In many areas of science one aims to estimate latent sub-population mean curves based only on observations of aggregated population curves. By aggregated curves we mean linear combination of functional data that cannot be observed individually. We assume that several aggregated curves with linear independent coefficients are available. More specifically, we assume each aggregated curve is an independent partial realization of a Gaussian process with mean modeled through a weighted linear combination of the disaggregated curves. We model the mean of the Gaussian processes as a smooth function approximated by a function belonging to a finite dimensional space ${\cal H}_K$ which is spanned by $K$ B-splines basis functions. We explore two different specifications of the covariance function of the Gaussian process: one that assumes a constant variance across the domain of the process, and a more general variance structure which is itself modelled as a smooth function, providing a nonstationary covariance function. Inference procedure is performed following the Bayesian paradigm allowing experts' opinion to be considered when estimating the disaggregated curves. Moreover, it naturally provides the uncertainty associated with the parameters estimates and fitted values. Our model is suitable for a wide range of applications. We concentrate on two different real examples: calibration problem for NIR spectroscopy data and an analysis of distribution of energy among different type of consumers.
In this paper we consider aggregated functional data composed by a linear combination of component curves and the problem of estimating these component curves. We propose the application of a bayesian wavelet shrinkage rule based on a mixture of a point mass function at zero and the logistic distribution as prior to wavelet coefficients to estimate mean curves of components. This procedure has the advantage of estimating component functions with important local characteristics such as discontinuities, spikes and oscillations for example, due the features of wavelet basis expansion of functions…
The present work describes simulation studies to compare the performances of bayesian wavelet shrinkage methods in estimating component curves from aggregated functional data. To do so, five methods were considered: the bayesian shrinkage rule under logistic prior by Sousa (2020), bayesian shrinkage rule under beta prior by Sousa et al. (2020), Large Posterior Mode method by Cutillo et al. (2008), Amplitude-scale invariant Bayes Estimator by Figueiredo and Nowak (2001) and Bayesian Adaptive Multiresolution Smoother by Vidakovic and Ruggeri (2001). Further, the so called Donoho-Johnstone test f…
Functional data, with basic observational units being functions (e.g., curves, surfaces) varying over a continuum, are frequently encountered in various applications. While many statistical tools have been developed for functional data analysis, the issue of smoothing all functional observations simultaneously is less studied. Existing methods often focus on smoothing each individual function separately, at the risk of removing important systematic patterns common across functions. We propose a nonparametric Bayesian approach to smooth all functional observations simultaneously and nonparametr…
Functional data are defined as realizations of random functions (mostly smooth functions) varying over a continuum, which are usually collected with measurement errors on discretized grids. In order to accurately smooth noisy functional observations and deal with the issue of high-dimensional observation grids, we propose a novel Bayesian method based on the Bayesian hierarchical model with a Gaussian-Wishart process prior and basis function representations. We first derive an induced model for the basis-function coefficients of the functional data, and then use this model to conduct posterior…
Disease mapping is an important statistical tool used by epidemiologists to assess geographic variation in disease rates and identify lurking environmental risk factors from spatial patterns. Such maps rely upon spatial models for regionally aggregated data, where neighboring regions tend to exhibit similar outcomes than those farther apart. We contribute to the literature on multivariate disease mapping, which deals with measurements on multiple (two or more) diseases in each region. We aim to disentangle associations among the multiple diseases from spatial autocorrelation in each disease. W…
Large scale Gaussian process (GP) regression is infeasible for larger data sets due to cubic scaling of flops and quadratic storage involved in working with covariance matrices. Remedies in recent literature focus on divide-and-conquer, e.g., partitioning into sub-problems and inducing functional (and thus computational) independence. Such approximations can be speedy, accurate, and sometimes even more flexible than an ordinary GPs. However, a big downside is loss of continuity at partition boundaries. Modern methods like local approximate GPs (LAGPs) imply effectively infinite partitioning an…
Beijing National Research Center for Information Science and Technology（北京国家信息科学与技术研究中心）；Tsinghua University（清华大学）
Using powerful posterior distributions is a popular approach to achieving better variational inference. However, recent works showed that the aggregated posterior may fail to match unit Gaussian prior, thus learning the prior becomes an alternative way to improve the lower-bound. In this paper, for the first time in the literature, we prove the necessity and effectiveness of learning the prior when aggregated posterior does not match unit Gaussian prior, analyze why this situation may happen, and propose a hypothesis that learning the prior may improve reconstruction loss, all of which are sup…
The Bayesian methods for linear inverse problems is studied using hierarchical Gaussian models. The problems are considered with different discretizations, and we analyze the phenomena which appear when the discretization becomes finer. A hierarchical solution method for signal restoration problems is introduced and studied with arbitrarily fine discretization. We show that the maximum a posteriori estimate converges to a minimizer of the Mumford--Shah functional, up to a subsequence. A new result regarding the existence of a minimizer of the Mumford--Shah functional is proved. Moreover, we st…
