Prediction of functional data with spatial dependence: a penalized approach

Abstract

This paper is focus on spatial functional variables whose observations are a set of spatially correlated sample curves obtained as realizations of a spatio-temporal stochastic process. In this context, as alternative to other geostatistical techniques (kriging, kernel smoothing, among others), a new method to predict the curves of temporal evolution of the process at unsampled locations and also the surfaces of geographical evolution of the variable at unobserved time points is proposed. In order to test the good performance of the proposed method, two simulation studies and an application with real climatological data have been carried out. Finally, the results were compared with ordinary functional kriging.

Appendix

Taking into account the following properties (Harville 1997)

$$\begin{aligned} vec(A)^{\prime }(D\otimes B)vec(C)&= trace(A^{\prime }BCD^{\prime }), \\ trace(A^{\prime }AB)&= trace(ABA^{\prime }), \end{aligned}$$

(6)

the Eq. (5) can be rewritten as

$$\begin{aligned} PSSE(y,\,\alpha )&= trace(C^{\prime }C\Psi )+ trace(Z^{\prime }ZA\Psi A^{\prime }) \\&\quad - 2trace(A\Psi C^{\prime }Z) + \lambda _{1} trace\left( A^{\prime } \left( \Delta _{d}^{U^{\prime }}\Delta _{d}^{U} \otimes I_{q}\right) A\right) \\&\quad + \lambda _{2} trace\left( A^{\prime } \left( I_{p} \otimes \Delta _{d}^{V^{\prime }}\Delta _{d}^{V}\right) A\right) + \lambda _{3} trace\left( A \left( \Delta _{d}^{T^{\prime }}\Delta _{d}^{T}\right) A^{\prime }\right) , \end{aligned}$$

(7)

with $\Psi =\int \phi ^{T} \phi ^{T}$ being the inner product matrix between the basis functions. Next step is to compute the derivatives of Eq. (3) with respect to A. By considering the following properties (Harville 1997),

$$\begin{aligned} \frac{\partial trace(XAX^{\prime })}{\partial X}&= X(A+A^{\prime }), \end{aligned}$$

(8)

$$\begin{aligned} \frac{\partial trace(X^{\prime }AX)}{\partial X}&= (A+A^{\prime })X, \end{aligned}$$

(9)

$$\begin{aligned} \frac{\partial trace(XA)}{\partial X}&= A^{\prime }, \end{aligned}$$

(10)

we have that

$$\begin{aligned} \frac{\partial trace(C^{\prime }C\Psi )}{\partial A}&= 0,\\ \frac{\partial trace(Z^{\prime }ZA\Psi A^{\prime })}{\partial A}&\mathop {=}\limits ^{(3)}Z^{\prime }ZA(\Psi + \Psi ^{\prime })\\&\mathop {=}\limits ^{(symmetry)}2Z^{\prime }ZA\Psi ,\\ \frac{\partial -2trace(A\Psi C^{\prime }Z)}{\partial A}&\mathop {=}\limits ^{(5)}{-}2Z^{\prime }C\Psi ^{\prime }, \\ \frac{\partial \lambda _{1} trace(A^{\prime } (\Delta _{d}^{U^{\prime }}\Delta _{d}^{U}\otimes I_{q}) A)}{\partial A}&\mathop {=}\limits ^{(4)}2\lambda _{1}\left( \Delta _{d}^{U^{\prime }}\Delta _{d}^{U} \otimes I_{q}\right) A, \\ \frac{\partial \lambda _{2} trace(A^{\prime } (I_{p} \otimes \Delta _{d}^{V^{\prime }}\Delta _{d}^{V}) A)}{\partial A}&\mathop {=}\limits ^{(4)}2\lambda _{2}\left( I_{p} \otimes \Delta _{d}^{V^{\prime }}\Delta _{d}^{V}\right) A, \\ \frac{\partial \lambda _{3} trace(A (\Delta _{d}^{T^{\prime }}\Delta _{d}^{T})A^{\prime })}{\partial A}&\mathop {=}\limits ^{(3)}2\lambda _{3}A\left( \Delta _{d}^{T^{\prime }}\Delta _{d}^{T}\right) . \end{aligned}$$

