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Learning Stationary Time Series using Gaussian Processes with Nonparametric Kernels
Felipe Tobar · Thang Bui · Richard Turner

Tue Dec 08 02:30 PM -- 03:00 PM (PST) @ Room 210 A

We introduce the Gaussian Process Convolution Model (GPCM), a two-stage nonparametric generative procedure to model stationary signals as the convolution between a continuous-time white-noise process and a continuous-time linear filter drawn from Gaussian process. The GPCM is a continuous-time nonparametric-window moving average process and, conditionally, is itself a Gaussian process with a nonparametric kernel defined in a probabilistic fashion. The generative model can be equivalently considered in the frequency domain, where the power spectral density of the signal is specified using a Gaussian process. One of the main contributions of the paper is to develop a novel variational free-energy approach based on inter-domain inducing variables that efficiently learns the continuous-time linear filter and infers the driving white-noise process. In turn, this scheme provides closed-form probabilistic estimates of the covariance kernel and the noise-free signal both in denoising and prediction scenarios. Additionally, the variational inference procedure provides closed-form expressions for the approximate posterior of the spectral density given the observed data, leading to new Bayesian nonparametric approaches to spectrum estimation. The proposed GPCM is validated using synthetic and real-world signals.

Author Information

Felipe Tobar (Universidad de Chile)

Felipe Tobar is an Assistant Professor at the Data & AI Initiative at Universidad de Chile. He holds Researcher positions at the Center for Mathematical Modeling and the Advanced Center for Electrical Engineering. Felipe received the BSc/MSc degrees in Electrical Engineering (U. de Chile, 2010) and a PhD in Signal Processing (Imperial College London, 2014), and he was an Associate Researcher in Machine Learning at the University of Cambridge (2014-2015). Felipe teaches Statistics and Machine Learning courses at undergraduate, graduate and professional levels. His research interests lie in the interface between Machine Learning and Statistical Signal Processing, including Gaussian processes, spectral estimation, approximate inference, Bayesian nonparametrics, and optimal transport.

Thang Bui (University of Cambridge)
Richard Turner (University of Cambridge)

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