Analysis of brain connectivity is important for understanding how information is processed by the brain. We propose a novel Bayesian vector autoregression (VAR) hierarchical model for analyzing brain connectivity in a resting-state fMRI data set with autism spectrum disorder (ASD) patients and healthy controls. Our approach models functional and effective connectivity simultaneously, which is new in the VAR literature for brain connectivity, and allows for both group- and single-subject inference as well as group comparisons. We combine analytical marginalization with Hamiltonian Monte Carlo (…
Calibration and prediction for NIR spectroscopy data are performed based on a functional interpretation of the Beer-Lambert formula. Considering that, for each chemical sample, the resulting spectrum is a continuous curve obtained as the summation of overlapped absorption spectra from each analyte plus a Gaussian error, we assume that each individual spectrum can be expanded as a linear combination of B-splines basis. Calibration is then performed using two procedures for estimating the individual analytes curves: basis smoothing and smoothing splines. Prediction is done by minimizing the squa…
Carpooling is an integral component in smart carbon-neutral cities, in particular to facilitate homework commuting. We study an innovative carpooling service developed by the start-up Ecov which specialises in homework commutes in peri-urban and rural regions. When a passenger makes a carpooling request, a designated driver is not assigned as in a traditional carpooling service; rather the passenger waits for the first driver, from a population of non-professional drivers who are already en route, to arrive. We propose a two-stage Bayesian hierarchical model to overcome the considerable diffic…
Hierarchical Bayesian inference for community detection and connectivity of functional brain networks
ShanghaiTech University（上海科技大学）；Monash University（蒙纳士大学）；The Hong Kong Polytechnic University（香港理工大学）；University College London（伦敦大学学院）；CIFAR（加拿大高级研究所）
In this paper, we deal with the problem of curves clustering. We propose a nonparametric method which partitions the curves into clusters and discretizes the dimensions of the curve points into intervals. The cross-product of these partitions forms a data-grid which is obtained using a Bayesian model selection approach while making no assumptions regarding the curves. Finally, a post-processing technique, aiming at reducing the number of clusters in order to improve the interpretability of the clustering, is proposed. It consists in optimally merging the clusters step by step, which correspond…
Variational Autoencoders (VAEs) have experienced recent success as data-generating models by using simple architectures that do not require significant fine-tuning of hyperparameters. However, VAEs are known to suffer from over-regularization which can lead to failure to escape local maxima. This phenomenon, known as posterior collapse, prevents learning a meaningful latent encoding of the data. Recent methods have mitigated this issue by deterministically moment-matching an aggregated posterior distribution to an aggregate prior. However, abandoning a probabilistic framework (and thus relying…
Cyber-security analysts face an increasingly large number of alerts received on any given day. This is mainly due to the low precision of many existing methods to detect threats, producing a substantial number of false positives. Usually, several signature-based and statistical anomaly detectors are implemented within a computer network to detect threats. Recent efforts in User and Entity Behaviour Analytics modelling shed a light on how to reduce the burden on Security Operations Centre analysts through a better understanding of peer-group behaviour. Statistically, the challenge consists of a…
We propose a novel Bayesian nonparametric method for hierarchical modelling on a set of related density functions, where grouped data in the form of samples from each density function are available. Borrowing strength across the groups is a major challenge in this context. To address this problem, we introduce a hierarchically structured prior, defined over a set of univariate density functions, using convenient transformations of Gaussian processes. Inference is performed through approximate Bayesian computation (ABC), via a novel functional regression adjustment. The performance of the propo…
The distribution of spatially aggregated data from a stochastic process $X$ may exhibit a different tail behavior than its marginal distributions. For a large class of aggregating functionals $\ell$ we introduce the $\ell$-extremal coefficient that quantifies this difference as a function of the extremal spatial dependence in $X$. We also obtain the joint extremal dependence for multiple aggregation functionals applied to the same process. Explicit formulas for the $\ell$-extremal coefficients and multivariate dependence structures are derived in important special cases. The results provide a …
Snow density estimates as a function of depth are used for understanding climate processes, evaluating water accumulation trends in polar regions, and estimating glacier mass balances. The common and interpretable physically-derived differential equation models for snow density are piecewise linear as a function of depth (on a transformed scale); thus, they can fail to capture important data features. Moreover, the differential equation parameters show strong spatial autocorrelation. To address these issues, we allow the parameters of the physical model, including random change points over dep…
Disaggregation modelling, or downscaling, has become an important discipline in epidemiology. Surveillance data, aggregated over large regions, is becoming more common, leading to an increasing demand for modelling frameworks that can deal with this data to understand spatial patterns. Disaggregation regression models use response data aggregated over large heterogenous regions to make predictions at fine-scale over the region by using fine-scale covariates to inform the heterogeneity. This paper presents the R package disaggregation, which provides functionality to streamline the process of r…
Shang and Hyndman (2017) proposed a grouped functional time series forecasting approach as a combination of individual forecasts obtained using generalized least squares method. We modify their methodology using generalized exponential smoothing technique for the most disaggregated functional time series in order to obtain more robust predictor. We discuss some properties of our proposals basing on results obtained via simulation studies and analysis of real data related to a prediction of a demand for electricity in Australia in 2016.
