A Latent Variable Approach to Modelling Multivariate Geostatistical Skew-Normal Data

Abstract

In this paper we propose a spatial latent factor model to deal with multivariate geostatistical skew-normal data. In this model we assume that the unobserved latent structure, responsible for the correlation among different variables as well as for the spatial autocorrelation among different sites is Gaussian, and that the observed variables are skew-normal. For this model we provide some of its properties like its spatial autocorrelation structure and its finite dimensional marginal distributions. Estimation of the unknown parameters of the model is carried out by employing a Monte Carlo Expectation Maximization algorithm, whereas prediction at unobserved sites is performed by using closed form formulas and Markov chain Monte Carlo algorithms. Simulation studies have been performed to evaluate the soundness of the proposed procedures.

Appendix

In this appendix we report some distributional results regarding the observable processes $Y _{i}\left (\mathbf{x}\right )$. Let us first recall some definitions. Following, for instance, [2], we say that a random vector Y = (Y ₁, …, Y _n )^T has an extended skew-normal distribution with parameters $\boldsymbol{\mu }$, $\boldsymbol{\varSigma }$, $\boldsymbol{\alpha }$ and τ, and we write $\mathbf{Y} \sim \mbox{ ESN}_{n}(\boldsymbol{\mu },\boldsymbol{\varSigma },\boldsymbol{\alpha },\tau )$, if it has probability density function of the form

$$\displaystyle{f(\mathbf{y}) =\phi _{n}(\mathbf{y}-\boldsymbol{\mu };\boldsymbol{\varSigma }) \cdot \varPhi (\alpha _{0} +\boldsymbol{\alpha } ^{T}\mathbf{D}^{-1}(\mathbf{y}-\boldsymbol{\mu }))/\varPhi (\tau ),\ \ \ \ \mbox{ for}\ \ \ \mathbf{y} \in \mathbb{R}^{n},}$$

(10)

where $\boldsymbol{\mu }\in \mathbb{R}^{n}$ is a vector of location parameters, $\phi _{n}(\ \cdot \;\boldsymbol{\varSigma })$ is the n-dimensional normal density function with zero mean vector and (positive-definite) variance-covariance matrix $\boldsymbol{\varSigma }$ having elements σ _ij , Φ(⋅ ) is the scalar N(0,1) distribution function, $\mathbf{D} = \mbox{ diag}(\sigma _{11},\ldots,\sigma _{nn})^{1/2}$ is the diagonal matrix formed with the standard deviations of the scale matrix $\boldsymbol{\varSigma }$, $\boldsymbol{\alpha }\in \mathbb{R}^{n}$ is a vector of skewness parameters, and $\tau \in \mathbb{R}$ is an additional parameter. Moreover, $\alpha _{0} =\tau (1 +\boldsymbol{\alpha } ^{T}\mathbf{R}\boldsymbol{\alpha })^{1/2}$ where R is the correlation matrix associated to $\boldsymbol{\varSigma }$, that is, $\mathbf{R} = \mathbf{D}^{-1}\boldsymbol{\varSigma }\mathbf{D}^{-1}$. Clearly, this distribution extends the multivariate normal distribution through the parameter vector $\boldsymbol{\alpha }$, and for $\boldsymbol{\alpha }= 0$ it reduces to the latter. When τ = 0, also α ₀ = 0 and (10) reduces to

$$\displaystyle{ f(\mathbf{y}) = 2 \cdot \phi _{n}(\mathbf{y}-\boldsymbol{\mu };\boldsymbol{\varSigma }) \cdot \varPhi (\boldsymbol{\alpha }^{T}\mathbf{D}^{-1}(\mathbf{y}-\boldsymbol{\mu })),\ \ \ \ \mbox{ for}\ \ \ \mathbf{y} \in \mathbb{R}^{n}. }$$

(11)

In this case we simply say that Y has a skew-normal distribution and we write, more concisely, $\mathbf{Y} \sim \mbox{ SN}_{n}(\boldsymbol{\mu },\boldsymbol{\varSigma },\boldsymbol{\alpha })$.

According to [13] and [14], we say that the n-dimensional random vector Y = (Y ₁, …, Y _n )^T has a multivariate closed skew-normal distribution, and we write $\mathbf{Y} \sim \mbox{ CSN}_{n,m}(\boldsymbol{\mu },\boldsymbol{\varSigma },\mathbf{D}_{c},\boldsymbol{\nu },\boldsymbol{\varDelta })$, if it has probability density function of the form

$$\displaystyle{ f(\mathbf{y}) = \frac{1} {\varPhi _{m}(\mathbf{0};\boldsymbol{\nu },\boldsymbol{\varDelta }+\mathbf{D}_{c}^{T}\boldsymbol{\varSigma }\mathbf{D}_{c})} \cdot \phi _{n}(\mathbf{y};\boldsymbol{\mu },\boldsymbol{\varSigma }) \cdot \varPhi _{m}(\mathbf{D}_{c}^{T}(\mathbf{y}-\boldsymbol{\mu });\boldsymbol{\nu },\boldsymbol{\varDelta }),\ \ \ \ \mbox{ for}\ \ \ \mathbf{y} \in \mathbb{R}^{n}, }$$

