linear.model.MLE {PrevMap} | R Documentation |
Maximum Likelihood estimation for the geostatistical linear Gaussian model
Description
This function performs maximum likelihood estimation for the geostatistical linear Gaussian Model.
Usage
linear.model.MLE(
formula,
coords = NULL,
data,
ID.coords = NULL,
kappa,
fixed.rel.nugget = NULL,
start.cov.pars,
method = "BFGS",
low.rank = FALSE,
knots = NULL,
messages = TRUE,
profile.llik = FALSE,
SPDE = FALSE,
mesh = NULL,
SPDE.analytic.hessian = FALSE
)
Arguments
formula |
an object of class " |
coords |
an object of class |
data |
a data frame containing the variables in the model. |
ID.coords |
vector of ID values for the unique set of spatial coordinates obtained from |
kappa |
shape parameter of the Matern covariance function. |
fixed.rel.nugget |
fixed value for the relative variance of the nugget effect; default is |
start.cov.pars |
if |
method |
method of optimization. If |
low.rank |
logical; if |
knots |
if |
messages |
logical; if |
profile.llik |
logical; if |
SPDE |
logical; if |
mesh |
an object obtained as result of a call to the function |
SPDE.analytic.hessian |
logical; if |
Details
This function estimates the parameters of a geostatistical linear Gaussian model, specified as
Y = d'\beta + S(x) + Z,
where Y
is the measured outcome, d
is a vector of coavariates, \beta
is a vector of regression coefficients, S(x)
is a stationary Gaussian spatial process and Z
are independent zero-mean Gaussian variables with variance tau2
. More specifically, S(x)
has an isotropic Matern covariance function with variance sigma2
, scale parameter phi
and shape parameter kappa
. In the estimation, the shape parameter kappa
is treated as fixed. The relative variance of the nugget effect, nu2=tau2/sigma2
, can be fixed though the argument fixed.rel.nugget
; if fixed.rel.nugget=NULL
, then the variance of the nugget effect is also included in the estimation.
Locations with multiple observations.
If multiple observations are available at any of the sampled locations the above model is modified as follows. Let Y_{ij}
denote the random variable associated to the measured outcome for the j-th individual at location x_{i}
. The linear geostatistical model assumes the form
Y_{ij} = d_{ij}'\beta + S(x_{i}) + Z{i} + U_{ij},
where S(x_{i})
and Z_{i}
are specified as mentioned above, and U_{ij}
are i.i.d. zer0-mean Gaussian variable with variance \omega^2
. his model can be fitted by specifing a vector of ID for the unique set locations thourgh the argument ID.coords
(see also create.ID.coords
).
Low-rank approximation.
In the case of very large spatial data-sets, a low-rank approximation of the Gaussian spatial process S(x)
can be computationally beneficial. Let (x_{1},\dots,x_{m})
and (t_{1},\dots,t_{m})
denote the set of sampling locations and a grid of spatial knots covering the area of interest, respectively. Then S(x)
is approximated as \sum_{i=1}^m K(\|x-t_{i}\|; \phi, \kappa)U_{i}
, where U_{i}
are zero-mean mutually independent Gaussian variables with variance sigma2
and K(.;\phi, \kappa)
is the isotropic Matern kernel (see matern.kernel
). Since the resulting approximation is no longer a stationary process, the parameter sigma2
is adjusted by a factorconstant.sigma2
. See adjust.sigma2
for more details on the the computation of the adjustment factor constant.sigma2
in the low-rank approximation.
Value
An object of class "PrevMap".
The function summary.PrevMap
is used to print a summary of the fitted model.
The object is a list with the following components:
estimate
: estimates of the model parameters; use the function coef.PrevMap
to obtain estimates of covariance parameters on the original scale.
covariance
: covariance matrix of the ML estimates.
log.lik
: maximum value of the log-likelihood.
y
: response variable.
D
: matrix of covariates.
coords
: matrix of the observed sampling locations.
ID.coords
: set of ID values defined through the argument ID.coords
.
method
: method of optimization used.
kappa
: fixed value of the shape parameter of the Matern function.
knots
: matrix of the spatial knots used in the low-rank approximation.
const.sigma2
: adjustment factor for sigma2
in the low-rank approximation.
fixed.rel.nugget
: fixed value for the relative variance of the nugget effect.
mesh
: the mesh used in the SPDE approximation.
call
: the matched call.
Author(s)
Emanuele Giorgi e.giorgi@lancaster.ac.uk
Peter J. Diggle p.diggle@lancaster.ac.uk
References
Diggle, P.J., Giorgi, E. (2019). Model-based Geostatistics for Global Public Health. CRC/Chapman & Hall.
Giorgi, E., Diggle, P.J. (2017). PrevMap: an R package for prevalence mapping. Journal of Statistical Software. 78(8), 1-29. doi: 10.18637/jss.v078.i08
Higdon, D. (1998). A process-convolution approach to modeling temperatures in the North Atlantic Ocean. Environmental and Ecological Statistics 5, 173-190.
See Also
shape.matern
, summary.PrevMap
, coef.PrevMap
, matern
, matern.kernel
, maxBFGS
, nlminb
.