Forecasting intermittent time series with Gaussian Processes and Tweedie likelihood

TL;DR

This paper addresses probabilistic forecasting of intermittent time series by treating a Gaussian Process as a latent function that governs the forecast distribution. It introduces two likelihoods: a negative binomial (NegBinGP) and a fully parameterized Tweedie (TweedieGP), the latter being novel for intermittence and capable of capturing zero mass and heavy tails. The authors show TweedieGP often yields the most accurate high-quantile forecasts and provide a scalable variational training framework with a careful truncation of the Tweedie density’s infinite sum. They demonstrate robustness across thousands of series and release an open implementation to enable further development of probabilistic methods for intermittent demand. The work highlights the importance of correctly modeling the zero-inflated, heavy-tailed nature of intermittent data and offers practical tools for inventory planning under uncertainty.

Abstract

We adopt Gaussian Processes (GPs) as latent functions for probabilistic forecasting of intermittent time series. The model is trained in a Bayesian framework that accounts for the uncertainty about the latent function. We couple the latent GP variable with two types of forecast distributions: the negative binomial (NegBinGP) and the Tweedie distribution (TweedieGP). While the negative binomial has already been used in forecasting intermittent time series, this is the first time in which a fully parameterized Tweedie density is used for intermittent time series. We properly evaluate the Tweedie density, which has both a point mass at zero and heavy tails, avoiding simplifying assumptions made in existing models. We test our models on thousands of intermittent count time series. Results show that our models provide consistently better probabilistic forecasts than the competitors. In particular, TweedieGP obtains the best estimates of the highest quantiles, thus showing that it is more flexible than NegBinGP.

Figures (8)

• Figure 1: Sample trajectories (left panel) and credible intervals (right panel) of the GP prior. The single trajectories fluctuate around the mean, while the credible intervals are flat and symmetric. Samples from $f$ can be negative.
• Figure 2: Prior (left) and posterior (right) distribution of the softplus of the GP latent function computed with a Tweedie likelihood. The shaded areas represent the 90% and 95% credible intervals. The vertical line divides train and test data. The data consist of the 1000-th time series from the Auto data set. Fig. \ref{['fig:predicitve_forecast']} (below) shows the prediction intervals for the test data.
• Figure 3: Example of TweedieGP forecasting six steps ahead. Left: posterior distribution of the GP latent function passed through the softplus. Right: forecast distribution using the Tweedie likelihood. The shaded regions represents $90\%$ and $95\%$ prediction intervals respectively. The figures complement the model fit in Fig. \ref{['fig:prior_posterior_latent']}.
• Figure 4: The Tweedie distribution is flexible and possibly bimodal. The two distributions have the same mean ($\mu=1$), but the right one has higher dispersion ($\phi$), implying a larger mass in 0 and in a longer right tail.
• Figure 5: Proportion of zeros and median demand size in the different data sets; the y-axis of the median demand size is in $\log_{10}$ scale.