an initial set of weights to optimize and example linear systems (e.g., a predictor and corrector linear system) based on a minimum variance condition and the set of constraints. For example, the linear system may be based on the following three conditions:
Where A is the constraint matrix, B is the constrained population matrix, x is a matrix of the unknown weights, and H is a Hessian matrix as previously described.
In this manner, a predictor linear system is created to determine how to adjust an arbitrary set of weights to minimize variance given an equality constraint and a non-negative constraint. An example predictor linear system is shown below:
Where Dz, for a column slackvector z (e.g., defined as z=ATx−B, where z>0), is a diagonal matrix with values equal to z, Dx, for a column Lagrange multiplier vector x (e.g., based on the inequality constraint), is a diagonal matrix with values equal to x, rd, rp, and rcz are residual vectors based on the population data, and dx, dy, and dz are vectors to related to an amount of adjustment to apply to an arbitrary set of weights to lead to the optimal solution. Additionally, the solutions (e.g., dx, dy, and dz) may be corrected to satisfy other constraints in the predictor step. In this manner, the solutions from the predictor linear equation are used to solve a second corrector linear system. Solving the corrector linear system adjusts the predictor solutions to satisfy constraints. An example corrector linear system is shown below:
Where d′x, d′y, and d′z are the corrector solutions used to adjust the initial set of weights toward an optimal solution. Additionally,
where m is the number of inequality constraints, and a can be defined by solving the following linear system:
In some examples, the linear system may be based on a bound constraint (e.g., the solutions must be above a lower bound, L, and an upper bound, U). In this manner, the linear system that satisfies the minimum variance, equality constraint, and bound constraint is:
Where DszL is, for a column Lagrange multiplier vector szL (e.g., based on the lower bound), is a diagonal matrix with values equal to szL, DszU is, for a column Lagrange multiplier vector szU (e.g., based on the upper bound), is a diagonal matrix with values equal to szU.
The example weight determiner 214 solves the linear equation by taking the inverse of the matrix on the left (e.g., defined herein as the Jacobian matrix) analytically such that the Jacobian does not need to be stored in memory 212. Alternatively, the method shown below may be used to invert and N×N matrix
As the number of constraints and panelists increases, the Jacobian increases exponentially. For example, for a linear equation containing a bound constraint which has 10,000 panelists with 100 constraints, the Jacobian will have 9×108 entries, far exceeding the memory available on a computer. The example weight determiner 214 finds a solution to the linear system analytically, such that neither the Jacobian nor the Hessian needs to be stored into memory or computing using conventional techniques (e.g., taking the inverse of the Jacobian numerically and multiplying by the residual vectors). In this manner, the example weight determiner 214 is able to solve large scale problems that were previously unsolvable by a computer, as further describe in
The example reporter 216 of
The example identifier 302 is a number, name, and/or code used to identify a panelist. In some examples, the identifier 302 may be a device identifier (e.g., a number and/or code associated with a device, etc.), a web browser unique identifier (e.g., a cookie), a user identifier (e.g., a user name, a login ID, etc.), identification information stored in an HTML5 datastore (where HTML is an abbreviation for hypertext markup language), and/or any other identifier that the AME 108 stores in association with demographic information about panelists in the example sample 102. In this manner, when the AME 108 receives the example identifier 302, the AME 108 may obtain demographic information corresponding to a panelist based on the identifier 302 that the AME 108 receives from a device associated with the panelist. In some examples, the identifier 302 may be encrypted (e.g., hashed) at the device so that only an intended final recipient of the example identifier 302 can decrypt the hashed identifier. For example, if the example identifier 302 is a cookie that is set in the device associated with the panelist by the AME 108, the example identifier 302 can be hashed so that only the AME 108 can decrypt the identifier 302. If the identifier 302 is an international mobile equipment identity (e.g., IMEI) number, the device can hash the example identifier 302 so that only a wireless carrier (e.g., the database proprietor) can decrypt the hashed identifier 302 to recover the IMEI for use in accessing demographic information corresponding to the panelist in the example sample 102. By hashing the example identifier 302, an intermediate party (e.g., an intermediate server or entity on the Internet) receiving the beacon request cannot directly identify a user of the example sample 102.
