Kriging with external drift in a Birnbaum–Saunders geostatistical model

Abstract

Spatial models to describe dependent georeferenced data are applied in different fields and, particularly, are used to analyze earth and environmental data. Most of these applications are carried out under Gaussian spatial models. The Birnbaum–Saunders distribution is a unimodal and positively skewed model which has received considerable attention in several areas, including earth and environmental sciences. In addition, theoretical arguments have been provided to justify its usage in the data modeling from these sciences, at least in the same settings where the lognormal distribution can be employed. This paper presents kriging with external drift based on a Birnbaum–Saunders spatial model. The maximum likelihood method is considered to estimate its parameters. The results obtained in the paper are illustrated by an experimental data set related to agricultural management. Specifically, in this illustration, the spatial variability of magnesium content in the soil as a function of calcium content is analyzed.

Appendix: multivariate BS and log-BS distributions

A continuous random variable $T_i > 0$ has a BS distribution with shape ($\alpha _i>0$) and scale ($\varrho _i >0$) parameters, denoted by $T_i \sim \text{BS}(\alpha_i, \varrho_i)$, for $i = 1, \ldots ,n$, if (see details in Leiva 2016, pp. 18–20)

$$\begin{aligned} Z_i= \frac{1}{\alpha _i}\left( \sqrt{{T_i}/{\varrho _i}}-\sqrt{{\varrho _i}/{T_i}}\right) \sim \text {N}(0,\,1), \end{aligned}$$

(16)

such that

$$\begin{aligned} T_i = \varrho _i\left( {\alpha _i Z_i}/{2}+\sqrt{\left( {\alpha _i Z_i}/{2}\right) ^2+1}\right) ^2. \end{aligned}$$

Let the random vector $\underline{T} = (T_1,\ldots ,T_n)^\top \in {\mathbb {R}}^{n}_{+}$ have a BS$_n$ distribution, with shape vector $\underline{\alpha } = (\alpha _1, \ldots , \alpha _n)^\top \in {\mathbb {R}}^{n}_{+}$, scale vector $\underline{\varrho }=(\varrho _1,\ldots ,\varrho _n)^\top \in {\mathbb {R}}^{n}_{+}$, and association matrix ${\varvec{{\varSigma }}} \in {\mathbb {R}}^{n\times n}_{+}$, with $\text{rk}({\varvec{{\varSigma}}}) = n$ and $\text{rk}({\varvec{{\varSigma}}})$ being the rank of the matrix ${\varvec{{\varSigma }}}$. This is denoted by

$$\begin{aligned} \underline{T} \sim \text{BS}_n(\underline{\alpha}, \underline{\varrho}, {\varvec{{\varSigma}}}). \end{aligned}$$

Note that ${\varvec{{\varSigma }}} = (\sigma _{ij})$ is the $n\times n$ variance-covariance matrix of the random vector $\underline{Z} = (Z_1, \ldots, Z_n)^\top \sim \text{N}_n(\underline{0}_{n\times 1}, {\varvec{{\varSigma}}})$, with $\sigma _{ij} = 1$, if $i=j$, whereas $\sigma_{ij} = \text{Cor}(Z_i, Z_j)$, if $i\ne j$, that is, the coefficient of correlation between $Z_i$ and $Z_j$ as given in (16), for $i,j=1,\ldots ,n$. Thus, if $\underline{T} \sim \text{BS}_n(\underline{\alpha}, \underline{\varrho}, {\varvec{{\varSigma}}})$, then its probability density function is given by

$$\begin{aligned} f_{\underline{T}}(\underline{t}; \underline{\alpha }, \underline{\varrho}, {\varvec{{\varSigma}}}) =& \phi_n\left( \frac{1}{\alpha_1} \left( \sqrt{\frac{t_1}{\varrho_1}} - \sqrt{\frac{\varrho_1}{t_1}}\right), \ldots, \right. \nonumber \\ &\left. \frac{1}{\alpha_n} \left( \sqrt{\frac{t_n}{\varrho_n}} - \sqrt{\frac{\varrho_n}{t_n}}\right); {\varvec{{\varSigma}}}\right) \nonumber \\ &\prod_{i=1}^n \frac{1}{2\alpha_i\varrho_i} \left( \sqrt{\frac{\varrho_i}{t_i}} + \sqrt{\frac{\varrho_i^3}{t_i^3}}\right), \end{aligned}$$

