Structured priors for multivariate time series

A class of prior distributions for multivariate autoregressive models is presented. This class of priors is built taking into account the latent component structure that characterizes a collection of autoregressive processes. In particular, the state-space representation of a vector autoregressive process leads to the decomposition of each time series in the multivariate process into simple underlying components. These components may have a common structure across the series. A key feature of the proposed priors is that they allow the modeling of such common structure. This approach also takes into account the uncertainty in the number of latent processes, consequently handling model order uncertainty in the multivariate autoregressive framework. Posterior inference is achieved via standard Markov chain Monte Carlo (MCMC) methods. Issues related to inference and exploration of the posterior distribution are discussed. We illustrate the methodology analyzing two data sets: a synthetic data set with quasi-periodic latent structure, and seasonally adjusted US monthly housing data consisting of housing starts and housing sales over the period 1965 to 1974.     

Bayesian Identification of Seasonal Autoregressive Models

Abstract

A very important and essential phase of time series analysis is identifying the model orders. This article develops an approximate Bayesian procedure to identify the orders of seasonal autoregressive processes. Using either a normal-gamma prior density or a noninformative prior, which is combined with an approximate conditional likelihood function, the foundation of the proposed technique is to derive the joint posterior mass function of the model orders in an easy form. Then one may inspect the posterior mass function and choose the orders with the largest posterior probability to be the suitable orders of the time series being analyzed. A simulation study, with different priors mass functions, is carried out to test the adequacy of the proposed technique and compare it with some non-Bayesian automatic criteria. The analysis of the numerical results supports the adequacy of the proposed technique in identifying the orders of the autoregressive processes.     

Bayesian identification of double seasonal autoregressive time series models

ABSTRACT

Seasonal autoregressive (SAR) models have been modified and extended to model high frequency time series characterized by exhibiting double seasonal patterns. Some researchers have introduced Bayesian inference for double seasonal autoregressive (DSAR) models; however, none has tackled the problem of Bayesian identification of DSAR models. Therefore, in order to fill this gap, we present a Bayesian methodology to identify the order of DSAR models. Assuming the model errors are normally distributed and using three priors, i.e. natural conjugate, g, and Jeffreys’ priors, on the model parameters, we derive the joint posterior mass function of the model order in a closed-form. Accordingly, the posterior mass function can be investigated and the best order of DSAR model is chosen as a value with the highest posterior probability for the time series being analyzed. We evaluate the proposed Bayesian methodology using simulation study, and we then apply it to real-world hourly internet amount of traffic dataset.     

Bayesian Identification of Multivariate Autoregressive Processes

Identification is one of the most important stages of a time series analysis. This paper develops a direct Bayesian technique to identify the order of multivariate autoregressive processes. By employing the conditional likelihood function and a matrix normal-Wishart prior density, or Jeffrey' vague prior, the proposed identification technique is based on deriving the exact posterior probability mass function of the model order in a convenient form. Then one may easily evaluate the posterior probabilities of the model order and choose the order that maximizes the posterior mass function to be the suitable order of the time series data being analyzed. Assuming the bivariate autoregressive processes, a numerical study, with different prior mass functions, is carried out to assess the efficiency of the proposed technique. The analysis of the numerical results supports the adequacy of the proposed technique in identifying the orders of multivariate autoregressive processes.     

Kullback-Leibler divergence to evaluate posterior sensitivity to different priors for autoregressive time series models

In this paper we use the Kullback-Leibler divergence to measure the distance between the posteriors of the autoregressive (AR) model coefficients, aiming to evaluate mathematically the sensitivity of the coefficients posterior to different types of priors, i.e. Jeffreys’, g, and natural conjugate priors. In addition, we evaluate the impact of the posteriors distance in Bayesian estimates of mean and variance of the model coefficients by generating a large number of Monte Carlo simulations from the posteriors. Simulation study results show that the coefficients posterior is sensitive to prior distributions, and the posteriors distance has more influence on Bayesian estimates of variance than those of mean of the model coefficients. Same results are obtained from the application to real-world time series datasets.     

Quantile Regression for Location‐Scale Time Series Models with Conditional Heteroscedasticity

This paper considers quantile regression for a wide class of time series models including autoregressive and moving average (ARMA) models with asymmetric generalized autoregressive conditional heteroscedasticity errors. The classical mean‐variance models are reinterpreted as conditional location‐scale models so that the quantile regression method can be naturally geared into the considered models. The consistency and asymptotic normality of the quantile regression estimator is established in location‐scale time series models under mild conditions. In the application of this result to ARMA‐generalized autoregressive conditional heteroscedasticity models, more primitive conditions are deduced to obtain the asymptotic properties. For illustration, a simulation study and a real data analysis are provided.     

