Algorithms for Generalized Cluster-wise Linear Regression

TL;DR

This paper introduces a generalized version of cluster-wise linear regression (CLR) that handles multiple observations per entity, proposing various algorithms including exact, heuristic, metaheuristic, and modified methods, tested on retail SKU clustering data.

Contribution

It extends CLR to a generalized form with multiple observations per entity and develops multiple algorithms for its solution, including an exact method and heuristics.

Findings

Algorithms perform well on real-world retail data.

Metaheuristic and heuristic algorithms are effective for large instances.

The generalized CLR effectively clusters SKUs based on seasonal effects.

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 Sp{\"a}th \cite{Spath1979} 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.