Approximate pairwise likelihood inference in SGLM models with skew normal latent variables

Abstract

Spatial generalized linear mixed models are commonly employed for modeling discrete spatial responses that are acquired on a continuous area. A standard assumption in these models is that the latent variables are normally distributed, however skewed residuals appear in some spatial generalized linear mixed models. In this study, we consider a closed skew Gaussian random field for the spatial latent variables in the spatial generalized linear mixed models and present a new approximate pairwise likelihood approach to estimate parameters. In order to introduce a new algorithm to obtain the pairwise maximum likelihood estimates for the parameters, we use a linearization method in the composite marginal likelihood and EM algorithm. Also, techniques to calculate parameter estimates and spatial prediction in this class of skew models are proposed. The performance of the proposed model and method are illustrated through a simulation study, and applied the Tehran air quality index data set.

Introduction

Spatial generalized linear mixed (SGLM) models are commonly used to model non-Gaussian discrete spatial responses. In these models, the spatial dependence of the data can be considered as random effects or latent variables. The most common assumption for the latent variables is to use a multivariate normal distribution. Inference of the model parameters and spatial prediction has been studied intensely for the normal assumption, see [1], [2], [3], [4], [5] and [6]. The SGLM model has recently received a great deal of attention, Smith et al. [7] proposed Poisson cokriging as a SGLM model and extended this methodology to predict the latent variables. Walder and Hanks [8] studied this model by using the Laplace moving average random fields and Bayesian approach.

In most of these studies, the latent variables are modeled by the multivariate normal distribution. Hosseini et al. [9], Hosseini and Mohammadzadeh [10] and Hosseini and Karimi [11] showed an erroneous normal assumption has influence on the estimation of the model parameters and the accuracy of spatial prediction in SGLM models. They used more flexible distributions such as the multivariate skew normal (SN) and the closed skew normal (CSN). Vicente et al. [12] defined nonlinear regression models with the SN errors and discussed classical and Bayesian approaches for these models. Karimi and Mohammadzadeh [13] discussed spatial regression models with closed skew normal correlated errors and showed that this modeling increases the accuracy of the parameter estimates. Alodat and Shakhatreh [14] modified the non-linear regression with the SN error terms and showed that it has better performance than the model with the normal error terms.

Azzalini and Dalla-Valle [15] introduced the SN distribution as an extension of the Normal distribution; it is contained in the new family of skew-normal distributions presented in [16]. The CSN distribution is an extension to the SN distribution and normal distribution, [17], [18], [19]. SN and CSN are more flexible because they include the normal distribution as a special case with an extra parameter to regulate skewness. Kim and Mallick [20] suggested a model based on SN distribution to handle skewed spatial data. Karimi et al. [21], Karimi and Mohammadzadeh [22] and Alodat and Al-Rawwash [23] defined a new closed skew Gaussian random field (CSG) and Rimstad and Omre [24] introduced an extension of this CSG random field. They defined a new parameterization of the CSN distribution. Tagle et al. [25] presented a new spatio-temporal model for non-Gaussian data by using skew-t distribution and showed that the skew-t model is more accurate than the Gaussian-model.

In this paper, we use the CSG random field proposed in Rimstad and Omre [24] and introduce a new SGLM model with skew spatial latent variables. We consider the marginal composite likelihood method introduced by Varin et al. [4] to estimate the model parameters. Their method has a high computational capability compared to the original likelihood model, especially for high dimensions. Also, we obtain an approximate CSN distribution for the conditional distribution of the latent variables given the spatial responses using a linearization method. Finally, the main contribution of the current work is to present a new approximate pairwise likelihood inference method for the resulting new proposed SGLM model.

This paper is organized as follows: In Section 2 the closed skew normal are defined. The CSG random field and SGLMM with closed skew normal latent variables are introduced in Section 3. The proposed approximate pairwise likelihood method, as well as the conditional distribution at unsampled locations for prediction is described in Section 4. Section 5 presents a simulation study to implement and evaluate the proposed model. Section 6 shows an application to Tehran Air Quality Index (AQI) data during March 2019 to March 2020. Closing remarks are given in Section 7.