# Compute Covariance Matrix Observations are averages over congruent rectangular plots that like in a lattice. For extensive observations one needs to multiply the matrix by the $area^2$ where $area$ is the common area of each plot. Various different classes of covariance functions, generalised covariance functions and their derivatives wrt parameters are built into this library. These include the Cauchy and Mat\'ern covariance functions as well as specific sub models such as the Bessel$_0$, Exponential, Bessel$_1$, spline and logarithmic covariance functions. ```r computeV(info, class = "ldt", params, rel.tol = .Machine$double.eps^0.25, abs.tol = rel.tol, cat.level = 0, K = NULL) ``` ## Arguments - `info`: Result of the precompute stage - `class`: The class of covariance functions,"ldt", "bess0", "exp", "bess1", "power", "powerNI", "matern", "spline", "cauchy". Can also be used to compute the derivatives of the covariance matrices for specific models, for example "dbess0", "dexp", "dexp2", "dbess1", "dpowerNI". Can also be used for any isotropic function K, simply define a function K in the workspace that has two arguments, distance and a vector of parameters. Then call `computeV` with `class="special"`. - `params`: Parameters that go with a specific class of models, for the "matern" class it requires an inverse range parameter and a smoothness parameter, for example `params=c(1,0.5)`, this corresponds to the case when `class="exp", params=c(1)`. - `rel.tol`: Relative Tolerance for one dimensional numerical integration - `abs.tol`: Absolute Tolerance for one dimensional numerical integration - `cat.level`: Controls level of time output, takes values 0, 0.5, 1 - `K`: If class="misc" pass your own covariance function K here, see example below ## Examples ```r ## Example for extensive variables - variances of combined plots library(spatialCovariance) nrows <- 1 ncols <- 2 rowwidth <- 1.1 colwidth <- 1.2 rowsep <- 0 colsep <- 0 info <- precompute(nrows,ncols,rowwidth,colwidth,rowsep,colsep) V <- computeV(info,class="matern",params=c(1,1)) info2 <- precompute(nrows=1,ncols=1,rowwidth=rowwidth,colwidth=colwidth*2,0,0) V2 <- computeV(info2,class="matern",params=c(1,1)) c(1,1) (rowwidth * (2*colwidth))^2 * V2 ## Bring in anisotropy V info <- precompute.update(info,aniso=2) ## geometric anisotropy update V <- computeV(info,class="matern",params=c(1,1)) V info <- precompute.update(info,aniso=5) ## geometric anisotropy update V ## Second Example - define your own covariance function, here we use a ## spherical one library(spatialCovariance) K <- function(d,params) { frac <- d/params ret <- rep(0,length(d)) ind <- which(frac<1) if(length(ind)) ret[ind] <- (1 - 2/pi*(frac[ind]*sqrt(1 - frac[ind]^2) + asin(frac[ind]))) return(ret) } dVals <- seq(0,10,l=1001) plot(dVals,K(dVals,8),type="l") lines(dVals,K(dVals,7),col=2) nrows <- 1 ncols <- 3 rowwidth <- 2 colwidth <- 2 rowsep <- 0 colsep <- 0 info <- precompute(nrows,ncols,rowwidth,colwidth,rowsep,colsep) V <- computeV(info,class="misc",params=c(8),K=K) V ## Now if we have a low value of theta_2 we should see that the first ## and third plot are independent as there is a 2 unit gap between ## them, so that term in V will be zero V <- computeV(info,class="misc",params=c(1),K=K) V ## If theta_2 gets a little bigger than 2 then we should see no ## non-zero entries in V V <- computeV(info,class="misc",params=c(2.005),K=K) V ## Check V is positive definite eigen(V)$values ## should all be positive ``` ## Author(s) David Clifford

computeV function