arXiv subjects

Mattias Villani

Publications and source records attributed to Mattias Villani.

At least 19 recordsLinked to original sources

arXiv2026

Spectral divide-and-conquer MCMC for long stationary time series

The temporal dependence inherent in time series models poses a fundamental challenge for distributed Bayesian inference, as it precludes embarrassingly parallel algorithms based on naive independence assumptions in the time domain. We propose a frequency-domain framework for scalable Bayesian inference in stationary time series that exploits the asymptotic independence underlying the Whittle likelihood. To exploit parallel computing resources, we develop a distributed fast Fourier transform and integrate it with embarrassingly parallel Markov chain Monte Carlo (MCMC) algorithms within a modern cluster-computing framework. This enables the analysis of time series that exceed the memory capacity of a single computational node or for which computation time is a bottleneck. The proposed methodology is compatible with a broad class of existing divide-and-conquer algorithms for independent data by applying them to frequency-domain rather than time-domain partitions. We establish that the error of the spectral divide-and-conquer MCMC posterior approximation relative to the exact time-domain posterior converges to zero in probability in a shrinking neighbourhood of the full-data Whittle posterior mode. The corresponding convergence rate is also derived. Across several experiments, we demonstrate that our approach provides accurate approximations to the full-data Whittle posterior. The proposed method is shown to outperform the current state-of-the-art time-domain divide-and-conquer methodology, particularly for highly persistent processes. The methodology is further illustrated by fitting a semi-long range model to a long meteorological time series.

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arXiv2026

Time-Varying Multi-Seasonal ARMA Models

We propose an ARMA model that allows for multiple seasonal periods and time varying parameters in both regular and seasonal components, building upon previous work for pure AR processes and the conditional likelihood. The model is parameterized to ensure stability and invertibility at each time point. The parameter evolution is governed by dynamic shrinkage processes, enabling extended periods of essentially constant parameters, gradual changes, and abrupt shifts. The model includes a stochastic volatility component to account for potentially heterogeneous noise, also modeled by a dynamic shrinkage process. A Gibbs sampler is developed using the exact likelihood, with separate updating steps for the latent errors and the unobserved pre-sample history of the process. The time-varying AR and MA parameters are sampled jointly using a fast posterior sampler based on the extended Kalman filter. The model and the efficiency of the Gibbs sampler are evaluated using simulated and real data. A case study on monthly air passenger data in the US during 1990-2024 reveals significant changes in seasonality during the Covid-19 pandemic.

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arXiv2025

Time-Varying Multi-Seasonal AR Models

We propose a seasonal AR model with time-varying parameter processes in both the regular and seasonal parameters. The model is parameterized to guarantee stability at every time point and can accommodate multiple seasonal periods. The time evolution is modeled by dynamic shrinkage processes to allow for long periods of essentially constant parameters, periods of rapid change, and abrupt jumps. A Gibbs sampler is developed with a particle Gibbs update step for the AR parameter trajectories. We show that the near-degeneracy of the model, caused by the dynamic shrinkage processes, makes it challenging to estimate the model by particle methods. To address this, a more robust, faster and accurate approximate sampler based on the extended Kalman filter is proposed. The model and the numerical effectiveness of the Gibbs sampler are investigated on simulated data. An application to more than a century of monthly US industrial production data shows interesting clear changes in seasonality over time, particularly during the Great Depression and the recent Covid-19 pandemic. Keywords: Bayesian inference; Extended Kalman filter; Particle MCMC; Seasonality.

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arXiv2025

Variational inference for steady-state BVARs

The steady-state Bayesian vector autoregression (BVAR) makes it possible to incorporate prior information about the long-run mean of the process. This has been shown in many studies to substantially improve forecasting performance, and the model is routinely used for forecasting and macroeconomic policy analysis at central banks and other financial institutions. Steady-steady BVARs are estimated using Gibbs sampling, which is time-consuming for the increasingly popular large-scale BVAR models with many variables. We propose a fast variational inference (VI) algorithm for approximating the parameter posterior and predictive distribution of the steady-state BVAR, as well as log predictive scores for model comparison. We use simulated and real US macroeconomic data to show that VI produces results that are very close to those from Gibbs sampling. The computing time of VI can be orders of magnitude lower than Gibbs sampling, in particular for log predictive scores, and VI is shown to scale much better with the number of time series in the system.

