Spatial-depth functional estimation of ocean temperature from non-separable covariance models

Abstract

Spatial-depth functional regression is applied for the estimation of ocean temperature, with projection onto the eigenvectors of the empirical covariance operator of the functional response (i.e., onto the Empirical Orthogonal Functions in space and depth). Moment-based estimation is performed to approximate the regression operators in the subspace generated by the empirical eigenvectors associated with nonnull eigenvalues. In addition, Bayesian estimation is performed to approximate the regression operators in the subspace generated by the empirical eigenvectors associated with almost null eigenvalues. The cross-validation results obtained, together with the spatial-depth residual correlation analysis carried out on a real data set for the South Atlantic area, to the east of Argentina and the Falkland Islands, represent an improvement on those provided by the wavelet-based approach recently proposed in Fernández-Pascual (Stoch Environ Res Risk Assess 30:523–557, 2016).

Appendix

In this section we show the computations to be developed to obtain the Bayes estimators of the coefficients of the regression operators with respect to the eigenvectors associated with the smallest eigenvalues of the covariance operator of the response.

For each $j=M+1, \ldots , H,$ the equation system (24) can be rewritten as

$$\begin{aligned} 0= & {} -\frac{1}{2\rho _{\widehat{\varphi }_{j},\widehat{\varphi }_{j}}}\sum _{t=1}^{T} -2\varvec{\Psi }^{*}_{j}\mathbf {X}_{t}\left( \varvec{\Psi }^{*}_{j}\mathbf {Y}_{t}-\lambda _{j}(\mathcal {K}_{1}) \varvec{\Psi }^{*}_{j}\mathbf {X}_{t}-\lambda _{j}(\mathcal {K}_{2})\varvec{\Psi }^{*}_{j}\mathbf {Z}_{t}^{1}-\lambda _{j}(\mathcal {K}_{3})\varvec{\Psi }^{*}_{j}\mathbf {Z}_{t}^{2}\right) \nonumber \\+ & {} \frac{a_{j1}-1}{\lambda _{j}(\mathcal {K}_{1})}-\frac{b_{j1}-1}{1-\lambda _{j}(\mathcal {K}_{1})} \nonumber \\ 0= & {} -\frac{1}{2\rho _{\widehat{\varphi }_{j},\widehat{\varphi }_{j}}}\sum _{t=1}^{T} -2\varvec{\Psi }^{*}_{j}\mathbf {Z}_{t}^{1}\left( \varvec{\Psi }^{*}_{j}\mathbf {Y}_{t}-\lambda _{j}(\mathcal {K}_{1}) \varvec{\Psi }^{*}_{j}\mathbf {X}_{t}-\lambda _{j}(\mathcal {K}_{2})\varvec{\Psi }^{*}_{j}\mathbf {Z}_{t}^{1}-\lambda _{j}(\mathcal {K}_{3})\varvec{\Psi }^{*}_{j}\mathbf {Z}_{t}^{2}\right) \nonumber \\+ & {} \frac{a_{j2}-1}{\lambda _{j}(\mathcal {K}_{2})}-\frac{b_{j2}-1}{1-\lambda _{j}(\mathcal {K}_{2})} \nonumber \\ 0= & {} -\frac{1}{2\rho _{\widehat{\varphi }_{j},\widehat{\varphi }_{j}}}\sum _{t=1}^{T} -2\varvec{\Psi }^{*}_{j}\mathbf {Z}_{t}^{2}\left( \varvec{\Psi }^{*}_{j}\mathbf {Y}_{t}-\lambda _{j}(\mathcal {K}_{1}) \varvec{\Psi }^{*}_{j}\mathbf {X}_{t}-\lambda _{j}(\mathcal {K}_{2})\varvec{\Psi }^{*}_{j}\mathbf {Z}_{t}^{1}-\lambda _{j}(\mathcal {K}_{3})\varvec{\Psi }^{*}_{j}\mathbf {Z}_{t}^{2}\right) \nonumber \\+ & {} \frac{a_{j3}-1}{\lambda _{j}(\mathcal {K}_{3})}-\frac{b_{j3}-1}{1-\lambda _{j}(\mathcal {K}_{3})}. \end{aligned}$$

(26)