In the example illustration of
Based on the information from the population data 106 of
The example sample calibrator 112 determines the optimal weights (e.g., the weights that satisfy the constraints and minimize variance) that, when applied to the sample, accurately represent the population, without requiring a processor to store and/or calculate a matrix whose dimensions are larger than the constraint matrix 400 of
While example manners of implementing the example sample calibrator 112 of
Flowcharts representative of example machine readable instructions for implementing the example sample calibrator 112 of
As mentioned above, the example processes of
The example machine readable instructions illustrated in
At block 504, the example population demographics determiner 208 determines the total number users in the example population 100 that belong to each demographic group based on the received population data 106. For example, the population demographics determiner 208 may determine that there are 575,000 males and 492,000 females in a population of television users. The example panelist demographic determiner 202 determines demographics for each panelist based on the received panelist data 104 (block 506). For example, the panelist demographic determiner 202 may determine that a first panelist is male, a second panelist is female, etc. At block 508, the example sample calibrator 112 calibrates the example panelist data 104 to match the population data 106 based on a set of constraints while minimizing variance, as further described in
The example machine readable instructions illustrated in
At block 602, the example constraint determiner 204 creates a constraint matrix based on the received demographic data from the sample of panelists, as previously described in
At block 608, the example weight determiner 214 determines a predictor linear system and a corrector linear system based on the stored constraint matrix and the constrained population matrix stored in the example memory 212. In some examples, the linear systems are further based on other constraints (e.g., a non-negative constraint, a bound constraint, etc.). The other constraints may be based on a preset setting or a preference from a user and/or administrator depending on the purpose of the linear system.
At block 610, the example weight determiner 214 creates an initial set of weights. In some examples, the initial set of weights is a matrix (e.g., (c, 1)) in which every weight is 1. Alternatively, the initial values in the initial set of weights may be any numbers. The initial values are irrelevant since multiple iterations of linear systems (e.g., applying the initial values to a predictor and corrector linear system) will adjust (e.g., converge) the initial set of weights to the optimal values.
At block 612, the example weight determiner 214 solves the linear systems analytically to find a set of adjustments that, when applied to the initial set of weights data, brings the initial set of weights one step closer to an optimal set of weights. The linear systems include a predictor linear system and a corrector linear system as further described in
Once the adjustments have been determined by solving the linear equations, the example weight determiner 214 adjusts the initial set of weights based on the determined adjustments (block 614). As previously described, the adjustments bring the initial set of weights closer to the optimal set of weights. At block 616, the example weight determiner 214 determines if the adjusted set of weights are satisfactory. If the set of weights are not satisfactory, the example weight determiner 214 runs an additional iteration by solving the predictor and corrector linear system using the adjusted set of weights. In this manner, the set of weights can continue to converge (e.g., get closer) to the optimal set of weights. In some examples, the example weight determiner 214 may determine that the set of weights are satisfactory based on a number (e.g., count of iterations). For example, the example weight determiner 214 may run 10,000 iterations before it determines that the set of weights are satisfactory. Alternatively, the example weight determiner 214 may determine that the set of weights is satisfactory based on a threshold tolerance level (e.g., when the corrector solutions are less than 10-16). The determination of what constitutes as satisfactory may be determined based on user and/or manufacture settings. If the example weight determiner 214 determines that the weights are satisfactory, the process continues to block 510 of
In some examples, the example weight determiner 214 applies the weights to the panelists. For example, if the example weight determiner 214 determines that an optimal weight for a panelist is 25.1, the example weight determiner 214 may credit demographics associated with the panelist 25.1 times for a single media exposure event. In some examples, once the weights are determined and/or applied to the example sample 102, the example reporter 216 may output a report regarding the optimal solution. The report may further include the panelist data, the population data, the constraints, the calculations, etc.
