# 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).