Inference and prediction with arma processes

An essential ingredient of any time series analysis is the estimation of the model parameters and the forecasting of future observations. This investigation takes a Bayesian approach to the analysis of time series by making inferences of the model parameters from the posterior distribution and forecasting from the predictive distribution.

The foundation of the approach is to approximate the condi-tional likelihood by a normal-gamma distribution on the parameter space. The techniques illustrated with many examples of ARMA processes.     

Disaggregation and Forecasting: A Bayesian Analysis

The problem of temporal disaggregation of time series is analyzed by means of Bayesian methods. The disaggregated values are obtained through a posterior distribution derived by using a diffuse prior on the parameters. Further analysis is carried out assuming alternative conjugate priors. The means of the different posterior distributions are shown to be equivalent to some sampling theory results. Bayesian prediction intervals are obtained. Forecasts for future disaggregated values are derived assuming a conjugate prior for the future aggregated value.     

A bivariate first-order signed integer-valued autoregressive process

Bivariate integer-valued time series occur in many areas, such as finance, epidemiology, business etc. In this article, we present bivariate autoregressive integer-valued time-series models, based on the signed thinning operator. Compared to classical bivariate INAR models, the new processes have the advantage to allow for negative values for both the time series and the autocorrelation functions. Strict stationarity and ergodicity of the processes are established. The moments and the autocovariance functions are determined. The conditional least squares estimator of the model parameters is considered and the asymptotic properties of the obtained estimators are derived. An analysis of a real dataset from finance and a simulation study are carried out to assess the performance of the model.     

Asymptotic Inefficiency of Mean-Correction on Parameter Estimation for a Periodic First-Order Autoregressive Model

A common practice in time series analysis is to fit a centered model to the mean-corrected data set. For stationary autoregressive moving-average (ARMA) processes, as far as the parameter estimation is concerned, fitting an ARMA model without intercepts to the mean-corrected series is asymptotically equivalent to fitting an ARMA model with intercepts to the observed series. We show that, related to the parameter least squares estimation of periodic ARMA models, the second approach can be arbitrarily more efficient than the mean-corrected counterpart. This property is illustrated by means of a periodic first-order autoregressive model. The asymptotic variance of the estimators for both approaches is derived. Moreover, empirical experiments based on simulations investigate the finite sample properties of the estimators.     

Bayesian Hierarchical Kernelized Probabilistic Matrix Factorization

We present a Bayesian analysis framework for matrix-variate normal data with dependency structures induced by rows and columns. This framework of matrix normal models includes prior specifications, posterior computation using Markov chain Monte Carlo methods, evaluation of prediction uncertainty, model structure search, and extensions to multidimensional arrays. Compared with Bayesian probabilistic matrix factorization, which integrates a Gaussian prior for single row of the data matrix, our proposed model, namely Bayesian hierarchical kernelized probabilistic matrix factorization, imposes Gaussian Process priors over multiple rows of the matrix. Hence, the learned model explicitly captures the underlying correlation among the rows and the columns. In addition, our method requires no specific assumptions like independence of latent factors for rows and columns, which obtains more flexibility for modeling real data compared to existing works. Finally, the proposed framework can be adapted to a wide range of applications, including multivariate analysis, times series, and spatial modeling. Experiments highlight the superiority of the proposed model in handling model uncertainty and model optimization.     

Bayesian principal component analysis with mixture priors

A central issue in principal component analysis (PCA) is that of choosing the appropriate number of principal components to be retained. Bishop (1999a) suggested a Bayesian approach for PCA for determining the effective dimensionality automatically on the basis of the probabilistic latent variable model. This paper extends this approach by using mixture priors, in that the choice dimensionality and estimation of principal components are done simultaneously via MCMC algorithm. Also, the proposed method provides a probabilistic measure of uncertainty on PCA, yielding posterior probabilities of all possible cases of principal components.     

Multivariate autoregressive time semes modeling: one scalar autoregressive model at-A-time

Multivariate (or interchangeably multichannel) autoregressive (MCAR) modeling of stationary and nonstationary time series data is achieved doing things one channel at-a-time using only scalar computations on instantaneous data. The one channel at-a-time modeling is achieved as an instantaneous response multichannel autoregressive model with orthogonal innovations variance. Conventional MCAR models are expressible as linear algebraic transformations of the instantaneous response orthogonal innovations models. By modeling multichannel time series one channel at-a-time, the problems of modeling multichannel time series are reduced to problems in the modeling of scalar autoregressive time series. The three longstanding time series modeling problems of achieving a relatively parsimonious MCAR representation, of multichannel stationary time series spectral estimation and of the modeling of nonstationary covariance time series are addressed using this paradigm.     