The example machine readable instructions illustrated in
Where B1-B9 are analytical blocks used to calculate the predictor adjustments.
At block 700, the weight determiner 214 separates components of the Jacobian matrix into blocks, as shown below:
In this example, −H, AT, A, and 0 become E, I and 0 become F, DZ and 0 become G, and Dx becomes J. As shown above, the block representation of the Jacobian is represented by M1.
At block 702, the example Jacobian (e.g., M1) is inverted using block-wise inversion, as shown below:
Since E is a matrix including the Hessian matrix H, E may be further broken up to reduce the dimensions of E. In this example, each component of M1−1 contains (E−FJ−1G)−1, therefore (E−FJ−1G)−1 may be separated into blocks in a similar fashion to block 700. At block 704, each element of (E−FJ−1G)−1 is separated into blocks as shown below:
Where the separation of (E−FJ−1G)−1 is labeled as M2−1.
At block 706, M2−1 is inverted using block-wise inversion, as shown below:
where W=(J2−G2E2−1F2)−1, F2=AT, G2=A.
In this example E2 is the only component containing the Hessian matrix (e.g., E2=−(H+Dy)). Since H is defined as diag(p)−ppT, where
and n represents frequency counts and N represents the total sum of the frequency counts, then H+Dy=[diag(p)−ppT]+[diag(y)]→diag(p+y)−ppT. Therefore, by using Sherman-Morrison formula, the inverse of H+Dy is calculated without computing the large Hessian H resulting in an expression for E2 whose dimensions (e.g., [n, 1]) are smaller than a threshold size (e.g., [c, n]) (e.g., the size of the constraint matrix) (block 708), as shown below:
Once the reduced expression for E2 is calculated, it is substituted for E2 (block 710). At block 712, M2−1 is replaced for (E−FJ−1G)−1 in M1−1, giving our final expression for the inverse of the Jacobian and the analytical blocks, as shown below:
By calculating the each element of M1−1, the following values for B1-B9 are acquired:
B1=L
B2=−VATW
B3=−LDx−1
B4=−WAV
B5=W
B6=WAVDx−1
B7=−Dx−1DzL
B8=Dx−1Dz(VATW)
B9=Dx−1+Dx−1(DzL)Dx−1
Where L=V+VATW AV,
W=Q−1, and Q=−AV AT
In this manner, the dimensions of V are (n, n) (e.g., because the number of entries in g depend on the number of entries in n), the dimensions for Q are (c, c) (e.g., −AcxnVnxnAnxcT=Qcxc), the dimensions for W are (c, c), the dimensions for L are (n, n) (e.g., Vnxn+VxnxAnxcTWcxcAcxnVnxn=Lnxn), and the dimensions of Dx and Dz are (n, n). Since V is always multiplied by some vector, i, creating a new vector, a subroutine may be applied to determine the product of V and the vector without storing V into the example memory 212, as shown below:
Where the first term may be expressed as element-product of two column vector, the product of (wTd) is a dot product of two vectors (e.g., creating a scalar), and the denominator is a scalar. Therefore, none of the calculations involve matrices whose dimensions are larger than (c, n). Additionally, the only inverse that needs to be calculated based on these results is (Q−1)d, which is small (e.g., (c, c)) and can be solved using standard optimizing software without taking the inverse.
At block 714, the weight determiner 214 multiplies the analytical blocks (e.g., B1-B9) to the residual vectors (e.g., rd, rp, rc) representative of the example population 100 resulting in the predictor adjustments, as shown below:
As previously described, determining the corrector adjustments (e.g., to adjust the predictor adjustments based on constraints) involves finding the inverse of a Jacobian matrix associated with a linear system based on the predictor adjustments, as shown below (block 716):
Where B1-B9 are analytical blocks used to calculate the corrector adjustments.
At block 718, the weight determiner 214 separates components of the Jacobian matrix into blocks, as shown below:
In this example, −H, AT, A, and 0 become E, I and 0 become F, DZ and 0 become G, and Dx becomes J. As shown above, the block representation of the Jacobian is represented by M1.
