Diffusion-aware Censored Gaussian Processes for Demand Modelling

TL;DR

This work tackles the challenge of recovering latent true demand from censored aggregate observations caused by limited supply and substitution across similar products. It introduces Diffusion-aware Censored Gaussian Processes (DCGP), which fuse a Tobit-like likelihood with a diffusion process on a product graph to model how unsatisfied demand transfers among substitutes, all within a Gaussian-process framework for time-aware, multi-product data. The authors develop scalable inference via a state-space GP formulation and CVI to handle the non-Gaussian diffusion likelihood, and they learn diffusion hyperparameters such as the diffusion lengthscale $\ell_{\text{diff}}$ and sink probability $\pi_{\text{diff}}$ alongside GP hyperparameters. Empirical results on artificial data and real-world datasets (supermarket sales, bike-sharing, and EV charging) show that DCGP more accurately estimates the latent true demand and provides improved out-of-sample predictions compared to standard censored GPs and non-censored GPs, validating the usefulness of incorporating substitution dynamics in censored demand modelling.

Abstract

Inferring the true demand for a product or a service from aggregate data is often challenging due to the limited available supply, thus resulting in observations that are censored and correspond to the realized demand, thereby not accounting for the unsatisfied demand. Censored regression models are able to account for the effect of censoring due to the limited supply, but they don't consider the effect of substitutions, which may cause the demand for similar alternative products or services to increase. This paper proposes Diffusion-aware Censored Demand Models, which combine a Tobit likelihood with a graph diffusion process in order to model the latent process of transfer of unsatisfied demand between similar products or services. We instantiate this new class of models under the framework of GPs and, based on both simulated and real-world data for modeling sales, bike-sharing demand, and EV charging demand, demonstrate its ability to better recover the true demand and produce more accurate out-of-sample predictions.

Figures (6)

• Figure 1: Diffusion process demo for 3 time-series using the transition matrix on the top-left (node 1 sends 80% of its unsatisfied demand to node 3 and 20% to the sink node, and so on). The true demand (top-right) that is above the supply (dashed line) gets transferred to the other nodes (incl. sink node) over the 2 diffusion steps (bottom row).
• Figure 2: GP posteriors obtained by the different models for the artificial dataset (Dataset A) depicted in (a). The green-shaded areas indicate time indexes where at least one of the time-series is subject to censoring. As shown in (d), the proposed approach achieves the best approximation to the true demand.
• Figure 3: Datasets A, B, and C, used for the artificial (independent) time-series experiments in Section  \ref{['subsection:artificial_data']}.
• Figure 4: Sampled spatial location (top left), transition matrix used for the diffusion process (top right), and 4 selected time-series sampled (bottom).
• Figure 5: Map of the Frederiksberg commune in Copenhagen (green shaded area), which we partition into 9 sub-areas (clusters) whose centroids are represented by the purple points.