Birnbaum–Saunders functional regression models for spatial data

Abstract

With the advancement of technology, data are often recorded continuously and instantaneously. Since the early nineties, this kind of observations have been described by models for functional data. Usually a large set of records for each individual in the sample become in a curve (by using some smoothing method) which is considered as a realization of a random function. In functional regression models these curves are used to establish whether there is a relation with an scalar response (functional regression model with scalar response). If two or more sets of curves are obtained for each individual, more complex functional regression models can be established. In particular, in geosciences, where spatial statistics is a primary tool, functional regression is becoming more frequent. Therefore, it is of interest to develop methodologies for spatially correlated functional data. Also in geosciences, as well as in other areas, it is common that the response variables follow positive skew distributions (for example, those obtained in studies about the level of chemical elements in soil or air). Hence, the standard geostatistical assumption of Gaussian errors, or at least of symmetry, is inappropriate. This type of variables, in non-spatial contexts, have been successfully described by the Birnbaum–Saunders distribution, becoming its modeling a very active research field. However, the use of this distribution in the treatment of geostatistical data has only been applied under stationarity. This paper develops a Birnbaum–Saunders model for geostatistical data considering a non-stationary process using functional covariates. The corresponding parameters are estimated by maximum likelihood and their performance is evaluated through Monte Carlo simulations. We illustrate the proposed model with two geo-referenced data sets, which shows its potential applications and a better performance in relation to the Gaussian model.

Appendix 1

1.1 Score vector

The elements of the $K_{\beta }\times 1$ score vector given in (13) are detailed as

$$\begin{aligned} \frac{\partial \ell ({\varvec{\theta }})}{\partial \alpha }&= \frac{\partial }{\partial \alpha }\left( -n\log (\alpha )-\frac{2}{\alpha ^2}{\varvec{V}}^\top {\varvec{\varGamma }}^{-1}{\varvec{V}}\right) = -\frac{n}{\alpha } + \frac{4}{\alpha ^3}\varvec{V}^\top {\varvec{\varGamma }}^{-1}{\varvec{V}},\\ \frac{\partial \ell ({\varvec{\theta }})}{\partial \beta _0}&= \frac{\partial }{\partial {\beta _0}}\left( -\frac{2}{\alpha ^2}{\varvec{V}}^\top {\varvec{\varGamma }}^{-1}{\varvec{V}} + \sum _{i=1}^{n}\log \left( \cosh \left( \frac{y_i - \mu _i}{2}\right) \right) \right) \\&= \frac{2}{\alpha ^2}{\varvec{W}}^\top {\varvec{\varGamma }}^{-1}{\varvec{V}} -\frac{1}{2}\sum _{i=1}^{n}\tanh \left( \frac{y_i -\mu _i}{2}\right) ,\\ \frac{\partial \ell ({\varvec{\theta }})}{\partial {\varvec{b}}_{j}}&=\frac{\partial }{\partial { {\varvec{b}}_j}}\left( -\frac{2}{\alpha ^2}{\varvec{V}}^\top {\varvec{\varGamma }}^{-1}{\varvec{V}} + \sum _{i=1}^{n}\log \left( \cosh \left( \frac{y_i - \mu _i}{2}\right) \right) \right) \\&= \frac{2}{\alpha ^2}({\varvec{z}}_j^\top {\varvec{W}})^\top {\varvec{\varGamma }}^{-1}{\varvec{V}} -\frac{1}{2}\sum _{i=1}^{n}z_{ij}\tanh \left( \frac{y_i -\mu _i}{2}\right) , \;{j = 1, \ldots , K_{\beta };} \\ \frac{\partial \ell ({\varvec{\theta }})}{\partial \phi }&=-\frac{1}{2}\text {tr}\left( {\varvec{\varGamma }}^{-1}\frac{\partial {\varvec{\varGamma }}}{\partial \phi }\right) +\frac{2\phi }{\alpha ^2}{\varvec{V}}^\top {\varvec{\varGamma }}^{-1}\frac{\partial {\varvec{\varGamma }}}{\partial \phi }{\varvec{\varGamma }}^{-1}{\varvec{V}}. \end{aligned}$$

where $\text {tr}({\varvec{A}})$ denotes the trace of ${\varvec{A}}$; ${\varvec{W}} = (W_1, \ldots , W_n)^\top$, with $W_i = \cosh ((y_i-\mu _i)/2)$, for $i = 1, \ldots , n$; ${\varvec{V}} = (V_1, \ldots , V_n)^\top$ is defined in (12) and $\mu _i={\varvec{z}}_i^\top {\varvec{\xi }}$, with ${\varvec{z}}_i$ being the ith row of the matrix ${\varvec{Z}}$ defined in (11).

