Missing Value Estimation for Hierarchical Time Series: A Study of Hierarchical Web Traffic

Hierarchical time series (HTS) is a special class of multivariate time series where many related time series are organized in a hierarchical tree structure and they are consistent across hierarchy levels. HTS modeling is crucial and serves as the basis for business planning and management in many areas such as manufacturing inventory, energy and traffic management. However, due to machine failures, network disturbances or human maloperation, HTS data suffer from missing values across different hierarchical levels. In this paper, we study the missing value estimation problem under hierarchical web traffic settings, where the user-visit traffic are organized in various hierarchical structures, such as geographical structure and website structure. We develop an efficient algorithm, HTSImpute, to accurately estimate the missing value in multivariate noisy web traffic time series with specific hierarchical consistency in HTS settings. Our HTSImpute is able to (1) utilize the temporal dependence information within each individual time series; (2) exploit the intra-relations between time series through hierarchy; (3) guarantee the satisfaction of hierarchical consistency constraints. Results on three synthetic HTS datasets and three real-world hierarchical web traffic datasets demonstrate that our approach is able to provide more accurate and hierarchically consistent estimations than other baselines.