Then, A satisfies the matrix system of linear equations given by

$$\begin{aligned} Z^{\prime }ZA\Psi + \lambda _{1}\left( \Delta _{d}^{U^{\prime }}\Delta _{d}^{U} \otimes I_{q}\right) A + \lambda _{2}\left( I_{p} \otimes \Delta _{d}^{V^{\prime }}\Delta _{d}^{V}\right) A +\lambda _{3}A \left( \Delta _{d}^{T^{\prime }}\Delta _{d}^{T}\right) = Z^{\prime }C\Psi ^{\prime }. \end{aligned}$$

(11)

In order to get the solution to A,  the Kronecker product is used to express Eq. (9) in conventional matrix algebra

$$\begin{aligned} vec(Z^{\prime }ZA\varPsi )&\mathop {=}\limits ^{(1)}\,(\varPsi \otimes (Z^{\prime }Z))vec(A),\\ vec\left( \lambda _{1}\left( \Delta _{d}^{U^{\prime }}\Delta _{d}^{U} \otimes I_{q}\right) A\right)&= vec\left( \lambda _{1}\left( \Delta _{d}^{U^{\prime }}\Delta _{d}^{U} \otimes I_{q}\right) A I_{r}\right) \\&\mathop {=}\limits ^{(1)}\lambda _{1}\left( \Delta _{d}^{U^{\prime }}\Delta _{d}^{U} \otimes I_{q} \otimes I_{r}\right) vec(A),\\ vec\left( \lambda _{2}\left( I_{p} \otimes \Delta _{d}^{V^{\prime }}\Delta _{d}^{V}\right) A\right)&= vec\left( \lambda _{2}\left( I_{p} \otimes \Delta _{d}^{V^{\prime }}\Delta _{d}^{V}\right) A I_{r}\right) \\&\mathop {=}\limits ^{(1)}\lambda _{2}\left( I_{p} \otimes \Delta _{d}^{V^{\prime }}\Delta _{d}^{V} \otimes I_{r}\right) vec(A),\\ vec\left( \lambda _{3}A \left( \Delta _{d}^{T^{\prime }}\Delta _{d}^{T}\right) \right)&= vec\left( \lambda _{3}\left( I_{p} \otimes I_{q}\right) A \left( \Delta _{d}^{T^{\prime }}\Delta _{d}^{T}\right) \right) \\&\mathop {=}\limits ^{(1)}\lambda _{3}\left( I_{p} \otimes I_{q} \otimes \Delta _{d}^{T^{\prime }}\Delta _{d}^{T}\right) vec(A). \\ \end{aligned}$$

Then, Eq. (9) can be re-written as follows

$$\begin{aligned} \left[ \varPsi \otimes (Z^{\prime }Z) + PEN_{d}^{U,V,T}\right] vec(A)= vec(Z^{\prime }C\varPsi ^{\prime }), \end{aligned}$$

where $PEN_{d}^{U,V,T}$ is a P-spline penalty developed by Eilers et al. (2006), which is given by

$$\begin{aligned} PEN_{d}^{U,V,T}&= \lambda _{1}\left( \Delta _{d}^{U^{\prime }}\Delta _{d}^{U}\otimes I_{q}\otimes I_{r}\right) + \lambda _{2}\left( I_{p}\otimes \Delta _{d}^{V^{\prime }}\Delta _{d}^{V} \otimes I_{r}\right) \\&\quad + \lambda _{3}\left( I_{p}\otimes I_{q} \otimes \Delta _{d}^{T^{\prime }}\Delta _{d}^{T}\right) . \end{aligned}$$

Finally, A is given by

$$\begin{aligned} vec(A)=\left[ \varPsi \otimes (Z^{\prime }Z) + PEN_{d}^{U,V,T}\right] ^{-1} vec(Z^{\prime }C\varPsi ^{\prime }). \end{aligned}$$