Chapter 9 List of symbols and notation

$$\mathbf{s} = (s_1, s_2)$$ location in space.

$$\mathbf{s}_i = (s_{1i}, s_{2i})$$ location in space of observation $$i$$.

$$\bD$$ is the domain of the study region.

$$\Re^d$$ is a real coordinate space of dimension $$d$$.

$$U(\mathbf{s})$$ is a stochastic process.

$$u(\textbf{s}_i)$$ is a realization of $$U(\mathbf{s})$$ at location $$\textbf{s}_i$$.

$$n$$ is the number of sampling locations.

$$h$$ is a distance (usually, between two points).

$$y_i$$ is an observation at location $$\textbf{s}_i$$.

$$\mu$$ is the mean or intercept of the model.

$$\mu_i$$ is the mean of the spatial process at location $$\textbf{s}_i$$.

$$e_i$$ is the error term.

$$\sigma^2_e$$ is the variance of the error term.

$$\Sigma$$ is a variance-covariance matrix.

$$\mathbf{F}_i$$ is a matrix of covariates at location $$\textbf{s}_i$$.

$$\mathbf{\beta}$$ is a vector of coefficients of covariates.

$$\beta_j$$ is the coefficient of covariate $$j$$.

$$\kappa$$ is the scale parameter of Matérn covariance.

$$\nu$$ is the smoothness parameter of Matérn covariance.

$$\parallel . \parallel$$ denotes the Euclidean distance.

$$K_\nu(\cdot)$$ is the modified Bessel function of the second kind.

$$\sigma^2_u$$ is the marginal variance of Matérn process.

$$\mathbf{u}$$ is a sample from a Matérn process.

$$\mathbf{z}$$ is a vector of $$n$$ samples from a standard Gaussian distribution.

$$\mathbf{R}$$ is the Cholesky decomposition of the covariance.

$$E(\cdot)$$ denotes the expectation.

$$Cor(\cdot, \cdot)$$ denotes the correlation.

$$Cor_M(\cdot, \cdot)$$ denotes the correlation of a Matérn process.

$$\mathbf{I}$$ is the identity matrix.

$$\lambda(s)$$ is the intensity of a point process at location $$s$$.

$$S(s)$$ is a continuous spatial Gaussian process (usually with a Matérn covariance).