At block 720, the example Jacobian (e.g., M1) is inverted using block-wise inversion, as shown below:
Since E is a matrix including the Hessian matrix H, E may be further broken up to reduce the dimensions of E. In this example, each component of M1−1 contains (E−FJ−1G)−1, therefore (E−FJ−1G)−1 may be separated into blocks in a similar fashion to block 718. At block 722, each element of (E−FJ−1G)−1 is separated into blocks as shown below:
Where the separation of (E−FJ−1G)−1 is labeled as M2−1.
At block 724, M2−1 is inverted using block-wise inversion, as shown below:
where W=(J2−G2E2−1F2)−1, F2=AT, G2=A.
In this example E2 is the only component containing the Hessian matrix (e.g., E2=−(H+Dy)). Since H is defined as diag(p)−ppT where
and n represents frequency counts and N represents the total sum of the frequency counts, then H+Dy=[diag(p)−ppT]+[diag(y)]→diag(p+y)−ppT. Therefore, by using Sherman-Morrison formula, the inverse of H+Dy is calculated without computing the large Hessian H resulting in an expression for E2 whose dimensions (e.g., [n, 1]) are smaller than a threshold size (e.g., [c, n]) (e.g., the size of the constraint matrix) (block 726), as shown below:
Once the reduced expression for E2 is calculated, it is substituted for E2 (block 728). At block 730, M2−1 is replaced for (E−FJ−1G)−1 in M1−1, giving our final expression for the inverse of the Jacobian and the analytical, as shown below:
By calculating the each element of M1−1, the following values for B1-B9 are acquired:
B1=L
B2=−VATW
B3=−LDx−1
B4=−WAV
B5=W
B6=WAVDx−1
B7=−Dx−1DzL
B8=Dx−1Dz(VATW)
B9=Dx−1+Dx−1(DzL)Dx−1
Where L=V+V ATW AV,
W=Q−1, and Q=−AV AT
In this manner, the dimensions of V are (n, n) (e.g., because the number of entries in g depend on the number of entries in n), the dimensions for Q are (c, c) (e.g., −AcxnVnxnAnxcT=Qcxc), the dimensions for W are (c, c), the dimensions for L are (n, n) (e.g., Vnxn+VxnxAnxcTWcxcAcxnVnxn=Lnxn), and the dimensions of Dx and Dz are (n, n). Since V is always multiplied by some vector, i, creating a new vector, a subroutine may be applied to determine the product of V and the vector without storing V into the example memory 212, as shown below:
Where the first term may be expressed as element-product of two column vector, the product of (wTd) is a dot product of two vectors (e.g., creating a scalar), and the denominator is a scalar. Therefore, none of the calculations involve matrices whose dimensions are larger than (c, n). Additionally, the only inverse that needs to be calculated based on these results is (Q−1)d, which is small (e.g., (c, c)) and can be solved using standard optimizing software without taking the inverse.
At block 732, the weight determiner 214 multiplies the analytical blocks (e.g., B1-B9) to the residual vectors (e.g., rd, rp, rc−Dzdx−Dxdz) representative of the example population 100 resulting in the predictor adjustments, as shown below:
The solution to the corrector linear equation is a set of adjustments that are applied to the arbitrary set of weights, as previously described in
While the example manners describe a particular technique to find set of optimal weights based on the equality constraint and the non-negative constraint, the technique may be used for linear systems accommodating any set of constraints. For example, if the predictor linear system includes an upper and lower bound constraint, the linear system will be slightly different, but the technique will be the same. In the upper and lower bound example, the analytical blocks may be determined based on the following predictor linear system:
In this manner, the example Jacobian will be broken up into the following blocks:
At this point, the steps of
B11=L
B12=−VATW
B13=−LDsz1−1
B14=LDsz2−1
B21=−WAV
B22=W
B23=WAVDsz1−
B24=−WAVDsz2−1
B31=−Dsz1−1L
B32=Dsz1−1(VATW)
B33−Dsz1−1+Dsz1−1LDsz1−1
B34=−Dsz1−1LDsz2−1
B41=Dsz2−1L
B42=−Dsz2−1(VATW)
B43=−Dsz2−1LDsz1−1
B44=Dsz2−1+Dsz2−1LDsz2−1
W=Q−1, Q=−AV AT Dsz1=diag(sz1), Dsz2=diag(sz2),
and wi=pigi. In some examples, a corrector linear solution may be constructed for the bounded problem in a similar manner.
