Bayesian model selection in linear mixed effects models with autoregressive(p) errors using mixture priors

In this article, we apply the Bayesian approach to the linear mixed effect models with autoregressive(p) random errors under mixture priors obtained with the Markov chain Monte Carlo (MCMC) method. The mixture structure of a point mass and continuous distribution can help to select the variables in fixed and random effects models from the posterior sample generated using the MCMC method. Bayesian prediction of future observations is also one of the major concerns. To get the best model, we consider the commonly used highest posterior probability model and the median posterior probability model. As a result, both criteria tend to be needed to choose the best model from the entire simulation study. In terms of predictive accuracy, a real example confirms that the proposed method provides accurate results.     

Model Selection for Vector Autoregressive Processes via Adaptive Lasso

Determination of the best subset is an important step in vector autoregressive (VAR) modeling. Traditional methods either conduct subset selection and parameter estimation separately or compute expensively. In this article, we propose a VAR model selection procedure using adaptive Lasso, for it is computational efficient and can select subset and estimate parameters simultaneously. By proper choice of tuning parameters, we can choose the correct subset and obtain the asymptotic normality of the non zero parameters. Simulation studies and real data analysis show that adaptive Lasso performs better than existing methods in VAR model fitting and prediction.     

Bayesian temporal density estimation with autoregressive species sampling models

We propose a novel Bayesian nonparametric (BNP) model, which is built on a class of species sampling models, for estimating density functions of temporal data. In particular, we introduce species sampling mixture models with temporal dependence. To accommodate temporal dependence, we define dependent species sampling models by modeling random support points and weights through an autoregressive model, and then we construct the mixture models based on the collection of these dependent species sampling models. We propose an algorithm to generate posterior samples and present simulation studies to compare the performance of the proposed models with competitors that are based on Dirichlet process mixture models. We apply our method to the estimation of densities for the price of apartment in Seoul, the closing price in Korea Composite Stock Price Index (KOSPI), and climate variables (daily maximum temperature and precipitation) of around the Korean peninsula.     

Separated hypotheses testing for autoregressive models with non-negative residuals

Normal residual is one of the usual assumptions in autoregressive model but sometimes in practice we are faced with non-negative residuals. In this paper, we have derived modified maximum likelihood estimators of parameters of the residuals and autoregressive coefficient. Also asymptotic distribution of modified maximum likelihood estimators in both stationary and non-stationary models are computed. So that, we can derive asymptotic distribution of unit root, Vuong's and Cox's tests statistics in stationary situation. Using simulation, it shows that Akaike information criterion and Vuong's test work to select the optimal autoregressive model with non-negative residuals. Sometimes Vuong's test select two competing models as equivalent models. These models may be suitable or unsuitable equivalent models. So we consider Cox's test to make inference after model selection. Kolmogorov–Smirnov test confirms our results. Also we have computed tracking interval for competing models to choosing between two close competing models when Vuong's test and Cox's test cannot detect the differences.     

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.     

Penalized regression models with autoregressive error terms

Penalized regression methods have recently gained enormous attention in statistics and the field of machine learning due to their ability of reducing the prediction error and identifying important variables at the same time. Numerous studies have been conducted for penalized regression, but most of them are limited to the case when the data are independently observed. In this paper, we study a variable selection problem in penalized regression models with autoregressive (AR) error terms. We consider three estimators, adaptive least absolute shrinkage and selection operator, bridge, and smoothly clipped absolute deviation, and propose a computational algorithm that enables us to select a relevant set of variables and also the order of AR error terms simultaneously. In addition, we provide their asymptotic properties such as consistency, selection consistency, and asymptotic normality. The performances of the three estimators are compared with one another using simulated and real examples.     

Diagnostic checks for integer-valued autoregressive models using expected residuals

Integer-valued time series models make use of thinning operators for coherency in the nature of count data. However, the thinning operators make residuals unobservable and are the main difficulty in developing diagnostic tools for autocorrelated count data. In this regard, we introduce a new residual, which takes the form of predictive distribution functions, to assess probabilistic forecasts, and this new residual is supplemented by a modified usual residuals. Under integer-valued autoregressive (INAR) models, the properties of these two residuals are investigated and used to evaluate the predictive performance and model adequacy of the INAR models. We compare our residuals with the existing residuals through simulation studies and apply our method to select an appropriate INAR model for an over-dispersed real data.     

