SpatialTools: maxlik.cov.st – R documentation

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maxlik.cov.st

Determines maximum likelihood estimates of covariance parameters

Description

Estimates covariance parameters of spatio-temporal covariance functions using maximum likelihood or restricted maximum likelihood. See cov.st for more details of covariance functions to be estimated. The covariance function is reparameterized slightly to speedup computation. Specifically, the variance parameter for the hidden process, sp.par[1], is profiled out and the error.var parameter is parameterized as sp.par[1] * error.ratio.

Usage

maxlik.cov.st(X, y, coords, time, sp.type = "exponential", 
	    range.par = stop("specify range.par argument"), 
	    error.ratio = stop("specify error.ratio argument"), 
	    smoothness = 0.5, t.type = "ar1", t.par = .5, D = NULL, T = NULL, 
	    reml = TRUE, lower = NULL, upper = NULL, control = list(trace = TRUE), 
	    optimizer="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.
`time`	A numeric vector of length n containing the time at which the responses were observed.
`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 variance of the error term.
`t.type`	A character vector indicating the spatial covariance type. Only `ar1` is currently available.
`t.par`	A value specifying the temporal dependence parameter of the ar1 process.
`D`	The Euclidean distance matrix for the coords matrix. Must be of size n \times n.
`T`	The Euclidean distance matrix for the time 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 a lower bound of .001 for `sp.par[2]`, 0 for `error.ratio`, and .001 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.

Details

When doing the numerical optimization, 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].

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`.

Author(s)

Joshua French

Examples

#Generate locations and observed times
coords <- matrix(rnorm(40), ncol = 2)
time <- rep(1:2, each = 10)

#Calculate distance matrix for time vector
T <- dist1(matrix(time))

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

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

#generate covariance matrix for spatio-temporal data
V <- exp(-dist1(coords)) * .25^T

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

maxlik.cov.st(X = X, y = y, coords = coords, time = time,
    sp.type = "exponential", range.par = 1, error.ratio = 0, 
    t.type = "ar1", t.par = .5, reml = TRUE)

SpatialTools

Tools for Spatial Data Analysis

v1.0.4

GPL (>= 2)

Authors

Joshua French <joshua.french@ucdenver.edu>

Initial release