Bayesian forecasting for AR(1) models with normal coefficients

This paper presents a Bayesian solution to the problem of time series forecasting, for the case in which the generating process is an autoregressive of order one, with a normal random coefficient. The proposed procedure is based on the predictive density of the future observation. Conjugate priors are used for some parameters, while improper vague priors are used for others.     

On a mixture vector autoregressive model

The authors show how to extend univariate mixture autoregressive models to a multivariate time series context. Similar to the univariate case, the multivariate model consists of a mixture of stationary or nonstationary autoregressive components. The authors give the first and second order stationarity conditions for a multivariate case up to order 2. They also derive the second order stationarity condition for the univariate mixture model up to arbitrary order. They describe an EM algorithm for estimation, as well as a diagnostic checking procedure. They study the performance of their method via simulations and include a real application.     

Modeling Interval Time Series with Space–Time Processes

We consider interval-valued time series, that is, series resulting from collecting real intervals as an ordered sequence through time. Since the lower and upper bounds of the observed intervals at each time point are in fact values of the same variable, they are naturally related. We propose modeling interval time series with space–time autoregressive models and, based on the process appropriate for the interval bounds, we derive the model for the intervals’ center and radius. A simulation study and an application with data of daily wind speed at different meteorological stations in Ireland illustrate that the proposed approach is appropriate and useful.     

Adaptive Shrinkage in Bayesian Vector Autoregressive Models

Vector autoregressive (VAR) models are frequently used for forecasting and impulse response analysis. For both applications, shrinkage priors can help improving inference. In this article, we apply the Normal-Gamma shrinkage prior to the VAR with stochastic volatility case and derive its relevant conditional posterior distributions. This framework imposes a set of normally distributed priors on the autoregressive coefficients and the covariance parameters of the VAR along with Gamma priors on a set of local and global prior scaling parameters. In a second step, we modify this prior setup by introducing another layer of shrinkage with scaling parameters that push certain regions of the parameter space to zero. Two simulation exercises show that the proposed framework yields more precise estimates of model parameters and impulse response functions. In addition, a forecasting exercise applied to U.S. data shows that this prior performs well relative to other commonly used specifications in terms of point and density predictions. Finally, performing structural inference suggests that responses to monetary policy shocks appear to be reasonable.     

A transitional Markov switching autoregressive model

ABSTRACT

This paper is concerned with properties of a transitional Markov switching autoregressive (TMSAR) model, together with its maximum-likelihood estimation and inference. We extend existing MSAR models by allowing dependence of AR parameters on hidden states at time points prior to the current time t. A stationary solution is given and expressions for the theoretical autocovariance function are derived. Two time series are analyzed and the new model outperforms two existing MSAR models in terms of maximized log-likelihood, residual correlations, and one-step-ahead forecasting performance. The new model also gives more regime changes in agreement with real events.     

Multichannel electroencephalographic analyses via dynamic regression models with time-varying lag–lead structure

Multiple time series of scalp electrical potential activity are generated routinely in electroencephalographic (EEG) studies. Such recordings provide important non-invasive data about brain function in human neuropsychiatric disorders. Analyses of EEG traces aim to isolate characteristics of their spatiotemporal dynamics that may be useful in diagnosis, or may improve the understanding of the underlying neurophysiology or may improve treatment through identifying predictors and indicators of clinical outcomes. We discuss the development and application of non-stationary time series models for multiple EEG series generated from individual subjects in a clinical neuropsychiatric setting. The subjects are depressed patients experiencing generalized tonic–clonic seizures elicited by electroconvulsive therapy (ECT) as antidepressant treatment. Two varieties of models—dynamic latent factor models and dynamic regression models—are introduced and studied. We discuss model motivation and form, and aspects of statistical analysis including parameter identifiability, posterior inference and implementation of these models via Markov chain Monte Carlo techniques. In an application to the analysis of a typical set of 19 EEG series recorded during an ECT seizure at different locations over a patient's scalp, these models reveal time-varying features across the series that are strongly related to the placement of the electrodes. We illustrate various model outputs, the exploration of such time-varying spatial structure and its relevance in the ECT study, and in basic EEG research in general.