Bayesian analysis of multivariate threshold autoregressive models with missing data

In some fields, we are forced to work with missing data in multivariate time series. Unfortunately, the data analysis in this context cannot be carried out in the same way as in the case of complete data. To deal with this problem, a Bayesian analysis of multivariate threshold autoregressive models with exogenous inputs and missing data is carried out. In this paper, Markov chain Monte Carlo methods are used to obtain samples from the involved posterior distributions, including threshold values and missing data. In order to identify autoregressive orders, we adapt the Bayesian variable selection method in this class of multivariate process. The number of regimes is estimated using marginal likelihood or product parameter-space strategies.     

Double threshold autoregressive conditionally heteroscedastic model building by genetic algorithms

Asymmetric behaviour in both mean and variance is often observed in real time series. The approach we adopt is based on double threshold autoregressive conditionally heteroscedastic (DTARCH) model with normal innovations. This model allows threshold nonlinearity in mean and volatility to be modelled as a result of the impact of lagged changes in assets and squared shocks, respectively. A methodology for building DTARCH models is proposed based on genetic algorithms (GAs). The most important structural parameters, that is regimes and thresholds, are searched for by GAs, while the remaining structural parameters, that is the delay parameters and models orders, vary in some pre-specified intervals and are determined using exhaustive search and an Asymptotic Information Criterion (AIC) like criterion. For each structural parameters trial set, a DTARCH model is fitted that maximizes the (penalized) likelihood (AIC criterion). For this purpose the iteratively weighted least squares algorithm is used. Then the best model according to the AIC criterion is chosen. Extension to the double threshold generalized ARCH (DTGARCH) model is also considered. The proposed methodology is checked using both simulated and market index data. Our findings show that our GAs-based procedure yields results that comparable to that reported in the literature and concerned with real time series. As far as artificial time series are considered, the proposed procedure seems to be able to fit the data quite well. In particular, a comparison is performed between the present procedure and the method proposed by Tsay [Tsay, R.S., 1989, Testing and modeling threshold autoregressive processes. Journal of the American Statistical Association, Theory and Methods, 84, 231–240.] for estimating the delay parameter. The former almost always yields better results than the latter. However, adopting Tsay's procedure as a preliminary stage for finding the appropriate delay parameter may save computational time specially if the delay parameter may vary in a large interval.     

Combining Bayesian method and Kalman smoother for detection additive outlier patches in autoregressive time series

ABSTRACT

This article proposes a development of detecting patches of additive outliers in autoregressive time series models. The procedure improves the existing detection methods via Gibbs sampling. We combine the Bayesian method and the Kalman smoother to present some candidate models of outlier patches and the best model with the minimum Bayesian information criterion (BIC) is selected among them. We propose that this combined Bayesian and Kalman method (CBK) can reduce the masking and swamping effects about detecting patches of additive outliers. The correctness of the method is illustrated by simulated data and then by analyzing a real set of observations.     

Maximum a-posteriori estimation of autoregressive processes based on finite mixtures of scale-mixtures of skew-normal distributions

This article investigates maximum a-posteriori (MAP) estimation of autoregressive model parameters when the innovations (errors) follow a finite mixture of distributions that, in turn, are scale-mixtures of skew-normal distributions (SMSN), an attractive and extremely flexible family of probabilistic distributions. The proposed model allows to fit different types of data which can be associated with different noise levels, and provides a robust modelling with great flexibility to accommodate skewness, heavy tails, multimodality and stationarity simultaneously. Also, the existence of convenient hierarchical representations of the SMSN random variables allows us to develop an EM-type algorithm to perform the MAP estimates. A comprehensive simulation study is then conducted to illustrate the superior performance of the proposed method. The new methodology is also applied to annual barley yields data.     

Nonlinear semiparametric autoregressive model with finite mixtures of scale mixtures of skew normal innovations

We propose data generating structures which can be represented as the nonlinear autoregressive models with single and finite mixtures of scale mixtures of skew normal innovations. This class of models covers symmetric/asymmetric and light/heavy-tailed distributions, so provide a useful generalization of the symmetrical nonlinear autoregressive models. As semiparametric and nonparametric curve estimation are the approaches for exploring the structure of a nonlinear time series data set, in this article the semiparametric estimator for estimating the nonlinear function of the model is investigated based on the conditional least square method and nonparametric kernel approach. Also, an Expectation–Maximization-type algorithm to perform the maximum likelihood (ML) inference of unknown parameters of the model is proposed. Furthermore, some strong and weak consistency of the semiparametric estimator in this class of models are presented. Finally, to illustrate the usefulness of the proposed model, some simulation studies and an application to real data set are considered.     