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arXiv2025

Calibrated Bayesian inference for random fields on large irregular domains using the debiased spatial Whittle likelihood

Bayesian inference for stationary random fields is computationally demanding. Whittle-type likelihoods in the frequency domain based on the fast Fourier Transform (FFT) have several appealing features: i) low computational complexity of only $\mathcal{O}(n \log n)$, where $n$ is the number of spatial locations, ii) robustness to assumptions of the data-generating process, iii) ability to handle missing data and irregularly spaced domains, and iv) flexibility in modelling the covariance function via the spectral density directly in the spectral domain. It is well known, however, that the Whittle likelihood suffers from bias and low efficiency for spatial data. The debiased Whittle likelihood is a recently proposed alternative with better frequentist properties. We propose a methodology for Bayesian inference for stationary random fields using the debiased spatial Whittle likelihood, with an adjustment from the composite likelihood literature. The adjustment is shown to give a well-calibrated Bayesian posterior as measured by coverage properties of credible sets, without sacrificing the quasi-linear computation time. We apply the method to simulated data and two real datasets.

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arXiv2024

Spectral domain likelihoods for Bayesian inference in time-varying parameter models

Inference for locally stationary processes is often based on some local Whittle-type approximation of the likelihood function defined in the frequency domain. The main reasons for using such a likelihood approximation is that i) it has substantially lower computational cost and better scalability to long time series compared to the time domain likelihood, particularly when used for Bayesian inference via Markov Chain Monte Carlo (MCMC), ii) convenience when the model itself is specified in the frequency domain, and iii) it provides access to bootstrap and subsampling MCMC which exploits the asymptotic independence of Fourier transformed data. Most of the existing literature compares the asymptotic performance of the maximum likelihood estimator (MLE) from such frequency domain likelihood approximation with the exact time domain MLE. Our article uses three simulation studies to assess the finite-sample accuracy of several frequency domain likelihood functions when used to approximate the posterior distribution in time-varying parameter models. The methods are illustrated on an application to egg price data.

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arXiv2024

Modeling local predictive ability using power-transformed Gaussian processes

A Gaussian process is proposed as a model for the posterior distribution of the local predictive ability of a model or expert, conditional on a vector of covariates, from historical predictions in the form of log predictive scores. Assuming Gaussian expert predictions and a Gaussian data generating process, a linear transformation of the predictive score follows a noncentral chi-squared distribution with one degree of freedom. Motivated by this we develop a noncentral chi-squared Gaussian process regression to flexibly model local predictive ability, with the posterior distribution of the latent GP function and kernel hyperparameters sampled by Hamiltonian Monte Carlo. We show that a cube-root transformation of the log scores is approximately Gaussian with homoscedastic variance, making it possible to estimate the model much faster by marginalizing the latent GP function analytically. A multi-output Gaussian process regression is also introduced to model the dependence in predictive ability between experts, both for inference and prediction purposes. Linear pools based on learned local predictive ability are applied to predict daily bike usage in Washington DC.

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arXiv2024

Joint estimation of the predictive ability of experts using a multi-output Gaussian process

A multi-output Gaussian process (GP) is introduced as a model for the joint posterior distribution of the local predictive ability of set of models and/or experts, conditional on a vector of covariates, from historical predictions in the form of log predictive scores. Following a power transformation of the log scores, a GP with Gaussian noise can be used, which allows faster computation by first using Hamiltonian Monte Carlo to sample the hyper-parameters of the GP from a model where the latent GP surface has been marginalized out, and then using these draws to generate draws of joint predictive ability conditional on a new vector of covariates. Linear pools based on learned joint local predictive ability are applied to predict daily bike usage in Washington DC.

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arXiv2023

Local Prediction Pools

We propose local prediction pools as a method for combining the predictive distributions of a set of experts conditional on a set of variables believed to be related to the predictive accuracy of the experts. This is done in a two step process where we first estimate the conditional predictive accuracy of each expert given a vector of covariates$\unicode{x2014}$or pooling variables$\unicode{x2014}$and then combine the predictive distributions of the experts conditional on this local predictive accuracy. To estimate the local predictive accuracy of each expert, we introduce the simple, fast, and interpretable caliper method. Expert pooling weights from the local prediction pool approaches the equal weight solution whenever there is little data on local predictive performance, making the pools robust and adaptive. We also propose a local version of the widely used optimal prediction pools. Local prediction pools are shown to outperform the widely used optimal linear pools in a macroeconomic forecasting evaluation, and in predicting daily bike usage for a bike rental company.

