# Covariance matrix for spatio-temporal model It computes the spatio-temporal covariance matrix for balanced data, i.e., when we have the same temporal indexes per location. To compute the spatial correlation it provides 5 functions: exponential, gaussian, matern, spherical and power exponential. To compute the temporal correlation is used an autocorrelation function of an AR(1) process. ```r CovarianceM(phi, rho, tau2, sigma2, distSpa, disTemp, kappa, type.S) ``` ## Arguments - `phi`: value of the spatial scaling parameter. - `rho`: value of the time scaling parameter. - `tau2`: value of the the nugget effect parameter. - `sigma2`: value of the partial sill. - `distSpa`: $n x n$ spatial distance matrix without considering repetitions. - `disTemp`: $T x T$ temporal distance matrix without considering repetitions. - `kappa`: parameter for all spatial covariance functions. In the case of exponential, gaussian and spherical function $\kappa$ is equal to zero. For the power exponential function $\kappa$ is a number between 0 and 2. For the matern correlation function is upper than 0. - `type.S`: type of spatial correlation function: '`exponential`' for exponential, '`gaussian`' for gaussian, '`matern`' for matern, '`pow.exp`' for power exponential and '`spherical`' for spherical function, respectively. See the analytical form of these functions in `EffectiveRange`. ## Returns The function returns the $nT x nT$ spatio-temporal covariance matrix for balanced data. ## Examples ```r # Initial parameter values phi <- 5; rho <- 0.45 tau2 <- 0.80; sigma2 <- 2 # Simulating data n1 <- 10 # Number of spatial locations n2 <- 5 # Number of temporal index set.seed(1000) x.co <- round(runif(n1,0,10),5) # X coordinate y.co <- round(runif(n1,0,10),5) # Y coordinate coord <- cbind(x.co,y.co) # Cartesian coordinates without repetitions time <- as.matrix(seq(1,n2)) # Time index without repetitions # Covariance matrix Ms <- as.matrix(dist(coord)) # Spatial distances Mt <- as.matrix(dist(time)) # Temporal distances Cov <- CovarianceM(phi,rho,tau2,sigma2,distSpa=Ms,disTemp=Mt,kappa=0,type.S="exponential") ``` ## Author(s) Katherine L. Valeriano, Victor H. Lachos and Larissa A. Matos

CovarianceM function