(17)

with $\underline{t} \in {\mathbb {R}}^{n}_{+}$, where $\phi _n$ is the n-variate N(0, 1) probability density function. If $\underline{T} \sim {{\text {BS}}_n}(\underline{\alpha }, \underline{\varrho }, {\varvec{{\varSigma }}})$, properties hold:

(i) For $i,\,j=1,\ldots ,n$, we have

$$\begin{aligned} {{\text {Cov}}(T_i, T_j)}\approx & {} \varrho _i\varrho _j \left( \alpha _i\alpha _j\left( 2+\frac{3}{4}\alpha _i^2\right. +\,\frac{3}{4}\alpha _j^2+\frac{9}{32} \alpha _i^2\alpha _j^2\right) \\ &\left. \times \rho _{ij}+\frac{1}{2}\alpha _i^2\alpha _j^2\rho _{ij}^2+\frac{3}{16}\alpha _i^2\alpha _j^2\rho _{ij}^3\right) ; \end{aligned}$$

(ii) For the partitions

$$\begin{aligned} \underline{T}= & {} \left( \underline{T_1}^\top ,\underline{T_2}^\top \right) ^\top ,\;\;\underline{\alpha }=\left( \underline{\alpha }_1^\top ,\underline{\alpha }_2^\top \right) ^\top ,\;\;\underline{\varrho }=\left( \underline{\varrho }_1^\top ,\underline{\varrho }_2^\top \right) ^\top ,\\ {\varvec{{\varSigma }}}= & {} \left( \begin{array}{ll}{\varvec{{\varSigma }_{\rm 11}}} &{}\quad {\varvec{{\varSigma }_{\rm 12}}}\\ {\varvec{{\varSigma }_{\rm 21}}} &{}\quad {\varvec{{\varSigma }_{\rm 22}}}\\ \end{array}\right) , \end{aligned}$$

where $\underline{T_1}, \underline{\alpha }_1,\underline{\varrho }_1$ are $q\times 1$ vectors, $\underline{T_2}, \underline{\alpha }_2,\underline{\varrho }_2$ are $p\times 1$ vectors, ${\varvec{{\varSigma }_{11}}}$ is an $n_1\times n_1$ matrix, ${\varvec{{\varSigma }_{22}}}$ is an $n_2\times n_2$ matrix, and ${\varvec{{\varSigma }_{12}}}={\varvec{{\varSigma }_{21}}}^\top$ is an $n_1\times n_2$ matrix, with $n_1+n_2=n$; then, $\underline{T}_l \sim \text{BS}_{n_l}(\underline{\alpha }_l,\underline{\varrho}_l; {\varvec{{\varSigma}}}_{ll})$, for $l=1,2$;

(iii) If $c>0$, then $c\underline{T} \sim \text{BS}_n(\underline{\alpha}, c\underline{\varrho};{\varvec{{\varSigma}}})$ and

$$\begin{aligned} \left( c\underline{T_1}^\top , \underline{T_2}^\top \right) ^\top \sim {{\text {BS}}_n}\left( \underline{\alpha }, \left( c\underline{\varrho }_1^\top ,\underline{\varrho }_2^\top \right) ^\top ;{\varvec{{\varSigma }}}\right) , \end{aligned}$$

whereas an analogous property applies to $c \underline{T_2}$; and (iv) Considering the notation $\underline{A}^{-1} = (1/A_i)$, where $\underline{A} =(A_i)$ is a vector and $A_i$ its elements, then

$$\begin{aligned} \underline{T}^{-1}= \left( \left( \underline{T_1}^{-1}\right)^\top, \left( \underline{T_2}^{-1}\right)^\top \right)^\top \sim \text{BS}_n(\underline{\alpha}, \underline{\varrho}^{-1}; {\varvec{{\varSigma}}}) \end{aligned}$$

and

$$\begin{aligned} \left( \left( \underline{T_1}^{-1}\right) ^\top, \underline{T_2}^\top \right)^\top \sim \text{BS}_n\left( \underline{\alpha}, \left( \left( \underline{\varrho}_1^{-1}\right)^\top, \underline{\varrho}_2^\top \right)^\top; {\varvec{{\varSigma}}}_1\right)^\top, \end{aligned}$$

