Bayesian analysis of mixture modelling using the multivariate t distribution

A finite mixture model using the multivariate t distribution has been shown as a robust extension of normal mixtures. In this paper, we present a Bayesian approach for inference about parameters of t-mixture models. The specifications of prior distributions are weakly informative to avoid causing nonintegrable posterior distributions. We present two efficient EM-type algorithms for computing the joint posterior mode with the observed data and an incomplete future vector as the sample. Markov chain Monte Carlo sampling schemes are also developed to obtain the target posterior distribution of parameters. The advantages of Bayesian approach over the maximum likelihood method are demonstrated via a set of real data.     

Likelihood-Based Inference in Autoregressive Models with Scaled t-Distributed Innovations by Means of EM-Based Algorithms

This article applies the EM-based (ECM and ECME) algorithms to find the maximum likelihood estimates of model parameters in general AR models with independent scaled t-distributed innovations whenever the degrees of freedom are unknown. The ECME, sharing advantages with both EM and Newton–Raphson algorithms, is an extension of ECM, which itself is an extension of the EM algorithm. The ECM and ECME algorithms, which are analytically quite simple to use, are then compared based on the computational running time and the accuracy of estimation via a simulation study. The results demonstrate that the ECME is efficient and usable in practice. We also show how our method can be applied to the Wolfer's sunspot data.     

Varying-time random effects models for longitudinal data: unmixing and temporal interpolation of remote-sensing data

Remote sensing is a helpful tool for crop monitoring or vegetation-growth estimation at a country or regional scale. However, satellite images generally have to cope with a compromise between the time frequency of observations and their resolution (i.e. pixel size). When concerned with high temporal resolution, we have to work with information on the basis of kilometric pixels, named mixed pixels, that represent aggregated responses of multiple land cover. Disaggreggation or unmixing is then necessary to downscale from the square kilometer to the local dynamic of each theme (crop, wood, meadows, etc.).

Assuming the land use is known, that is to say the proportion of each theme within each mixed pixel, we propose to address the downscaling issue through the generalization of varying-time regression models for longitudinal data and/or functional data by introducing random individual effects. The estimators are built by expanding the mixed pixels trajectories with B-splines functions and maximizing the log-likelihood with a backfitting-ECME algorithm. A BLUP formula allows then to get the ‘best possible’ estimations of the local temporal responses of each crop when observing mixed pixels trajectories. We show that this model has many potential applications in remote sensing, and an interesting one consists of coupling high and low spatial resolution images in order to perform temporal interpolation of high spatial resolution images (20 m), increasing the knowledge on particular crops in very precise locations.

The unmixing and temporal high-resolution interpolation approaches are illustrated on remote-sensing data obtained on the South-Western France during the year 2002.     

Maximum likelihood estimation in vector autoregressive models with multivariate scaled t-distributed innovations using EM-based algorithms

This article is concerned with the likelihood-based inference of vector autoregressive models with multivariate scaled t-distributed innovations by applying the EM-based (ECM and ECME) algorithms. The ECM and ECME algorithms, which are analytically quite simple to use, are applied to find the maximum likelihood estimates of the model parameters and then compared based on the computational running time and the accuracy of estimation via a simulation study. The results demonstrate that the ECME is efficient and usable in practice. We also show how the method can be applied to a multivariate dataset.     

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.     

ML estimation for factor analysis: EM or non-EM?

To obtain maximum likelihood (ML) estimation in factor analysis (FA), we propose in this paper a novel and fast conditional maximization (CM) algorithm, which has quadratic and monotone convergence, consisting of a sequence of CM log-likelihood (CML) steps. The main contribution of this algorithm is that the closed form expression for the parameter to be updated in each step can be obtained explicitly, without resorting to any numerical optimization methods. In addition, a new ECME algorithm similar to Liu’s (Biometrika 81, 633–648, 1994) one is obtained as a by-product, which turns out to be very close to the simple iteration algorithm proposed by Lawley (Proc. R. Soc. Edinb. 60, 64–82, 1940) but our algorithm is guaranteed to increase log-likelihood at every iteration and hence to converge. Both algorithms inherit the simplicity and stability of EM but their convergence behaviors are much different as revealed in our extensive simulations: (1) In most situations, ECME and EM perform similarly; (2) CM outperforms EM and ECME substantially in all situations, no matter assessed by the CPU time or the number of iterations. Especially for the case close to the well known Heywood case, it accelerates EM by factors of around 100 or more. Also, CM is much more insensitive to the choice of starting values than EM and ECME.     

Two-Piece location-scale distributions based on scale mixtures of normal family

In this work, we study the maximum likelihood (ML) estimation problem for the parameters of the two-piece (TP) distribution based on the scale mixtures of normal (SMN) distributions. This is a family of skewed distributions that also includes the scales mixtures of normal class, and is flexible enough for modeling symmetric and asymmetric data. The ML estimates of the proposed model parameters are obtained via an expectation-maximization (EM)-type algorithm.     

On modelling data from degradation sample paths over time

This paper is mainly concerned with modelling data from degradation sample paths over time. It uses a general growth curve model with Box‐Cox transformation, random effects and ARMA(p, q) dependence to analyse a set of such data. A maximum likelihood estimation procedure for the proposed model is derived and future values are predicted, based on the best linear unbiased prediction. The paper compares the proposed model with a nonlinear degradation model from a prediction point of view. Forecasts of failure times with various data lengths in the sample are also compared.     

EM-based algorithms for autoregressive models with t-distributed innovations

This paper considers the estimation of parameters of AR(p) models for time series with t-distribution via EM-based algorithms. The paper develops asymptotic properties for the estimation to show that the estimators are efficient. Also testing theory for the estimators is considered. The robustness of the estimators and various tests to deviations from an assumed model is investigated. The study shows that the algorithms have equal estimation efficiency even if the error distribution is miss-specified or perturbed by outliers. Interestingly, the estimators from these algorithms performed better than that of the Modified Maximum Likelihood (MML) considered in Tiku et al. (2000 Tiku, M. L., Wong, W. K., Vaughan, D. C., Bian, G. (2000). Time series models in non-normal situations: Symmetric innovations. Journal of Time Series Analysis, 21: 571–596. [Google Scholar]).