Testing the type of non-separability and some classes of space-time covariance function models

Abstract

In statistical space-time modeling, the use of non-separable covariance functions is often more realistic than separable models. In the literature, various tests for separability may justify this choice. However, in case of rejection of the separability hypothesis, none of these tests include testing for the type of non-separability of space-time covariance functions. This is an important and further significant step for choosing a class of models. In this paper a method for testing positive and negative non-separability is given; moreover, an approach for testing some well known classes of space-time covariance function models has been proposed. The performance of the tests has been shown using real and simulated data.

Appendices

Appendix 1: Test for the type of non-separability

If the non-separability assumption is reasonable, the type of non-separability have to be investigated through the following steps:

• 1. Estimation of the space-time covariance surface ($\widehat{C}({\mathbf{h}},u)$ for a set of space-time lags $({\mathbf{h}},u)$) and of the corresponding empirical space-time correlation function ($\widehat{\rho }({\mathbf{h}},u)$).

• 2. Computation of the non-separability index ratio $\widehat{r}({\mathbf{h}},u)$ (i.e. from definition (12), the ratio computed between the empirical space-time correlation and the product of the corresponding sample spatial and temporal marginals).

• 3. Graphical representation of the sample non-separability ratio through box-plots, classified for spatial lags and temporal lags. The inspection of box-plots could help to decide for a right tailed test (18) or a left tailed test (19).

• 4. Fix the null hypothesis, i.e. the type of the test: a right tailed test for testing the negative non-separability, a left tailed test for testing the positive non-separability.

• 5. Selection of spatial and temporal lags ($\Lambda$) with sample non-separability ratios much greater/less than one. The test statistic is always computed for lags characterized by the same type of non-separability, in such a way that compensations among terms of the test statistic with different signs are avoided.

• 6. Computation of the test statistic, which requires the estimation of ${\mathbf {\Sigma }}, \widehat{{\mathbf {G}}}, {\mathbf {f}}(\widehat{{\mathbf {G}}})$ and the construction of the contrast matrix A (Li et al. 2007), and of the corresponding p value.

• 7. If the p value associated with the test statistic is less than 0.05 the null hypothesis is rejected.

As explained in Li et al. (2007), De Iaco et al. (2016), the analyst can compute the tests on a set of spatial and temporal lags chosen on the basis of different reasons. In general, the analyst can start with few pairs of spatial locations (spatial lags), which are spread out over the domain and are representative of the spatio-temporal correlation of the data set. In some empirical cases, the selection of the pairs of spatial locations might be based on intrinsic characteristics of the phenomenon under study (some examples for wind data are in the above mentioned papers). Moreover, the lags or pairs of locations can be chosen by taking into account the pairs of points with the smallest or, alternatively, with the largest ratio between the east-west component and the north-south component of the spatial lag, as well as the pairs of points with the shortest distance $\Vert {\mathbf{h}}\Vert$. Regarding the temporal lags, it is common to use short temporal lags characterized by strong correlation. In addition to the above mentioned insights, the analyst can select, for the test on the type on non-separability, spatial and temporal lags for which the sample non-separability ratios are much greater/less than one. For this aim, an useful support is given by the inspection of box-plots of the sample non-separability ratios, classified by spatial and temporal lags.

Appendix 2: Proof of Proposition 4.1

Given $d({\mathbf{h}}_1,u_2)=C({\mathbf {h}}_1, u_2)C({\mathbf {0}}, 0) - C({\mathbf {h}}_1, 0)C({\mathbf {0}}, u_2)$, it is easy to show that

$$\begin{aligned} d({\mathbf{h}}_i,u_j)={\sigma ^4\over 4}[\rho _{s,1}({\mathbf{h}}_i)-\rho _{s,2}({\mathbf{h}}_i) ][\rho _{t,1}(u_j)-\rho _{t,2}(u_j)], \qquad i,j=1,2. \end{aligned}$$

(57)

Then for any set of spatial lags ${\mathbf {h}}_1$ and ${\mathbf {h}}_2$ and temporal lags $u_1$ and $u_2$ the following properties are satisfied:

$$\begin{aligned}&{d({\mathbf {h}}_1,u_1) \over {d({\mathbf {h}}_2, u_1) }}- {d({\mathbf {h}}_1,u_2) \over {d({\mathbf {h}}_2, u_2) }} =0, \qquad \forall \, u_1, u_2, \end{aligned}$$

(58)

$$\begin{aligned}&{d({\mathbf {h}}_1,u_1) \over {d({\mathbf {h}}_1, u_2) }}- {d({\mathbf {h}}_2,u_1) \over {d({\mathbf {h}}_2, u_2) }}=0, \qquad \forall \, {\mathbf {h}}_1,{\mathbf {h}}_2. \end{aligned}$$

