A Kronecker-based covariance specification for spatially continuous multivariate data

Abstract

We propose a covariance specification for modeling spatially continuous multivariate data. This model is based on a reformulation of Kronecker’s product of covariance matrices for Gaussian random fields. The structure holds for different choices of covariance functions with parameters varying in their usual domains. In comparison with classical models from the literature, we used the Matérn correlation function to specify the marginal covariances. We also assess the reparametrized generalized Wendland model as an option for efficient calculation of the Cholesky decomposition, improving the model’s ability to deal with large data sets. The reduced computational time and flexible generalization for increasing number of variables, make it an attractive alternative for modelling spatially continuous data. The proposed model is fitted to a soil chemistry properties dataset, and adequacy measures, forecast errors and estimation times are compared with the ones obtained based on classical models. In addition, the model is fitted to a North African temperature dataset to illustrate the model’s flexibility in dealing with large data. A simulation study is performed considering different parametric scenarios to evaluate the properties of the maximum likelihood estimators. The simple structure and reduced estimation time make the proposed model a candidate approach for multivariate analysis of spatial data.

Appendices

Appendix: Derivatives of the covariance matrix

Based on matrix properties (Wand 2002; Bonat et al. 2020), the score functions with respect to the $\pmb {\beta }$ and $\pmb {\lambda }$ parameters, are given, respectively, by:

$$\begin{aligned}&{\mathcal {U}}_{\pmb {\beta }} = {\mathbf {D}}^\top \varvec{\Sigma }(\pmb {\lambda })^{-1}{\mathbf {r}}(\pmb {\beta }),\nonumber \\&{\mathcal {U}}_{\pmb {\lambda }} = -\dfrac{1}{2} \left\{ \varvec{\Sigma }(\pmb {\lambda })^{-1}-\varvec{\Sigma }(\pmb {\lambda })^{-1} {\mathbf {r}}(\pmb {\beta }){\mathbf {r}}(\pmb {\beta })^\top \varvec{\Sigma }(\pmb {\lambda })^{-1}\right\} \dfrac{\partial \varvec{\Sigma }(\pmb {\lambda })}{\partial \pmb {\lambda }}, \end{aligned}$$

(11)

with ${\mathbf {r}}(\pmb {\beta }) = ({\mathbf {y}}-\pmb {\mu }(\pmb {\beta }))$, ${\mathbf {D}}=\frac{\partial \pmb {\mu }(\pmb {\beta })}{\partial \pmb {\beta }} = \text {Bdiag}({\mathbf {X}}_1, \dots , {\mathbf {X}}_p)$ and the inverse calculation was described earlier.

We achieve the maximum likelihood estimator of $\pmb {\beta }$ by solving ${\mathcal {U}}_{\pmb {\beta }}$, which results in:

$$\begin{aligned} \hat{\pmb {\beta }} = \left( {\mathbf {D}}^\top \varvec{\Sigma }(\pmb {\lambda })^{-1}{\mathbf {D}}\right) ^{-1} \left( {\mathbf {D}}^\top \varvec{\Sigma }(\pmb {\lambda })^{-1}{\mathbf {y}}\right) . \end{aligned}$$

Making similar calculations, we find the Fisher information matrix which, for $\pmb {\beta }$, is given by:

$$\begin{aligned} {\mathcal {F}}_{\pmb {\beta }} = {\mathbf {D}}^\top \varvec{\Sigma }(\pmb {\lambda })^{-1} {\mathbf {D}}. \end{aligned}$$

For $\pmb {\lambda }$, the $(i,j)^{th}$ entry of the Fisher information matrix, is given by:

$$\begin{aligned}{}[{\mathcal {F}}_{\pmb {\lambda }}]_{ij} =\dfrac{1}{2}\text {tr}\left[ {\mathbf {W}}_{\lambda _i}\varvec{\Sigma }(\pmb {\lambda }){\mathbf {W}}_{\lambda _j}\varvec{\Sigma }(\pmb {\lambda })\right] , \end{aligned}$$

where ${\mathbf {W}}_{\lambda _i}=-\partial \varvec{\Sigma }(\pmb {\lambda })^{-1}/\partial \lambda _i$.

Considering $\hat{\pmb {\theta }} = (\hat{\pmb {\beta }}^\top , \hat{\pmb {\lambda }}^\top )^\top$ the maximum likelihood estimator of $\pmb {\theta }$ parameter, the asymptotic distribution of $\hat{\pmb {\theta }}$ is $\hat{\pmb {\theta }} \sim N(\pmb {\theta }, {\mathcal {F}}_\theta ^{-1})$, where ${\mathcal {F}}_{\pmb {\theta }} = \left( \begin{array}{cc} {\mathcal {F}}_{\pmb {\beta }} &{} \pmb {0} \\ \pmb {0} &{} {\mathcal {F}}_{\pmb {\lambda }} \end{array}\right)$ denotes the Fisher information matrix of $\pmb {\theta }$. This result is compatible with the increasing domain regime (Cressie 1993).

