Spatial Random Effects class

This is the central class definition of the FRK package, containing the model and all other information required for estimation and prediction.

Details

The spatial random effects (SRE) model is the model employed in Fixed Rank Kriging, and the SRE object contains all information required for estimation and prediction from spatial data. Object slots contain both other objects (for example, an object of class Basis) and matrices derived from these objects (for example, the matrix \(S\)) in order to facilitate computations.

Slots

f

formula used to define the SRE object. All covariates employed need to be specified in the object BAUs

data

the original data from which the model's parameters are estimated

basis

object of class Basis used to construct the matrix \(S\)

BAUs

object of class SpatialPolygonsDataFrame, SpatialPixelsDataFrame of STFDF that contains the Basic Areal Units (BAUs) that are used to both (i) project the data onto a common discretisation if they are point-referenced and (ii) provide a BAU-to-data relationship if the data has a spatial footprint

S

matrix constructed by evaluating the basis functions at all the data locations (of class Matrix)

S0

matrix constructed by evaluating the basis functions at all BAUs (of class Matrix)

D_basis

list of distance-matrices of class Matrix, one for each basis-function resolution

Ve

measurement-error variance-covariance matrix (typically diagonal and of class Matrix)

Vfs

fine-scale variance-covariance matrix at the data locations (typically diagonal and of class Matrix) up to a constant of proportionality estimated using the EM algorithm

Vfs_BAUs

fine-scale variance-covariance matrix at the BAU centroids (typically diagonal and of class Matrix) up to a constant of proportionality estimated using the EM algorithm

Qfs_BAUs

fine-scale precision matrix at the BAU centroids (typically diagonal and of class Matrix) up to a constant of proportionality estimated using the EM algorithm

Z

vector of observations (of class Matrix)

Cmat

incidence matrix mapping the observations to the BAUs

X

design matrix of covariates at all the data locations

G

list of objects of class Matrix containing the design matrices for random effects at all the data locations

G0

list of objects of class Matrix containing the design matrices for random effects at all BAUs

K_type

type of prior covariance matrix of random effects. Can be "block-exponential" (correlation between effects decays as a function of distance between the basis-function centroids), "unstructured" (all elements in K are unknown and need to be estimated), or "neighbour" (a sparse precision matrix is used, whereby only neighbouring basis functions have non-zero precision matrix elements).

mu_eta

updated expectation of the basis-function random effects (estimated)

mu_gamma

updated expectation of the random effects (estimated)

S_eta

updated covariance matrix of random effects (estimated)

Q_eta

updated precision matrix of random effects (estimated)

Khat

prior covariance matrix of random effects (estimated)

Khat_inv

prior precision matrix of random effects (estimated)

alphahat

fixed-effect regression coefficients (estimated)

sigma2fshat

fine-scale variation scaling (estimated)

sigma2gamma

random-effect variance parameters (estimated)

fs_model

type of fine-scale variation (independent or CAR-based). Currently only "ind" is permitted

info_fit

information on fitting (convergence etc.)

response

A character string indicating the assumed distribution of the response variable

link

A character string indicating the desired link function. Can be "log", "identity", "logit", "probit", "cloglog", "reciprocal", or "reciprocal-squared". Note that only sensible link-function and response-distribution combinations are permitted.

mu_xi

updated expectation of the fine-scale random effects at all BAUs (estimated)

Q_posterior

updated joint precision matrix of the basis function random effects and observed fine-scale random effects (estimated)

log_likelihood

the log likelihood of the fitted model

method

the fitting procedure used to fit the SRE model

phi

the estimated dispersion parameter (assumed constant throughout the spatial domain)

k_Z

vector of known size parameters at the observation support level (only applicable to binomial and negative-binomial response distributions)

k_BAU

vector of known size parameters at the observed BAUs (only applicable to binomial and negative-binomial response distributions)

include_fs

flag indicating whether the fine-scale variation should be included in the model

include_gamma

flag indicating whether there are gamma random effects in the model

normalise_wts

if TRUE, the rows of the incidence matrices \(C_Z\) and \(C_P\) are normalised to sum to 1, so that the mapping represents a weighted average; if false, no normalisation of the weights occurs (i.e., the mapping corresponds to a weighted sum)

fs_by_spatial_BAU

if TRUE, then each BAU is associated with its own fine-scale variance parameter

obsidx

indices of observed BAUs

simple_kriging_fixed

logical indicating whether one wishes to commit to simple kriging at the fitting stage: If TRUE, model fitting is faster, but the option to conduct universal kriging at the prediction stage is removed

References

Zammit-Mangion, A. and Cressie, N. (2017). FRK: An R package for spatial and spatio-temporal prediction with large datasets. Journal of Statistical Software, 98(4), 1-48. doi:10.18637/jss.v098.i04.

See also

SRE for details on how to construct and fit SRE models.