1.2 Information matrix

For the BS functional regression model the Hessian matrix presented in (15) has elements expressed as

$$\begin{aligned} \ddot{{\varvec{\ell }}}(\alpha )&= \frac{n}{\alpha ^2} - \frac{12}{\alpha ^4}{\varvec{V}}^\top \varvec{\varGamma }^{-1}{\varvec{V}}\\ \ddot{{\varvec{\ell }}}(\alpha \beta _0)&= \frac{4}{\alpha ^3}{\varvec{W}}^\top \varvec{\varGamma }^{-1}{\varvec{V}}\\ \ddot{{\varvec{\ell }}}(\alpha \beta _0)&= \frac{4}{\alpha ^3}{\varvec{W}}^\top \varvec{\varGamma }^{-1}{\varvec{V}}\\ \ddot{{\varvec{\ell }}}(\alpha {\varvec{b}}_j)&= -\frac{4}{\alpha ^3}({\varvec{z}}_j^\top {\varvec{W}})^\top \varvec{\varGamma }^{-1}{\varvec{V}}\\ \ddot{{\varvec{\ell }}}(\alpha \phi )&= -\frac{4\phi }{\alpha ^3}{\varvec{V}}^\top \varvec{\varGamma }^{-1}\frac{\partial \varvec{\varGamma }}{\partial \phi }{\varvec{\varGamma }}^{-1}{\varvec{V}}\\ \ddot{{\varvec{\ell }}}(\beta _0) =&-\frac{1}{\alpha ^2}({\varvec{V}}^\top \varvec{\varGamma }^{-1}{\varvec{V}}) - \frac{1}{\alpha ^2}{\varvec{W}}^\top \varvec{\varGamma }^{-1}{\varvec{W}}\\&+\frac{1}{4}\sum _{i=1}^{n}\left( \text {sech}\left( \frac{y_i -\mu _i}{2}\right) \right) ^2\\ \ddot{{\varvec{\ell }}}(\beta _0{\varvec{b}}_j) =&-\frac{1}{\alpha ^2}({\varvec{V}}^\top \odot {\varvec{z}}_j^\top )\varvec{\varGamma }^{-1}{\varvec{V}} - \frac{1}{\alpha ^2}{\varvec{W}}^\top \varvec{\varGamma }^{-1}({\varvec{W}} \odot {\varvec{z}}_j)\\&+\frac{1}{4}\sum _{i=1}^{n}\left( \text {sech}\left( \frac{y_i -\mu _i}{2}\right) \right) ^{2}{\varvec{z}}_{ij}, \; j = 1, \ldots , K_{\beta };\\ \ddot{{\varvec{\ell }}}({\varvec{b}}_j) = \frac{{\partial }^2 \ell ({\varvec{\theta }})}{\partial {\varvec{b}}_l\partial \varvec{b_j}} =&-\frac{1}{\alpha ^2}({\varvec{V}}^\top \odot {\varvec{z}}_l^\top \odot {\varvec{z}}_j^\top )\varvec{\varGamma }^{-1}{\varvec{V}} - \frac{1}{\alpha ^2}({\varvec{W}}^\top \odot {\varvec{z}}_l^\top )\varvec{\varGamma }^{-1}({\varvec{W}} \odot {\varvec{z}}_j)\\&+\frac{1}{4}\sum _{i=1}^{n}\left( \text {sech}\left( \frac{y_i -\mu _i}{2}\right) \right) ^{2}{\varvec{z}}_{il}{\varvec{z}}_{ij}, \; l,j = 1, \ldots , K_{\beta };\\ \ddot{{\varvec{\ell }}}(\phi ) = -&\frac{1}{2}\text {tr}\left( -{\varvec{\varGamma }}^{-1}\frac{\partial {\varvec{\varGamma }}}{\partial \phi }{\varvec{\varGamma }}^{-1}\frac{\partial {\varvec{\varGamma }}}{\partial \phi } + {\varvec{\varGamma }}^{-1}\frac{\partial ^2{\varvec{\varGamma }}}{\partial \phi ^2}\right) +\\&\frac{2}{\alpha ^2}{\varvec{V}}^\top \left( -{\varvec{\varGamma }}^{-1}\frac{\partial {\varvec{\varGamma }}}{\partial \phi }{\varvec{\varGamma }}^{-1}\frac{\partial {\varvec{\varGamma }}}{\partial \phi }{\varvec{\varGamma }}^{-1} \right. \\&\left. + {\varvec{\varGamma }}^{-1} \left( \frac{\partial ^2{\varvec{\varGamma }}}{\partial \phi ^2} {\varvec{\varGamma }}^{-1} - \frac{\partial {\varvec{\varGamma }}}{\partial \phi }{\varvec{\varGamma }}^{-1}\frac{\partial {\varvec{\varGamma }}}{\partial \phi }{\varvec{\varGamma }}^{-1} \right) \right) {\varvec{V}}\\ \ddot{{\varvec{\ell }}}(\beta _0\phi ) = -&\frac{2}{\alpha ^2}{\varvec{W}}^\top \varvec{\varGamma }^{-1} \frac{\partial {\varvec{\varGamma }}}{\partial \phi }\varvec{\varGamma }^{-1}{\varvec{V}}\\ \ddot{{\varvec{\ell }}}({\varvec{b}}_j \phi ) = -&\frac{2}{\alpha ^2} ({\varvec{W}}^\top \odot {\varvec{z}}_j^\top ) \varvec{\varGamma }^{-1} \frac{\partial {\varvec{\varGamma }}}{\partial \phi }\varvec{\varGamma }^{-1}{\varvec{V}}, \; j = 1, \ldots , K_{\beta } \end{aligned}$$