The maximum likelihood estimates of $\pmb {\lambda }$ can be found through Newton’s scoring iterative algorithm (Bonat et al. 2020):

$$\begin{aligned} \pmb {\lambda }^{i+1}=\pmb {\lambda }^i-\alpha {\mathcal {F}}_{\pmb {\lambda }}^{-1}{\mathcal {U}}_{\pmb {\lambda }}(\tilde{\pmb {\theta }}; {\mathbf {y}}), \end{aligned}$$

where $\tilde{\pmb {\theta }}=(\hat{\pmb {\beta }}^\top , \pmb {\lambda }^\top )^\top$ and $\alpha$ controls the step length.

Now, let $\rho _r$, for $r = 1,\dots , p(p-1)/2$, denoting the correlation parameters of $\varvec{\Sigma }_{b}$, $\sigma _{i}^2$, $\phi _i$ and $\nu _i$, denoting the variance, scale and smoothness parameters of the marginal-covariance matrix, $\Sigma _{ii}$, for $i=1,\dots ,p$.

The partial derivative of the matrix-valued covariance function $\varvec{\Sigma }$, with respect to each correlation parameter, $\rho _r$, is given by:

$$\begin{aligned}&\dfrac{\partial \varvec{\Sigma }}{\partial \rho _r} = \text {Bdiag}\left( \tilde{\varvec{\Sigma }}_{11}, \tilde{\varvec{\Sigma }}_{22}, \dots , \tilde{\varvec{\Sigma }}_{pp}\right) \left( \dfrac{\partial \varvec{\Sigma }_{b}}{\partial \rho _r} \otimes {\mathbf {I}}\right) \nonumber \\&\quad \text {Bdiag}\left( \tilde{\varvec{\Sigma }}_{11}^\top , \tilde{\varvec{\Sigma }}_{22}^\top , \dots , \tilde{\varvec{\Sigma }}_{pp}^\top \right) . \end{aligned}$$

To obtain the partial derivative with respect to variance parameter, $\sigma _{i}^2$, we will use matrix properties, that is,

$$\begin{aligned}&\dfrac{\partial \varvec{\Sigma }}{\partial \sigma _{i}^2} = \text {Bdiag}\left( {\mathbf {0}}, \ldots , \dfrac{\partial \tilde{\varvec{\Sigma }}_{ii}}{\partial \sigma _{i}^2}, \ldots , {\mathbf {0}}\right) \left( \varvec{\Sigma }_{b} \otimes {\mathbf {I}}\right) \nonumber \\&\quad \text {Bdiag}\left( \tilde{\varvec{\Sigma }}_{11}^\top , \tilde{\varvec{\Sigma }}_{22}^\top , \ldots , \tilde{\varvec{\Sigma }}_{pp}^\top \right) \nonumber \\&\quad + \text {Bdiag}\left( \tilde{\varvec{\Sigma }}_{11}, \tilde{\varvec{\Sigma }}_{22}, \ldots , \tilde{\varvec{\Sigma }}_{pp}\right) \left( \varvec{\Sigma }_{b} \otimes {\mathbf {I}}\right) \nonumber \\&\quad \text {Bdiag}\left( {\mathbf {0}}, \ldots , \dfrac{\partial \tilde{\varvec{\Sigma }}_{ii}^\top }{\partial \sigma _{i}^2}, \ldots , {\mathbf {0}}\right) . \end{aligned}$$

(12)

An analogous procedure to the Eq. (12) can be used to obtain the derivatives with respect to $\phi _i$ and $\nu _i$. Thus, to obtain the derivatives of $\varvec{\Sigma }$ with respect to each parameter, we must calculate the parcial derivatives in (12). Using the result of partial derivatives of Cholesky’s factorization (Särkkä 2013; Bonat and Jørgensen 2016), follows:

$$\begin{aligned} \dfrac{\partial \tilde{\varvec{\Sigma }}_{ii}}{\partial \sigma _{i}^2} = \tilde{\varvec{\Sigma }}_{ii} \Phi \left( \tilde{\varvec{\Sigma }}_{ii}^{-1} \dfrac{\partial \Sigma _{ii}}{\partial \sigma _i^2}\tilde{\varvec{\Sigma }}_{ii}^{-1}\right) , \\ \dfrac{\partial \tilde{\varvec{\Sigma }}_{ii}}{\partial \phi _i} = \tilde{\varvec{\Sigma }}_{ii} \Phi \left( \tilde{\varvec{\Sigma }}_{ii}^{-1} \dfrac{\partial \Sigma _{ii}}{\partial \phi _i}\tilde{\varvec{\Sigma }}_{ii}^{-1}\right) ,\\ \dfrac{\partial \tilde{\varvec{\Sigma }}_{ii}}{\partial \nu _i} = \tilde{\varvec{\Sigma }}_{ii} \Phi \left( \tilde{\varvec{\Sigma }}_{ii}^{-1} \dfrac{\partial \Sigma _{ii}}{\partial \nu _i}\tilde{\varvec{\Sigma }}_{ii}^{-1}\right) , \\ \end{aligned}$$

where $\Phi (.)$ is the strictly lower triangular part of the argument and half of its diagonal.

Appendix: Computational results

Fig. 7

Estimation time for MatSimpler model considering different sample sizes and number of variables

Table 8 Elapsed estimation time (Elps.Time), number of iterations required (N.Iter) and average time per iteration (Time.by.Iter) for each model, considering simulated data from the MatConstr model for different sample sizes

Fig. 8

Average time per iteration (Time.by.Iter) for each model, considering simulated data from the MatConstr model for different sample sizes