Precision matrix for an AR1 process

Description

Functions for creating precision matricies and observations of an AR1 process

Usage

Q.AR1(M, sigma, rho, sparse=FALSE, vcov=FALSE)

r.AR1(n, M, sigma, rho)

Arguments

Value

Q.AR1 returns either a precision or variance-covariance function with a AR1 structure.

r.AR1 retrurns a matrix with n rows which are the n observations of a Gaussian Markov random field AR1 process.

Examples

require("ggplot2")
# simulate AR1 GMRF
obs <- r.AR1(100, M=30, sigma=1, rho=.98)
# resulting matrix is n x M
dim(obs)
# subtract off the first time point to more easily observe correlation
obs_adj <- obs - obs[,1]
# move objects to a data frame
ar1_df <- data.frame(obs=c(t(obs_adj)), realization=rep(1:100, each=30),
                     time=rep(1:30, 100))
# plot each realization
ggplot(data=ar1_df, aes(time, obs, group=realization, color=realization)) +
    geom_line()

Precision matrix for a IID process

Description

Functions for creating precision matricies and observations of a independent identically distributed GMRF process.

Usage

Q.iid(M, sigma, sparse=FALSE,  vcov=FALSE)

r.iid(n, M, sigma)

Arguments

Value

Q.iid returns either a precision or variance-covariance function with iid structure.

r.iid retrurns a matrix with n rows which are the n observations of a Gaussian Markov random field iid process.

Examples

require("leaflet")
require("sp")

# simulate iid data and attach to spatial polygons data frame
US.df@data$data <- c(r.iid(1, M=nrow(US.graph), sigma=1))

# color palette of data
pal <- colorNumeric(palette="YlGnBu", domain=US.df@data$data)

# see map
map1<-leaflet() %>%
    addProviderTiles("CartoDB.Positron") %>%
    addPolygons(data=US.df, fillColor=~pal(data), color="#b2aeae",
                fillOpacity=0.7, weight=0.3, smoothFactor=0.2) %>%
    addLegend("bottomright", pal=pal, values=US.df$data, title="", opacity=1)
map1

Precision matrix for a pCAR process

Description

Functions for creating precision matricies and observations of a Leroux CAR(lCAR) process as defined in MacNab 2011. The matrix defines the precision of estimates when observations share connections which are conditionally auto-regressive(CAR).

Usage

Q.lCAR(graph, sigma, rho, sparse=FALSE, vcov=FALSE)

r.lCAR(n, graph, sigma, rho)

Arguments

Value

Q.lCAR returns either a precision or variance-covariance function with a lCAR structure.

r.lCAR retrurns a matrix with n rows which are the n observations of a Gaussian Markov random field lCAR process.

References

Y.C. MacNab On Gaussian Markov random fields and Bayesian disease mapping. Statistical Methods in Medical Research. 2011.

Examples

require("leaflet")
require("sp")

# simulate lCAR data and attach to spatial polygons data frame
US.df@data$data <- c(r.lCAR(1, graph=US.graph, sigma=1, rho=.99))

# color palette of data
pal <- colorNumeric(palette="YlGnBu", domain=US.df@data$data)

# see map
map1<-leaflet() %>%
    addProviderTiles("CartoDB.Positron") %>%
    addPolygons(data=US.df, fillColor=~pal(data), color="#b2aeae",
                fillOpacity=0.7, weight=0.3, smoothFactor=0.2) %>%
    addLegend("bottomright", pal=pal, values=US.df$data, title="", opacity=1)
map1

Modified Precision matrix for a BYM process

Description

EXPIREMENTAL. Functions for creating precision matricies and observations of a modified BYM process as defined in MacNab 2011. The matrix defines the precision of estimates when observations share connections which are conditionally auto-regressive(CAR). Because the precision matrix is not symetric the process is not a true GMRF.

Usage

Q.mBYM(graph, sigma, rho, vcov=FALSE)

r.mBYM(n, graph, sigma, rho)

Arguments

Value

Q.mBYM returns either a precision or variance-covariance function with a modified BYM structure.

r.mBYM retrurns a matrix with n rows which are the n observations of a pseudo Gaussian Markov random field of a modified BYM process.