Priors and component structures in autoregressive time series models

New approaches to prior specification and structuring in autoregressive time series models are introduced and developed. We focus on defining classes of prior distributions for parameters and latent variables related to latent components of an autoregressive model for an observed time series. These new priors naturally permit the incorporation of both qualitative and quantitative prior information about the number and relative importance of physically meaningful components that represent low frequency trends, quasi-periodic subprocesses and high frequency residual noise components of observed series. The class of priors also naturally incorporates uncertainty about model order and hence leads in posterior analysis to model order assessment and resulting posterior and predictive inferences that incorporate full uncertainties about model order as well as model parameters. Analysis also formally incorporates uncertainty and leads to inferences about unknown initial values of the time series, as it does for predictions of future values. Posterior analysis involves easily implemented iterative simulation methods, developed and described here. One motivating field of application is climatology, where the evaluation of latent structure, especially quasi-periodic structure, is of critical importance in connection with issues of global climatic variability. We explore the analysis of data from the southern oscillation index, one of several series that has been central in recent high profile debates in the atmospheric sciences about recent apparent trends in climatic indicators.     

The specification of vector autoregressive moving average models

In this paper we propose a new identification method based on the residual white noise autoregressive criterion (Pukkila et al., 1990) to select the order of VARMA structures. Results from extensive simulation experiments based on different model structures with varying number of observations and number of component series are used to demonstrate the performance of this new procedure. We also use economic and business data to compare the model structures selected by this order selection method with those identified in other published studies.     

Integer-valued autoregressive processes with periodic structure

In this paper the periodic integer-valued autoregressive model of order one with period T, driven by a periodic sequence of independent Poisson-distributed random variables, is studied in some detail. Basic probabilistic and statistical properties of this model are discussed. Moreover, parameter estimation is also addressed. Specifically, the methods of estimation under analysis are the method of moments, least squares-type and likelihood-based ones. Their performance is compared through a simulation study.     

Sparse Bayesian variable selection in multinomial probit regression model with application to high-dimensional data classification

Here we consider a multinomial probit regression model where the number of variables substantially exceeds the sample size and only a subset of the available variables is associated with the response. Thus selecting a small number of relevant variables for classification has received a great deal of attention. Generally when the number of variables is substantial, sparsity-enforcing priors for the regression coefficients are called for on grounds of predictive generalization and computational ease. In this paper, we propose a sparse Bayesian variable selection method in multinomial probit regression model for multi-class classification. The performance of our proposed method is demonstrated with one simulated data and three well-known gene expression profiling data: breast cancer data, leukemia data, and small round blue-cell tumors. The results show that compared with other methods, our method is able to select the relevant variables and can obtain competitive classification accuracy with a small subset of relevant genes.     

First-order random coefficient integer-valued autoregressive processes

A first-order random coefficient integer-valued autoregressive (RCINAR(1)) model is introduced. Ergodicity of the process is established. Moments and autocovariance functions are obtained. Conditional least squares and quasi-likelihood estimators of the model parameters are derived and their asymptotic properties are established. The performance of these estimators is compared with the maximum likelihood estimator via simulation.     

A Bernoulli autoregressive moving average model applied to rainfall occurrence

Abstract

In this article, we propose a new model for binary time series involving an autoregressive moving average structure. The proposed model, which is an extension of the GARMA model, can be used for calculating the forecast probability of an occurrence of an event of interest in cases where these probabilities are dependent on previous observations in the near term. The proposed model is used to analyze a real dataset involving a series that contains only data 0 and 1, indicating the absence or presence of rain in a city located in the central region of São Paulo state, Brazil.     

Sample path properties of an explosive double autoregressive model

This article studies sample path properties of an explosive double autoregressive (DAR) model. After suitable renormalization, it is shown that the sample path converges weakly to a geometric Brownian motion. This further strengthens our understanding of sample paths of nonstationary DAR processes. The obtained results can be extended to nonstationary random coefficient autoregressive (RCA) models. Simulation studies are carried out to support our results.