The processor platform 800 of the illustrated example includes a processor 812. The processor 812 of the illustrated example is hardware. For example, the processor 812 can be implemented by integrated circuits, logic circuits, microprocessors or controllers from any desired family or manufacturer.
The processor 812 of the illustrated example includes the example memory 212 (e.g., a cache). The example processor 812 of
The processor platform 800 of the illustrated example also includes an interface circuit 820. The interface circuit 820 may be implemented by any type of interface standard, such as an Ethernet interface, a universal serial bus (USB), and/or a PCI express interface.
In the illustrated example, one or more input devices 822 are connected to the interface circuit 820. The input device(s) 822 permit(s) a user to enter data and commands into the processor 812. The input device(s) can be implemented by, for example, a sensor, a microphone, a camera (still or video), a keyboard, a button, a mouse, a touchscreen, a track-pad, a trackball, isopoint and/or a voice recognition system.
One or more output devices 824 are also connected to the interface circuit 820 of the illustrated example. The output devices 824 can be implemented, for example, by display devices (e.g., a light emitting diode (LED), an organic light emitting diode (OLED), a liquid crystal display, a cathode ray tube display (CRT), a touchscreen, a tactile output device, and/or speakers). The interface circuit 820 of the illustrated example, thus, typically includes a graphics driver card, a graphics driver chip or a graphics driver processor.
The interface circuit 820 of the illustrated example also includes a communication device such as a transmitter, a receiver, a transceiver, a modem and/or network interface card to facilitate exchange of data with external machines (e.g., computing devices of any kind) via a network 826 (e.g., an Ethernet connection, a digital subscriber line (DSL), a telephone line, coaxial cable, a cellular telephone system, etc.).
The processor platform 800 of the illustrated example also includes one or more mass storage devices 828 for storing software and/or data. Examples of such mass storage devices 828 include floppy disk drives, hard drive disks, compact disk drives, Blu-ray disk drives, RAID systems, and digital versatile disk (DVD) drives.
The coded instructions 832 of
From the foregoing, it would be appreciated that the above disclosed methods, apparatus, and articles of manufacture determine a minimum variance solution for a constrained weight problem in a scalable and memory efficient manner. Using the examples disclosed herein, linear systems are used to adjust a set of weights to an optimal set of weights (e.g., whose variance is minimized) based on a set of constraints. In some examples, the optimal set of weights is a set of weights that is unique, satisfies all constraints, and minimizes variance. In some examples, the linear systems are solved analytically so that a matrix whose dimensions are larger than the constraint matrix does not need to be calculated or stored into memory. In some examples, blockwise inversion and Sherman-Morrison formula are used to determine the optimal solution. In some example, the optimal set of weights is applied to a sample to represent a population.
Conventional techniques to solve a linear system are not scalable or memory efficient. In fact, there is not enough available memory to solve medium-to-large scale problems using conventional techniques. The methods, apparatus, and articles of manufacture disclosed herein eliminate the need to store a potentially large linear equation associated with the optimal weights by solving the predictor and corrector linear systems analytically to adjust a set of weights to a desired set of weights by minimizing variance such that the amount of memory required to solve the linear system is dramatically reduced. The methods, apparatus, and articles of manufacture provide a technical solution acquiring such weights that could previously not be acquired.