References

Y.C. MacNab On Gaussian Markov random fields and Bayesian disease mapping. Statistical Methods in Medical Research. 2011.

Examples

## Not run: 
require("leaflet")
require("sp")

# simulate mBYM data and attach to spatial polygons data frame
US.df@data$data <- c(r.mBYM(1, graph=US.graph, sigma=1, rho=.99))

# color palette of data
pal <- colorNumeric(palette="YlGnBu", domain=US.df@data$data)

# see map
map1<-leaflet() %>%
    addProviderTiles("CartoDB.Positron") %>%
    addPolygons(data=US.df, fillColor=~pal(data), color="#b2aeae",
                fillOpacity=0.7, weight=0.3, smoothFactor=0.2) %>%
    addLegend("bottomright", pal=pal, values=US.df$data, title="", opacity=1)
map1

## End(Not run)

Precision matrix for a pCAR process

Description

Functions for creating precision matricies and observations of a proper CAR(pCAR) process as defined in MacNab 2011. The matrix defines the precision of estimates when observations share connections which are conditionally auto-regressive(CAR).

Usage

Q.pCAR(graph, sigma, rho, sparse=FALSE, vcov=FALSE)

r.pCAR(n, graph, sigma, rho)

Arguments

Value

Q.pCAR returns either a precision or variance-covariance function with a pCAR structure.

r.pCAR retrurns a matrix with n rows which are the n observations of a Gaussian Markov random field pCAR process.

References

Y.C. MacNab On Gaussian Markov random fields and Bayesian disease mapping. Statistical Methods in Medical Research. 2011.

Examples

require("leaflet")
require("sp")

# simulate pCAR data and attach to spatial polygons data frame
US.df@data$data <- c(r.pCAR(1, graph=US.graph, sigma=1, rho=.99))

# color palette of data
pal <- colorNumeric(palette="YlGnBu", domain=US.df@data$data)

# see map
map1<-leaflet() %>%
    addProviderTiles("CartoDB.Positron") %>%
    addPolygons(data=US.df, fillColor=~pal(data), color="#b2aeae",
                fillOpacity=0.7, weight=0.3, smoothFactor=0.2) %>%
    addLegend("bottomright", pal=pal, values=US.df$data, title="", opacity=1)
map1

Spatial Polygons Data Frame of Counties for Several States

Description

Spatial Polygons data frame with 475 counties from the US states Louisiana, Texas, Mississippi, & Arkansas. FIPS codes for the state and county are provided in the data frame.

Matrix of Shared Boundaries Between US.df Counties

Description

A 475x475 matrix where the index corresponds to a row in the US.df Spatial Polygons data frame and the index of the matrix at row i column j is 1 when US.df[i,] and US.df[j,] share a border and 0 when they do not.

Simulate correlated data from a precision matrix.

Description

Takes in a square precision matrix, which ideally should be sparse and using Choleski factorization simulates data from a mean 0 process where the inverse of the precision matrix represents the variance-covariance of the points in the process. The resulting simulants represent samples of a Gaussian Markov random field (GMRF).

Usage

sim.AR(n, Q)

Arguments

Value

Matrix object, matrix where each row is a single obsrevation from a GMRF with covariance structure Q^-1.

Examples

require("ggplot2")

# simulate 2D ar1 process
# pairwise correlation
rho <- .95
# pairwise variance
sigma <- .5

# 2 dimensions of simulations
years <- 20
ages <- 10

# kronnecker product to get joint covariance
Q2D <- kronecker(Q.AR1(M=years, sigma, rho), Q.AR1(M=ages, sigma, rho))

# simulate the data and place it in a data frame
Q2D.df <- data.frame(obs=c(sim.AR(1, Q2D)), age=rep(1:ages, years),
                     year=rep(1:years, each=ages))

# graph results
ggplot(data=Q2D.df, aes(year, obs, group=age, color=age)) + geom_line()