Estimation of covariance functions by a fully data-driven model selection procedure and its application to Kriging spatial interpolation of real rainfall data

Abstract

In this paper, we propose a data-driven model selection approach for the nonparametric estimation of covariance functions under very general moments assumptions on the stochastic process. Observing i.i.d replications of the process at fixed observation points, we select the best estimator among a set of candidates using a penalized least squares estimation procedure with a fully data-driven penalty function, extending the work in Bigot et al. (Electron J Stat 4:822–855, 2010). We then provide a practical application of this estimate for a Kriging interpolation procedure to forecast rainfall data.

Appendix

In this section we recall some of the inequalities used in the proof of our results. The next theorem is the Proposition 4.3 given in Bigot et al. (2010), which is a $k$-variate extension of the Corollary 5.1 in Baraud (2000) (which is recovered for the particular case $k=1$).

Theorem 5

Given $N,k\in N$ , let $\widetilde{\mathbf {A}}\in \mathbb {R} ^{Nk\times Nk}\backslash \{0\}$ be a non-negative definite and symmetric matrix, and let ${\varepsilon }_{1},\ldots ,{\varepsilon }_{N}$ be i.i.d random vectors in $\mathbb {R}^{k}$, with $\mathbb {E}(\mathbf { \varepsilon }_{1})=\mathbf {0}$ and $\mathbb {V}({\varepsilon }_{1})= \varvec{\Phi }$. Denote ${\varepsilon }=({\varepsilon } _{1}^{\top },\ldots ,{\varepsilon }_{N}^{\top })^{\top }$, $\zeta ( {\varepsilon })=\sqrt{{\varepsilon }^{\top }\widetilde{\mathbf { A}}{\varepsilon }}$, and $\delta _{*}^{2}=\frac{\mathrm {Tr}\left( \widetilde{\mathbf {A}}(\mathbf {I}_{N}\otimes \varvec{\Phi })\right) }{ \mathrm {Tr}\left( \widetilde{\mathbf {A}}\right) }$. Then, for all $p\ge 2$, such that $\mathbb {E}\Vert {\varepsilon }_{1}\Vert _{l_{2}}^{p}<\infty$ it holds that for all $x>0$,

$$\begin{aligned} \mathbb {P}\left( \zeta ({\varepsilon })\ge \delta _{*}^{2} \mathrm {Tr}\left( \widetilde{\mathbf {A}}\right) +2\delta _{*}^{2}\sqrt{ \mathrm {Tr}\left( \widetilde{\mathbf {A}}\right) \rho \left( \widetilde{ \mathbf {A}}\right) x}+\delta _{*}^{2}\rho \left( \widetilde{\mathbf {A}} \right) x\right) \le C(p)\frac{\mathbb {E}\Vert {\varepsilon } _{1}\Vert _{l_{2}}^{p}\mathrm {Tr}\left( \widetilde{\mathbf {A}}\right) }{ \delta _{*}^{p}\rho \left( \widetilde{\mathbf {A}}\right) x^{\frac{p}{2}}} , \end{aligned}$$

where $\rho \left( \widetilde{\mathbf {A}}\right)$ is the spectral norm of $\widetilde{\mathbf {A}}$.

The following result is the Corollary 4.2 that appears in Bigot et al. (2010), which constitutes also a natural extension of Corollary 3.1 in Baraud (2000), providing a similar bound as in Gendre (2008).

Theorem 6

Let $q>0$ be given such that there exists $p>2(1+q)$ satisfying $\mathbb {E}\Vert \varepsilon _{i}\Vert _{l_{2}}^{p}<\infty$. Then, for some constants $K(\theta )>1$ we have that

$$\begin{aligned} \left( \mathbb {E}\Vert \mathbf {f}-\widetilde{\mathbf {f}}\Vert _{N}^{2q}\right) ^{\frac{1}{q}}\le 2^{\left( \frac{1}{q}-1\right) _{+}} \left[ K(\theta )\inf _{m\in \mathcal {M}}\left( \Vert \mathbf {f}-\mathbf {P} _{m}\mathbf {f}\Vert _{N}^{2}+\frac{\delta _{m}^{2}D_{m}}{N}\right) +\frac{ \Delta _{p}}{N}\delta _{sup}^{2}\right] , \end{aligned}$$

where

$$\begin{aligned} \Delta _{p}^{q}=C(p,q,\theta )\mathbb {E}\Vert {\varepsilon }_{i}\Vert _{l_{2}}^{p}\left( \sum _{m\in \mathcal {M}}\delta _{m}^{-p}D_{m}^{-\left( \frac{p}{2}-1-q\right) }\right) . \end{aligned}$$

Proposition 7

(Hermite Hadamard’s Inequality) For all convex functions $f\!:\![a,b]\!\rightarrow \! \mathbb {R}$ is known that:

$$\begin{aligned} f\left( \frac{a+b}{2}\right) \le \frac{1}{b-a}\int \limits _{a}^{b}f(x)dx\le \frac{ f(a)+f(b)}{2}. \end{aligned}$$

Now we recall two moment inequalities for sum of independent centered random variables, which are repeatedly used throughout this paper.

Theorem 8

(Rosenthal’s Inequality) Let $U_{1},U_{2},\ldots U_{n}$ be independent centered random variables with values in $\mathbb {R}$. Then for any $p\ge 2$ we have:

$$\begin{aligned} \mathbb {E}\left[ \left| \sum _{i=1}^{n}U_{i}\right| ^{p}\right] \le C(p)\left( \sum _{i=1}^{n}\mathbb {E}[|U_{i}|^{p}]+\left( \sum _{i=1}^{n} \mathbb {E}[U_{i}^{2}]\right) ^{\frac{p}{2}}\right) . \end{aligned}$$

For the proof of this inequality, we refer to Petrov (1995). The next result explores the case where $p\in [1,2]$. To our knowledge the result is due to Bahr and Esseen (1965).

Theorem 9

Let $U_{1},U_{2},\ldots ,U_{n}$ be independent centered random variables with values $\mathbb {R}$. For any $p$ with $p\in [1,2]$ it holds that:

$$\begin{aligned} \mathbb {E}\left[ \left| \sum _{i=1}^{n}U_{i}\right| ^{p}\right] \le 8\sum _{i=1}^{n}\mathbb {E}[|U_{i}|^{p}]. \end{aligned}$$