Model implementation Sample Clauses

Model implementation. First we discuss the choice of the spatial kernel vector (s). Several types of kernel functions have been suggested, including Gaussian kernels (▇▇▇▇▇▇ et al. 2001), harmonic functions (e.g., ▇▇▇▇▇▇ et al. 2007) and bisquare functions (▇▇▇▇ et al. 2012). In the proposed model we have considered a Gaussian kernel specified as k(s) = exp{ -(s - ck)’ -1 (s - ck)/2}, k = 1, … , p (3.10) where ck denotes the center of the kernel and  determines the shape. The number of kernels p, their locations and shapes must be chosen. These choices are often based on the presence of prior information such as smoothness and spatial dependence related to the spatial process under study (▇▇▇▇▇▇ et al. 2001). If we choose spherically shaped kernels, i.e.,  = kI2 on R2 and k > 0, and the centers belong to a regular grid over an unbounded domain, the resulting spatial process approximates a covariance function of a stationary isotropic process when the number of kernels p is very large. Alternatively, a geometrically anisotropic process may be obtained if we choose non- spherical Gaussian kernels. One way to assess the shape of S is to perform variogram analyses for different directions (see, e.g., ▇▇▇▇▇▇▇ 1993). Figure 3.1a shows an example of such analysis for the region of the tropical Atlantic. The variogram suggests that isotropy is a plausible assumption for Mt(s) at small distances since the patterns are not largely different from each other. If the analysis is confirmed at several time points suggests, equally-spaced Gaussian kernels can be used (see Figure 3.1b) We recommend to always perform sensitivity analyses for different choices of p. Figure 3.1 - Example of kernel specification for gridded data for the region of the tropical Atlantic. (a) Empirical variogram for a given time t for four different directions (black solid: 0, red dashed: 45, gray dotted 90, blue dashed 135). The variogram was analyzed using the robust estimator by ▇▇▇▇▇▇▇ (1993) and indicates no anisotropy. (b) Spherically-shaped 36 equally-spaced Gaussian kernels. Red crosses indicate the centers of the kernels.