The methods, apparatus, and articles of manufacture disclosed herein are able to solve a technical problem associated with solving large-scaled panelist-to-total population problems on a computer. Since a sample panel usually include more than 10,000 panelists, determining a set of weights based on a large number of panelists is desirable. Conventional techniques, however, cannot be executed by modern computers due to lack of memory. The methods, apparatus, and articles of manufacture disclosed herein are able to determine weights for a large number of panelists without storing large matrices associated with conventional techniques. In this manner, weights for large panels may be determined to represent a population of users.
Although certain example methods, apparatus and articles of manufacture have been described herein, the scope of coverage of this patent is not limited thereto. On the contrary, this patent covers all methods, apparatus and articles of manufacture fairly falling within the scope of the claims of this patent.
Claims
1. A non-transitory computer readable storage medium comprising instructions which, when executed, cause a machine to at least:
- store a constraint matrix and a population matrix into a memory, the constraint matrix representing demographics associated with a plurality of panelists in a population, the population matrix representing second demographics of the population;
- determine a linear system based on a set of constraints including the constraint matrix and the population matrix;
- determine a solution to the linear system by determining an inverse of a first matrix of the linear system, the determination of the inverse of the first matrix including: separating components of the first matrix into a first set of blocks; inverting the first matrix using blockwise inversion to generate a second matrix; separating a first element of the second matrix into a second set of blocks; inverting the second matrix using blockwise inversion to generate a third matrix, a second element of the third matrix having dimensions larger than the constraint matrix; reducing the dimensions of the second element to a size less than or equal to the dimensions of the constraint matrix, the reduced second element being a third element; replacing the second element with the third element in the second matrix; and replacing the first element with the second matrix; and
- output a resultant set of weights based on the solution, the resultant set of weights satisfying the set of constraints.
2. The computer readable storage medium of claim 1, wherein the linear system is a predictor linear system, the instructions to cause the machine to:
- determine a first solution to the predictor linear system;
- determine a corrector linear system based on the first solution;
- solve the corrector linear system based on the set of constraints to determine an adjusted solution; and
- adjust an initial set of weights based on an inverse of a fourth matrix of the corrector linear system, wherein the resultant set of weights is further based on the adjusted initial set of weights.
3. The computer readable storage medium of claim 2, wherein the instructions cause the machine to calculate a second solution to the corrector linear system multiple times until the adjusted initial set of weight converges to the resultant set of weights, the multiple calculations based on prior solutions to the corrector linear system.
4. The computer readable storage medium of claim 1, wherein the instructions cause the machine to apply the resultant set of weights to data associated with the panelists.
5. The computer readable storage medium of claim 4, wherein the instructions cause the machine to apply the resultant set of weights by:
- receiving a media exposure event from a panelist from the plurality of panelists with a weight from the set of weights; and
- crediting demographics of the panelist for the media exposure event based on the weight.
6. The computer readable storage medium of claim 1, wherein the resultant set of weights is an optimal solution to an equality constraint, the equality constraint being an equation including the constraint matrix, the population matrix, and the set of weights.
7. The computer readable storage medium of claim 6, wherein:
- the set of constraints further includes at least one of a non-negative constraint, a bound constraint, and the equality constraint; and
- the optimal solution is a non-negative solution with minimum variance.
8. The computer readable storage medium of claim 1, wherein the instructions cause the machine to reduce the dimensions of the second element using Sherman-Morrison formula.
9. The computer readable storage medium of claim 8, wherein the instructions cause the machine to conserve the memory by eliminating a need to store the linear system.
10. An apparatus for determining a plurality of weights for a plurality of panelists representative of a population, the apparatus comprising:
- memory to store a constraint matrix and a population matrix into a memory, the constraint matrix representing demographics associated with a plurality of panelists in a population, the population matrix representing second demographics of the population;
- a weight determiner to: determine a linear system based on a set of constraints including the constraint matrix and the population matrix; determine a solution to the linear system by determining an inverse of a first matrix of the linear system, the weight determiner to determine the inverse of the first matrix by: separating components of the first matrix into a first set of blocks; inverting the first matrix using blockwise inversion to generate a second matrix; separating a first element of the second matrix into a second set of blocks; inverting the second matrix using blockwise inversion to generate a third matrix, a second element of the third matrix having dimensions larger than the constraint matrix; reducing the dimensions of the second element to a size less than or equal to the dimensions of the constraint matrix, the reduced second element being a third element; replacing, in the third matrix, the second element with the third element in the second matrix; and replacing the first element of the second matrix with the second matrix; and output a resultant set of weights based on the solution, the resultant set of weights satisfying the set of constraints.
