maxlik.cov.sp: Determines maximum likelihood estimates of covariance parameters

Description

Estimates covariance parameters of spatial covariance functions using maximum likelihood or restricted maximum likelihood. See cov.sp for more details of covariance functions to be estimated.

Usage

maxlik.cov.sp(X, y, coords, sp.type = "exponential", 
	    range.par = stop("specify range.par argument"), 
	    error.ratio = stop("specify error.ratio argument"), 
	    smoothness = 0.5,  
	    D = NULL, reml = TRUE, lower = NULL, upper = NULL, 
	    control = list(trace = TRUE), optimizer="nlminb")

Value

Returns a list with the following elements:

sp.type: The covariance form used.
sp.par: A vector containing the estimated variance of the hidden process and the spatial dependence.
error.var: The estimated error variance.
smoothness: The smoothness of the matern covariance function.
par: The final values of the optimization parameters. Note that these will not necessarily match sp.par, error.var, and smoothness because of the reparameterization.
convergence: Convergence message from nlminb.
message: Message from nlminb.
iterations: Number of iterations for optimization to converge.
evaluations: Evaluations from nlminb.

Arguments

X: A numeric matrix of size \(n \times k\) containing the design matrix of the data locations.
y: A vector of length \(n\) containing the observed responses.
coords: A numeric matrix of size \(n \times d\) containing the locations of the observed responses.
sp.type: A character vector specifying the spatial covariance type. Valid types are currently exponential, gaussian, matern, and spherical.
range.par: An initial guess for the spatial dependence parameter.
error.ratio: A value non-negative value indicating the ratio error.var/sp.par[1].
smoothness: A positive number indicating the smoothness of the matern covariance function, if applicable.
D: The Euclidean distance matrix for the coords matrix. Must be of size \(n \times n\).
reml: A boolean value indicating whether restricted maximum likelihood estimation should be used. Defaults to TRUE.
lower: A vector giving lower bounds for the covariance parameters sp.par[2], error.ratio, and smoothness (when the model is matern). Order matters! If not given defaults to an upper bound of Inf for sp.par[2], 1 for error.ratio, and 10 for smoothness.
upper: A vector giving upper bounds for the covariance parameters sp.par[2], error.ratio, and smoothness (when the model is matern). Order matters! If not given defaults to an upper bound of Inf for sp.par[2], 1 for error.ratio, and 10 for smoothness.
control: A list giving tuning parameters for the nlminb function. See nlminb for more details.
optimizer: A vector describing the optimization function to use for the optimization. Currently, only nlminb is an acceptable value.

Author

Joshua French

Details

When doing the numerical optimizaiton, the covariance function is reparameterized slightly to speedup computation. Specifically, the variance parameter for the process of interest,sp.par[1], is profiled out, and the error.var parameter is parameterized as sp.par[1] * error.ratio, where error.ratio = error.var/sp.par[1].

Examples

Run this code

#generate 20 random (x, y) coordinates
coords <- matrix(rnorm(20), ncol = 2)

#create design matrix
X <- cbind(1, coords)

#create mean for observed data to be generated
mu <- X %*% c(1, 2, 3)

#generate covariance matrix
V <- exp(-dist1(coords))

#generate observe data
y <- rmvnorm(mu = mu, V = V)

#find maximum likelihood estimates of covariance parameters
maxlik.cov.sp(X = X, y = y, coords = coords,
    sp.type = "exponential", range.par = 1, error.ratio = 0, 
    reml = TRUE)

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