Anastasios Panagiotelis

Fan Cheng, Anastasios Panagiotelis, Rob J Hyndman

Analyzing high-dimensional data with manifold learning algorithms often requires searching for the nearest neighbors of all observations. This presents a computational bottleneck in statistical manifold learning when observations of probability distributions rather than vector-valued variables are available or when data size is large. We resolve this problem by proposing a new method for approximation in statistical manifold learning. The novelty of our approximation is the strongly consistent distance estimators based on independent and identically distributed samples from probability distributions. By exploiting the connection between Hellinger/total variation distance for discrete distributions and the L2/L1 norm, we demonstrate that the proposed distance estimators, combined with approximate nearest neighbor searching, could largely improve the computational efficiency with little to no loss in the accuracy of manifold embedding. The result is robust to different manifold learning algorithms and different approximate nearest neighbor algorithms. The proposed method is applied to learning statistical manifolds of electricity usage. This application demonstrates how underlying structures in high dimensional data, including anomalies, can be visualized and identified, in a way that is scalable to large datasets.

Nathaniel Tomasetti, Catherine S. Forbes, Anastasios Panagiotelis

Variational Bayesian (VB) methods produce posterior inference in a time frame considerably smaller than traditional Markov Chain Monte Carlo approaches. Although the VB posterior is an approximation, it has been shown to produce good parameter estimates and predicted values when a rich classes of approximating distributions are considered. In this paper we propose Updating VB (UVB), a recursive algorithm used to update a sequence of VB posterior approximations in an online setting, with the computation of each posterior update requiring only the data observed since the previous update. An extension to the proposed algorithm, named UVB-IS, allows the user to trade accuracy for a substantial increase in computational speed through the use of importance sampling. The two methods and their properties are detailed in two separate simulation studies. Two empirical illustrations of the proposed UVB methods are provided, including one where a Dirichlet Process Mixture model with a novel posterior dependence structure is repeatedly updated in the context of predicting the future behaviour of vehicles on a stretch of the US Highway 101.