Thus, for each $j=M+1, \ldots , H,$ the Bayes estimator of the parameter vector $(\lambda _{j}(\mathcal {K}_{1}),\lambda _{j}(\mathcal {K}_{2}),\lambda _{j}(\mathcal {K}_{3})),$ the generalised maximum likelihood estimator given by the mode of the posterior, is obtained as the solution to the following system of non-linear equations:

$$\begin{aligned} 0= & \, \frac{\alpha _{T,1,2} }{\rho _{\widehat{\varphi }_{j},\widehat{\varphi }_{j}}}-\frac{\lambda _{j}(\mathcal {K}_{1})}{\rho _{\widehat{\varphi }_{j},\widehat{\varphi }_{j}}}\beta _{T,2}- \frac{\lambda _{j}(\mathcal {K}_{2})}{\rho _{\widehat{\varphi }_{j},\widehat{\varphi }_{j}}}\alpha _{T,2,3}-\frac{\lambda _{j}(\mathcal {K}_{3})}{\rho _{\widehat{\varphi }_{j},\widehat{\varphi }_{j}}}\alpha _{T,2,4} +\frac{a_{j1}-1}{\lambda _{j}(\mathcal {K}_{1})}-\frac{b_{j1}-1}{1-\lambda _{j}(\mathcal {K}_{1})}\nonumber \\ 0= & \, \frac{\alpha _{T,1,3} }{\rho _{\widehat{\varphi }_{j},\widehat{\varphi }_{j}}}-\frac{\lambda _{j}(\mathcal {K}_{2})}{\rho _{\widehat{\varphi }_{j},\widehat{\varphi }_{j}}}\beta _{T,3} -\frac{\lambda _{j}(\mathcal {K}_{1})}{\rho _{\widehat{\varphi }_{j},\widehat{\varphi }_{j}}}\alpha _{T,2,3}-\frac{\lambda _{j}(\mathcal {K}_{3})}{\rho _{\widehat{\varphi }_{j},\widehat{\varphi }_{j}}}\alpha _{T,3,4} +\frac{a_{j2}-1}{\lambda _{j}(\mathcal {K}_{2})}-\frac{b_{j2}-1}{1-\lambda _{j}(\mathcal {K}_{2})}\nonumber \\ 0= & \, \frac{\alpha _{T,1,4} }{\rho _{\widehat{\varphi }_{j},\widehat{\varphi }_{j}}}-\frac{\lambda _{j}(\mathcal {K}_{3})}{\rho _{\widehat{\varphi }_{j},\widehat{\varphi }_{j}}}\beta _{T,4} -\frac{\lambda _{j}(\mathcal {K}_{1})}{\rho _{\widehat{\varphi }_{j},\widehat{\varphi }_{j}}}\alpha _{T,2,4}-\frac{\lambda _{j}(\mathcal {K}_{2})}{\rho _{\widehat{\varphi }_{j},\widehat{\varphi }_{j}}}\alpha _{T,3,4} +\frac{a_{j3}-1}{\lambda _{j}(\mathcal {K}_{3})}-\frac{b_{j3}-1}{1-\lambda _{j}(\mathcal {K}_{3})}, \end{aligned}$$

(27)

where

$$\begin{aligned} \alpha _{T,1,2}= & {} \sum _{t=1}^{T}\varvec{\Psi }^{*}_{j}\mathbf {X}_{t}\varvec{\Psi }^{*}_{j}\mathbf {Y}_{t}\quad \alpha _{T,2,3} =\sum _{t=1}^{T}\varvec{\Psi }^{*}_{j}\mathbf {Z}_{t}^{1}\varvec{\Psi }^{*}_{j}\mathbf {X}_{t}\quad \alpha _{T,2,4} =\sum _{t=1}^{T}\varvec{\Psi }^{*}_{j}\mathbf {Z}_{t}^{2}\varvec{\Psi }^{*}_{j}\mathbf {X}_{t} \nonumber \\ \alpha _{T,1,3}= & {} \sum _{t=1}^{T}\varvec{\Psi }^{*}_{j}\mathbf {Z}_{t}^{1}\varvec{\Psi }^{*}_{j}\mathbf {Y}_{t}\quad \alpha _{T,3,4} =\sum _{t=1}^{T}\varvec{\Psi }^{*}_{j}\mathbf {Z}_{t}^{2}\varvec{\Psi }^{*}_{j}\mathbf {Z}_{t}^{1}\quad \nonumber \\ \alpha _{T,1,4}= & {} \sum _{t=1}^{T}\varvec{\Psi }^{*}_{j}\mathbf {Z}_{t}^{2}\varvec{\Psi }^{*}_{j}\mathbf {Y}_{t} \nonumber \\ \beta _{T,2}= & {} \sum _{t=1}^{T} [\varvec{\Psi }^{*}_{j}\mathbf {X}_{t}]^{2}\quad \beta _{T,3}=\sum _{t=1}^{T} [\varvec{\Psi }^{*}_{j}\mathbf {Z}_{t}^{1}]^{2}\quad \beta _{T,4}=\sum _{t=1}^{T} [\varvec{\Psi }^{*}_{j}\mathbf {Z}_{t}^{2}]^{2}. \end{aligned}$$

(28)

The non-linear system of Eq. (27), in the parameters $\lambda _{j}(\mathcal {K}_{1}),$ $\lambda _{j}(\mathcal {K}_{2})$ and $\lambda _{j}(\mathcal {K}_{3}),$ for each $j=M+1, \ldots , H,$ can be solved by applying the Multivariate Newton–Raphson Method.