(59)

Appendix 3: Proof of Proposition 4.2

Part 1 (if)

For the proof, consider that because of (28) the following properties are satisfied for any set of spatial lags ${\mathbf {h}}_1$, ${\mathbf {h}}_2$, and ${\mathbf {h}}_3$

$$\begin{aligned}&\gamma ({\mathbf {h}}_1,u)-\gamma ({\mathbf {h}}_2, u)=[\gamma ({\mathbf {h}}_1,0) - \gamma ({\mathbf {h}}_2,0)][1-k\gamma ({\mathbf {0}},u)], \qquad \forall \, u, \end{aligned}$$

(60)

$$\begin{aligned}&\gamma ({\mathbf {h}}_3,u)-\gamma ({\mathbf {h}}_2,u)=[\gamma ({\mathbf {h}}_3,0) - \gamma ({\mathbf {h}}_2,0)][1-k\gamma ({\mathbf {0}},u)], \qquad \forall \, u, \end{aligned}$$

(61)

or equivalently

$$\begin{aligned} {\gamma ({\mathbf {h}}_3,u)-\gamma ({\mathbf {h}}_2,u) \over {\gamma ({\mathbf {h}}_3,0) - \gamma ({\mathbf {h}}_2,0)}}- {\gamma ({\mathbf {h}}_1,u)-\gamma ({\mathbf {h}}_2,u) \over {\gamma ({\mathbf {h}}_1,0) - \gamma ({\mathbf {h}}_2,0)}}=0, \qquad \forall \, u. \end{aligned}$$

(62)

Recalling the relationship between $\gamma$ and C, the expression (62) can be rewritten as:

$$\begin{aligned} {C({\mathbf {h}}_3,u)-C({\mathbf {h}}_2,u) \over {C({\mathbf {h}}_3,0) - C({\mathbf {h}}_2,0)}}- {C({\mathbf {h}}_2,u)-C({\mathbf {h}}_1,u) \over {C({\mathbf {h}}_2,0) - C({\mathbf {h}}_1,0)}}=0, \qquad \forall \, u. \end{aligned}$$

(63)

Analogously, for the proof of the property in (30).

Part 2 (only if)

Let C be a space-time covariance function. The ratio of increments in (29), where the spatial marginal $C({\mathbf {h}} ,0)$ is interpreted as an independent variable, is constant with respect to this last variable and depends solely on u. This implies that the space-time covariance C is a linear function of the spatial marginal. Analogously, the incremental ratio in (30), where the temporal marginal $C(\mathbf 0 ,u)$ is interpreted as an independent variable, is constant with respect to this last variable and depends solely on ${\mathbf {h}}$. This implies that the space-time covariance function C is a linear function of the temporal marginal. In conclusion, the space-time covariance function C is a linear function of both marginals, thus the most general form that ensures this feature is given by the construction (27).

Appendix 4: Test for some classes of covariance function models

The appropriateness of a class of space-time covariance function models can be investigated through the following steps:

• 1. Estimation of the space-time covariance surface ($\widehat{C}({\mathbf{h}},u)$ for a set of space-time lags $({\mathbf{h}},u)$) and of the corresponding empirical space-time correlation function ($\widehat{\rho }({\mathbf{h}},u)$).

• 2. Fix the class of models to be tested on the basis of empirical evidences (such as type of non-separability, behavior at the origin and to infinity) and the corresponding null hypothesis.

• 3. Selection of spatial and temporal lags ($\Lambda$). Note that the integrated product, Gneiting and Cressie class of models have to be tested at least on one triplet of spatial lags ${\mathbf {h}}_{i_1}, {\mathbf {h}}_{i_2}, {\mathbf {h}}_{i_3}$, such that $||{\mathbf {h}}_{i_1}||^{2\gamma }- ||{\mathbf {h}}_{i_2}||^{2\gamma } = ||{\mathbf {h}}_{i_2}||^{2\gamma }-||{\mathbf {h}}_{i_3}||^{2\gamma }$, ${i_1}\ne {i_2}\ne {i_3}$ or at least on one triplets of temporal lags ${u}_{j_1}, {u}_{j_2}, {u}_{j_3}$, such that $u_{j_1}^{2\alpha }-u_{j_2}^{2\alpha }=u_{j_2}^{2\alpha }-u_{j_3}^{2\alpha }$, ${j_1}\ne {j_2}\ne {j_3}$.

• 4. Computation of the test statistic which requires the estimation of ${\mathbf {\Sigma }}, \widehat{{\mathbf {G}}}, {\mathbf {f}}(\widehat{{\mathbf {G}}})$ and the construction of the contrast matrix A (Li et al. 2007).

• 5. If the p value associated with the test statistic is less than 0.05 the null hypothesis is rejected.