Zheng Zhao

Zheng Zhao

This thesis is mainly concerned with state-space approaches for solving deep (temporal) Gaussian process (DGP) regression problems. More specifically, we represent DGPs as hierarchically composed systems of stochastic differential equations (SDEs), and we consequently solve the DGP regression problem by using state-space filtering and smoothing methods. The resulting state-space DGP (SS-DGP) models generate a rich class of priors compatible with modelling a number of irregular signals/functions. Moreover, due to their Markovian structure, SS-DGPs regression problems can be solved efficiently by using Bayesian filtering and smoothing methods. The second contribution of this thesis is that we solve continuous-discrete Gaussian filtering and smoothing problems by using the Taylor moment expansion (TME) method. This induces a class of filters and smoothers that can be asymptotically exact in predicting the mean and covariance of stochastic differential equations (SDEs) solutions. Moreover, the TME method and TME filters and smoothers are compatible with simulating SS-DGPs and solving their regression problems. Lastly, this thesis features a number of applications of state-space (deep) GPs. These applications mainly include, (i) estimation of unknown drift functions of SDEs from partially observed trajectories and (ii) estimation of spectro-temporal features of signals.

Zheng Zhao, Rui Gao, Simo Särkkä

This paper is concerned with regularized extensions of hierarchical non-stationary temporal Gaussian processes (NSGPs) in which the parameters (e.g., length-scale) are modeled as GPs. In particular, we consider two commonly used NSGP constructions which are based on explicitly constructed non-stationary covariance functions and stochastic differential equations, respectively. We extend these NSGPs by including $L^1$-regularization on the processes in order to induce sparseness. To solve the resulting regularized NSGP (R-NSGP) regression problem we develop a method based on the alternating direction method of multipliers (ADMM) and we also analyze its convergence properties theoretically. We also evaluate the performance of the proposed methods in simulated and real-world datasets.

Zheng Zhao, Muhammad Emzir, Simo Särkkä

This paper is concerned with a state-space approach to deep Gaussian process (DGP) regression. We construct the DGP by hierarchically putting transformed Gaussian process (GP) priors on the length scales and magnitudes of the next level of Gaussian processes in the hierarchy. The idea of the state-space approach is to represent the DGP as a non-linear hierarchical system of linear stochastic differential equations (SDEs), where each SDE corresponds to a conditional GP. The DGP regression problem then becomes a state estimation problem, and we can estimate the state efficiently with sequential methods by using the Markov property of the state-space DGP. The computational complexity scales linearly with respect to the number of measurements. Based on this, we formulate state-space MAP as well as Bayesian filtering and smoothing solutions to the DGP regression problem. We demonstrate the performance of the proposed models and methods on synthetic non-stationary signals and apply the state-space DGP to detection of the gravitational waves from LIGO measurements.

Zheng Zhao, Toni Karvonen, Roland Hostettler et al.

The paper is concerned with non-linear Gaussian filtering and smoothing in continuous-discrete state-space models, where the dynamic model is formulated as an Itô stochastic differential equation (SDE), and the measurements are obtained at discrete time instants. We propose novel Taylor moment expansion (TME) Gaussian filter and smoother which approximate the moments of the SDE with a temporal Taylor expansion. Differently from classical linearisation or Itô--Taylor approaches, the Taylor expansion is formed for the moment functions directly and in time variable, not by using a Taylor expansion on the non-linear functions in the model. We analyse the theoretical properties, including the positive definiteness of the covariance estimate and stability of the TME Gaussian filter and smoother. By numerical experiments, we demonstrate that the proposed TME Gaussian filter and smoother significantly outperform the state-of-the-art methods in terms of estimation accuracy and numerical stability.

Zheng Zhao, Simo Särkkä, Ali Bahrami Rad

In this article, we propose a novel ECG classification framework for atrial fibrillation (AF) detection using spectro-temporal representation (i.e., time varying spectrum) and deep convolutional networks. In the first step we use a Bayesian spectro-temporal representation based on the estimation of time-varying coefficients of Fourier series using Kalman filter and smoother. Next, we derive an alternative model based on a stochastic oscillator differential equation to accelerate the estimation of the spectro-temporal representation in lengthy signals. Finally, after comparative evaluations of different convolutional architectures, we propose an efficient deep convolutional neural network to classify the 2D spectro-temporal ECG data. The ECG spectro-temporal data are classified into four different classes: AF, non-AF normal rhythm (Normal), non-AF abnormal rhythm (Other), and noisy segments (Noisy). The performance of the proposed methods is evaluated and scored with the PhysioNet/Computing in Cardiology (CinC) 2017 dataset. The experimental results show that the proposed method achieves the overall F1 score of 80.2%, which is in line with the state-of-the-art algorithms.