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arXiv2022

Dynamic Mixture of Experts Models for Online Prediction

A mixture of experts models the conditional density of a response variable using a mixture of regression models with covariate-dependent mixture weights. We extend the finite mixture of experts model by allowing the parameters in both the mixture components and the weights to evolve in time by following random walk processes. Inference for time-varying parameters in richly parameterized mixture of experts models is challenging. We propose a sequential Monte Carlo algorithm for online inference and based on a tailored proposal distribution built on ideas from linear Bayes methods and the EM algorithm. The method gives a unified treatment for mixtures with time-varying parameters, including the special case of static parameters. We assess the properties of the method on simulated data and on industrial data where the aim is to predict software faults in a continuously upgraded large-scale software project.

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arXiv2022

Spectral Subsampling MCMC for Stationary Multivariate Time Series with Applications to Vector ARTFIMA Processes

Spectral subsampling MCMC was recently proposed to speed up Markov chain Monte Carlo (MCMC) for long stationary univariate time series by subsampling periodogram observations in the frequency domain. This article extends the approach to multivariate time series using a multivariate generalisation of the Whittle likelihood. To assess the computational gains from spectral subsampling in challenging problems, a multivariate generalisation of the autoregressive tempered fractionally integrated moving average model (ARTFIMA) is introduced and some of its properties derived. Bayesian inference based on the Whittle likelihood is demonstrated to be a fast and accurate alternative to the exact time domain likelihood. Spectral subsampling is shown to provide up to two orders of magnitude additional speed-up, while retaining MCMC sampling efficiency and accuracy, compared to spectral methods using the full dataset. Keywords: Bayesian, Markov chain Monte Carlo, Semi-long memory, Spectral analysis, Whittle likelihood.

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arXiv2022

Bayesian Optimization of Hyperparameters from Noisy Marginal Likelihood Estimates

Bayesian models often involve a small set of hyperparameters determined by maximizing the marginal likelihood. Bayesian optimization is a popular iterative method where a Gaussian process posterior of the underlying function is sequentially updated by new function evaluations. An acquisition strategy uses this posterior distribution to decide where to place the next function evaluation. We propose a novel Bayesian optimization framework for situations where the user controls the computational effort, and therefore the precision of the function evaluations. This is a common situation in econometrics where the marginal likelihood is often computed by Markov chain Monte Carlo (MCMC) or importance sampling methods, with the precision of the marginal likelihood estimator determined by the number of samples. The new acquisition strategy gives the optimizer the option to explore the function with cheap noisy evaluations and therefore find the optimum faster. The method is applied to estimating the prior hyperparameters in two popular models on US macroeconomic time series data: the steady-state Bayesian vector autoregressive (BVAR) and the time-varying parameter BVAR with stochastic volatility. The proposed method is shown to find the optimum much quicker than traditional Bayesian optimization or grid search.

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arXiv2022

A Multilayered Block Network Model to Forecast Large Dynamic Transportation Graphs: an Application to US Air Transport

Dynamic transportation networks have been analyzed for years by means of static graph-based indicators in order to study the temporal evolution of relevant network components, and to reveal complex dependencies that would not be easily detected by a direct inspection of the data. This paper presents a state-of-the-art probabilistic latent network model to forecast multilayer dynamic graphs that are increasingly common in transportation and proposes a community-based extension to reduce the computational burden. Flexible time series analysis is obtained by modeling the probability of edges between vertices through latent Gaussian processes. The models and Bayesian inference are illustrated on a sample of 10-year data from four major airlines within the US air transportation system. Results show how the estimated latent parameters from the models are related to the airline's connectivity dynamics, and their ability to project the multilayer graph into the future for out-of-sample full network forecasts, while stochastic blockmodeling allows for the identification of relevant communities. Reliable network predictions would allow policy-makers to better understand the dynamics of the transport system, and help in their planning on e.g. route development, or the deployment of new regulations.

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arXiv2022

Robust Real-Time Delay Predictions in a Network of High-Frequency Urban Buses

Providing transport users and operators with accurate forecasts on travel times is challenging due to a highly stochastic traffic environment. Public transport users are particularly sensitive to unexpected waiting times, which negatively affect their perception on the system's reliability. In this paper we develop a robust model for real-time bus travel time prediction that depart from Gaussian assumptions by using Student-$t$ errors. The proposed approach uses spatiotemporal characteristics from the route and previous bus trips to model short-term effects, and date/time variables and Gaussian processes for long-run forecasts. The model allows for flexible modeling of mean, variance and kurtosis spaces. We propose algorithms for Bayesian inference and for computing probabilistic forecast distributions. Experiments are performed using data from high-frequency buses in Stockholm, Sweden. Results show that Student-$t$ models outperform Gaussian ones in terms of log-posterior predictive power to forecast bus delays at specific stops, which reveals the importance of accounting for predictive uncertainty in model selection. Estimated Student-$t$ regressions capture typical temporal variability between within-day hours and different weekdays. Strong spatiotemporal effects are detected for incoming buses from immediately previous stops, which is in line with many recently developed models. We finally show how Bayesian inference naturally allows for predictive uncertainty quantification, e.g. by returning the predictive probability that the delay of an incoming bus exceeds a given threshold.

