Bayesian data assimilation for estimating instantaneous reproduction numbers during epidemics: Applications to COVID-19.

Estimating the changes of epidemiological parameters, such as instantaneous reproduction number, Rt, is important for understanding the transmission dynamics of infectious diseases. Current estimates of time-varying epidemiological parameters often face problems such as lagging observations, averaging inference, and improper quantification of uncertainties. To address these problems, we propose a Bayesian data assimilation framework for time-varying parameter estimation. Specifically, this framework is applied to estimate the instantaneous reproduction number Rt during emerging epidemics, resulting in the state-of-the-art 'DARt' system. With DARt, time misalignment caused by lagging observations is tackled by incorporating observation delays into the joint inference of infections and Rt; the drawback of averaging is overcome by instantaneously updating upon new observations and developing a model selection mechanism that captures abrupt changes; the uncertainty is quantified and reduced by employing Bayesian smoothing. We validate the performance of DARt and demonstrate its power in describing the transmission dynamics of COVID-19. The proposed approach provides a promising solution for making accurate and timely estimation for transmission dynamics based on reported data.

Kernelized Sparse Bayesian Matrix Factorization.

Extracting low-rank and/or sparse structures using matrix factorization techniques has been extensively studied in the machine learning community. Kernelized matrix factorization (KMF) is a powerful tool to incorporate side information into the low-rank approximation model, which has been applied to solve the problems of data mining, recommender systems, image restoration, and machine vision. However, most existing KMF models rely on specifying the rows and columns of the data matrix through a Gaussian process prior and have to tune manually the rank. There are also computational issues of existing models based on regularization or the Markov chain Monte Carlo. In this article, we develop a hierarchical kernelized sparse Bayesian matrix factorization (KSBMF) model to integrate side information. The KSBMF automatically infers the parameters and latent variables including the reduced rank using the variational Bayesian inference. In addition, the model simultaneously achieves low-rankness through sparse Bayesian learning and columnwise sparsity through an enforced constraint on latent factor matrices. We further connect the KSBMF with the nonlocal image processing framework to develop two algorithms for image denoising and inpainting. Experimental results demonstrate that KSBMF outperforms the state-of-the-art approaches for these image-restoration tasks under various levels of corruption.

Image Denoising Based on Nonlocal Bayesian Singular Value Thresholding and Stein's Unbiased Risk Estimator.

Singular value thresholding (SVT)- or nuclear norm minimization (NNM)-based nonlocal image denoising methods often rely on the precise estimation of the noise variance. However, most existing methods either assume that the noise variance is known or require an extra step to estimate it. Under the iterative regularization framework, the error in the noise variance estimate propagates and accumulates with each iteration, ultimately degrading the overall denoising performance. In addition, the essence of these methods is still least squares estimation, which can cause a very high mean-squared error (MSE) and is inadequate for handling missing data or outliers. In order to address these deficiencies, we present a hybrid denoising model based on variational Bayesian inference and Stein's unbiased risk estimator (SURE), which consists of two complementary steps. In the first step, the variational Bayesian SVT performs a low-rank approximation of the nonlocal image patch matrix to simultaneously remove the noise and estimate the noise variance. In the second step, we modify the conventional SURE full-rank SVT and its divergence formulas for rank-reduced eigen-triplets to remove the residual artifacts. The proposed hybrid BSSVT method achieves better performance in recovering the true image compared with state-of-the-art methods.

AdOn HDP-HMM: An Adaptive Online Model for Segmentation and Classification of Sequential Data.

Recent years have witnessed an increasing need for the automated classification of sequential data, such as activities of daily living, social media interactions, financial series, and others. With the continuous flow of new data, it is critical to classify the observations on-the-fly and without being limited by a predetermined number of classes. In addition, a model should be able to update its parameters in response to a possible evolution in the distributions of the classes. This compelling problem, however, does not seem to have been adequately addressed in the literature, since most studies focus on offline classification over predefined class sets. In this paper, we present a principled solution for this problem based on an adaptive online system leveraging Markov switching models and hierarchical Dirichlet process priors. This adaptive online approach is capable of classifying the sequential data over an unlimited number of classes while meeting the memory and delay constraints typical of streaming contexts. In this paper, we introduce an adaptive "learning rate" that is responsible for balancing the extent to which the model retains its previous parameters or adapts to new observations. Experimental results on stationary and evolving synthetic data and two video data sets, TUM Assistive Kitchen and collated Weizmann, show a remarkable performance in terms of segmentation and classification, particularly for sequences from evolutionary distributions and/or those containing previously unseen classes.

Doubly Nonparametric Sparse Nonnegative Matrix Factorization Based on Dependent Indian Buffet Processes.

Sparse nonnegative matrix factorization (SNMF) aims to factorize a data matrix into two optimized nonnegative sparse factor matrices, which could benefit many tasks, such as document-word co-clustering. However, the traditional SNMF typically assumes the number of latent factors (i.e., dimensionality of the factor matrices) to be fixed. This assumption makes it inflexible in practice. In this paper, we propose a doubly sparse nonparametric NMF framework to mitigate this issue by using dependent Indian buffet processes (dIBP). We apply a correlation function for the generation of two stick weights associated with each column pair of factor matrices while still maintaining their respective marginal distribution specified by IBP. As a consequence, the generation of two factor matrices will be columnwise correlated. Under this framework, two classes of correlation function are proposed: 1) using bivariate Beta distribution and 2) using Copula function. Compared with the single IBP-based NMF, this paper jointly makes two factor matrices nonparametric and sparse, which could be applied to broader scenarios, such as co-clustering. This paper is seen to be much more flexible than Gaussian process-based and hierarchial Beta process-based dIBPs in terms of allowing the two corresponding binary matrix columns to have greater variations in their nonzero entries. Our experiments on synthetic data show the merits of this paper compared with the state-of-the-art models in respect of factorization efficiency, sparsity, and flexibility. Experiments on real-world data sets demonstrate the efficiency of this paper in document-word co-clustering tasks.

Dynamic Infinite Mixed-Membership Stochastic Blockmodel.

Directional and pairwise measurements are often used to model interactions in a social network setting. The mixed-membership stochastic blockmodel (MMSB) was a seminal work in this area, and its ability has been extended. However, models such as MMSB face particular challenges in modeling dynamic networks, for example, with the unknown number of communities. Accordingly, this paper proposes a dynamic infinite mixed-membership stochastic blockmodel, a generalized framework that extends the existing work to potentially infinite communities inside a network in dynamic settings (i.e., networks are observed over time). Additional model parameters are introduced to reflect the degree of persistence among one's memberships at consecutive time stamps. Under this framework, two specific models, namely mixture time variant and mixture time invariant models, are proposed to depict two different time correlation structures. Two effective posterior sampling strategies and their results are presented, respectively, using synthetic and real-world data.

Fitting multiple connected ellipses to an image silhouette hierarchically.

In this paper, we seek to fit a model, specified in terms of connected ellipses, to an image silhouette. Some algorithms that have attempted this problem are sensitive to initial guesses and also may converge to a wrong solution when they attempt to minimize the objective function for the entire ellipse structure in one step. We present an algorithm that overcomes these issues. Our first step is to temporarily ignore the connections, and refine the initial guess using unconstrained Expectation-Maximization (EM) for mixture Gaussian densities. Then the ellipses are reconnected linearly. Lastly, we apply the Levenberg-Marquardt algorithm to fine-tune the ellipse shapes to best align with the contour. The fitting is achieved in a hierarchical manner based upon the joints of the model. Experiments show that our algorithm can robustly fit a complex ellipse structure to a corresponding shape for several applications.