with

$$\begin{aligned} {\varvec{{\varSigma }}}_1=\left( \begin{array}{ll} {\varvec{{\varSigma }_{\rm 11}}} &{}\quad -\,{\varvec{{\varSigma }_{\rm 12}}}\\ -\varvec{{\varSigma }}_{\rm 21} &{}\quad {\varvec{{\varSigma }_{\rm 22}}}\\ \end{array}\right) , \end{aligned}$$

which applies analogously to $\underline{T_2}^{-1}$. Let

$$\begin{aligned} \underline{T} = (T_1, \ldots, T_n)^\top \sim \text{BS}_n(\underline{\alpha}, \underline{\varrho}, {\varvec{{\varSigma}}}), \end{aligned}$$

$Y_i = \log (T_i)$ and $\mu _i = \log (\varrho _i)$, for $i=1,\ldots ,n$, with $-\,\infty< Y_i < +\,\infty$ and $-\,\infty<\mu _i < +\,\infty$. Then, the random vector $\underline{Y}=(Y_1,\ldots ,Y_n)^\top$ has a log-BS$_n$ distribution, with shape and location vectors, respectively,

$$\begin{aligned} \underline{\alpha } = (\alpha _1,\ldots ,\alpha _n)^\top ,\;\; \underline{\mu } = (\mu _1,\ldots ,\mu _n)^\top , \end{aligned}$$

and association matrix ${\varvec{{\varSigma }}}$ as given in (17). This is denoted by $\underline{Y} \sim \text{log-BS}_n(\underline{\alpha}, \underline{\mu}, {\varvec{{\varSigma}}})$ and its probability density function is defined as

$$\begin{aligned} f_{\underline{Y}}(\underline{y}; \underline{\alpha }, \underline{\varrho }, {\varvec{{\varSigma }}} )=& {} \phi _n\left( \frac{2}{\alpha _1} \sinh \left( \frac{y_1-\mu _1}{2}\right) ,\ldots ,\right. \nonumber \\&\left. \frac{2}{\alpha _n} \sinh \left( \frac{y_n-\mu _n}{2}\right) ;{\varvec{{\varSigma }}}\right) \\&\prod _{i=1}^n \frac{1}{\alpha _i}\cosh \left( \frac{y_i-\mu _i}{2}\right) ,\;\;\underline{y} \in {\mathbb {R}}^{n},\nonumber \end{aligned}$$

(18)

where $\phi _n$ is given in (17). Let $\underline{Y} \sim \text{log-BS}_n(\underline{\alpha}, \underline{\mu}, {\varvec{{\varSigma}}})$. Thus, the following properties hold:

• (i) $Y_i \sim \text{log-BS}(\alpha_i,\mu_i)$  , for $i=1,\ldots ,n$;

• (ii) If $\underline{a}=(a_1,\ldots ,a_n)^\top$ is a constant vector, $\underline{Y}+\underline{a}\sim \text{log-BS}_n(\underline{\alpha}, \underline{\mu} + \underline{a}; {\varvec{{\varSigma}}})$;

• (iii) $\text{E}(\underline{Y}) = \underline{\mu}$;

• (iv) If $\underline{\alpha }=\alpha \underline{1}_{n\times 1}^\top$ and $\underline{\mu }=\underline{0}_{n\times 1}$, then $\text{Cov}(\underline{Y}) \approx \alpha^2{\varvec{{\varSigma}}}$;

• (v) If $\underline{Y} \sim \text{log-BS}_n(\alpha \underline{1}_{n\times 1}, \underline{\mu}; {\varvec{{\varSigma}}})$, then

  $$\begin{aligned} \frac{4}{\alpha ^2}\underline{V}^\top {\varvec{{\varSigma }}}^{-1}\underline{V} \end{aligned}$$

  has a $\chi ^2$ distribution with n degrees of freedom, where $\underline{V}= (V_1,\ldots ,V_n)^\top$, with

  $$\begin{aligned} V_i=\sinh \left( \frac{y_i-\underline{x}_i^\top {\underline{\varrho }}}{2}\right) ,\quad i=1,\ldots ,n. \end{aligned}$$