Total Unduplicated Reach and Frequency Optimization at Procter & Gamble

The Procter & Gamble Company (P&G) is a consumer goods corporation that employs over 90,000 people and has operations in roughly 80 countries worldwide. Products in P&G’s 10-category portfolio of products are sold in over 180 countries. The Consumer Research Analytics group at P&G empowers internal clients by using analytics to ensure that the products in P&G’s portfolio of products are not just well received by consumers but become the products of choice for the maximum number of consumers, thereby maximizing P&G’s market share. One of the most frequently used analytical approaches for managing a product line is Total Unduplicated Reach and Frequency analysis. We replaced the previous enumerative approach with integer programming coupled with cuts to the unit hypercube to dramatically speed up the analysis. As a result, P&G achieved higher utilization of its system, improvements to existing products, and more thorough analyses for product line planning and other applications.

On subcopula estimation for discrete models

PurposeTo discuss subcopula estimation for discrete models.Design/methodology/approachThe convergence of estimators is considered under the weak convergence of distribution functions and its equivalent properties known in prior works.FindingsThe domain of the true subcopula associated with discrete random variables is found to be discrete on the interior of the unit hypercube. The construction of an estimator in which their domains have the same form as that of the true subcopula is provided, in case, the marginal distributions are binomial.Originality/valueTo the best of our knowledge, this is the first time such an estimator is defined and proved to be converged to the true subcopula.

Multivariate radial symmetry of copula functions: finite sample comparison in the i.i.d case

Given a d-dimensional random vector X = (X 1, . . ., X d ), if the standard uniform vector U obtained by the component-wise probability integral transform (PIT) of X has the same distribution of its point reflection through the center of the unit hypercube, then X is said to have copula radial symmetry. We generalize to higher dimensions the bivariate test introduced in [11], using three different possibilities for estimating copula derivatives under the null. In a comprehensive simulation study, we assess the finite-sample properties of the resulting tests, comparing them with the finite-sample performance of the multivariate competitors introduced in [17] and [1].

Improved error estimates for an exponentially convergent quadratures

Calculation of the multidimensional cubatures in unit hypercube is a complex problem of numerical methods, and its application value is great. This paper compares various calculation methods: product of regular one-dimensional grid formulae, classical Monte Carlo method using pseudorandom points and the Sobol sequences. It is suggested to use not any Sobol sequences, but only the ones with magic numbers N equal to powers of 2. In addition, the shifted Sobol points are proposed: all coordinates of the magic Sobol points are simultaneously increased by 1/(2N). Comparisons on the test showed that the latter method is significantly more accurate than all the others.

Resistant Sets in the Unit Hypercube

Ideal matrices and clutters are prevalent in combinatorial optimization, ranging from balanced matrices, clutters of T-joins, to clutters of rooted arborescences. Most of the known examples of ideal clutters are combinatorial in nature. In this paper, rendered by the recently developed theory of cuboids, we provide a different class of ideal clutters, one that is geometric in nature. The advantage of this new class of ideal clutters is that it allows for infinitely many ideal minimally nonpacking clutters. We characterize the densest ideal minimally nonpacking clutters of the class. Using the tools developed, we then verify the replication conjecture for the class.

Matrix Completion in the Unit Hypercube via Structured Matrix Factorization

Several complex tasks that arise in organizations can be simplified by mapping them into a matrix completion problem. In this paper, we address a key challenge faced by our company: predicting the efficiency of artists in rendering visual effects (VFX) in film shots. We tackle this challenge by using a two-fold approach: first, we transform this task into a constrained matrix completion problem with entries bounded in the unit interval [0,1]; second, we propose two novel matrix factorization models that leverage our knowledge of the VFX environment. Our first approach, expertise matrix factorization (EMF), is an interpretable method that structures the latent factors as weighted user-item interplay. The second one, survival matrix factorization (SMF), is instead a probabilistic model for the underlying process defining employees' efficiencies. We show the effectiveness of our proposed models by extensive numerical tests on our VFX dataset and two additional datasets with values that are also bounded in the [0,1] interval.

EVALUATION OF PAIRWISE DISTANCES AMONG POINTS FORMING A REGULAR ORTHOGONAL GRID IN A HYPERCUBE

Cartesian grid is a basic arrangement of points that form a regular orthogonal grid (ROG). In some applications, it is needed to evaluate all pairwise distances among ROG points. This paper focuses on ROG discretization of a unit hypercube of arbitrary dimension. A method for the fast enumeration of all pairwise distances and their counts for a high number of points arranged into high-dimensional ROG is presented. The proposed method exploits the regular and collapsible pattern of ROG to reduce the number of evaluated distances. The number of unique distances is identified and frequencies are determined using combinatorial rules. The measured computational speed-up compared to a naïve approach corresponds to the presented theoretical analysis. The proposed method and algorithm may find applications in various fields. The paper shows application focused on the behaviour of various distance measures with the motivation to find the lower bounds on the criteria of point distribution uniformity in Monte Carlo integration.

New perspective on sampling-based motion planning via random geometric graphs

Roadmaps constructed by many sampling-based motion planners coincide, in the absence of obstacles, with standard models of random geometric graphs (RGGs). Those models have been studied for several decades and by now a rich body of literature exists analyzing various properties and types of RGGs. In their seminal work on optimal motion planning, Karaman and Frazzoli conjectured that a sampling-based planner has a certain property if the underlying RGG has this property as well. In this paper, we settle this conjecture and leverage it for the development of a general framework for the analysis of sampling-based planners. Our framework, which we call localization–tessellation, allows for easy transfer of arguments on RGGs from the free unit hypercube to spaces punctured by obstacles, which are geometrically and topologically much more complex. We demonstrate its power by providing alternative and (arguably) simple proofs for probabilistic completeness and asymptotic (near-)optimality of probabilistic roadmaps (PRMs) in Euclidean spaces. Furthermore, we introduce three variants of PRMs, analyze them using our framework, and discuss the implications of the analysis.