In addition, $\ddot{{\varvec{\ell }}}({\varvec{b}}_j\alpha ) = \ddot{{\varvec{\ell }}}(\alpha {\varvec{b}}_j)$, $\ddot{{\varvec{\ell }}}(\phi \alpha ) = \ddot{{\varvec{\ell }}}( \alpha \phi )$, $\ddot{{\varvec{\ell }}}({\varvec{b}}_j\beta _0) = (\ddot{{\varvec{\ell }}}(\beta _0{\varvec{b}}_j))^\top$, $\ddot{{\varvec{\ell }}}(\phi \beta _0) = \ddot{{\varvec{\ell }}}( \beta _0\phi )$ and $\ddot{{\varvec{\ell }}}(\phi {\varvec{b}}_j) = \ddot{{\varvec{\ell }}}({\varvec{b}}_j \phi )$.

Therefore, following Garcia-Papani et al. (2018a), the expected Fisher information matrix given in (16), is formed by the elements

$$\begin{aligned} E(-\ddot{{\varvec{\ell }}}(\alpha ))&= \frac{2n}{\alpha ^2}\\ E(-\ddot{{\varvec{\ell }}}(\alpha \beta _0))&= 0\\ E(-\ddot{{\varvec{\ell }}}(\alpha \varvec{b_j}))&= 0\\ E(-\ddot{{\varvec{\ell }}}(\alpha \phi ))&= \frac{1}{\alpha }\text {tr}\left( {\varvec{\varGamma }}^{-1}\frac{\partial {\varvec{\varGamma }}}{\partial \phi }\right) \\ E(-\ddot{{\varvec{\ell }}}(\beta _0))&= \frac{1}{\alpha ^2}{\varvec{\varGamma }}^{-1}\\ E(-\ddot{{\varvec{\ell }}}(\beta _0\varvec{b_j}))&= \frac{1}{\alpha ^2}{\varvec{1}}^\top {\varvec{\varGamma }}^{-1}{\varvec{z}}_j, \; j = 1, \ldots , K_{\beta }\\ E(-\ddot{{\varvec{\ell }}}(\beta _0\phi ))&= 0\\ E(-\ddot{{\varvec{\ell }}}(\varvec{b_j}\beta _0))&= \frac{1}{\alpha ^2}{\varvec{z}}_j^\top {\varvec{\varGamma }}^{-1}{\varvec{1}}, \; j = 1, \ldots , K_{\beta }\\ E(-\ddot{{\varvec{\ell }}}(\varvec{b_j}))&= \frac{1}{\alpha ^2}{\varvec{z}}_l^\top {\varvec{\varGamma }}^{-1}{\varvec{z}}_j, \; l,j = 1, \ldots , K_{\beta }\\ E(-\ddot{{\varvec{\ell }}}(\varvec{b_j}\phi ))&= 0 \\ E(-\ddot{{\varvec{\ell }}}(\phi ))&= \frac{1}{2}\text {tr}\left( {\varvec{\varGamma }}^{-1}\frac{\partial {\varvec{\varGamma }}}{\partial \phi }{\varvec{\varGamma }}^{-1}\frac{\partial {\varvec{\varGamma }}}{\partial \phi }\right) . \end{aligned}$$