MRF {gamlss.spatial}

Markov Random Fields Fitting Functions

Description

The functions MRF() and MRFA() fit a Gaussian Markov Random Fields (MRF) model. They are used by the functions mrf() and mrfa() respectively to fit a MRF additive term within GAMLSS

Usage

MRF(y, x, precision = NULL, neighbour = NULL, polys = NULL, 
            area = NULL, weights = rep(1, length(y)), sig2e = 1, 
            sig2b =             1, sig2e.fix = FALSE, 
            sig2b.fix = FALSE, penalty = FALSE, 
            delta = c(0.01, 0.01), shift = c(0, 0))

MRFA(y, x, precision = NULL, neighbour = NULL, polys = NULL, 
           area = NULL, weights = rep(1, length(y)), 
           lambda = NULL, df = NULL, start = 10)

Arguments

y

response variable

x

a factor containing the areas

precision

the precision matrix if set

neighbour

an object containing the neighbour information for the area if set

polys

the polygon information if set

area

this argument is here to allow more areas than the levels of the factor x, see example below.

weights

prior weights

sig2e

starting values for the error variance

sig2b

starting values for the random field variance

sig2e.fix

whether sig2e is fixed in the fitting, default equals FALSE

sig2b.fix

whether sig2B is fixed in the fitting, default equals FALSE

penalty

whether quadratic penalty is required to help convergence in for flat likelihoods, this is equivalent of putting a normal prior distribution for the log-sigmas e.g. logsig2e-N(shift, 1/delta)

delta

the precision of the prior

shift

the mean of the prior

lambda

smoothing parameter for MRFA function

start

starting value for the smoothing parameter lambda for MRFA function

df

for fixing the degrees of freedom (only in MRFA())

Details

There are two functions for fitting Markov random fields: i) MRF()) which uses the Q-function (marginal likelihood) for estimating the sig2e and sig2b parameters and ii) MRFA() which estimates the smoothing parameter lambda=sig2e/sig2b using the "alternating" method.

Value

a fitted MRF object

Author(s)

Fernanda De Bastiani, Mikis Stasinopoulos, Robert Rigby and Vlasios Voudouris.

Maintainer: Fernanda <[e-mail]>

References

De Bastiani, F. Rigby, R. A., Stasinopoulos, D. M., Cysneiros, A. H. M. A. and Uribe-Opazo, M. A. (2016) Gaussian Markov random spatial models in GAMLSS. Journal of Applied Statistics, pp 1-19.

Rigby, R. A. and Stasinopoulos D. M. (2005). Generalized additive models for location, scale and shape,(with discussion), Appl. Statist., 54, part 3, pp 507-554.

Rigby, R. A., Stasinopoulos, D. M., Heller, G. Z., and De Bastiani, F. (2019) Distributions for modeling location, scale, and shape: Using GAMLSS in R, Chapman and Hall/CRC. An older version can be found in https://www.gamlss.com/.

Rue and Held (2005) Gaussian markov random fields: theory and applications, Chapman & Hall, USA.

Stasinopoulos D. M. Rigby R.A. (2007) Generalized additive models for location scale and shape (GAMLSS) in R. Journal of Statistical Software, Vol. 23, Issue 7, Dec 2007, https://www.jstatsoft.org/v23/i07/.

Stasinopoulos D. M., Rigby R.A., Heller G., Voudouris V., and De Bastiani F., (2017) Flexible Regression and Smoothing: Using GAMLSS in R, Chapman and Hall/CRC.

(see also https://www.gamlss.com/).

See Also

mrf

Examples

library(mgcv)
data(columb)
data(columb.polys)
vizinhos=polys2nb(columb.polys)
precisionC <- nb2prec(vizinhos,x=columb$district)
# MRFA
 m1<-MRFA(columb$crime, columb$district, polys=columb.polys)
m11<-MRFA(columb$crime, columb$district, precision=precisionC)
m12<-MRFA(columb$crime, columb$district,  neighbour=vizinhos)
draw.polys(columb.polys, m12, scheme="heat",swapcolors=TRUE)
## Not run: 
# MRF 
  m2<-MRF(columb$crime, columb$district, polys=columb.polys)
 m21<-MRF(columb$crime, columb$district, precision=precisionC)
 m22<-MRF(columb$crime, columb$district, neighbour=vizinhos)
AIC(m1, m11,m12,m2, m21, m22, k=0)
draw.polys(columb.polys, m12, scheme="heat",swapcolors=TRUE)
# removing one area
columb2 <- columb[-5,]
# creating new precision matrix
precisionC2 <- nb2prec(vizinhos,x=columb$district,area=columb$district)
# MRFA 
# new data but declaring  area
m11<-MRFA(columb2$crime, columb2$district, polys=columb.polys, area=columb$district)
# new data old polys
m112<-MRFA(columb2$crime, columb2$district, polys=columb.polys)
# new data old precision old area
m111<-MRFA(columb2$crime, columb2$district, precision=precisionC,area=columb$district)
# new data old neighbour old area
m121<-MRFA(columb2$crime, columb2$district,  neighbour=vizinhos,area=columb$district)
# new data new precision old area
m113<-MRFA(columb2$crime, columb2$district, precision=precisionC2,area=columb$district)
AIC(m11,m112,m111,m121,m113, k=0)
m11<-MRFA(columb2$crime, columb2$district, polys=columb.polys, area=columb$district)
# new data old polys
m112<-MRFA(columb2$crime, columb2$district, polys=columb.polys)
# new data old precision old area
m111<-MRFA(columb2$crime, columb2$district, precision=precisionC,area=columb$district)
# new data old neighbour old area
m121<-MRFA(columb2$crime, columb2$district,  neighbour=vizinhos,area=columb$district)
# new data new precision old area
m113<-MRFA(columb2$crime, columb2$district, precision=precisionC2,area=columb$district)
AIC(m11,m112,m111,m121,m113, k=0)
draw.polys(columb.polys, fitted(m11))

## End(Not run)