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arXiv2022

Bayesian Prediction with Covariates Subject to Detection Limits

Missing values in covariates due to censoring by signal interference or lack of sensitivity in the measuring devices are common in industrial problems. We propose a full Bayesian solution to the prediction problem with an efficient Markov Chain Monte Carlo (MCMC) algorithm that updates all the censored covariate values jointly in a random scan Gibbs sampler. We show that the joint updating of missing covariate values can be at least two orders of magnitude more efficient than univariate updating. This increased efficiency is shown to be crucial for quickly learning the missing covariate values and their uncertainty in a real-time decision making context, in particular when there is substantial correlation in the posterior for the missing values. The approach is evaluated on simulated data and on data from the telecom sector. Our results show that the proposed Bayesian imputation gives substantially more accurate predictions than naïve imputation, and that the use of auxiliary variables in the imputation gives additional predictive power.

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arXiv2021

Bayesian Modeling of Effective and Functional Brain Connectivity using Hierarchical Vector Autoregressions

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 (HMC) to obtain highly efficient posterior sampling. The results from more simplified covariance settings are, in general, overly optimistic about functional connectivity between regions compared to our results. In addition, our modeling of heterogeneous subject-specific covariance matrices is shown to give smaller differences in effective connectivity compared to models with a common covariance matrix to all subjects.

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arXiv2020

Spatial 3D Matérn priors for fast whole-brain fMRI analysis

Bayesian whole-brain functional magnetic resonance imaging (fMRI) analysis with three-dimensional spatial smoothing priors has been shown to produce state-of-the-art activity maps without pre-smoothing the data. The proposed inference algorithms are computationally demanding however, and the proposed spatial priors have several less appealing properties, such as being improper and having infinite spatial range. We propose a statistical inference framework for whole-brain fMRI analysis based on the class of Matérn covariance functions. The framework uses the Gaussian Markov random field (GMRF) representation of possibly anisotropic spatial Matérn fields via the stochastic partial differential equation (SPDE) approach of Lindgren et al. (2011). This allows for more flexible and interpretable spatial priors, while maintaining the sparsity required for fast inference in the high-dimensional whole-brain setting. We develop an accelerated stochastic gradient descent (SGD) optimization algorithm for empirical Bayes (EB) inference of the spatial hyperparameters. Conditionally on the inferred hyperparameters, we make a fully Bayesian treatment of the brain activity. The Matérn prior is applied to both simulated and experimental task-fMRI data and clearly demonstrates that it is a more reasonable choice than the previously used priors, using comparisons of activity maps, prior simulation and cross-validation.

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arXiv2020

Physiological Gaussian Process Priors for the Hemodynamics in fMRI Analysis

Background: Inference from fMRI data faces the challenge that the hemodynamic system that relates neural activity to the observed BOLD fMRI signal is unknown. New Method: We propose a new Bayesian model for task fMRI data with the following features: (i) joint estimation of brain activity and the underlying hemodynamics, (ii) the hemodynamics is modeled nonparametrically with a Gaussian process (GP) prior guided by physiological information and (iii) the predicted BOLD is not necessarily generated by a linear time-invariant (LTI) system. We place a GP prior directly on the predicted BOLD response, rather than on the hemodynamic response function as in previous literature. This allows us to incorporate physiological information via the GP prior mean in a flexible way, and simultaneously gives us the nonparametric flexibility of the GP. Results: Results on simulated data show that the proposed model is able to discriminate between active and non-active voxels also when the GP prior deviates from the true hemodynamics. Our model finds time varying dynamics when applied to real fMRI data. Comparison with Existing Method(s): The proposed model is better at detecting activity in simulated data than standard models, without inflating the false positive rate. When applied to real fMRI data, our GP model in several cases finds brain activity where previously proposed LTI models does not. Conclusions: We have proposed a new non-linear model for the hemodynamics in task fMRI, that is able to detect active voxels, and gives the opportunity to ask new kinds of questions related to hemodynamics.

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