# For the following questions, answer T (true) or F (false). (a) For a Matérn covariance model, the larger ν (the smoothness parameter) is, the smoother the spatial data is over spac... For the following questions, answer T (true) or F (false). (a) For a Matérn covariance model, the larger ν (the smoothness parameter) is, the smoother the spatial data is over space. (b) When ν = 0.5, the Matérn covariance function is the same as the exponential covariance function. (c) If a certain spatial data set is assumed to be geometrically anisotropic, it is stationary. (d) When performing Kriging, kriging weights (i.e. weight value assigned to each observation) must be non-negative. (e) For a geometrically anisotropic random field, its variance does not change over its spatial domain. (f) The unit of spatial range is the same as the unit of the spatial distance. (g) Suppose we have spatial data, Z1, Z2, ..., Zn, observed from n locations, s1, s2, ..., sn, respectively. If there is no spatial dependence in this data set, predicted value for the spatial variable using Kriging method is equal to n1 (Z1 + · · · + Zn). (h) If we perform Kriging with a spatial regression model, Z(s) = β0 + β1X(s) + e(s) (with unknown parameters β0 and β1), it is ordinary Kriging. (i) A Gaussian random field with zero mean is always isotropic. (j) If we use Gaussian random field model for spatial data, its Kriging predictors also follow Gaussian distribution.

#### Understand the Problem

The question is asking to determine whether each of the statements related to Matérn covariance models and Kriging is true or false. Each statement must be evaluated in the context of spatial statistics and covariance functions without the need for explanation.

#### Answer

T, T, F, F, F, T, F, F, F, T

(a) T, (b) T, (c) F, (d) F, (e) F, (f) T, (g) F, (h) F, (i) F, (j) T

##### Answer for screen readers

(a) T, (b) T, (c) F, (d) F, (e) F, (f) T, (g) F, (h) F, (i) F, (j) T

#### More Information

The Matérn covariance model's smoothness parameter ν determines the smoothness of spatial data. When ν = 0.5, it corresponds to the exponential covariance function. Geometrical anisotropy does not imply stationarity. Kriging weights can be negative, and variance in anisotropic fields can change spatially. Gaussian random fields may not always be isotropic.

#### Tips

A common mistake is assuming kriging weights must always be non-negative, which is not a requirement for the method. Additionally, geometric anisotropy does not imply stationarity, as these concepts are distinct.

#### Sources

- The Matérn Model: A Journey through Statistics - arxiv.org
- Bayesian Smoothing with Gaussian Processes - jstatsoft.org