11. The apparatus of claim 10, wherein the linear system is a predictor linear system, the weight determiner to:
- determine a first solution to the predictor linear system;
- determine a corrector linear system based on the first solution;
- solve the corrector linear system based on the set of constraints to determine an adjusted solution; and
- adjust an initial set of weights based on an inverse of a fourth matrix of the corrector linear system, wherein the resultant set of weights is further based on the adjusted initial set of weights.
12. The apparatus of claim 11, wherein a second solution to the corrector linear system is calculated multiple times until the adjusted initial set of weight converges to the resultant set of weights, the multiple calculations based on prior solutions to the corrector linear system.
13. The apparatus of claim 10, wherein the weight determiner is to apply the resultant set of weights to data associated with the panelists.
14. The apparatus of claim 13, wherein the weight determiner is to apply the resultant set of weights by:
- receiving a media exposure event from a panelist from the plurality of panelists with a weight from the set of weights; and
- crediting demographics of the panelist for the media exposure event based on the weight.
15. The apparatus of claim 10, wherein the resultant set of weights is an optimal solution to an equality constraint, the equality constraint being an equation including the constraint matrix, the population matrix, and the set of weights.
16. The apparatus of claim 15, wherein:
- the set of constraints further includes at least one of a non-negative constraint, a bound constraint, and the equality constraint; and
- the optimal solution is a non-negative solution with minimum variance.
17. The apparatus of claim 10, wherein the weight determiner is to reduce the dimensions of the second element using Sherman-Morrison formula.
18. A method for determining a plurality of weights for a plurality of panelists representative of a population, the method comprising:
- storing a constraint matrix and a population matrix into a memory, the constraint matrix representing demographics associated with a plurality of panelists in a population, the population matrix representing second demographics of the population;
- determining, by executing an instruction with a processor, a linear system based on a set of constraints including the constraint matrix and the population matrix;
- determining, by executing an instruction with the processor, a solution to the linear system by determining an inverse of a first matrix of the linear system, the determination of the inverse of the first matrix including: separating components of the first matrix into a first set of blocks; inverting the first matrix using blockwise inversion to generate a second matrix; separating a first element of the second matrix into a second set of blocks; inverting the second matrix using blockwise inversion to generate a third matrix, a second element of the third matrix having dimensions larger than the constraint matrix; reducing the dimensions of the second element to a size less than or equal to the dimensions of the constraint matrix, the reduced second element being a third element; replacing the second element with the third element in the second matrix; and replacing the first element with the second matrix; and
- outputting a resultant set of weights based on the solution, the resultant set of weights satisfying the set of constraints.
19. The method of claim 18, wherein the linear system is a predictor linear system, further including:
- determining a first solution to the predictor linear system;
- determining a corrector linear system based on the first solution;
- solving the corrector linear system based on the set of constraints to determine an adjusted solution; and
- adjusting an initial set of weights based on an inverse of a fourth matrix of the corrector linear system, wherein the resultant set of weights is further based on the adjusted initial set of weights.
20. The method of claim 19, further including calculating a second solution to the corrector linear system multiple times until the adjusted initial set of weight converges to the resultant set of weights, the multiple calculations based on prior solutions to the corrector linear system.
Type: Application
Filed: Nov 2, 2020
Publication Date: Feb 18, 2021
Inventors: Michael Sheppard (Holland, MI), Jonathan Sullivan (Hurricane, UT), Peter Lipa (Tuscon, AZ), Alejandro Terrazas (Santa Cruz, CA), Matt Reid (Alameda, CA), Paul Donato (New York, NY)
Application Number: 17/087,374