Algorithms for Generalized Clusterwise Linear Regression

Abstract

Cluster-wise linear regression (CLR), a clustering problem intertwined with regression, is to find clusters of entities such that the overall sum of squared errors from regressions performed over these clusters is minimized, where each cluster may have different variances. We generalize the CLR problem by allowing each entity to have more than one observation, and refer to it as generalized CLR. We propose an exact mathematical programming based approach relying on column generation, a column generation based heuristic algorithm that clusters predefined groups of entities, a metaheuristic genetic algorithm with adapted Lloyd's algorithm for K-means clustering, a two-stage approach, and a modified algorithm of Späth [26] for solving generalized CLR. We examine the performance of our algorithms on a stock keeping unit (SKU) clustering problem employed in forecasting halo and cannibalization effects in promotions using real-world retail data from a large supermarket chain. In the SKU clustering problem, the retailer needs to cluster SKUs based on their seasonal effects in response to promotions. The seasonal effects are the results of regressions with predictors being promotion mechanisms and seasonal dummies performed over clusters generated. We compare the performance of all proposed algorithms for the SKU problem with real-world and synthetic data.

Classification and Regression via Integer Optimization for Neighborhood Change

“…As noted previously, the basic version of CRIO that we implemented is not highly scalable, which limited us to regression models over five features and five clusters for 236 census tracts in this analysis; quite simply, this is the computational price to be paid for global optimality that previous approaches lacked. Future work should look at incorporating one or more of the scalability enhancements for the CRIO methodology (Bertsimas and Shioda 2007;Zhu, Li, and Kong 2012;Park et al 2017) that involve a range of techniques from pre-clustering to symmetry breaking to column generation methods for optimization.…”

“…Future work should look at incorporating one or more of the scalability enhancements for the CRIO methodology (Bertsimas and Shioda 2007; Zhu, Li, and Kong 2012; Park et al. 2017) that involve a range of techniques from pre‐clustering to symmetry breaking to column generation methods for optimization.…”

“…Follow-on work by Zhu, Li, and Kong (2012) explored a similar MILP approach and explored symmetry breaking approaches to enhancing scalability. Recent work by Park et al (2017) switched back to a less computationally efficient squared error objective, but also provided column generation techniques to improve the scalability of his mathematical programming framework.…”

“…Recent work by Park et al. (2017) switched back to a less computationally efficient squared error objective, but also provided column generation techniques to improve the scalability of his mathematical programming framework.…”

Algoritmo greedy para predecir el índice de servicio de pavimento basado en agrupación y regresión lineal

“…El segundo en[18] donde proponen el uso de CLR para determinar la relación entre la motivación académica y la intención de abandonar estudios en instituciones de educación superior alemanas. Y el tercero en[19] donde proponen un CLR generalizado donde una entidad puede tener más de una observación e implementan y comparan tres algoritmos en un problema de mantenimiento de existencia de unidades empleado para pronosticar efectos de halo y canibalización en promociones utilizando datos reales de minoristas de una gran cadena de supermercados.En 2018 se seleccionan cuatro trabajos, a saber: En[20] formulan el problema CLR como un problema de optimización no convexo y no suave utilizando la función de error de regresión al cuadrado. En este problema, la función objetivo se representa como una diferencia de funciones convexas y se definen condiciones de optimalidad.…”