(12)

where: m is an integer greater than 0; $\boldsymbol{\mu }\in \mathbb{R}^{n}$; $\boldsymbol{\varSigma }\in \mathbb{R}^{n\times n}$ is a positive-definite matrix; $\mathbf{D}_{c} \in \mathbb{R}^{n\times m}$ is an n × m matrix; $\boldsymbol{\nu }\in \mathbb{R}^{m}$ is a vector; $\boldsymbol{\varDelta }\in \mathbb{R}^{m\times m}$ is a positive-definite matrix; and $\phi _{n}(\ \cdot \;\boldsymbol{\mu },\boldsymbol{\varSigma })$ and $\varPhi _{n}(\ \cdot \;\boldsymbol{\mu },\boldsymbol{\varSigma })$ are the probability density function and the cumulative distribution function, respectively, of the n-dimensional normal distribution with mean vector $\boldsymbol{\mu }$ and variance-covariance matrix $\boldsymbol{\varSigma }$.

Though, as we have already noticed, the multivariate finite-dimensional marginal distributions of the multivariate spatial process $\left (Y _{1}\left (\mathbf{x}\right ),\ldots,Y _{m}\left (\mathbf{x}\right )\right )^{T}$, for $\mathbf{x} \in \mathbb{R}^{2}$, are not skew-normal (in the sense of [2]), it is possible to show that they are closed skew-normal, according to the definition of [13]. This implies that, for any given i = 1, …, m, each univariate spatial process $Y _{i}\left (\mathbf{x}\right )$ has all its finite-dimensional marginal distributions that are closed skew-normal. To see this (see also [24]), consider n spatial locations x ₁, …, x _n , and the corresponding n-dimensional random vector Y = (Y _i (x ₁), …, Y _i (x _n ))^T. Recalling that for any given $\mathbf{x} \in \mathbb{R}^{2}$ we can write $Y _{i}(\mathbf{x}) =\beta _{i} + Z_{i}(\mathbf{x}) +\omega _{i}S_{i}(\mathbf{x})$, the vector Y can be written as $\mathbf{Y} =\beta _{i}\mathbf{1}_{n} + \mathbf{Z} + \mathbf{D}_{\omega }\mathbf{S} = \mathbf{W} + \mathbf{V}$, where $\mathbf{W} =\beta _{i}\mathbf{1}_{n} + \mathbf{Z}$, V = D _ω S, Z = (Z _i (x ₁), …, Z _i (x _n ))^T, S = (S _i (x ₁), …, S _i (x _n ))^T and D _ω is the n × n diagonal matrix with ω _i on the diagonal. Now, since S _i (x), for $\mathbf{x} \in \mathbb{R}^{2}$, are independently and identically distributed as CSN_1, 1(0, 1, α _i , 0, 1), according to Theorem 3 of [14], we have that $\mathbf{S} \sim \mbox{ CSN}_{n,n}(0,\mathbf{I}_{n},\mathbf{D}_{\alpha },0,\mathbf{I}_{n})$, where D _α is the n × n diagonal matrix with α _i on the diagonal. On the other hand, since Z follows a multivariate normal distribution with mean 0 and covariance matrix $\boldsymbol{\varSigma }_{Z}$ with entries given by $\mathrm{Cov}\big[Z_{i}(\mathbf{x}),Z_{i}(\mathbf{x} + \mathbf{h})\big] =\varsigma _{ i}^{2}\rho (\mathbf{h})$, we also have that $\mathbf{Z} \sim \mbox{ CSN}_{n,1}(0,\boldsymbol{\varSigma }_{Z},0,0,1)$. Moreover, being W distributed as a multivariate normal with mean β _i 1 _n and covariance matrix $\boldsymbol{\varSigma }_{Z}$, we can write that $\mathbf{W} \sim \mbox{ CSN}_{n,1}(\beta _{i}\mathbf{1}_{n},\boldsymbol{\varSigma }_{Z},0,0,1)$, and using Theorem  of [14] we can also write that $\mathbf{V} \sim \mbox{ CSN}_{n,n}(0,\mathbf{D}_{\omega ^{2}},\mathbf{D}_{\alpha /\omega },0,\mathbf{I}_{n})$, where $\mathbf{D}_{\omega ^{2 }}$ is the n × n diagonal matrix with ω _i ² on the diagonal, and D _α∕ω is the n × n diagonal matrix with α _i ∕ω _i on the diagonal. Thus, considering that $\mathbf{Y} = \mathbf{W} + \mathbf{V}$, we can conclude, using Theorem 4 of [14], that $\mathbf{Y} \sim \mbox{ CSN}_{n,n+1}(\beta _{i}\mathbf{1}_{n},\boldsymbol{\varSigma }_{Z} +\omega _{ i}^{2}\mathbf{I}_{n},\mathbf{D}^{{\ast}},0,\boldsymbol{\varDelta }^{{\ast}})$, for some matrices D ^∗ and $\boldsymbol{\varDelta }^{{\ast}}$.